Citation for published version (APA): Klauser, A. M. (2012). Adjacent spin operator correlations in the Heisenberg spin chain

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1 UvA-DARE (Digital Academic Repository) Adjacent spin operator correlations in the Heisenberg spin chain Klauser, A.M. Link to publication Citation for published version (APA): Klauser, A. M. (2012). Adjacent spin operator correlations in the Heisenberg spin chain General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 18 Dec 2018

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3 Adjacent spin operator correlations in the Heisenberg spin chain ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op donderdag 5 april 2012, te 10:00 uur door Antoine Marc Klauser geboren te Chêne-Bougeries, Zwitserland

4 Promotiecommissie Co-promotor: Dr. J.-S. Caux Promotores: Prof. dr. J. van den Brink Prof. dr. C. J. M. Schoutens Overige leden: Dr. N. J. van Druten Prof. dr. J. M. Maillet Prof. dr. M. M. Mostovoy Prof. dr. B. Nienhuis Dr. H. Rønnow Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

5 Contents 1 Introduction One dimensional physics Spin models on a lattice Spin chain regimes Integrability and diffraction Alternative methods Luttinger Liquid Conformal Field Theory Density-Matrix Renormalization Group Quantum Monte Carlo Spin chains in nature Contents of the thesis Spectrum of the Heisenberg spin chain and classification of its eigenstates Metallic atom chain model Diagonalization Solutions Bethe Ansatz and scattering equations Bethe equations and Orbach parametrization Bethe-Takahashi equations and complex solutions Spin of the eigenstates and infinite rapidities Catalog of excitations Excitations at h = Excitations at h >

6 3 Introduction to the Algebraic Bethe Ansatz Monodromy matrix and commutation relations Trace identity and Bethe equations Local spin operators Norm and Scalar product Norm of an eigenstate Scalar product Probing the Heisenberg spin chain Inelastic neutron scattering Neutron sources and apparatus Microscopic scheme Resonant inelastic x-ray scattering Ultrashort Lifetime Expansion Spin chain Adjacent spin operators form factors and correlation functions S j S j Form factor Homogeneous limit Reduced H for string solutions Dynamical structure factor and total spin sectors Sum Rules Numerical evaluation of the S ++ (q, ω) dynamical structure factor Sj zsz j Form factor Homogeneous limit Fourier transform Dynamical structure factor and total spin sectors Sum Rules Numerical evaluation of the S 4z (q, ω) dynamical structure factor Spin-exchange correlation function Dynamical structure factor and total spin sectors Sum Rules Integrated intensity First frequency moment Numerical results of the spin-exchange DSF and comparison between RIXS and INS

7 7 Conclusion 87 Bibliography 89 Summary 93 Samenvatting 96 List of publications 98 Acknowledgments 99 5

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9 List of Figures 1.1 Phase diagram of the XXZ spin chain in a magnetic field h at zero temperature and with J = Combinations of quantum numbers corresponding to the ground state and several excited states (see text for description) in the N = 16 spin chain For N = 128, M = 64, we show in (a) the DesCloizeaux-Pearson spectrum of 1 spinon and in (b), (c), (d) the densities of states corresponding of states containing 2,4 and 6 spinons. The... represents any additional static quasiparticle like, 1s 2,1s 3 which does not contribute to the density of states For N = 128, M = 16, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label For N = 128, M = 32, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label For N = 128, M = 48, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label Experimental setup of time-of-flight spectroscopy This sketch outlines the 1D magnetic active part of Sr 2 CuO 3. We show here a layer of the crystal and in the bulk, each chain is isolated from the layers above and below by Strontium atoms Mechanism by which double-spin flip transitions are created in the indirect magnetic RIXS process S ++ (q, ω) DSF in a N = 400 XXX spin chain and (a) M = 50, (b) 100, (c) 150 and (d)

10 Separated graphs of the S ++ (q, ω) DSF for the three main types of excitation with N = 400 and M = 150. The type of excitation plotted is specified in the label and is described and explained in section Contributions by type of excitation to the S ++ (q, ω) DSF for N = 400 and M = 200. The graphs are labeled by the type of excitations that are represented Sc 4z (q, ω) DSF for a N = 400 XXX spin chain and (a) M = 50, (b) 100, (c) 150 and (d) Separated figures of the Sc 4z (q, ω) DSF with N = 400 corresponding to types of excitation in the labels. The magnetization is M = 150 in (a), (b) and (c) and M = 200 for (d) and (e) (a) Spin-exchange DSF S exch (q, ω) and (b) single spin DSF S zz (q, ω) for N = 400 sites in h = 0. The intensity around q = π is markedly different in the two cases and the signal of the spin-exchange DSF around q = π 2, 3π 2 is enhanced Fixed momentum profiles of the biased spin-exchange DSF S exch (q, ω)/(cos 2 (q/2)) (plain) and S zz (q, ω) (dashed), each normalized to its own sum rule. 86 8

11 Chapter 1 Introduction 1.1 One dimensional physics In a first view, working on one dimensional models might look like a way to reduce the size of the coordinate space and therefore to simplify the physics. Indeed the behavior of free particles in one dimension is very similar to multi-dimensional systems but with a smaller phase space to handle. Actually, the reason for studying the 1D world appears as soon as an interaction is switched on and reveals drastic differences with the cases of higher dimensions. An excellent example of exotic behavior, is the spin-charge separation of electrons. While in higher dimensions the spin and the charge are usually carried by the motion of one particle, in one dimension they do not follow the same dynamics and they evolve independently [1, 2]. In most two and three dimension systems, the Fermi liquid theory successfully describes the behavior of interacting fermions [3]. It fails however to explain the properties of 1D Fermi systems. The reason is that whereas the Fermi liquid describes the highly correlated system with quasiparticles with essentially the same properties as the free fermions [4], in one dimension the actual single-particle propagator does not have any pole [5]. Therefore no single-particle excitation, such as described by the Fermi liquid theory, exists and one has to look for combined particle modes. Intuitively, it is not difficult to understand why the interactions produce such a difference. In two or higher dimensions, the concept of almost free quasiparticles is still valid and means that a particle can propagate in space although this space is populated with obstacles such as static particles. Indeed, it is always possible to exchange two particle positions while avoiding any contact. On the contrary, in a one-dimensional system, this is not possible. Indeed, a particle on a line already populated with still standing particles, will not be able to move along the line without coming into close contact and thus interacting with the particles. Qualitatively, the only possibility is to bump all the particles and therefore create a collective excitation. Furthermore, taking into account the influence of the neighbors as a mean-field, each particle is surrounded by no more than 9

12 1.2 two others, compared to two and three dimensions. Any ordered state is then less likely to be found and moreover, long distance order can only exist at zero temperature [6]. In consequence, no finite temperature phase transition can happen. In the ground state, however, the modification of microscopic parameters can lead to phase transitions and to long range orders. In summary, one dimensional systems are strongly interacting and exhibit a collective behavior. Furthermore, because the temperature does not yield any phase transition, all the interesting physics takes place in the vicinity of the ground state, at zero temperature [7]. 1.2 Spin models on a lattice In the perspective of understanding the magnetic properties of matter, microscopic models which recreate magnetic degrees of freedom of atoms in crystal lattices are studied. Following Weiss s theory published in 1907 [8] which says that each atom experiences a force proportional to the alignment of the other atoms of the lattice, Ising showed in 1925 [9] that there exists no possibly large enough classical force to produce ferromagnetism on an atomic chain. But in 1928 [10], Heisenberg brought a new quantum mechanical interpretation of Weiss s molecular force by calculating the effects of the exchange between neighbors on the lattice and his results successfully reproduce a ferromagnetic behavior. Defining the simplest model of spin 1/2 on a one-dimensional lattice, one gives to each site j of the lattice a spin S j = σ j /2, with σ j = (σj x, σy j, σz j ) the vector of Pauli matrices acting on the jth lattice site. The three components of the spin, S j = (Sj x, Sy j, Sz j ), therefore satisfy the commutation relations [S α j, S β k ] = iδ j,kε αβγ S γ j (1.1) with ε αβγ being the Levi-Civita symbol and δ j,k the Kronecker delta which ensures that two spins on different sites commute. Supposing that the spins only interact with their nearest neighbors via spin-exchange process [10, 11], the evolution of the spin chain is dictated by the general Hamiltonian [12, 13, 14] H 0 = J ( Sj x Sj+1 x + S y j Sy j+1 + Sj z Sj+1 z 1 ). (1.2) 4 j Here J is a coupling constant and is an anisotropy parameter that allows to vary the coupling along the z-axis. This one axis anisotropic Hamiltonian refers to the general XXZ model and if one chooses = 1 it becomes the Heisenberg (XXX) model. The constant term 4 is added on each site in order that the magnetic saturated state (with all spins aligned along z-axis) has zero energy. Two different types of behavior occur depending of the sign of the coupling, supposing positive (the case of < 0 is discussed later). If J < 0, the energy is minimized by a configuration with all the spins pointing in the same direction. The spin 10

13 1.3 chain has therefore a ferromagnetic ground state. On the other hand, when J > 0, the neighboring spins tend to point in opposite directions and the spin chain is antiferromagnetic. To understand the effect of the anisotropy parameter, it is helpful to first notice that one can write the first two terms of the Hamiltonian Sj xsx j+1 + Sy j Sy j+1 = ( 1 2 S + j S j+1 + S j j+1) S+ with the spin lowering or spin raising operators: S j = Sj x isy j, S+ j = Sj x + isy j. One can then interpret (1.2) in term of z-spin hard core bosonic particles (known as quantum lattice gas [15, 16, 7]): the first two terms are equivalent to a hopping process for a spin up or down, and the last term Sj zsz j+1 represents thus an interaction term between two neighbor z-spin values. From this point of view, the prefactor allows one to tune the interaction and in the limit of = 0, the model can be understood as free fermionic particles via the Wigner-Jordan transformation [17]. In the other limit of, the spins become static scalar values corresponding to the classical Ising model. We complete the Hamiltonian with a term of Zeeman interaction between the spins and a magnetic field along the z-axis h H = H 0 j hs z j. (1.3) This coupling with the magnetic field allows one to tune the magnetization spins. Different extensions of the XXZ model are possible [4]. One can, for instance, include frustration by adding a next-nearest neighbor interaction term with a coupling constant of opposite sign. The model can also include phonons-spin coupling in the system and in this case the well-known spin-peierls transition can occur. However, a great interest stands in the understanding of the original XXZ model because of its simplicity which hides a great underlying richness. 1.3 Spin chain regimes The completed Hamiltonian (1.2, 1.3) introduced here-above contains three coupling constants J, and h, however, we can reduce by symmetry the analysis to two parameters. The transformation, defined by S x j ( 1) j S x j, S y j ( 1)j S y j, Sz j S z j, (1.4) conserves the commutation relations (1.1) and changes the coupling constants J, J,. Therefore by putting J = 1 and varying < <, we access all the regimes with the use of the unitary transformation (1.4). For instance the Heisenberg ferromagnetic model with J = 1, = 1 is equivalent, through the transformation, to the case J = 1, = 1. The properties and phases of the ground state have been extensively studied in [18, 19, 20] and we can divide the parameter space into three distinct regions [21]. As shown in figure 1.3, for < 1 and h > h sat. with h sat. = + 1, the magnetization of the spin chain is saturated and all the spins are aligned in 11

14 Ferromagnetic critical field spin-flop field J=1 h 1 Critical 0.5 Antiferro Δ Figure 1.1: Phase diagram of the XXZ spin chain in a magnetic field h at zero temperature and with J = 1. a ferromagnetic order. Besides, for values > 1 and h < h spin-flop, the spin chain is characterized by a finite staggered magnetization ( j ( 1)j Sj z 0) and consequently there is an antiferromagnetic order along the z-axis. As a consequence of the absence of Goldstone bosons in 1D [22], the excitations of these two long range ordered regimes are gapped. When h = h spin-flop, the antiferromagnetic order is brought from the z-axis into the plane perpendicular to the field. In the region in between, when the magnetic field exceeds the value of the spin-flop transition h spin-flop (see [20, 23] for implicit formulation) or if 1, the antiferromagnetic order disappears and the system enters the critical phase. Moreover, in this region of the parameter space, 1 1 and h sat. h h spin-flop, the energy level are gapless and, in consequence, the quantum fluctuations break down all possible long range order. This means that all kinds of correlations between spins go to zero when the distance increases. Although we are discussing here only the fundamental spin-1/2 model, it is interesting to notice that the phase diagram described here above, depends on the value of the spin S. For instance, Haldane showed [24, 25] that for integer S the corresponding XXZ model becomes gapped and therefore shows completely different ground state properties than half-integer spins which, like the spin-1/2 case, have a gapless ground state. Eventually, in the limit S one recover the classical Heisenberg spin model with anisotropy. 1.4 Integrability and diffraction The exact meaning of integrability varies a lot if one considers classical mechanics or quantum systems [7]. The Classical integrability is, contrarily to its quantum counterpart, a very clear and well-understood concept. In the description of a 12

15 1.4 n-body problem, one defines the Hamiltonian H(q, p) with q and p, n-vectors of coordinates and momenta. If there exist n quantities L 1,..., L n which are first integrals of motion: {L i, H} = 0, i = 1,..., n, (1.5) with {...,...} the Poisson brackets and which are in involution i.e.: {L i, L i } = 0, i, j, (1.6) then it is possible to change coordinates into action-angle ones. After a transformation from the canonical coordinates (q, p) to the action-angle coordinates ω, L, with ω a set of n cyclic coordinates, the new equations of motion are dl i = H = 0 dt ω i dω i dt = H = Cst i (L). (1.7) L i Here, the first equation vanishes because L are constants of motion and, in consequence, the Hamiltonian is independent of the angular coordinates ω. In the second line, the partial derivative of the Hamiltonian must be constant in ω i. Therefore, the equations can be explicitly integrated and the solutions have a periodic motion on the torus phase-space. Therefore, the condition for the integrability of a n-body classical system is to contain n first integrals of motion that allow one to solve the system exactly in action-angle coordinates. Additionally, due to this cyclic trajectory in the phasespace, ergodicity is absent of these integrable systems. The notion of integrability then clearly divides classical models into two classes with significantly different physics. However, as pointed by Weigert in [26], the concept of integrability in quantum systems is ambiguous. Indeed, it is unclear how to translate the notion of independent integrals of motion into quantum mechanics and taking classical limit of the systems is not helpful for this problem. Caux and Mossel review in [27] all the existent definitions for the quantum integrability and give a new basis for a categorization of the models. In comparison with the classical mechanics where the systems are either integrable or not, it seems that the notion of quantum integrability is not so direct and allows multiple degrees of integrability. Instead of trying to define the integrability of a system such as the XXZ spin chain, we analyze the scattering between particles as a consequence of the interaction and we conclude which are the possible conditions on the solution. Considering a one-dimensional periodic system of n bodies in interaction with a finite range potential, we suppose that the size of the system L is much larger than the potential range. Therefore, when the distances between the particles is large enough, the asymptotic form of the wave function is a product of plane waves defined by n asymptotic momenta k 1,..., k n. For n = 2, we can deduce what happens when particles are close and scatter. Through the laws of conservation, 13

16 1.5 the result of this collision is either exchange or the conservation of the particle momenta. Supposing they carry a momentum k 1, k 2 and energy ɛ 1, ɛ 2 respectively, we illustrate the process here {k 1, k 2 } collision {k 1, k 2 } or {k 1, k 2 } collision {k 2, k 1 }. (1.8) The solution is therefore a combination of two terms with the original order and the exchanged momenta : Ψ(x 1, x 2 ) = Ae i(k1x1+k2x2) + Be i(k1x2+k2x1), x 1, x 2 [0 : L[ (1.9) with A and B two phases which contain the dephasing due to the scattering and ensure that the wave function is symmetric or antisymmetric depending of the statistics of the particles. For n > 2, the solution can not be generalized to all kinds of interaction with finite range like for n = 2. A two body scattering with the same result as illustrated in (1.8) is still possible when two of the n particles get close enough to each other. However, other scattering processes including more than two simultaneous interacting bodies may happen. The total momentum and energy are still conserved but these two conditions are not enough anymore to restrict the scattering to a direct and exchange process. Indeed, in addition to these two outcomes, the particle momenta can also have a diffusive scattering where they take n new other values [0 : 2π[ which satisfy the conservation laws. We can sketch these m-body (m n) collisions by collision k 1, k 2,..., k m k 1, k 2,..., k m with k k m = k k m and ɛ ɛ m = ɛ ɛ m. (1.10) The consequence of these higher body scattering is drastic because, unlike the case n = 2, the wave function cannot be defined with a unique set of n conserved momenta. However if the diffusive collision, where the number of scattering particles is higher than the number of conservation laws, are absent from the model, or if the number of conservation laws is equal to the number of particles, the solution wave function takes the Bethe Ansatz form Ψ(x 1,..., x n ) = A(P )e i(k P (1)x 1+k P (2) x k P (n) x n), x 1,..., x n [0 : L[ P (1.11) with P the permutations of the symmetric group S n. In the XXZ spin chain model with the Hamiltonian (1.2), the interaction occurs only between two particles, two down spins on neighboring sites. The scattering is then non-diffusive and therefore the model can be exactly solved by the Bethe Ansatz. As the interaction is ultralocal, meaning only between two neighboring sites, the hypothesis of asymptotic plane wave solutions stays true until the spin chain reaches the filling of one down spin per two sites. Beyond this filling, we simply rotate the spins upside down to have no more than an average of one spin down for two sites. We can apply therefore the solution as well for the ferromagnetic case, where the density of down spins is low, as for the antiferromagnetic regime in zero magnetic field where the ground state is half-filled with down spins. 14

17 Alternative methods In one dimensional integrable models and particularly the spin chain, besides the use of the Bethe Ansatz which results in an exact solution, multiple other methods provide information on the correlations along the spin chain. We describe hereafter the Luttinger Liquid Theory and the Conformal Field Theory, two commonly used analytical approaches in addition to two numerical techniques: the Density- Matrix Renormalization Group and the Quantum Monte Carlo which have greatly contributed to the understanding of the spin chain Luttinger Liquid Trying to catch the phenomenology of the collective excitations in one dimensional physics, the Luttinger Liquid describes the system with a free boson formulation. As mentioned earlier, the XXZ spin chain can be mapped to a system of spin-less fermions in interaction using the Wigner-Jordan transformation. Using that the spectrum of these particles is linear at the Fermi surface, the problem is reformulated in terms of free bosonic quasiparticles with the Hamiltonian H = 1 2π [ dx uk ( x θ(x)) 2 + u K ( xφ(x)) 2] (1.12) with φ(x) and θ(x) two bosonic operators which commute canonically [φ(x), x θ(x )] = iπδ(x x ). Although the Luttinger liquid describes only the low energy behavior of a one dimensional system, the method is non perturbative and applies to all interaction strengths. A lot of one-dimensional gapless models can be characterized by (1.12) but the parameters u and K are not universal and must be evaluated for each microscopic model. Any correlation function of local operators (such as S z,,+ j in the spin chain models) can then be written in terms of φ(x) and θ(x) and their asymptotic behavior explicitly calculated. For instance, Lake et al. in [28] obtained good agreement the response function of neutron scattering in the spin chain. Additionally, one can find also the calculation of a four spin correlation function in the anisotropic spin chain [29]. Good introduction to the bosonization and Luttinger liquid theory can be found in [30, 31, 4] Conformal Field Theory During second and higher order phase transitions, at criticality, the typical fluctuation lengths of a system become infinite and there is thus an invariance under scaling transformation. The Conformal Field Theory is a method which uses the properties of the critical phase of two dimensional systems. For models with local interactions, an immediate consequence of a scale invariance is the conformal invariance i.e.: the system is invariant under transformations which preserve angles between intersecting curves. The conformal transformation group is large enough 15

18 1.6 to extract informations about the correlations of the system exploiting the conformal symmetries. For instance, the correlation functions in the Heisenberg model at zero temperature are scaling invariant for long distance and the exponent of the spin correlation can be related to the exponential decay of the correlations of a model at the finite temperature. From a general point of view, the Conformal Field Theory is a complementary method which allows one to export known results from one model geometry to another. References [32, 33] give a nice introduction and more specific details Density-Matrix Renormalization Group This method developed during the last twenty years permitted to push further the precision of numerical computations in low dimensional systems. The simulation, inspired by the renormalization group method, is to split the Hilbert space in two blocks where only the low energy states are computed and kept. A good introduction can be found in the recent reviews [34, 35]. A great advantage of this technique is that, by the clever selection of low energy states, very big system sizes can be reached and this allows one to get rid of the boundary effects. The calculation of static correlation functions over the selected states is then straightforward and recent developments allowed calculation of short time dynamic correlation function and finite temperature computation. [36] is a good example of Density-Matrix Renormalization Group calculation in the Heisenberg spin chain where the authors verify numerically a specific log correction to the spin correlation function. Using a time evolution algorithm, Bouillot et al. [37] calculated the dynamic spin correlations in a Heisenberg spin ladder model at zero temperature Quantum Monte Carlo Well known, the Quantum Monte Carlo (QMC) method is a statistical method used to evaluate path integrals for the evaluation of observables. Working on finite systems but wanting to avoid boundary effects, one considers rather large sizes. However, the size of the Hilbert space grows exponentially with the number of sites. The QMC allows one to evaluate observables with a small sample of the configurations in the Hilbert space. The Hirsh-Fye algorithm is a famous implementation of QMC and probably one of the simplest [38]. Nevertheless, there are several difficulties and disadvantages in the QMC. For fermionic systems, the antisymmetry of the wave function yields the famous sign problem. It originates from the fact that for fermions the weight of a configuration (given by the exponential of minus the action) can be negative. Moreover, observables are calculated in imaginary time which make the evaluation of dynamic quantities very difficult and the statistical error over the quantities computed goes like 1 s with s the sample size (basically proportional to the computation time). Nevertheless before the development of more efficient numerical tools, QMC has been widely used in 1D system and one can find in [39, 40, 41, 42] few examples of spin chain calculations. 16

19 Spin chains in nature Although one dimensional models might seem artificial in a three dimension world, it is possible to find in nature cases where the degrees of freedom are almost reduced to 1D and which realize quasi one-dimensional models. For magnetic systems, most of the examples are found in atomic crystalline materials which contain a stack of spin chains. The latter are in general insulators where electrons are localized on the atoms. The spin lattice is realized when the outer shell electron is unpaired and within Mott-insulator regime, its spin can be super-exchanged with other neighboring atoms. A one-dimensional pattern can appear then in strongly anisotropic insulators i.e.: if the spin-exchange coupling is much larger in one direction than in the others. To avoid the effect of any inter chain coupling, we can tune the temperature of the system high enough to decouple chains from each other but lower than the coupling along the chain in order to have an effective realization of a 1D spin chain at low temperature. Multiple examples of crystal insulators using 3d electrons of copper as spin degree of freedom exist. For instance the crystals KCuF 3, SrCuO 3, CuPzN, LiCuVO 4 are known 1D spin chain realization which have been probed with neutron scattering [43, 28, 44, 45, 46, 47, 48] (see chapter 4). More examples can be found in [49]. However there are also other kinds of spin chain realizations. Organic conductors can provide a 1D spin chain setup and, for instance, exhibit a spin Peierls transition due to phonon-spin coupling [50]. Moreover, compounds with e.g. Ti can realize spin chains with next nearest-neighbor coupling leading to a frustration effect [47]. Additionally, cold-atom constructions are able now to represent an effective Hubbard model. An optical lattice is filled up with Rb atoms and a pseudo-spin-1/2 is realized by two Zeeman levels of each atom in a magnetic field. At half-filling and in the limit of strong atomic repulsion, the system becomes a Mott-insulator with superexchange processes between the pseudo-spin of neighboring sites. The construction of a 1D XXZ spin chain is then possible [51] and with the control of single spin along the chain, [52, 53], one can hope that such a realization will soon be able to probe the dynamics of the spin chain. 1.7 Contents of the thesis We start the thesis by describing how to solve the Heisenberg spin chain with the coordinate Bethe Ansatz and we create a catalog of the principal excitations. We give then a short introduction to the algebraic Bethe Ansatz formalism and the local representation of spin operators. With these tools, we present our original results: We show that the response function of the K-edge Resonant Inelastic X- ray Scattering experiment on a Heisenberg spin chain is the spin-exchange correlation function. 17

20 1.7 In the finite spin chain, a determinant expression for the correlation function of the two adjacent spin operators, S j S j+1 and Sz j Sz j+1, is given and the corresponding sum rules and f-sum rules are calculated. We evaluate them numerically and we analyze the contribution of each type of excitations. The spin-exchange correlation function is computed with the determinant expressions and the response function is compared with a Inelastic Neutron Scattering measurement in the same spin chain. In a more detailed description, the thesis contains the following points. In chapter 2, we follow the method introduced by Bethe to solve the isotropic Heisenberg spin chain model and we give the Bethe equations and Bethe-Takahashi equations which allow one to compute eigenstates. We then give a catalog of the basic excitations above the ground state and we show their densities of states. We introduce in chapter 3 the algebraic Bethe Ansatz. Within this formalism, we explain how to construct the algebraic forms for the Bethe states and the local spin operators. Afterwards we give the formula for the norm of a Bethe state and the explicit reduction of the formula for states containing complex string configurations. Eventually, we recall the determinant expression for the scalar product between two states. In chapter 4, the two experimental setups: Inelastic Neutron Scattering and Resonant Inelastic X-ray Scattering are presented. We first enumerate the necessary ingredients for an inelastic neutron scattering measurement and we recall that the response signal is proportional to the single spin operator dynamical structure function. For the Resonant Inelastic X-ray Scattering, we describe the K-edge scattering process in detail and we show how to determine the response function. Being more specific, we give the result for the spin chain model which is the spin exchange dynamical structure factor. Using the formalism of chapter 3, we construct in chapter 5 first the determinant formula for the form factor of the operators S j S j+1 and Sz j Sz j+1. With these results, we give an expression for the two corresponding correlation functions and the sum rules: the integrated intensities and the first frequency moments. Afterwards, we compute, the map in momentum and energy of the two dynamical structure factors at different magnetization and we analyze the contribution of each type of excitations. The chapter 5 is based on the content of the preprint paper cond-mat/ (2012) (see the list of publications). In chapter 6, we use the previous results for the Sj zsz j+1 form factor in order to compute the spin-exchange dynamical structure factor and we compute also the corresponding sum rules. With a numerical evaluation, we compare then the response functions of the Resonant Inelastic X-ray Scattering and of the Inelastic Neutron Scattering. The results of this chapter have been published previously in the paper Phys. Rev. Lett. 106, (2011) (see the list of publications). 18

21 Chapter 2 Spectrum of the Heisenberg spin chain and classification of its eigenstates We describe in this chapter the method introduced by Bethe to solve the isotropic Heisenberg spin chain model. This introduction follows first the main lines of the intuitive development given in [54] to find the eigenstates of the system. We then explain and describe the classification of the basic excitations above the ground state and we compute their densities of states. 2.1 Metallic atom chain model Addressing the problem of the atomic chain with two levels which interact, Bethe [55] determines the eigenstates and energy levels of the system. The two levels can represent the spin- 1 2 of the outer shell electrons in the metal atoms. Considering the lattice spacing between two atoms to be large, the hopping of the outer shell electron towards the neighboring sites is weak. However, these little electronic motions give rise to an interaction between spins on neighboring sites: the spin superexchange [56, 11]. We describe therefore the same spin dynamics as the Hubbard model in the Mott-Hubbard limit: 1. Two aligned neighbor spins carry an energy equal to the coupling constant J and two opposed neighbors lower the energy by J. 2. Neighbor spins can exchange their z-spin eigenvalue with an amplitude J/2. 3. All the individual spins interact with an external magnetic field along z-axis h via the Zeeman effect. 19

22 2.2 This description for a periodic N-sites chain results in a dynamics dictated by the Hamiltonian [10] N H = J S j S j+1 JN 4 hsz j (2.1) j=1 where S j is the spin vector on the site j which has the three components Sj x, S y j and Sz j. Each component acts on the N j=1 C 2 Hilbert space and is defined as Sj α = j 1 i=1 1 2 σα N i=j+1 1 with σα the Pauli matrices. The sign of J allows one to differentiate between the ferromagnetic (J < 0) and antiferromagnetic (J > 0) chain and in order to avoid boundary effects, the system is chosen periodic. The constant term JN/4 is added in order to have the ferromagnetic ground state as reference state with energy zero. An important property of this Hamiltonian is the conservation of the total spin and total z-component of the spin i.e.: [H, S] = 0, [H, S z ] = 0 (2.2) with total spin operators S z = j Sz j and S = j S j and we write their respective eigenvalues S tot z and S tot. 2.2 Diagonalization The conservation of the magnetization by the Hamiltonian allows one to describe the eigenstates of the system by a superposition of basis states with a fixed Sz tot = N 2 M. With 0 being the so-called pseudo-vacuum state where all the spins point upwards, we can write a state of M localized down flipped spins as j 1 j 2... j M = S j 1 S j 2... S j M 0 where S j i is the spin lowering operator on the site j i. Therefore, the most general expression for a state of Sz tot magnetization is M = {j} Ψ(j 1, j 2,..., j M ) j 1 j 2... j M. (2.3) The sum runs over the ( N M) sets {j} where j1 < j 2 <... < j M. The goal here is to diagonalize the system or equivalently to find Ψ({j}) such that H {j} Ψ({j}) {j} = E {j} Ψ({j}) {j} (2.4) with E the energy of the eigenstate. If we look at the action of the Hamiltonian on one state of a given {j} we have H j 1... j M = J = J 2 N [ ( 1 ( S j 2 S+ j+1 + ) S+ j S j+1 + Sj z Sj+1 z 1 ) ] hsj z {j} 4 { ( )} # +# N {j } J + h 2 2 M {j}. (2.5) j=1 {j } 20

23 2.3 The sum of off-diagonal elements runs over all the sets {j } which corresponds to all the possible configurations where one of the positions in {j} is moved left or right: {j 1,..., j M } = {j 1,..., j α 1, j α ± 1, j α+1,..., j M } α if j α ± 1 j α±1. The diagonal elements of H contain the number of opposed neighbor spins (# +# ) which is nothing else than the number of possible transitions from{j} to {j }. In (2.4), all basis states being orthogonal, the prefactor of a state must be equal on both side and with (2.5), we write the equality for the prefactor of the state with j 1... j M : {j } ( ) Ψ({j }) Ψ({j}) = 2E + h(n 2M) Ψ({j}). (2.6) J Before going further, it is important to notice that the sum includes only the allowed configurations where 0 j 1 < j 2 <... < j M N + 1. (2.7) The periodicity condition imposes j M N < j 1 and one must then identify Ψ(0j 2... j M ) = Ψ(j 2... j M N), Ψ(j 1... j M 1 (N + 1)) = Ψ(1j 1... j M 1 ). (2.8) 2.3 Solutions For M = 1, the equations (2.6) simplify to Ψ(j + 1) + Ψ(j 1) 2Ψ(j) = 2E + h(n 2) Ψ(j). (2.9) J A solution of this equation is the plane wave amplitude: Ψ(j) = e ijk with 0 k < 2π and the energy E J = cos(k) 1 h ( ) N J 2 1, k = 2π 1 N, 2π 2,..., 2π (2.10) N where the value of k are imposed by periodic boundary conditions and taken in the first Brillouin zone. In the case of two down flipped spins, M = 2, the equations (2.6) split into two cases: if j = j 2 then Ψ(j 1 1, j 2 ) + Ψ(j 1, j 2 + 1) 2Ψ(j 1, j 2 ) = and if j < j 2, 2E + h(n 4) Ψ(j 1, j 2 )(2.11) J Ψ(j 1 1, j 2 ) + Ψ(j 1 + 1, j 2 ) + Ψ(j 1, j 2 1) + Ψ(j 1, j 2 + 1) ( ) 2E + h(n 4) = + 4 Ψ(j 1, j 2 ). (2.12) J 21

24 2.4 Making an Ansatz in the form of a two-particle plane wave Ψ(j 1, j 2 ) = A 12 e ik1j1+ik2j2 + A 21 e ik1j2+ik2j1 (0 k 1, k 2 < 2π), we find that E J = cos(k 1) + cos(k 2 ) 2 h ( ) N J 2 2. (2.13) If we define scattering phases by A ab A ba = e iψ a,b, from the coupled equations (2.11) and (2.12) the dephasing reads e iψ a,b = ei(ka+k b) 2e ika + 1 e i(ka+k b) 2e ik b + 1. (2.14) Finally the solution wave function up to a global arbitrary phase is Ψ(j 1, j 2 ) = e ik1j1+ik2j2+ i 2 ψ12 e ik1j2+ik2j1+ i 2 ψ21. (2.15) Trying to solve (2.6) for a general M becomes very complicated since one needs to write an equation for each case where two down spins are neighbors. A better way is to proceed with the decoupling of the equations (2.6). First one needs to expand the definition of the Ψ(j 1... j M ) to the domain 0 j 1 j 2... j M N + 1 (2.16) where two coordinates can coincide. In consequence, one can rewrite (2.6) into M α=1 ( ) Ψ(j 1,..., j α 1,..., j M ) + Ψ(j 1,..., j α + 1,..., j M ) 2Ψ(j 1,..., j M ) = 2E + h(n 2M) Ψ(j 1,..., j M ). (2.17) J But in order to keep the validity of the equations (2.6), one must impose that the terms added in (2.17) vanish. For instance, if two down spins are neighbors, then j α + 1 = j α+1 and one must constrain Ψ(j 1,..., j α, j α,..., j M ) + Ψ(j 1,..., j α + 1, j α + 1,..., j M ) = 2Ψ(j 1,..., j α, j α + 1,..., j M ). (2.18) With this last condition applied to all the value of α = 1,..., N, the sets of equations (2.17) and (2.6) are equivalent. 2.4 Bethe Ansatz and scattering equations The equations (2.17) are factorisable in each coordinate and therefore satisfied by the multiple plane wave function: e i(k1j1+...+k M j M ) with 0 k i < 2π and the resulting energy is M E J = (cos(k i ) 1) h ( ) N J 2 M. (2.19) i=1 22

25 2.4 Bethe Ansatz for the overall wave function is a combination of all the energy degenerate plane wave functions i.e.: Ψ(j 1,..., j M ) = M i=1 k P (i)j i (2.20) P S M ( 1) [P ] A(P )e i where the sum runs over all the permutations P of the order M symmetric group S M and ( 1) [P ] is the sign of the permutation that we include from now on in the factor A(P ) for the sake of brevity. Each rearrangement of the {k α } by a permutation carries a weight A(P ) which we calculate explicitly by the use of (2.18). Plugging the Bethe Ansatz in the equations gives ( ) A(P ) e i(k P (α)+k P (α+1) ) e ik P (α+1) + 1 P S M e ik P (1)j i(k P (α) +k P (α+1) )j α+...+ik P (M) j M = 0. (2.21) As one sums over all S M, one can act with transposition of the element α and α+1, P α,α+1, in front of P and the overall sum will stay the same since (P P α,α+1 ) S M. We can then write ( ) A(P ) e i(k P (α)+k P (α+1) ) e ik P (α+1) + 1 P S M e ik P (1)j i(k P (α) +k P (α+1) )j α+...+ik P (M) j M = ( ) A(P P α,α+1 ) e i(k P (α+1)+k P (α) ) e ik P (α) + 1 P S M e ik P (1)j i(k P (α+1) +k P (α) )j α+...+ik P (M) j M. (2.22) This equality must be true for all {j} and α such that j α+1 = j α +1. Moreover, one can identify on both sides the terms in the sum which have the same combination of k P (i) and j i i.e., after simplification ( A(P ) and we find the equality ) e i(k P (α)+k P (α+1) ) e ik P (α+1) + 1 = A(P P α,α+1 ) ( e i(k P (α+1)+k P (α) ) e ik P (α) + 1 ) (2.23) A(P )e iψ P (α+1),p (α) = A(P P α,α+1 ) (2.24) with e iψ α,β the scattering phase defined in (2.14). All permutations can be expressed as a product of transpositions and therefore every P in S M can be connected to another P by one transposition. Summing over all permutations is then equivalent to the sum of all possible transposition paths in S M and using the equality above for all P and α, the solution, up to a phase, is found to be [54] A(P ) = ( 1) [P ] e i 2 23 α<β ψ P (α),p (β). (2.25)

26 2.5 The Bethe wave function then reads Ψ(j 1,..., j M ) = ( 1) [P ] e i M i=1 k P (i)j i+ i 2 α<β ψ P (α),p (β). (2.26) P S M The fermionic character in such a wave function, consequence of (2.18), appears in the fact that if k α = k α+1, the wave function vanishes. As we consider a finite spin chain, the boundary conditions play a non negligible role in the solutions of the system. However, in order to avoid boundary effects, we choose the periodic boundary conditions and we see that this condition allows one to determine the set of rapidities {k}. We impose the periodicity condition over the wave function following a simple reasoning. For a given permutation P, if we take the coordinate of a down spin and we translate it along the spin chain by N sites, we must recover the same amplitude of the wave function. If we extend the definition of Ψ to values j i = 0, 1,..., 2N, this condition reads Ψ(..., j α 1, j α, j α+1,...) = Ψ(..., j α 1, j α+1,..., j M, j α + N). (2.27) Since one moves the spin with rapidity k α along the chain, every scattering with one of the M 1 down spins with rapidity k β adds a phase e iψ P (α),p (β). Additionally, the wave function takes a phase e ik P (α)n due to the space translation and we can rewrite the translated wave function as Ψ(j 1,..., j α 1, j α+1,..., j M, j α + N) = e ik P (α)n ( 1) M 1 e i β α ψ P (α),p (β) Ψ(j 1,..., j M ). (2.28) With (2.27) applied to all α = 0,..., M, we can then omit the permutation and write the periodicity condition for all the rapidities as e ikαn = ( 1) M 1 e i β α ψ P (α),p (β). (2.29) These equations are known as scattering equations since the term e i β α ψ P (α),p (β) can be represented as a product of 2 particles S-matrices. 2.5 Bethe equations and Orbach parametrization Taking the logarithm of (2.29), we have a set of explicit equations for {k} namely the Bethe equations: k α = 2π N I α 1 N ψ P (α),p (β) (2.30) with, for N even, I 1,..., I M a set of distinct integers for odd M and half-integers for even M. The set of quantum numbers {I α } is equivalent through these equations to 24 β α

27 2.6 the set of rapidities {k α } to parametrize an eigenstate. The existence of solutions for (2.30) has been discussed in [18] and for the thermodynamic limit in [57]. It has been found that the set of quantum numbers for the ground state is {I 0 k = k M+1 2 }, k = 1,..., M. Orbach s change of variable [14] quite simplifies the scattering term of the Bethe equations (2.30). By the parametrization k α = π 2atan(2λ α ) with the variables λ α C, the Bethe equations read atan(2λ α ) = π N I α + 1 atan(λ α λ β ). (2.31) N The maximum value of a quantum number is calculated by taking the limit of one rapidity going to infinity in (2.30): ( ) lim atan(2λ α ) 1 N atan(λ α λ β ) = π π(m 1). (2.32) λ a N 2 2N l=1 The boundary under which the quantum numbers must stay is therefore I 1 = N M+1 2. The solutions for real rapidites are conveniently described by the sets of quantum numbers which follow these simple rules: 1. They belong either to integers or half integers depending on M. 2. They must be all distinct to have no coinciding rapidities (and therefore vanishing wave function). 3. Their values are bounded by the equations (2.30). There exists states with coinciding quantum numbers but this represent few extraordinary cases. Therefore, generally, all the possible combinations of I α which fulfill these three conditions correspond to an eigenstate of the system with the energy and momentum E = J M 1/2 1/4 + λ 2 h( N α 2 M), α=1 β 2π P = πm + N M I α (mod 2π). (2.33) A more complete description of the ground state and excited states for either zero or finite magnetization is given in sec.2.8 and moreover solutions including complex rapidities and infinite rapidities are discussed hereafter. α=1 2.6 Bethe-Takahashi equations and complex solutions The solutions of (2.30) are not restricted to real rapidities only but, as Bethe mentioned in his seminal work [55], there exist self-conjugate complex solutions 25

28 2.7 which can be interpreted as bound states of down spins which also satisfy the Bethe equations. Bethe [55] made the conjecture, completed later by Takahashi et al. [58], that such complex rapidities form string structures which are λ j,a α = λ j α + i 2 (n j + 1 2a) + iδ j,a α (2.34) with a = 1,..., n j, n j being the string length and the string deviation δα nj,a. Here α is an index that runs from 1 to M j where M j is the number of n j strings (strings of length n j ) and j = 1,..., N s where N s is the total number of different possible lengths. With the string structure hypothesis, the set of eigenstates constructed was assumed to be unique and complete [58]. Unfortunately, it turned out that this hypothesis is not always right, although it can be shown that generally the string deviations vanish exponentially with system size. However exceptions do exist. For instance, in spin chains with more than 21 sites and filled with two down spins, the string hypothesis fails to predict the correct set of solutions and there are O( N) unconventional solutions made of coinciding quantum numbers [59]. Besides, the number of complex solutions is less that the one constructed from the string hypothesis and other kinds of complex solutions appear but the global difference over the Hilbert space remains O( N) [60, 61]. In the presence of string solutions with vanishing deviations the Bethe equations (2.31) become undetermined. The remedy is to rewrite the Bethe equations only in terms of the real centers λ nj α of the strings to obtain the Bethe-Takahashi equations [58]: N s M k Nθ nj (λ j α) Θ nj,n k (λ j α λ k β) = 2πIα j (2.35) where k=1 β=1 θ nj (λ) = 2atan (2λ/n j ) Θ nj,n k (λ) = (1 δ j,k )θ nj n k (λ) + 2θ nj n k +2(λ) θ nj+n k 2(λ) + θ nj+n k (λ) (2.36) and I j α are the set of quantum number corresponding to strings of length n j which take integer values if M j is odd and half-integer if the number of n j strings is even. The momentum of a state can then be generalized to include states containing strings and is written as function of the new set of quantum numbers {I j α}: N s P = π j=1 M j 2π N N s j=1 α=1 M Iα j (mod 2π). (2.37) 2.7 Spin of the eigenstates and infinite rapidities ( ) 2 In the isotropic spin chain the total spin operator j S j commutes with the Hamiltonian and therefore the eigenstates have a well-defined total spin value S tot. 26

29 2.8 Moreover, the eigenstates created with the Bethe Ansatz are highest-weight with respect to the global su(2) algebra i.e. the states constructed with M rapidities have the spin eigenvalues Sz tot = S tot = N 2 M [54]. In order to access eigenstates with Sz tot < S tot, one must act on a Bethe state with the global spin lowering operator Sq=0 = 1 N N j=1 S j. These lower weight states are actually also Bethe solutions but they include infinite rapidities. Indeed the Bethe equations allow rapidities to go to infinity (λ α, k α 0) and from (2.26), one notices that a state with M rapidities of which M tend to infinity can be written ( ) {λ 1,..., λ M M,,..., } = S M }{{} 0 {λ1,..., λ M M }. This eigenstate is no M longer highest-weight but has S tot = N 2 M +M and Sz tot = N 2 M. The norm of a state which contains one infinite rapidity can easily be expressed by commutation of spin operators as N({λ, } M ) = 1 {λ} M 1 S + i N S j {λ} M 1 i,j = 1 2δ i,j {λ} M 1 Si z {λ} M 1 N i,j = N 2M + 2 N({λ} M 1 ). (2.38) N We supposed that there is no other infinite rapidity in {λ} M 1 and therefore it is highest-weight. We used then that i S+ i {λ} M 1 = 0 and we replaced i Sz i by its eigenvalue. We proceed the same way for two infinite rapidities and the norm then is N({λ,, } M = 1 N 2 {λ} M 2 S + i S+ j S k S l {λ} M 2 = 1 N 2 i,j,k,l i,j,k,l ( 2δjk {λ} M 2 S + i Sz j S l {λ} M 2 + {λ} M 2 S + i S j S+ k S l {λ} M 2 ) N 2 (M 1) = N({λ, } M ) + 2 N N 2 {λ} M 2 S + i S j Sz k {λ} M 2 = i,j,k 2 (N 2M + 4) (N 2M + 3) N 2 N({λ} M 2 ). We give here only specific formulas which are useful for the calculation we achieve, but it is possible to give more general expressions. 2.8 Catalog of excitations (2.39) We give in this section a catalog of the excitations that occur in the XXX spin chain for a better understanding of the correlation. We divide the description of the ex- 27

30 2.8 Figure 2.1: Combinations of quantum numbers corresponding to the ground state and several excited states (see text for description) in the N = 16 spin chain. cited states into two parts: zero and non-zero magnetic field, since the polarization modifies drastically the combinatorics of the states above the ground state. The identification of the states is made with the use of the quantum number {Iα} j (see (2.31)), and the dynamics of the spin chain is described by the combinatorics of those numbers. The requirement of real solutions of the Bethe equations λ i <, imposes boundaries on quantum numbers which are [58] [ ] Iα j < I j = 1 N s N + 1 M k (2 min(n j, n k ) δ j,k ). (2.40) 2 k=1 We proceed therefore for zero and finite magnetic field with the description of the type of excitations above the ground state Excitations at h = 0 In the case of zero polarization, the ground state quantum numbers occupy all the allowed vacancies for finite real rapidities, i.e. {Iα 1 = α M+1 2 }, α = 1,..., M. The creation of excitations requires therefore to either move a rapidity to, create a string or to remove rapidities. String excitations The creation of a bound state by the formation of an n string state allows vacancies in the quantum numbers for real rapidities. The string state so created is nondispersive since there is only one accessible quantum number, but the holes created in the set {I 1 α}, denoted spinon quasiparticles [62], have a continuum spectrum of excitations. The presence of the complex rapidities imposes via the Bethe equations the number of spinons created and for each n string, there are 2n 2 spinons in the solution. As notation, we use, e.g. 2sp1s 2 for a state with 2 spinons and one 2 string or 6sp1s 2 1s 3 for 6 spinons accompanied by a 2 string and a 3 string. 28

31 2.8 4 spinon 3 ω [J] π k 2π (a) Spinon spectrum (b) 2sp... (c) 4sp... (d) 6sp... Figure 2.2: For N = 128, M = 64, we show in (a) the DesCloizeaux-Pearson spectrum of 1 spinon and in (b), (c), (d) the densities of states corresponding of states containing 2,4 and 6 spinons. The... represents any additional static quasiparticle like, 1s 2,1s 3 which does not contribute to the density of states. Spin raising and infinite rapidities We describe here two processes which create spinons by removing quantum numbers from the ground state. First, if one acts with a local spin raising operator, i.e.: S + j, the spin chain will change sector from M = N/2 to M = N/2 1 in an excited state with S tot = Sz tot = 1. Regarding the Bethe solutions, this is equivalent to removing one number from the set {Iα}. 1 The quantum numbers go then from integers to half integers or vice versa, and consequently each spin raising creates two spinons. We denote this excitation with 2sp, 4sp,... for two, four and higher number of spinons. Second, the Bethe equations allow one to send rapidities to infinity and then to remove the respective quantum numbers. Similarly to the spin raising operation, this creates a pair of spinons for each rapidity sent to infinity but contrarily to the spin lowering operation, the excited has Sz tot = 0 but S tot = 1. The infinite rapidity is actually equivalent to a global spin raising operation: {λ, } M = 1 N j S j {λ} M 1 where {λ, } M belongs then to a higher spin sector but 29

32 2.8 4 ω [J] 3 2 hole particle 2 string π 2π k (a) Spectra (b) 1p1h (c) 2p2h (d) 2h1s 2 (e) 2h (f) 1p3h Figure 2.3: For N = 128, M = 16, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label. with the same S tot z. These states are described in detail in section: 2.7 and they contribute to other spin sectors (see 5.2.4). The notation we use is 2sp, 4sp2,... for excitations of one, two, etc. number of infinite rapidities. For the three different kinds of excitations described above, the quasiparticles created, spinons, have always the same dispersion relation and therefore the spectra are identical even though the states are actually different. Although they always come in pairs, we show in figure 2.2(a) the 1sp spectrum calculated by Cloizeaux and Pearson in [63]. This allows one to understand more easily the density of states (DOS) shown in the three other graphs of figure 2.2. For instance, the 2sp... DOS represents all the linear combinations of two spinon spectra. The DOS is centered in q = π because the two spinons are accompanied by a 1 or 1s 2 of momentum π (see (2.37)). However, as already mentioned above, the infinite rapidities and, for M = N/2, the n string bound states are static and, consequently, their only contribution to the DOS is a shift of π in momentum. We include them by... in the description of figure

33 2.8 4 ω [J] 3 2 hole particle 2 string π 2π k (a) Spectra (b) 1p1h (c) 2p2h (d) 2h1s 2 (e) 2h (f) 1p3h Figure 2.4: For N = 128, M = 32, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label Excitations at h > 0 In presence of a magnetic field, the ground state of the spin chain is polarized, Sz tot > 0, and therefore there are N 2M vacancies in the quantum numbers. Similarly to the h = 0 case, the lowest energy state forms a Fermi like sea in the quantum numbers: {Iα 1 = α M+1 2 }, α = 1,..., M, although there are vacancies between the highest/lowest quantum numbers and the boundaries: M < I. j We discuss here three excitations that can occur above the ground state. Particle-hole Due to the presence of vacancies in the quantum numbers, it is possible to take a quantum number of the ground state and move it to higher value. One creates this way a pair of quasiparticles: a hole left in the Fermi sea and a particle evolving above the Fermi surface. We write in this article these excited states by ipih for i particle-hole pairs. 31

34 2.8 Spin raising Similarly to the h = 0 case, acting with a spin raising operator S + j on the spin chain, removes one rapidity and creates a hole in the Fermi sea. However, although the change in numbers of rapidities increases also the number of vacancies above the Fermi sea, we only count as quasiparticle the holes below the Fermi surface. Therefore, applying i spin raising operators on the ground state creates an excited state with i holes belonging to the sector M = M i and that we simply denote ih. In finite magnetic field, the creation of infinite rapidities is still admissible by the Bethe equations and gives similar hole quasiparticles but as discussed in 5.2.4, the form factors of such states decay like the inverse of the spin chain length. String excitations Excited states can be created by forming a string structure in the rapidities. In contrast to h = 0, we consider here hole quasiparticles and a n string leaves n holes in Fermi sea of rapidities. Moreover, the boundary I n is of O(N) and then much bigger than at h = 0. Therefore the string states have dynamics that contribute to the spectrum. The notation we use for a i string state is ih1s i. Having multiple strings in one solution is also allowed but does not carry significant signal for the calculation we are interested in. We show in the figures 2.3(a), 2.4(a) and 2.5(a), the artificial dispersion relations of 1p, 1h and 1s 2 quasiparticles. We call them artificial spectra because these quasiparticles are accompanied by others, forming the continuum of excitations, and it is physically impossible to isolate one of them. In the other graphs of figures 2.3, 2.4 and 2.5, the DOS for different type of excitations are plotted. The choice of the states is related to their contribution to the correlation function calculated in chapter 5; we have chosen the types of excitations which carry most of the signal. In this chapter we covered the diagonalization of the Heisenberg spin chain with the coordinate Bethe Ansatz. After the construction of eigenstates of the system, we have described the different kinds of excitations that occur above the ground state and we have shown their densities of state. 32

35 hole particle 2 string ω [J] π 2π k (a) Spectra (b) 1p1h (c) 2p2h (d) 2h1s 2 (e) 2h (f) 1p3h Figure 2.5: For N = 128, M = 48, we present (a) the artificial spectra of 1h, 1p and 1s 2 and (b),(c),(d),(e),(f) are the densities of states corresponding to the type of excitation mentioned in the label. 33

36 2.8 34

37 Chapter 3 Introduction to the Algebraic Bethe Ansatz We give in this chapter a short introduction to the algebraic Bethe Ansatz also known as the quantum inverse scattering method. This combination of the Bethe Ansatz principles (see chapter 2) and of the classical inverse scattering method is the result of the work from the so-called Leningrad school (Faddeev, Sklyanin, Takhtajan and others) [64, 65, 66, 67]. After the introduction we recall the formula for the norm of a Bethe state and we give an explicit development of the reduced Gaudin matrix for string configuration solutions. Eventually, we finish the chapter with the determinant expression for the scalar product between two states. 3.1 Monodromy matrix and commutation relations Trying to build up an algebraic formalism to describe the quantum integrability of a system, one naturally uses the conserved charges of the model as foundation of this construction. There exists a complete set of quantum operators I n, n = 1, 2,... in involution: [I m, I n ] = 0, m, n and which act on H the Hilbert space of the system. They commute with the Hamiltonian [H, I n ] = 0 and are therefore constants of motion. Grouping all these operators in the definition of one object τ(λ) = e cn n=1 n! (λ ξ) n I n, (3.1) with λ a spectral parameter of the so defined operator and ξ a parameter proper to the model. One can then recover all the set of constants of motions by the so called trace identities I n = 1 d n c n dλ n ln τ(λ) λ=ξ (3.2) and the commutation of all the conserved charges is equivalent to requiring [τ(λ), τ(µ)] = 0 λ, µ. (3.3) 35

38 3.1 The basic idea of the algebraic Bethe Ansatz formalism is to give a method to generate such a τ(λ) operator. Combining an additional general vector space V 0, called auxiliary space, with the Hilbert space of the spin chain, we introduce the monodromy matrix T (λ) which acts on the tensor product V 0 H and of which the trace over the auxiliary space is identified with the transfer matrix: tr 0 T (λ) = τ(λ) (3.4) where we take the trace over the auxiliary space. Following the classical formalism and using the Lax-matrices L j which describe the elementary lattice transition from site j to j + 1 [68], we construct the monodromy matrix : T (λ; ξ 1,..., ξ N ) = L N (λ, ξ N )L N 1 (λ, ξ N 1)... L 1 (λ, ξ 1 ) (3.5) where the L-matrix acts in the tensor product of the auxiliary vector space and the jth site Hilbert space. The monodromy matrix is function of the spectral parameters λ and inhomogeneity parameters ξ j attached to the jth site. Hereafter in the notation of the monodromy matrix, the inhomogeneity parameters ξ i are implicit and omitted for brevity. In the auxiliary space, the matrix reads ( ) A(λ) B(λ) T (λ) = (3.6) C(λ) D(λ) with the operators A,B, C and D acting on H the Hilbert space of the system. As we see more in detail hereafter, these four operators play a major role either for the creation of states with B and C or for the diagonalization of the system, A + D being the transfer matrix which commutes with the Hamiltonian. In order to formulate their commutation relations which will be helpful later, we define implicitly the quantum R-matrix with the commutation of two L-matrices on the same site j but with distinct auxiliary space and parameters (denoted by 1 and 2): R 12 (λ 1, λ 2 )L j,1 (λ 1, ξ j )L j,2 (λ 2, ξ j ) = L j,2 (λ 2, ξ j )L j,1 (λ 1, ξ j )R 12 (λ 1, λ 2 ). (3.7) A necessary condition for this equation is that the L-matrix corresponds to an integrable model. The R operator acts in the tensor product of two auxiliary vector spaces, V 1 V 2 and both are associated to a spectral parameter λ and µ respectively. By denoting the vector space where the operator acts with a subscript: R 12 (λ 1, λ 2 ), this last relation gives the commutation relation between two monodromy matrices of two spectral parameters and acting on two different auxiliary spaces (subscripts 1 and 2): R 12 (λ 1, λ 2 )T 1 (λ 1 )T 2 (λ 2 ) = T 2 (λ 2 )T 1 (λ 1 )R 12 (λ 1, λ 2 ). (3.8) If we consider now, the product of three monodromy matrices T 1 (λ 1 )T 2 (λ 2 )T 3 (λ 3 ), there is two different ways to intertwine the space 1 and 3 with the use of three R-matrices (omitting the spectral parameters for brevity): T 3 T 2 T 1 = R 12 R 13 R 23 T 1 T 2 T 3 R23 1 R 1 13 R 1 12 = R 23 R 13 R 12 T 1 T 2 T 3 R12 1 R 1 13 R (3.9) 36 [0]

39 3.2 The sufficient condition to satisfy this equality is to impose to the R-matrix the Yang-Baxter relation: R 12 (λ 1, λ 2 )R 13 (λ 1, λ 3 )R 23 (λ 2, λ 3 ) = R 23 (λ 2, λ 3 )R 13 (λ 1, λ 3 )R 12 (λ 1, λ 2 ). (3.10) In order to recover the fundamental spin models, V 0 is taken isomorphic to C 2 and we construct the L-matrix directly from the R-matrix: L j (λ, ξ j ) = R 0,j (λ, ξ j ) (3.11) where the subscripts indicate that the R-matrix acts in the tensor product of the auxiliary space with the jth site Hilbert space and where the quantum R-matrix is R(λ, µ) = b(λ, µ) c(λ, µ) 0 0 c(λ, µ) b(λ, µ) (3.12) with b(λ, µ) = φ(λ µ) φ(λ µ + η), c(λ, µ) = φ(η) φ(λ µ + η). (3.13) The definitions of variable η and the function φ depend on the anisotropy regime. For the XXX spin chain, they take the value η = i, φ(x) = x with x C. (3.14) Moreover, following from (3.12), the R-matrix has the property that if its two arguments are identical, it becomes the permutation operator which exchanges vector spaces 1 and 2: R 1,2 (λ, λ) = P 1,2. (3.15) From (3.5), the monodromy matrix can then be expressed as a product of R- matrices T (λ) = R 0,N (λ, ξ N )... R 0,1 (λ, ξ 1 ) (3.16) and to finally recover the Heisenberg model, we have to take the homogeneous limit which consists of taking all the inhomogeneity parameters to ξ j = η 2, j. Tracing then (3.8) over the auxiliary space 1 and 2 leads to the fundamental property of the τ(λ) matrix in (3.3) and with (3.6), (3.8) is also a compact form of the commutation relations between the operators A,B,C and D. For instance, we show here four of them explictly [B(λ), B(µ)] = 0, [C(λ), C(µ)] = 0, (3.17) A(λ)B(µ) = b 1 (λ, µ) ( B(µ)A(λ) c(λ, µ)b(λ)a(µ) ) (3.18) D(λ)B(µ) = b 1 (µ, λ) ( B(µ)D(λ) c(µ, λ)b(λ)d(µ) ). (3.19) 37

40 Trace identity and Bethe equations The τ(λ) matrix is actually identified as the transfer matrix which describes and provides, through the trace identity that we introduce in (3.2), the momentum operator and the Hamiltonian of the spin chain in the homogeneous limit (2.1): P = i log(τ(λ)) (3.20) λ= η 2,ξj= η 2 j H = φ(η) d 2 dλ log(τ(λ)) λ= JN η 2,ξj= η 2 j 4 hs z. (3.21) [ With the commutation relations (3.3) and τ(λ), ] j Sz j = 0 and with (3.2), the eigenstates of H are therefore given by the common eigenvectors of τ(λ) for a fixed λ and of the magnetization j Sz j. In order to construct such eigenvectors, we suppose the existence of a generating vector (pseudovacuum) such that A(λ) 0 = a(λ) 0, D(λ) 0 = d(λ) 0, C(λ) 0 = 0, B(λ) 0 0. (3.22) Considering the spin chain model, the pseudo-vacuum 0 corresponds to the state with all spins aligned up and following the defintion (3.12) we have the pseudovacuum eigenvalues a(λ) = 1, d(λ) = N j=1 b(λ, ξ j). We suppose then that one creates states of H with down spins by acting with the B operator or respectively C operator for dual state: M B(λ i ) 0 = {λ} i=1 M 0 C(λ i ) = {λ}. (3.23) i=1 We know from the commutation relation (3.19) that acting with an A or D operator on the {λ} gives A(µ) {λ} = Λ {λ} + D(µ) {λ} = Λ {λ} + M Λ n B(µ) n=1 M Λ n B(µ) n=1 M i=1,i n M i=1,i n B(λ i ) 0 (3.24) B(λ i ) 0 (3.25) with M Λ = a(µ) b 1 (µ, λ i ), i=1 M Λ = d(µ) b 1 (λ i, µ), i=1 Λn = a(λ n ) c(λn,µ) b(λ n,µ) M i=1,i n b 1 (λ n, λ i ) Λn = d(λ n ) c(µ,λn) b(µ,λ n) M i=1,i n b 1 (λ i, λ n ). (3.26) 38

41 3.3 In consequence, if a state is an eigenvector of (A+D)(λ) the off-diagonal part must vanish i.e. Λ j = Λ j for all j = 1,..., M. This condition leads to the same Bethe equations as for the coordinate Bethe Ansatz (2.29): a(λ j ) d(λ j ) i=1,i j b(λ j, λ i ) = 1, j = 1,..., M. (3.27) b(λ i, λ j ) Therefore, the states {λ} satisfying the algebraic Bethe equations are eigenstates of the Hamiltonian with the momentum and energy given in (2.33) and (2.37). Moreover, one deduces from (3.25) their transfer matrix eigenvalues: ( ) M M (A + D)(µ) {λ} = a(µ) b 1 (µ, λ i ) + d(µ) b 1 (λ i, µ) {λ}. (3.28) i=1 3.3 Local spin operators In order to compute correlation functions along the spin chain with the algebraic Bethe Ansatz, one has to address the problem of the representation of a local spin operator. For the calculation of spin operator matrix elements such as {µ} Sj α {λ}, a formulation of the spin operator as function of A, B, C and D would be helpful because their commutation relations can then be used to calculate the matrix element. However, the non-local character of these operators makes the task very non-trivial. For instance, the B operator as function of Pauli matrices reads [69] i=1 N B(λ) = Ω j σ j j=1 + j k l Ω j,k,l σ j σ k σ+ l + O ( σ (σ σ + ) 2) (3.29) with Ω j and Ω j,k,l diagonal operators acting on all sites but, respectively, j and j, k, l. The definition of the Pauli matrices are extended to the entire spin chain σj α = j 1 i=1 1 σα N i=j+1 1. The sum over all the sites make the spin lowering non-local and the higher terms including more spin-exchange operators make the operator also highly non diagonal. However following the demonstration in [69] that we summarize here, we show how local representations of the spin operator can be accessed. A more general formulation for the expressions of local spin and field operators can be found in [70]. We first give two properties that we use further. defined in (3.15) on a R matrix gives First, using the permutation P 0,j R 0,k (λ, ξ k )P 0,j = R j,k (λ, ξ j ). (3.30) Second, in [71], a new basis is introduced to make the monodromy matrix invariant under permutation. In this so called F -basis, T (λ) is completely symmetric under simultaneous permutations of the chain site j and of the corresponding ξ j. If 39

42 3.3 we write the F -basis transformation matrix F 0,1...N (λ; ξ 1,..., ξ N ), the monodromy matrix in the new basis reads (written with all parameters and arguments explicit) F 1...N (ξ 1,..., ξ N )T 0,1...N (λ; ξ 1,..., ξ N )F N (ξ 1,..., ξ N ) = T 0,1...N (λ; ξ 1,..., ξ N ) (3.31) and the symmetry under permutation reads with π any element of S N. T 0,π(1)...π(N) (λ; ξ π(1),..., ξ π(n) ) = T 0,1...N (λ; ξ 1,..., ξ N ) (3.32) We express then hereafter the local spin operator in terms of the A, B, C, D operators. Considering the trace over the vector space 0 of the product between a Pauli matrix σ α 0 acting in the auxiliary space and the monodromy matrix with the spectral parameter equal to ξ j, we then write T 0,1...N (ξ j ) in the F -basis and we perform a permutation on the chain sites: tr 0 ( σ α 0 T 0,1...N (ξ j ) ) = F N tr 0( σ α 0 T0,1...N (ξ j ) ) F 1...N = F N tr 0( σ α 0 T0,j j+1...n 1...j 1 (ξ j ) ) F 1...N. (3.33) We then transform back the monodromy matrix in the original basis and we express it as a product of R-matrices (3.16). The factor R 0,j (ξ j, ξ j ) being the permutation operator between the 0 space and the jth vector space and thanks to the cyclicity property of the trace, we act with this permutation over the Pauli matrix and all the R-matrices in the trace (3.30): tr 0 ( σ α 0 T0,j j+1...n 1...j 1 (ξ j ) ) = F j...j 1 tr 0 ( σ α 0 R 0,j 1 (ξ j, ξ j 1 )... P 0,j ) F 1 j...j 1 = F j...j 1 tr 0 ( σ α j R j,j 1 (ξ j, ξ j 1 )... R j,j+1 (ξ j, ξ j+1 )P 0,j ) F 1 j...j 1 = F j...j 1 σ α j R j,j 1 (ξ j, ξ j 1 )... R j,j+1 (ξ j, ξ j+1 )F 1 j...j 1 (3.34) where in the last step, we used that tr 0 (P 0,j ) = 1 j. We then use the Lemma 4.1 in [69] and the identity N j=1 (A + D)(ξ j) = 1 to rewrite F N F j...j 1 = N i=j (A + D)(ξ i) F 1 j...j 1 F 1...N = j 1 i=1 (A + D)(ξ i) R j,j 1 (ξ j, ξ j 1 )... R j,j+1 (ξ j, ξ j+1 ) = (A + D)(ξ j ). (3.35) With (3.6), one shows that this is equal to a linear combination of A, B, C, D: ( tr 0 σ α 0 T 0,1...N (ξ j ) ) B(ξ j ) for α = = C(ξ j ) for α = + (A D)(ξ j ) for α = z (3.36) Combining the results, we eventually give the expression of the three components 40

43 3.4 of the local spin operator σ j σ + j (A + D)(ξ i )B(ξ j ) = j 1 i=1 (A + D)(ξ i )C(ξ j ) = j 1 i=1 N i=j+1 N i=j+1 j 1 σj z = (A + D)(ξ i )(A D)(ξ j ) i=1 (A + D)(ξ i ) (A + D)(ξ i ) N i=j+1 (A + D)(ξ i ). (3.37) With these formulas in combination with the expressions for the state and dual state (3.23), we are able to express form factors of the spin chain in term of the operators A, B, C, D. This is done for for two specific operators in chapter Norm and Scalar product Norm of an eigenstate The norm of an eigenstate {λ}, known as the generalized Gaudin hypothesis [72, 73], reads N({λ}) = φ(η) M M α β φ(λ α λ β + η) φ(λ α λ β ) with the determinant of the so-called Gaudin matrix [54]: Φ a,b ({λ}) = δ a,b [N λa θ 1 (λ a ) det Φ({λ}) (3.38) ] M λa θ 2 (λ a λ k ) + λa θ 2 (λ a λ b ). (3.39) k=1 We should mention here that since C(λ) is not the hermitian conjugate of B(λ), the norm formula in (3.38) is generally a complex number (see [73] chapter 10.1). This definition is equal to the norm squared of the wave function, only for the Bose gas model,. However, in the calculation of dynamical structure factor, the norm squared gives the correct real number. Whereas this definition allows one to calculate the norm of an eigenstate with real rapidities, the determinant of the Gaudin matrix (3.39) becomes degenerate for an eigenstate which contains complex string rapidities. Indeed columns of the matrix have in consequence terms of order 1/δ, inverse of the string deviations. We manipulate here the columns and extract the leading order terms of O (1/δ) which cancel with the additional singular prefactors (φ(λ α λ β + η)) in (3.38), and we eventually reproduce a reduced expression independent of the string deviations [74, 61]. 41

44 3.4 Let s suppose that the eigenstate contains a n j string in the rapidities. In a general anisotropy case, the rapidities belonging to that n j string with a deviation δα j,a are denoted λ j,a α = λ j α η 2 (n j + 1 2a) + π 4 (1 ν j) + iδα j,a with ν j the parity of the string. One recovers the isotropic string structure in (2.34) if we put η = i and ν j = 1. We consider first only the n j rows and columns of the Gaudin matrix that correspond to the string rapidity. The elements of the submatrix are Φ nj α a,n j α a({λ}) k a,a±1 4 = N 1 + 4(λ j,a α ) (λ j,a k α λ j 2 α,k )2 1 + (λ j,a k,n,β α λ n β,k )2 }{{} Φ a,a ( implicit nj,α) 1 + O(δα j,a ) 1 + O(δα ( ) j,a ) ( ) (a + 1 a) δα j,a+1 δα j,a (a 1 a) δα j,a 1 δα j,a }{{} 1/δ aa+1 +1/δ aa 1 ( implicit n j,α) Φ nj α a,n j α (a±1) = 1/δ a,a±1. (3.40) From the definition we have δ a,a 1 = δ a 1,a and as we describe only the row and lines of the n j string, we keep hereafter the two indices α, n j implicit. With these notations, the n j string submatrix reads 1 δ + Φ δ Φ δ 12 δ δ + Φ δ Φ Φ δ 23 δ δ + Φ δ.... (3.41) We now operate successively on the Gaudin matrix two transformations which do not modify the value of its determinant : first, adding the first column to the second column and second, adding the first row to the second row. The resulting submatrix is 1 δ + Φ Φ 11 Φ Φ 1 11 δ + Φ Φ 11 1 δ + Φ Φ 24 + Φ Φ δ + Φ δ δ + Φ δ (3.42) In consequence, considering the entire modified Gaudin matrix Φ, we can now extract a term of O(1/δ) if we use the Laplace s formula for the determinant: det(φ ) = k,β,b( 1) [n k β b,n j α 1] Φ n k β b,n j α 1minor(Φ ) (nk β b,n j α 1) = 1 δ 12 minor(φ ) (nj α 1,n j α 1) + O(1) (3.43) 42

45 3.4 where minor(m) (m,n) is the determinant of the matrix M from which one removes the mth row and the nth column. In minor(φ ) (nj α 1,n j α 1) the first row and first column of the n j string submatrix are removed and the remainder is 1 δ + Φ Φ 22 1 δ + Φ Φ 14 + Φ δ + Φ δ δ + Φ δ Φ Φ 41 + Φ δ 34 δ δ + Φ δ.... (3.44) Doing the same two successive operations again, the determinant of the modified Gaudin matrix becomes det(φ ) = 1 δ 12 δ 23 minor(φ ) (nj α 1,n j α 1)(n j α 2,n j α 2) + O(1/δ) (3.45) where in this notation for the minor of Φ, one removes the two rows and two columns mentioned. This row-column manipulation is done until only one matrix element remains in the n j string submatrix i.e.: Φ n j α,n j α = n j a=1 Φ aa + 2 a 2 b Φ ab (implicit n j, α) and where the reduced Gaudin matrix determinant reads: det(φ ) = n j 1 i=1 1 δ ii+1 minor(φ ) (nj string submatrix) + O(1/δ nj 1 ) (3.46) where minor(φ ) (nj string submatrix) is the determinant of the reduced Gaudin matrix where all the first n j 1 rows and columns of the n j string submatrix have been cut off and ther remains only the element Φ n j α,n j α. During the operations to reduce the Gaudin matrix, the elements outside the string submatrices are also modified. We schematically show here how the reductions of the string submatrices change the entire Gaudin matrix. The original Gaudin matrix with two string submatrices reads explicitly Φ njα 1,n jα 1... Φ njα 1,n jα n j Φ njα1,n k β1... Φ njα1,n k βn k Φ njα n j,n jα 1... Φ njα n j,n jα n j Φ njαn j,n k β1... Φ njαn j,n k βn k Φ nk β 1,n jα 1... Φ nk β 1,n jα n j Φ nk β 1,n k β 1... Φ nk β 1,n k β n k Φ nk β n k,n jα 1... Φ nk β n k,n jα n j Φ nk β n k,n k β 1... Φ nk β n k,n k β n k. (3.47) The reduction of the first submatrix yields Φ nj nα,nα a=1 Φ nj n jα a,n k β1... a=1 Φ n jα a,n k βn k nj a=1 Φ n k β 1,n jα a Φ nk β 1,n k β 1... Φ nk β 1,n k β n k nj a=1 Φ n k β n k,n jα a Φ nk β n k,n k β 1... Φ nk β n k,n k β n k 43 (3.48)

46 3.4 and after reduction of the second submatrix, the Gaudin matrix becomes Φ nj,n k nα,nα a,b=1 Φ n jα a,n k β b nk,n j b,a=1 Φ n k β b,n jα a Φ. (3.49) n k β,n k β With the simplifications and n j n j a 2 a 2 2 Φ ab = Φ ab + a=1 Φ aa + a=1 n j b a=1 b {a,a±1} n j a=1 b n j Φ ab = N a=1 n j n k a=1 b=a (λ j,a α Φ ab = ) 2 n j a=1 b {a,a±1} n j a=1 k,β,b 1 + (λ j,a α Φ ab (3.50) 2 λ k,b β )2, (3.51) the elements of the Gaudin matrix become n j Φ n jα,n k β = δ njα,n k 4 β N 1 + 4(λ j,a α ) 2 a=1 n j + ( 1 δ njα,n k β a=1 k,β,b ) nj,n k a,b=1 1 + (λ j,a α 2 λa θ 2 (λ j,a α λ k,b β )2 λ k,b β ) = δ njα,n k β N λ j θ nj (λ j α) α n k λ j Θ nj,n k (λ j α λ k β),β α + ( ) 1 δ njα,n k β λ j Θ nj,n k (λ j α λ k β). (3.52) α The norm of a state including a string is eventually expressed using 3.38 and 3.46 as N s N({λ}) = φ(η) M N s M j j=1 α=1 a=1 M j,m k n j,n k j,k=1 α,β=1 a,b=1 ((n j,α,a) (n k,β,b)) n j N s = φ(η) M j,k=1 δ j,a+1 α 1 δ j,a α φ( η) M kn k det Φ ({λ}) φ(λ j,a α φ(λ j,a α φ(λ j,a α λ k,b β + η) λ k,b β ) λ k,b β + η) α,β,a,b φ(λ j,a α λ k,b β ) ((n j,α,a) {(n k,β,b),(n k,β,b+1)}) 44 det Φ ({λ}) (3.53)

47 3.4 with Φ the reduced Gaudin matrix (3.52). This result is not new and has already been calculated in [74] but the purpose was to give a clear and explicit development for the calculation for the reduced Gaudin matrix Scalar product We show here the expression for the scalar product which has been originally calculated by Slavnov [75]. This result has been also re-derived later by Kitanine et al. in a more direct way using the F -basis [69]. The scalar product between a state {µ} solution of the Bethe equations (3.27) and an arbitrary state {λ} is written with a determinant representation: {µ} {λ} = det H({µ}, {λ}) M j>k φ(µ k µ j )φ(λ j λ k ) (3.54) with the M M matrix φ(η) H ab = a(λ b ) φ(µ k λ b + η) d(λ b ) φ(µ k λ b η). φ(µ a λ b ) k a k a (3.55) In summary, we have given in this chapter an introduction to the construction of Bethe states and local spin operators within the formalism of algebraic Bethe Ansatz. In chapter 5, we compute through these latter results, the form factors of two adjacent spin operators and of the spin-exchange operator. In addition, we have recalled in the last section of the chapter how to compute the norm of a Bethe state and how to reduce the Gaudin matrix in case of states containing string structures. 45

48 3.4 46

49 Chapter 4 Probing the Heisenberg spin chain In order to know which quantities are measurable in the spin chain, we have to understand how the experimental setups probe the latter and which correlation functions are involved. For this purpose, we shortly explain in this chapter how two different observation methods based on neutron scattering or on x-ray scattering, provide a response function proportional to two different correlation functions. These two setups probe therefore different dynamics in the spin chain. 4.1 Inelastic neutron scattering The use of neutrons to probe the magnetic properties of matter is optimal: they do not interact with the charges of the system but they interact with the microscopic magnetic degrees of freedom through the dipole-dipole interaction. The dynamical correlation function can then be related at the first order to the cross section of the neutron scattering on the crystal. The neutrons needed to probe condensed matter have an energy of order of 1meV and are called slow neutrons. A key parameter to have an efficient measurement of a sample is the luminosity of the beam. The quality and the accuracy of the results are directly related to the counting rate of neutrons and the processes providing enough brightness are known to be nuclear reactions Neutron sources and apparatus The neutrons are produced by two different sources [76]. As a first way, the spallation sources are based on an accelerator which produces high energy proton pulsed beams colliding with heavy nuclei. The spallation term means that the de Broglie 47

50 4.1 Detectors Monochrom. crystal Time-of-flight Monochrom. Moderator Sample Figure 4.1: Experimental setup of time-of-flight spectroscopy. length of the protons is smaller than the size of the nuclides and the scattering processes produce intra-nuclei cascades which produce neutrons by evaporation of the nucleus. Due to the short time release of spallation neutrons, the time distribution of the source follows the proton beam pulses. A second possible source of neutrons are nuclear reactors. The Uranium fission reaction is self-sustained and releases a number of exceeding neutrons during each nuclear fission event. The nuclear fission that happens in a reactor produces a continuous beam of particle with a energy in order of 1MeV. The energies of neutrons coming out of the two sources mentioned here are orders of magnitude above the 1meV of the slow neutrons energy which are needed for the measurement. The beams must therefore be slowed down in a moderator where they thermalize within a few microseconds. The sources being isotropic, the neutrons are emitted in all directions and in order to reduce the angular divergence, one uses collimators made of neutron opaque material. They restrict then the outgoing beam to only a small solid angle. To control the magnitude of the wavevector of the neutrons, different kinds of monochromators are used. Mechanically moving slits which allow only the particles with a given velocity range to pass trough, are called time-of-flight monochromators whereas crystals which reflect the neutron via a momentum selective Bragg scattering are called monochromator crystals. In order to count the number of scattered particles, one uses detectors based on the ionization current produced by the collision with nuclei in a gas. The strong electric field applied along the gas tube, induces a chain reaction and the resulting strong amplification makes the neutrons reliably detectable. We give as example of neutron scattering experimental setup, a time of flight spectrometer such as used in [46]. In figure 4.1, the path of the neutron beam is sketched, starting from the moderator located right after the source, through monochromators, scattering on the sample and ending on the detectors. We omit in this drawing the collimators and the neutron guides which allow to keep the flux luminosity along the experimental setup. The setup allows one to determine the energy loss ω = ω out ω in and the relative momentum q = q out q in of the scattered neutrons. 48

51 Microscopic scheme The experimental realization allows one to make neutron scattering in a window of the measured momentum transfer q and energy transfer ω. With these two known quantities, the interesting part of the experiment is the measurement of the scattering section for a infinitesimal window of solid angle dω along q out and of the energy loss dω. This differential cross section can be divided between two distinct contributions [77]: d 2 σ dωdω = d2 σ dωdω + d2 σ coherent dωdω. (4.1) incoherent Whereas the incoherent part provides signal from the individual dynamics of the atoms and is related to diffusion, the coherent part yields information about collective excitations either elastic like the Bragg peak or inelastic like phonons and magnons excitations. Using neutron scattering in order to push forward the understanding of the highly correlated systems, we are naturally interested in the coherent part. Regarding condensed matter physics, neutron scattering on the crystal represents a small perturbation of the system. We can therefore use Fermi s Golden Rule to quantify the cross section of the coherent magnetic scattering: d 2 σ dωdω λis i,s f = q out q in λ f f l 2 e iq r l U sis f l λ i δ(ω + E i E f ) (4.2) where λ i,λ f are the initial and final state of the sample with the corresponding energies E i,e f and where U sis f l is the matrix element for the atomic scattering on site l between the initial and final neutron spin states, s i and s f. By the use of Van Hove s result [78], we can express the cross section for the scattering of unpolarized neutrons as a function of the spin correlator in the crystal: d 2 σ dωdω α,β {x,y,z} with the dynamical structure factor (δ α,β q αq β q 2 )Sαβ (q, ω) (4.3) S αβ (q, ω) = 1 2π dte iωt l e iq r l S0 α (0)S β l (t). (4.4) In the context of the Heisenberg spin chain, inelastic neutron scattering (INS) measurement was in agreement with the theory predictions for KCuF 3 [43, 28], CuPzN [44], Sr 2 CuO 3 [45, 46] and Cu 3 (CO 3 ) 2 (OH) 2 [79] We show in chapter 6 an example of calculation of the dynamical structure factor in the Heisenberg spin chain as it would be measured by INS. 49

52 Resonant inelastic x-ray scattering In view of the increase in brightness of the new generation synchrotrons and, consequently, of the capabilities to produce high flow x-ray beams, resonant inelastic x-ray scattering (RIXS) has developed rapidly into a new probe of matter. Tuning the scattered x-ray to a resonant mode of the substrate, RIXS benefits therefore from a high photon absorption rate. For instance, the synchrotron can be set up to make a so-called K-edge transition in transition metal ions (5 10keV ), i.e. the absorbed photon excites a 1s core electron into a 4p high orbital state. The 4p state being at high energy, far above the Fermi surface, this transition does not cause direct excitations in the electrons at the conduction band (3d). However, the hole left in 1s creates a localized Coulomb potential which interacts with the valence 3d electrons. The locality of the photon absorption is an assumption that is generally accepted and widely treated as an Anderson model [80, 81]. In addition, the high energy of the intermediate state induces a very short life time for the excitation and within a few femtoseconds, the 4p electron recombines with the 1s core hole and emits an outgoing photon [82]. The so-produced mechanism of indirect excitation can probe collective behaviors of the band electrons such as charge, orbital, or spin of which the energy and momentum are experimentally determined by the energy and momentum difference between the incoming and outgoing photons. The cross section for the resonant scattering process is given by the Kramers- Heisenberg relation [83, 84]. As a function of photon energy loss ω = ωin 0 ω0 out and momentum transfer q = q in q out the scattering cross section reads d 2 σ dωdω = A fi 2 δ(w E f + E i ), (4.5) res f where the scattering amplitude is A fi = ω res n f ˆD n n ˆD i ω 0 in ω res E n iγ. (4.6) Here i, n, f are the initial, intermediate and final states with respective energies E i, E n, E f. The resonance energy is ω res and the dipole operator is ˆD = j e iqrj s j p j + j e iqrj s j p j (4.7) where R j is the spatial coordinates of the core-hole and where j is an index running through the whole lattice and where q is the momentum operator acting on the photon Hilbert subspace. The operators s ( ) j and p ( ) j annihilate (or create) respectively a 1s and 4p electron Ultrashort Lifetime Expansion We use here the fact that the intermediate state is not a steady state and therefore has a very short lifetime. This means that in Eq. 4.6, the energy broadening Γ 50

53 4.2 Figure 4.2: This sketch outlines the 1D magnetic active part of Sr 2 CuO 3. We show here a layer of the crystal and in the bulk, each chain is isolated from the layers above and below by Strontium atoms. which is the inverse of the core-hole lifetime is very large and we suppose that Γ E n. We can then expand the scattering amplitude (4.6) in a power series [85]: ω res ( ) l E n A fi = f ω in ω res iγ ω in ω res iγ ˆD n n ˆD i = ω res ω in ω res iγ l l n f ˆD (H 0 + H c ) l ˆD i (ω in ω res iγ) l (4.8) where H 0 is the generic Hamiltonian of the band electrons and H c is the local Coulomb coupling between the 1s core-hole and the 3d electrons. The scattering amplitude can then be well approximated by the leading term of the series which is a form factor between the initial state and the final state. The determination of the leading term depends on the ratio between the core-hole potential and the band width [82] Spin chain Considering copper oxide metal at low enough temperature, the material becomes a Mott-insulator where the remaining low-energy degrees of freedom are spins. The superexchange process between electrons localized in two neighboring sites is equivalent to an antiferromagnetic spin-exchange interaction and we can then describe the spin dynamics of the 3d electrons with the antiferromagnetic spin-1/2 Heisenberg model. Motivated by the successful correspondence between predictions and experiments for the 2D Heisenberg antiferromagnet [86, 87, 88, 89], we set out the computation 51

54 4.2 Initial Intermediate Final 4p 3d J J J J J J 1s q in, ωin 0 q out, ωout 0 j 1 j j +1 j 1 j j +1 j 1 j j +1 Figure 4.3: Mechanism by which double-spin flip transitions are created in the indirect magnetic RIXS process. of the response function in the 1D case. As a possible realization, the magnetically active part of Sr 2 CuO 3 are chains of corner-sharing CuO 4 plaquettes (see illustration in figure 4.2). The exchange interaction J between adjacent Cu spins on the chains is extremely large ( mev); the interchain exchange interaction is about 5 orders of magnitude smaller. It is therefore a realization of a spin-1/2 antiferromagnetic Heisenberg chain over a wide temperature range, with high energy magnetic excitations that are in principle accessible to Cu K-edge RIXS, using today s energy resolutions. In this case a magnetic scattering process as sketched in Fig. 4.3 can occur [85, 86]. An x-ray incoming on site j produces a K-edges transition and creates a core-hole on the jth copper ion of the chain. In the intermediate state, the presence of the core-hole modifies the 3d on-site energy levels through the Coulomb interaction. In the Mott-insulating limit, this perturbation modifies the actual superexchange coupling J between the spins. If U is the original on-site double occupancy potential which is lowered by U c on the jth site and raised by U c on the (j 1)th and (j + 1)th sites, the perturbed superexchange coupling reads J c = 2t 2 ( 1 U U c + 1 U + U c ) (4.9) and therefore it modifies the spin-exchange interaction with the two neighbor spins. In this case, we can write explicitly the Hamiltonian and the core-hole coupling term: H 0 = J j S j S j+1 H c = j s j s j (J c J) (S j 1 S j + S j S j+1 ). (4.10) 52

55 4.2 With a value of Γ 3 4 ev [90] and J c J J 1 [86], keeping only the leading term l = 1 in (4.8) provides [ still a good approximation. Choosing the initial state energy to be zero, with H 0, ˆD ] = 0 and since there is no core-hole in the initial and final states, we can simplify the matrix element of (4.8) for l = 1 as f ˆD (H 0 + H c ) ˆD i J c J = i,j,k f e iqoutri s i p is j s j (S j 1 S j + S j S j+1 ) e iqinr k s k p k i = f j e iqrj (S j 1 S j + S j S j+1 ) i (4.11) where q = q in q out, we used s j s k i = δ j,k i and similarly for p j operators. With this last result, we can finally express the x-ray scattering cross section at resonance (4.5) to the leading order d 2 σ dωdω ω res J c 2 ( J (J res Γ iγ + ω S exch c J) 2 ) (q, ω) + O Γ 3. (4.12) It incorporates the spin-exchange dynamical structure factor S exch (q, ω) = 2π α 0 X q α 2 δ(ω ω α ), (4.13) where the spin chain ground state is 0, the excited states α, excitation energies ω α = E α E GS and the spin-exchange operator is defined X q = 1 e iqj (S j 1 S j + S j S j+1 ). (4.14) N j In the chapter 6, we show how to calculate this spin-exchange dynamical structure factor and we explicitly compute the relevant magnetic response function for indirect RIXS in the Heisenberg chain. 53

56 4.2 54

57 Chapter 5 Adjacent spin operators form factors and correlation functions Using the eigenstates constructed with the Bethe equations (chapter 2) and using the formulation of the eigenstates and local spin operators in an algebraic form (chapter 3), we construct an explicit determinant formula for the form factor of the operators S j S j+1 and Sz j Sz j+1. With these results we show how to calculate the two adjacent pairs correlation function. We compute as an example, the map in momentum and energy of the dynamical structure factor for the two form factors at different magnetization and we analyze the contribution of each kind of fundamental excitation. 5.1 S j S j+1 We describe in this section how, using the Algebraic Bethe Ansatz, one can represent and evaluate the S j S j+1 dynamical structure factor (DSF) defined as the Fourier transform of the S j S j+1 correlator: S ++ (q, ω) = 1 N N j,j = Form factor Using the result for the local spin operators (3.37): σ j = j 1 dte iq(j j )+iωt S j (t)s j+1 (t)s+ j (0)S+ j +1 (0). (5.1) (A + D)(ξ i )B(ξ j ) i=1 55 N i=j+1 (A + D)(ξ i ), (5.2)

58 5.1 we express the form factor of S j S j+1 between two eigenstates as {µ} M S j S j+1 {λ} M 2 = {µ} M σ j σ j+1 {λ} M 2 j 1 = {µ} M (A + D)(ξ i )B(ξ j ) i=1 j (A + D)(ξ i )B(ξ j+1 ) i=1 j 1 = {µ} M (A + D)(ξ i )B(ξ j )B(ξ j+1 ) i=1 N i=j+2 N i=j+2 N i=j+1 (A + D)(ξ i ) (A + D)(ξ i ) {λ} M 2 (A + D)(ξ i ) {λ} M 2. (5.3) Introducing φ j ({µ}) = j i=1 M k=1 b 1 (µ k, ξ i ), we evaluate the transfer matrix on the two eigenstates with the eigenvalues Λ+ Λ (3.25). The vacuum eigenvalue being d(ξ i ) = 0, i, we then have: j 1 {µ} [A + D] (ξ i ) = φ j 1 ({µ}) {µ} (5.4) N i=j+2 i=1 [A + D] (ξ i ) {λ} = j+1 i=1 k=1 M b(µ k, ξ i ) {λ} = φ 1 j+1 ({λ}) {λ}. (5.5) Using then the formula for the scalar product (3.54), we can write the form factor as a determinant: M 2 φ j 1 ({µ}) φ j+1 ({λ}) {µ} M B(ξ j )B(ξ j+1 ) B(λ i ) 0 = φ j 1({µ}) φ j+1 ({λ}) {µ} M {ξ j, ξ j+1, λ i } M = φ j 1({µ}) φ j+1 ({λ}) with the M M matrix i det H({µ} M, {ξ j+1, ξ j, λ i } M ) M j<k φ(µ k µ j ) M j>k φ(λ (5.6) k λ j ) λm 1 =ξ j,λ M =ξ j+1 ( φ(η) φ(µ a λ b ) k a φ(µ k λ b + η) d(λ b ) ) k a φ(µ k λ b η), b < M 1 φ(η) H ab = φ(µ a ξ j)φ(µ a ξ j+η) k φ(µ k ξ j + η), b = M 1 φ(η) φ(µ a ξ j+1)φ(µ a ξ j+1+η) k φ(µ k ξ j+1 + η), b = M. (5.7) 56

59 5.1 We then express the product explicitly as M M 1 φ(λ k λ j ) = φ(λ k ξ j ) j>k λm 1 =ξ j, λ M =ξ j+1 M 2 = φ(ξ j+1 ξ j ) k k M 2 φ(λ k ξ j ) k λ M 1 =ξ j+1 M 1 j>k M 2 φ(λ k ξ j+1 ) φ(λ k λ j ) j>k λm 1 =ξ j+1 φ(λ k λ j ). (5.8) Homogeneous limit The spin chain models correspond to the homogeneous case where ξ i η/2 i. This limit should be taken with care in order to obtain a finite expression for the ratio det H({λ}, {µ c m,n, ξ j, ξ j+1 })/φ(ξ j+1 ξ j ). We first simplify the H matrix in (5.7) by factorizing out of the determinant the products k φ(µ k ξ... + η) in columns M 1 and M and we define a H matrix: det(h) = k φ(λ k ξ j + η)φ(λ k ξ j+1 + η) det H (5.9) We use then l Hôpital s rule to have a well-defined homogeneous limit: lim ξ j,ξ j+1 η/2 det H φ(ξ j+1 ξ j ) = k φ(λ k η/2) 2 lim ξ j,ξ j+1 η/2 ξj det H ξj φ(ξ j+1 ξ j ). (5.10) Using Laplace s formula over the column M 1 of the determinant, we have ξj det H = ξi ( 1) a Ha(M 1) minor( H) (a,m 1) a = a ( 1) a ( ξi Ha(M 1) ) minor( H) (a,m 1) + a ( 1) a Ha(M 1) ξi minor( H) (a,m 1) (5.11) where minor(a) (a,b) is the determinant of the A without row a and column b. Because the columns M 1 and M of the matrix H are identical in the homogenous limit, the second term vanishes when ξ j+1 ξ j and we eventually only differentiate column M 1 of the H matrix. The resulting formula after the limit reads lim ξ j,ξ j+1 η/2 1 φ(ξ i+1 ξ i ) det H({µ} M, {ξ j+1, ξ j, λ i } M ) M 2 M k φ(λ k ξ j+1 ) j<k φ(µ k µ j ) M 2 j>k φ(λ k λ j ) M k = φ(µ k + η/2) 2 det H ({µ} M, {λ i } M 2) M 2 i φ(λ i η/2) 2 M j<k φ(µ k µ j ) M 2 j>k φ(λ k λ j ) (5.12) 57

60 5.1 with H ab = ( φ(η) φ(µ a λ b ) k a φ(µ k λ b + η) d(λ b ) ) k a φ(µ k λ b η), b < M 1 φ(η)φ(2µ a )φ 2 (µ a η/2)φ 2 (µ a + η/2), b = M 1 φ(η)φ 1 (µ a η/2)φ 1 (µ a + η/2), b = M. Finally, the form factor reads (5.13) {µ} M σ j σ j+1 {λ} M 2 = φ j 1({µ}) lim φ j+1 ({λ}) {µ} M B(ξ j )B(ξ j+1 ) {λ} M 2 ξ j,ξ j+1 η/2 = φ M j 1({µ}) k φ(µ k + η/2) 2 det H ({µ} M, {λ i } M 2) φ j 1 ({λ}) M 2 i φ(λ i + η/2) 2 M j<k φ(µ k µ j ) M 2 j>k φ(λ k λ j ). (5.14) Reduced H for string solutions If one considers {λ} states which include n-strings structures in a general anisotropy form: λ j,a α = λ j α η 2 (n j + 1 2a) + i π 4 (1 ν j) + iδα j,a. Here ν j is the parity of the string and if we put η = i and ν j = 1, one recovers the isotropic string structure introduced in (2.34). H contains either singular factors of order 1/δ or indeterminate factors δ/δ, with δ being a string deviation. The matrix can however be manipulated in order to cancel these terms and have well defined elements. In the section hereafter we have to use two different indexings of the rapidities. The notation with one index λ a (a = 1,..., M) is convenient to identify a rapidity which corresponds to a row or a column of the H matrix. On the other hand, the notation including three parameters λ k,a α (k = 1,..., N s, α = 1,..., M k and a = 1,..., n k ) is useful to cover a subset of rapidities that belong to the same string and which correspond to a submatrix of H. In the definition (5.13), the ill-defined factors originate from the factor d(λ k,b β ) = M a=1 = φ(λ a λ k,b β +η) φ(λ a λ k,b β φ(λ k,1 β φ(λ k,1 β η). If we write it explicitly, the product reads λk,b β + η) λk,b β a,λ a λ k,b 1 β a,λ a λ k,b+1 β = F b 1 i(δ k,b 1 β F b+1 with i(δ k,b β k,k β φ(λ k,k β η)... φ(λ φ(λ a λ k,b 1 β ) φ(λ a λ k,b+1 β ) i(δ β λ k,b β + η) λ k,b β η) k,b 1 β i(δ k,b+1 δ k,b β )(1 δ b,1) + δ b,1 n a,λ a {λ k,1 β,...,λk,n k β } δ k,b β )(1 δ b,1) + δ b,1 φ(λ a λ k,b β + η) φ(λ a λ k,b β η) δ k,b β )(1 δ + O(δ) b,n) + δ b,n δk,b+1 β )(1 δ b,n ) δ b,n + O(δ) (5.15) F b = a,λ a λ k,b β φ(λ a λ k,b β ), b = 0,..., n + 1 (5.16) 58

61 5.1 where we extend the value of the string index b and with implicit β and k indices. Defining the following values with implicit indices k and β δ a,b = iδ k,a β iδ k,b β λ k,b β,(c) = λ k β η 2 (n k + 1 2b) + i π 4 (1 ν k) + iδ k,c β J (c) a,b = G b = c 1 φ(µ a λ k,b β,(c) ) φ(µ c λ k,b β ) (5.17) and with the equalities φ(µ c λ k,b β ± η) = c φ(µ c λ k,b 1 φ(µ a λ k,b 1 β,(b) ) c a d ( ) J (b) a,b dµ J (b+1) a,b+1 a = G b 1 + O(δ) J (b) a,b 1 1 ( = δ b,b+1 J (b+1) a,b β,(b) ) J (b+1) a,b+1 J (b) a,b J (b) a,b+1 ), (5.18) the submatrix of H which contains the columns corresponding to a n k string reads ( H ab = φ(η)j (1) a,1 J (1) a,0 G 0 F 0 H ab = φ(η)j (2) a,2 H ab ( J (2) a,1 G 1 F 1 δ 1,2 F 2 J (1) a,2 G 2 δ 1,2 (2) J F 3 δ2,3 a,3 G 3 (... = φ(η)j (n k 1) a,n k 1 J (n k 1) a,n k 2 G n k 2 F n k 2 F nk ), b : (k, β, 1) ), b : (k, β, 2) δ nk 2,nk 1 ) δ n J (n k 1) k 1,n k a,n k G nk, b : (k, β, n k 1) ( H ab = φ(η)j (n k) a,n k J (n k) a,n k 1 G n k 1 + F ) n k 1 δ n k 1,n k J (n k) a,n k +1 G n k +1, F nk +1 b : (k, β, n k ). (5.19) In order to remove the factors δ c,c+1 and have well defined elements in the matrix, we act on the n k string submatrix with unitary operations which keep the determinant unchanged. As a first manipulation, we add the n k th column multiplied by Fn k 2δn k 2,n k 1 G nk F nk δ n k 1,n k G nk 1 to the (n k 1)th column. The (n k 1)th column, 59

62 5.1 b : (k, β, n k 1), becomes H (r) ab φ(η) = J (n k 1) a,n k 1 J (n k 1) a,n k 2 G n k 2 F n k 2δ n k 2,n k 1 F nk δ n k 1,n k + F n k 2δ n k 2,n k 1 F nk δ n k 1,n k J (n k) a,n k 1 J (n k) a,n k G nk J (n k 1) a,n k J (n k 1) a,n k 1 G n k +δ n k 2,n k 1 J (n k) a,n k +1 J (n F k) nk 1F nk 2G nk G nk +1 a,n k F nk +1F nk G nk 1 = J (n k 1) a,n k 1 J (n k 1) a,n k 2 G n k 2 + F n k 2 G nk δ n k 2,n k 1 d (J F nk dµ a,nk J a,nk 1) a +δ n k 2,n k 1 J (n k) a,n k +1 J (n F k) nk 1F nk 2G nk G nk +1 a,n k. (5.20) F nk F nk +1G nk 1 The (n k 1)th column is then multiplied by Fn k 3δn k 3,n k 2 G nk 1 F nk 1δ n k 2,n k 1 G nk and added to 2 the (n k 2)th column ( b : (k, β, n k 2) ) which becomes H (r) ab φ(η) = J (n k 2) a,n k 2 J (n k 2) a,n k 3 G n k 3 + F n k 3 F nk 1 + F n k 3F nk 2 F nk 1F nk G nk 1δ n k 3,n k 2 G nk G nk 1 G nk 2 d (J a,nk 1J a,nk 2) dµ a δ n k 3,n k 2 d (J dµ a,nk J a,nk 1) a + F n k 3F nk 2G nk G nk +1 δ n k 3,n k 2 J (n k) a,n F nk F nk +1G k +1 J (n k) a,n k. nk 2 (5.21) Continuing this procedure until the first column of the submatrix is reached, one F finally has to add the 2nd column times 0G 2 F 2(δ 1,2 )G 1 to the 1st column. This latter reads H (r) ab φ(η) = F 0F 1 G 1 n k i=0 G i G [ i+1 (1 δ i,n δ i,0 ) d (J F i F i+1 dµ a,i J a,i+1 ) a +(δ i,n + δ i,0 )J a,i J a,i+1 ], b : (k, β, 1). (5.22) The submatrix including the rapidities of the string structure is now well-defined and if we neglect the terms of O(δ), the reduced n k string submatrix is H (r) ab = φ(η) F 0F 1 n k G i G [ i+1 (δ i,n + δ i,0 ) J a,i J a,i+1 G 1 F i=0 i F i+1 ] +(1 δ i,n δ i,0 ) µa (J a,i J a,i+1 ), b : (k, β, 1) H (r) ab = φ(η)j a,i J a,i 1 G i 1, b : (k, β, i), i = 2,..., n k. (5.23) 60

63 Dynamical structure factor and total spin sectors In order to have the response of the system to an excitation of momentum q and energy ω, we compute the dynamical structure factor (DSF) defined as the Fourier transform of the space-time correlation function. With the Fourier transform (1/ N) N j=1 e iqj S j S j+1 = (1/ N) p eip S q ps p, the DSF reads: S ++ (q, ω) = 1 N = 2π N N j,j =1 e iq(j j ) dte iωt S j (t)s j+1 (t)s+ j (0)S+ j +1 (0) e iq GS Sq ps p α 2 δ(ω ω α ) (5.24) α p where GS Sq ps p α is the form factor between the normalized ground state and the whole set of normalized eigenstates. ω α = E α E GS is the relative energy of the state α. Since the operator Sq ps p turns down two spins, by conservation of the total z-spin, no sectors of M > M GS 2 are contributing to the DSF. Moreover using the Wigner-Eckart theorem (explicit formulation in 5.2), acting on the ground state twice with a spin lowering operator of total spin one can not generate states of S tot > SGS tot + 2. Therefore, if the ground state belongs to the sector M GS, only eigenstates of sector M GS 2 give a non-zero form factor. The DSF can be evaluated with the Algebraic Bethe Ansatz by taking the Fourier transform of the form factor (5.14) and summing over all the eigenstates of the sector M GS 2: S ++ (q, ω) = 1 j e iqj {µ} σ j σ j+1 {λ} 2 δ(ω ω µ ) N N({µ}) N({λ}) µ (5.25) with N({λ}) the norm of a state the state {λ} as defined in 3.4. Using the transfer matrix eigenvalue φ m ({µ}) = e iqµm, with q µ being the total momentum of the 61

64 5.1 {µ} state, we can give a final expression for the DSF: S ++ (q, ω) = µ δ(ω ω µ ) 1 N e i(q qµ+qλ)(j j ) j,j } {{ } N 2 δ q,qµ qλ M a φ(µ a + η/2) 2 det H 2 M 2 b φ(λ b + η/2) 2 M a>b φ(µ b µ a ) 2 M 2 a>b φ(λ b λ a ) 2 1 a b φ(µa µ b+η) φ(µ a µ b ) det Φ({µ}) a b φ(λa λ b+η) φ(λ a λ b ) det Φ({λ}) = N M a δ q,qµ q λ δ(ω ω µ ) φ(µ a + η/2) 2 M 2 µ b φ(λ b + η/2) 2 det H (r) 2 det Φ({µ}) 1 det Φ ({λ}) 1 M a>b (φ(µ a µ b ) 2 + φ(η) 2 ) nj,n k a,b=1 ((n j,α,a) {(n k,β,b),(n k,β,b+1)}) φ(λ nj,a α λ n k,b β + η) (5.26) with the values φ(λ) = λ and η = i corresponding to the isotropic spin chain. In order to make the result compatible with states {λ} which contain string configurations, we write the formula in the last equality including the reduced determinant (5.23) and the reduced Gaudin matrix Φ (3.52) with the simplified prefactor (see (3.53)) Sum Rules The sum rules, i.e. the integrated intensity and the first frequency moment, are of particular importance for the calculation since they provide checks independent of the Algebraic Bethe Ansatz and determinant representation. The integrated intensities allow one to control the efficiency of the algorithm by summing all the computed form factors. The idea of calculating the first frequency moment, called also f-sum rule, comes from [91]. At the time of this publication, the matrix elements in the spin chain were not accessible, and the authors combined the des Cloizeaux-Pearson spectrum with the f-sum rule to determine where the DSF center of mass at fixed momentum is located. Nowadays, as the form factor can be calculated, we mainly use the f-sum rule as a test at fixed momentum. Here we summarize the calculation leading to the result of the two sum rules. 62

65 5.1 Integrated intensity 2π 1 N 1 N N q=0 1 dωs ++ (q, ω) = 1 2π N = 1 N N GS S j S+ j S j+1 S+ j+1 GS j=1 j ( 1 GS = 1 4 ( 1 2 M N 2 Sz j ) + ) ( ) 1 2 Sz j+1 GS ) ( 1 E 0 N (5.27) where we used the Hellmann-Feynman theorem for the evaluation of j GS Sz j Sz j+1 GS. The result is valid for h 0 and as well for the isotropic as for the anisotropic spin chain. First frequency moment The results hereafter are restricted to the case of zero magnetic field and isotropic chain since we use several rotational symmetries in order to keep the result concise. 1 dωωs ++ (q, ω) 2π = 2 N (E 0 E µ ) µ GS 1 e iqj S j S j+1 µ N j=1 1 = e iq(j j ) [ [H, ] ] GS S j 2N S j+1, S + j S+ j +1 GS. (5.28) j,j With the intermediate result [ ] H, S j S j+1 = S j 1 Sz j S j+1 + S j Sz j+1s j+2 Sz j 1S j S j+1 S j S j+1 Sz j+2 (5.29) and using reflection and rotation symmetry of the spin chain, we find the first frequency moment to be 1 2π = J 2N N j=1 dωωs ++ (q, ω) 4 cos(2q) Sj z Sj+1S z j+2s z j+3 z 2 cos(2q) S + j S j+1 Sz j+2sj+3 z + (6 cos(2q) + 4 cos(q)) S + j Sz j+1sj+2s z j+3 (4 cos(q) 2 cos(2q)) S + j Sz j+1s j+2 Sz j+3 (2 2 cos(q)) S z j S z j+2 (4 + 2 cos(q)) S z j S z j+1. (5.30) 63

66 5.1 We denote here O the expectation value of the operator O at zero temperature. This result is only valid at h = 0 and for the isotropic spin chain. If we use the results of [92] to evaluate the correlators, the first frequency moment equals [ cos(2q) ln(2) (10ζ(5) 83 ) 5ζ(3) 65 ( 41 8 ζ(5) + ζ(3) 5 9 ) 5 ζ(3) + 1 ] 20 + cos(q) [ζ(3) 43 ] ln(2) 3 4 ζ(3) + 2 ln(2) 1 4 = cos(q) cos(2q). (5.31) Numerical evaluation of the S ++ (q, ω) dynamical structure factor q S ++ (q, ω) ( % ) M = 50 M = 100 M = 150 M = 200 2h sp p3h sp1s p4h dωs Table 5.1: Percentages of contributions of excitation types to the the S ++ (q, ω) DSF for N = 400 and M = 50, 100, 150 and 200. The last row gives the saturation of the integrated intensity. We show in this section the results of the numerical evaluation of the DSF in several magnetic fields in the isotropic spin chain and we analyze the contribution of the different kinds of excitations in all the cases. Considering a spin chain of length N = 400 and with fillings of M = 200, 150, 100, 50, we compute the value of the DSF by summing over the eigenstates contributions. The summation over intermediate states is performed to obtain quantitative results over the Brillouin zone and over an energy range that covers all the significant weight. This is done using the ABACUS algorithm [93] which sums intermediate state contributions in a close to optimal order. From a given set of quantum numbers, we solve the Bethe equations (2.31), and calculate the energy and momentum (2.33). The value of the form factor is then computed with the formula (5.26). Large families of excited states are summed over until satisfactory saturations of sum rules are achieved: for a complete map of the DSF, we consider that a integrated intensity above 90% is acceptable and we expect the 10% left to be some signal diffused in energy. This condition fixes the maximum size of the system around 400 sites for a reasonable computation time (O(100 cpu day). We check additionally with the f-sum rule that the saturation is homogeneous over all momenta. If the distribution of the weight does not look even, it would mean that 64

67 5.1 (a) (b) (c) (d) Figure 5.1: S ++ (q, ω) DSF in a N = 400 XXX spin chain and (a) M = 50, (b) 100, (c) 150 and (d) 200. the scan does not reach properly some kinds of states which would be problematic. In more specific studies, like an extraction of an exponent in ω or the analysis of the long distance time asymptotes, a higher saturation would be required and by lowering the size of the system to N = 200, we can easily afford a saturation of the integrated intensity above 99%. From a general point of view, the saturation could be improved by either a longer time of computation or a more efficient scanning. To obtain smooth curves in frequency ω, the delta function in (5.26) is broadened in energy to a scale commensurate with the level spacing. The resulting figure of the S ++ (q, ω) DSF is shown in figure 5.1 for the four different magnetizations. One observes that the shape is similar to the single spin raising S + (q, ω) DSF [94] but with a shift of π in momentum. There is also a significant difference in the distribution of the signal. For the two spin raising DSF, the weight is more homogeneously spread from low to high energy although for a single spin raising, it is mainly located at the lower boundary of the spectrum. S ++ (q, ω) being the propagator of a pair of neighbor spin up, as a consequence, it vanishes when the polarization of the spin chain tends to the saturation (M = 0). In table 5.1, one can observe also that as the magnetic field increases, the number of excitations with 65

68 5.1 one or two particle-hole created by the double spin raising decreases. In figure 5.2, we split the total signal of S ++ (q, ω) into the three main contributing type of excitations. The three graphs are quite similar although the signal becomes weaker as the number of particles in the excited state increases (the scales are different for each graph). We also notice that if the 2h type carries weight at low energy, the signal of the 1p3h and 2p4h vanishes at zero energy and contributes mainly at higher energy. Besides, the DSF at h = 0 (M = 200) shows a signal comparable in shape with the non-zero magnetic field figures and is mainly composed of 4 spinons excitations (table 5.1). We divide the S ++ (q, ω) DSF signal into the 4 spinons and 6 spinons excitation in figure 5.3. In addition to the fact that the 6sp1s 2 contribution is globally two orders of magnitude lower than for the 4sp, we notice that the 6sp1s 2 weight is essentially inexistent between q = π/2 and q = 3π/2. (a) 2h (b) 1p2h (c) 2p2h Figure 5.2: Separated graphs of the S ++ (q, ω) DSF for the three main types of excitation with N = 400 and M = 150. The type of excitation plotted is specified in the label and is described and explained in section

69 5.2 (a) 4sp (b) 6sp1s 2 Figure 5.3: Contributions by type of excitation to the S ++ (q, ω) DSF for N = 400 and M = 200. The graphs are labeled by the type of excitations that are represented. 5.2 S z j Sz j+1 We explain in this section how to represent and evaluate the S z j Sz j+1 dynamical structure factor (DSF) using the algebraic Bethe Ansatz. The DSF is defined as the Fourier transform of the S z j Sz j+1 correlator: S 4z (q, ω) = 1 N N j,j =1 dte iq(j j )+iωt S z j (t)s z j+1(t)s z j (0)Sz j +1(0). (5.32) Form factor The form factor of the operator 1 4 σz j σz j+1 with the operators A(ξ) and D(ξ): is expressed using the result in (3.37) σ z j = j 1 (A + D)(ξ i )(A D)(ξ j ) i=1 j 1 = 2 (A + D)(ξ i )D(ξ j ) i=1 N i=j+1 N i=j+1 (A + D)(ξ i ) (A + D)(ξ i ) + 1. (5.33) Here we used the identity j 1 i=1 (A + D)(ξ i)(a + D)(ξ j ) N i=j+1 (A + D)(ξ i) = 1. The product of two neighbor operators reads then j 1 σj z σj+1 z = 4 [A + D] (ξ i )D(ξ j )D(ξ j+1 ) i=1 67 N i=j+2 [A + D] (ξ i )+σ z j +σ z j+1 1. (5.34)

70 5.2 Using the eigenvalues of the transfer matrix over the two states similarly to (5.5), we have {λ} σ z j σ z j+1 {µ} = {λ} σ z j {µ} + {λ} σ z j+1 {µ} {λ} {µ} + 4 φ j 1({λ}) M φ j+1 ({µ}) 0 M C(λ a )D(ξ j )D(ξ j+1 ) B(µ b ) 0. (5.35) a=1 Whereas the determinant expressions for the first three terms are known [69, 61, 75], the last term needs to be expanded. As introduced in (3.25), one D(ξ) operator on the right state gives D(ξ j+1 ) recalling that M B(µ a ) 0 = Λ(ξ M j+1, {µ}) B(µ a ) 0 }{{} a=1 =0 M + Λ n (ξ j+1, {µ})b(ξ j+1 ) B(µ a ) 0 (5.36) a n a=1 Λ(ξ, {µ}) = d(ξ) n=1 M b 1 (ξ, µ a ) a=1 φ(η) Λ n (ξ, {µ}) = d(µ n ) b 1 (µ n, µ a ). (5.37) φ(ξ µ n ) Acting therefore with two D(ξ) operators gives = + D(ξ j )D(ξ j+1 ) M M n=1 m=1,m n M B(µ a ) 0 a=1 a n b=1 Λ m (ξ j, {µ a n, ξ j+1 }) Λ n (ξ j+1, {µ})b(ξ j )B(ξ j+1 ) a n,m B(µ a ) 0 M Λ n (ξ j, {µ a n, ξ j+1 }) Λ n (ξ j+1, {µ})b(ξ j ) B(µ a ) 0. (5.38) }{{} a n =0 n=1 The last term vanishes because d(ξ k ) = 0 for k = 1,..., N (see definitions in section 3.2). If we include the scalar product with the left state (3.54), that part of the form factor is then {λ} D(ξ j )D(ξ j+1 ) {µ} = M M n=1 m=1,m n Λ m (ξ j, {µ a n, ξ j+1 }) Λ n (ξ j+1, {µ}) det H({λ i }, {µ a m,n, ξ i, ξ i+1 }) a<b φ(λ b λ a ) a>b φ(µ. (5.39) b µ a ) µn=ξ j,µ m=ξ j+1 68

71 5.2 With the expansions and 1 b<a M we can rewrite {λ} D(ξ j )D(ξ j+1 ) {µ} = φ(µ b µ a ) µn=ξ j,µ m=ξ j+1 = φ(ξ j ξ j+1 ) φ(ξ j µ c ) φ(ξ j+1 µ n ) φ(µ n µ c ) c n φ(µ m µ n ) φ(ξ j µ m ) c m φ(ξ j+1 µ c ) φ(µ m µ c ) 1 b<a M φ(µ b µ a ) (5.40) Λ m (ξ j, {µ i n, ξ j+1 }) = b(µ m, µ n ) b(µ m, ξ j+1 ) Λ m (ξ j, {µ i }), (5.41) M d(µ n ) n=1 a n M d(µ m ) m=1,m n φ(µ n µ a + η) φ(ξ j µ a ) b m φ(µ m µ b + η) φ(ξ j+1 µ b ) ϕ 2 (η) φ(µ m µ n + η) b(µ m, ξ j+1 ) 1 a>b φ(λ b λ a ) det H({λ}, {µ c m,n, ξ j, ξ j+1 }) a>b φ(µ b µ a ) φ(ξ j ξ j+1 ) (5.42) with the matrix ( φ(η) φ(λ a µ b ) k a φ(λ k µ b + η) d(µ b ) ) k a φ(λ k µ b η), b m, n φ(η) H ab = φ(λ a ξ j+1)φ(λ a ξ j+1+η) k φ(λ k ξ j+1 + η), b = m φ(η) φ(λ a ξ j)φ(λ a ξ j+η) k φ(λ k ξ j + η), b = n. (5.43) Homogeneous limit The spin chain models correspond to the homogeneous case where ξ i η/2 i. Similarly to the calculation of the S j S j+1 form factor, the limit should be taken with care in order to obtain a finite expression for the ratio det H({λ}, {µ c m,n, ξ j, ξ j+1 })/φ(ξ j+1 ξ j ). We follow the same procedure as described in section to have a well defined homogeneous limit and the ratio becomes ξj det H({λ}, {µ c m,n, ξ j, ξ j+1 }) lim ξ j,ξ j+1 η/2 ξj φ(ξ j+1 ξ j ) 69 = k φ(λ k + η/2) 2 det H (5.44)

72 5.2 with φ(η) φ(λ a µ b )( k a φ(λ k µ b + η) det H d(µ = b ) ) k a φ(λ k µ b η), b m, n φ(η) φ(λ a η/2)φ(λ a+η/2), b = m φ(η)φ(2λ a) φ 2 (λ a η/2)φ 2 (λ, b = n. a+η/2) (5.45) The non-trivial part of the matrix element (5.35) reads then {λ} D(η/2)D(η/2) {µ} = i φ2 (λ i + η/2)φ 2 (µ i η/2) M j<k φ(λ k λ j ) j>k φ(µ k µ j ) where from the definition below det F nn = 0 and with M A n = d(µ n )φ(µ n η/2) φ(µ i µ n η) i=1 n=1 A n M m=1 det F nm (5.46) Bab n i = (1 δ b,n )d(µ b )φ(µ b + η/2) φ(µ i µ b η) φ(η) φ(µ n µ b η) φ(λ a η/2)φ(λ a + η/2) φ(η)φ(2λ a ) C a = φ 2 (λ a η/2)φ 2 (λ a + η/2) { Fab nm G n ab =, b m Bab n, b = m { ( φ(η) G n φ(λ ab = a µ b ) k a φ(λ k µ b + η) d(µ b ) ) k a φ(λ k µ b η), b n C a, b = n. (5.47) Using an application of Laplace s determinant formula (see chapter 9 appendix in [68]) and the fact that B n is a rank one matrix, we write the summation over determinants as two determinants: M det F nm = det(g n + B n ) det G n (5.48) m=1 and the complete form factor is eventually {λ} σj z σj+1 z {µ} = {λ} σj z {µ} + {λ} σj+1 {µ} z {λ} {µ} +4 φ j 1({λ}) i φ2 (λ i + η/2)φ 2 (µ i + η/2) M φ j 1 ({µ}) j<k φ(λ k λ j ) j>k φ(µ A n (det(g n + B n ) det G n ). k µ j ) 70 n=1 (5.49)

73 Fourier transform With (1/ N) N j=1 e iqj {λ} S z j Sz j+1 {µ} = (1/ N) p eip {λ} S z q ps z p {µ}, the Fourier transform of the form factor (5.49), we give here an explicit expression for the norm squared of the matrix element between normalized states. This formula which results from the same simplifications as in (5.26) is used for the further computation of the DSF: 2π N with p eip {λ} S z q ps z p {µ} 2 N({λ})N({µ}) Ns Mj,M k nj,n k j,k=0 α,β=1 = N φ(λ i + η/2) 16 φ(µ i + η/2) i δ(q + q λ q µ ) φ(µ j α,i µk β,j + η) Φ ({µ}) i,j=1,((n,α,i) (m,β,j 1)) ( e i(q λ q µ) + 1 ) F 1 + e i(q λ q µ) F 2 {λ} {µ} F 3 2 α>β (φ2 (λ α λ β ) + φ 2 (η)) Φ({λ}) 2 (5.50) F 1 = det(h({λ}, {µ}) 2P ({λ}, {µ})) F 2 = 4 φ(η/2 + λ i ) M A n (det(g n + B n ) det G n ) φ(η/2 + µ i i ) n=1 j>k F 3 = φ(µ k µ j ) j>k φ(λ k λ j ) i φ2 (η/2 + λ i )φ 2 (η/2 + µ i ) φ(η) H ab ({λ}, {µ}) = φ(λ k µ b + η) d(µ b ) φ(λ k µ b η) φ(λ a µ b ) k a k a P ab ({λ}, {µ}) = φ(η) k φ(µ k µ b + η) φ(λ a η/2)φ(λ a + η/2) (5.51) and with N({λ}) the norm of the {λ} state (see section 3.4). The function φ(λ) = λ an the variable η = i correspond to the isotropic spin chain. In case the state {µ} contains string configurations, this result includes the reduced Gaudin matrix Φ ({µ}) (3.52) and the simplified prefactor (see (3.53)). The operator S z j Sz j+1 being Hermitian, {λ} σz j σz j+1 {µ} = {µ} σz j σz j+1 {λ}, and if one computes the form factor between the ground state and an excited state, one must put the state which contains string configurations on the left (bra). Indeed, whereas the presence of string configuration in the left state does not cause any problem for the numerical evaluation of (5.51), string configurations in the right state (ket) yields undetermined factors in the G n and B n matrices similarly to the H matrix ( see section 5.1.3). In this case, however, the reduction method used for the H matrix is helpless and we did not find out how to treat the singular factors in order to have a well-defined determinant. 71

74 Dynamical structure factor and total spin sectors In order to have the response of the system to an excitation of momentum q and energy ω, we express the dynamical structure factor with the norm squared of the Fourier transformed form factor: S 4z (q, ω) = 2π N e ip GS Sq ps z p z α 2 δ(ω ω α ) (5.52) α p where GS Sq ps z p α z is the form factor between the normalized ground state and the whole set of normalized eigenstates and ω α = E α E GS is the relative energy. We explain hereafter that in order to cover all the contributing eigenstates α, we express the results in terms of the Sj zsz j+1 and S j S j+1 form factors. The constructed eigenstates which belong to the M th sector, λ M, have the spin eigenvalues Sz tot = S tot = N 2 M (see section 2.7). As the Sq ps z p z conserves the spin Sz tot and if we consider the form factor with a ground state with M reversed spins, the matrix elements are only non-zero for states with Sz tot = N/2 M. To access the states in different sectors but with Sz tot = N 2 M, we act with S q=0 on the eigenstate, and with the contributions of all sectors, the DSF reads: S 4z (q, ω) = M M =0 α M e ip GS M Sq ps z p(s z 0 ) )(M M α M 2 (5.53) p with GS M the normalized ground state and α M a normalized eigenstate including M rapidities. As the ground state is also highest-weight, i S+ i GS M = 0 and by commutation, we identify the only three contributing sectors e ip GS M Sq ps z p z α 2 = e ip GS M Sq ps z z p α M α M α p 1 + N 2M + 2 α M 1 p 2 + (N 2M + 3)(N 2M + 4) p e ip GS M Sq ps p z + Sq ps z p α M 1 α M 2 p 2 2 e ip GS M Sq ps p α M 2 2 (5.54) where the expression for Fourier transformed form factor is given in (5.51) and the S ++ (q, ω) DSF (5.26). The factors in front of the sums originate from the norms of the states including infinite rapidities (2.39). In the case of zero magnetic field, (M = N/2), the contribution of the sector M 1 is identically zero. Indeed, following the Wigner-Eckart theorem [95], the matrix elements can be expressed proportionally to a Clebsch-Gordan coefficient. We decompose the indexing of the Bethe states α into a state parameter, total 72

75 5.2 spin and spin-z numbers: β, S tot, Sz tot and we rewrite the form factor as: GS Sj z Sj+1 z α GS, 0, 0 Sj z β, s tot, s tot z β, s tot, s tot z S z j+1 α, S tot, 0 β,s tot,s tot z s t,s t z 0, 1; 0, 0 s tot, s tot z s tot, 1; s tot z, 0 S tot, 0 0, 1; 0, 0 1, 0 1, 1; 0, 0 S tot, 0 (5.55) with j 1, j 2 ; m 1, m 2 J, M the Clebsch-Gordan coefficient. Then the only non-zero contributing sectors are S tot = 0, 2. One notices from (5.54) that for large spin chains, in a finite magnetic field, the 1 contributions of the sectors M 1 and M 2 are in order of N and 1 N, so 2 negligible. In summary, to evaluate the S 4z (q, ω) DSF, one needs the Sq ps z p z form factor in sector S tot = N 2 M and in absence of magnetic field, one must add the contributions of Sq ps p form factor in sector S tot = Sum Rules We provide in this section the analytical results for the sum rules which are used as independent checks for the numerical evaluation of the dynamical structure factor. Integrated intensity 1 1 dωs 4z (q, ω) = 1 N 2π q N 2 q N j,j =1 e iq(j j ) GS S z j S z j+1s z j Sz j +1 GS = (5.56) This result stays valid for every magnetic field and for any anisotropy. First frequency moment One can express the first frequency moment as a sum of double commutators: 1 dωωs 4z (q, ω) = 1 (E 0 E µ ) GS e iqj Sj z S z j+1 µ 2 2π µ N j = J ) GS [[ Si x Si+1 x + S y i 2N Sy i+1, Sz j Sj+1 z ], S z j Sj z ] +1 GS. (5.57) i,j,j e iq(j j With the equalities [ S x i Si+1 x + S y i Sy i+1, Sz j Sj+1] z i = i(s x j 1S y j Sz j+1 + S z j S y j+1 Sx j+2) + i(s y j 1 Sx j S z j+1 + S z j S x j+1s y j+2 ) (5.58) 73

76 5.2 and for the isotropic spin chain with h = 0, we use translational and rotation symmetries of the different correlators to rewrite the first frequency moment as 1 2π dω ωs 4z (q, ω) = J 4N GS Sj x Sj+1 x + S y j Sy j+1 GS j + cos(2q) J GS S z ( j 1 S x N j Sj+1 x + S y j j+1) Sy S z j+2 GS. (5.59) 1 dωωs 4z (q, ω) = J 2π j If we evaluate numerically the several correlators with [62], the result reads [ ln(2) cos(2q) ( ln ζ(3) 5 3 ζ(3) ln ζ(3) ζ(5) + 5 ζ(5) ln 2 3 ) ] = J ( cos(2q)). (5.60) Numerical evaluation of the S 4z (q, ω) dynamical structure factor In this section, we numerically evaluate the DSF in several magnetic fields in the isotropic spin chain and we analyze the signal weight of the different kinds of excitations in all the cases. We choose to show in this section the connected correlator in order to emphasize the dynamic properties; the ground state contribution is important but completely static and constant. We define it by Sc 4z (q, ω) = S 4z (q, ω), (q, ω) (0, 0) and Sc 4z (q = 0, ω = 0) = 0. The process for the computation being the same for all the DSFs, the details for the procedure can be found in section q S 4z c (q, ω) ( % ) M = 50 M = 100 M = 150 M = 200 1p1h sp1s p2h sp p3h sp2 1s h1s sp2s dωs 4z c Table 5.2: Percentages of contributions of excitation types to the the Sc 4z (q, ω) DSF for N = 400 and M = 50, 100, 150 and 200. The last row gives the saturation of the integrated intensity. As pictured in figure 5.4, the Sc 4z (q, ω) DSF for h > 0 has a signal shape which is very similar to the single spin S zz (q, ω) DSF [61, 94] and as shown in table 5.2, the 74

77 5.2 (a) (b) (c) (d) Figure 5.4: Sc 4z (q, ω) DSF for a N = 400 XXX spin chain and (a) M = 50, (b) 100, (c) 150 and (d) 200 contribution of the 2 string states are of the same order. We notice however that on the contrary to the single spin DSF, the weight is concentrated at low energy. Another interesting feature that comes out of table 5.2 is that the contribution of the 1 and 2 particle-hole excitations to the sum rule is stable as the magnetic field decreases and the Fermi-sea shrinks. We show in figures 5.5(a), 5.5(b) and 5.5(c) more information about the contribution of each type of excitation. For the case M = 150 we split the Sc 4z (q, ω) DSF into the three main weight carrying types. We notice that the 1p1h states contribute mainly at low energy whereas the 2p2h states carry its highest signal in the upper part of the DSF map. We observe also that the weight of eigenstates with a single 2 string configuration is localized around q = π and ω = 2 and that this category of excitation is gapped in energy. The signal of the Sc 4z (q, ω) DSF at zero magnetic field takes the original form shown in figure 5.4. From the analysis of the computation results, we can identify where the different kinds of excited eigenstates supply weight. Most of the weight carried by the spin singlet eigenstates (S tot = 0) corresponds to 2sp1s 2 excitations (table 5.2). By definition, these excited states are located inside the 2-spinon spectrum 75

78 5.2 ( % ) Sc 4z (q, ω) S zz (q, ω) 2sp1s sp sp sp1 1s sp2 1s sp1 2s inside 2-s. spectrum above 2-s. spectrum Table 5.3: Percentages of weight of either the S 4z c (q, ω) or S zz (q, ω) DSF for N = 400 and M = N/2. The three first rows give the contribution of 2-, 4- and 6-spinon excitations. In two last rows is the ratio of integrated DSF which is either inside the 2-spinon spectrum or above it. π, i.e.: 2 sin(q) < ω 2-spinon < π sin(q/2) [62]. Almost all the rest of the weight is carried by 4-spinons states (6 and higher spinons states are negligible) and is then included in the 4-spinons spectrum: π 2 sin(q) < ω 4-spinon < 2π sin(q/4). We see on table 5.2 that among the 4-spinons states, the spin quintuplet eigenstates (with two infinite rapidities) contribute the most to the signal and singlet states e.g. 4sp2s 2 (q, ω) DSF into its mainly contributing eigenstates in figures 5.5(d) and 5.5(e). These illustrations show clearly the difference in the (q, ω) localization of the 2 spinon and 4 spinon signals. Moreover, the weight distribution of the 4sp2 is clearly broader in energy although the 2sp1s 2 signal is highly concentrated around q = π and zero energy. are insignificant. We divide the S 4z c From an experimental point of view, this analysis is not valuable since during measurement only the intensity at a point (q, ω) is known and the internal degrees of freedom like the number of spinons or the total spin are inaccessible. We then interpret the results with regard to their energy: if a measure is done under the frequency π sin(q/2), it could contain 2 and 4-spinons, and higher excitations and if the signal is measured above this line, it does not include 2-spinons. In comparison with the S zz (q, ω) corresponding to the response function of neutron scattering (see e.g. figure 6.1(b)), the S 4z (q, ω) DSF reveals higher energy area where the excitations can be clearly identified as higher than 2-spinons. In this perspective we compare the two DSFs in table 5.3 and we compare the ratio of the integrated intensity that is inside the 2-spinon spectrum or above. In this chapter, we showed how to obtain the correlation functions S ++ (q, ω) and Sc 4z (q, ω). With the numerical evaluations, we identified which are the states contributing the most over the integrated signal. In particular, we showed that the Sc 4z (q, ω) reveals the 4-spinons excitations and that a measure of this observable would permit to clearly identify their existence at a given momentum and frequency. 76

79 5.2 (a) 1p1h (b) 2p2h (c) 2h1s2 (d) 2sp1s2 (e) 4sp2 Figure 5.5: Separated figures of the Sc4z (q, ω) DSF with N = 400 corresponding to types of excitation in the labels. The magnetization is M = 150 in (a), (b) and (c) and M = 200 for (d) and (e). 77

80 5.2 78

81 Chapter 6 Spin-exchange correlation function In this chapter we use the previous results for the S 4z (q, ω) in order to calculate the spin-exchange dynamical structure factor which is directly related to RIXS measurement in the isotropic spin chain. We then compare a numerical evaluation of this correlation function with the single spin DSF S zz (q, ω), which is proportional to the INS cross section. 6.1 Dynamical structure factor and total spin sectors With the spin exchange operator defined in (4.14) the spin-exchange DSF is defined as X q = 1 e iqj (S j 1 S j + S j S j+1 ), (6.1) N j S exch. (q, ω) = 2π GS e iqj (S j 1 S j + S j S j+1 ) α 2 δ(ω ω α ) N α j 4π(1 + cos(q)) = GS e iqj S j S j+1 α 2 δ(ω ω α ) (6.2) N α j where the normalized ground state is GS, the normalized excited states α, excitation energies ω α = E α E GS. By expanding explicitly the spin operator inner 79

82 6.1 product, the DSF contains S exch. (q, ω) = 8π cos2 (q/2) δ(ω ω α ) N α e iq(i j) GS Si a Si+1 a α α Sj b Sj+1 b GS. (6.3) a,b=x,y,z i,j In the specific case of the isotropic spin chain in zero magnetic field, we can exploit the spin isotropy of the system to express the full spin-exchange dynamical structure factor as a function of the Si zsz i+1 form factors only. By globally rotating 0 Sj xsx j+1 α and 0 Sy j Sy j+1 α about the y and x axes, respectively, and using the fact that the ground state is a global su(2) singlet, one can show that only singlet excited states contribute to (6.2). Noticing that the spin-exchange operator conserves Stot: z e iqi [ Sj z ], S i S i+1 = 0 (6.4) i,j one finds that the only contributing α are of S z tot = 0 and by the Wigner-Eckart theorem, similarly to (5.55), α can only belong to total spin S tot = 0 or 2. Let s denote the states of these two sectors by α Stot=0,2. The singlet states are invariant under rotation and therefore e iφsa α 0 = α 0 with a = x, y, z. Then the expansion of the rotation about the y or x axis for the states of S tot = 2 reads e iφsy α 2 = e φ 2 (S + S ) α2 ( ) n φ 1 = 2 n! (S+ S ) n α 2 (6.5) n=0 ( ) n iφ 1 e iφsx α 2 = 2 n! (S+ + S ) n α 2 (6.6) n=0 with S {x,y,±} = i S{x,y,±} i. As the rotated states must have Stot z = 0, we can restrict the sum to only even exponent n and with [S +, S ] = 2S z, (S + ) 3 α 2 = (S ) 3 α 2 = 0 (6.7) one can write the intermediate result (S + ± S ) n α 2 = (±1) n/ n α , S z tot 0, (n = 2, 4, 6,...). (6.8) We can then rewrite the result identical for the two rotations e iφsx or y α 2 = α ( 1) m ( 2φ) 2m 1 4 2m! α , Stot z 0 m=1 ( 1 = ) 4 cos(2φ) α , Stot z 0. (6.9) 80

83 6.2 The rotation of the form factor around the y-axis is therefore GS Si x Si+1 x α = GS e iπ 2 Sy Si z Si+1e z iπ 2 Sy α { GS Si z = Sz i+1 α, S tot = GS Sz i Sz i+1 α, S tot = 2. (6.10) And we have the same result for the rotation of GS S y i Sy i+1 α about the x-axis. The spin-exchange DSF can then be expressed exclusively as a function of the z-component form factor: N S exch. (q, ω) 8π cos 2 (q/2) = 9 δ(ω ω α ) e iqj GS Sj z S z j+1 α 0 α 0 j + δ(ω ω α ) e iqj GS Sj z S z j+1 α 2 2 α 2 j + 4 ( δ(ω ω α )4 1 ) e iqj GS Sj z S z j+1 α α 2 j + 4 ( δ(ω ω α )4 1 ) 2 e iqj GS Sj z S z j+1 α 2 2. (6.11) 2 α 2 j 2 In this last formula, all contributions of total spin 2 sector vanish and the weight of the correlation is only due to singlet states: S exch (q, ω) = cos 2 (q/2) 72π N α S tot=0 2 e iqj GS Sj z Sj+1 z α δ(ω ω α ). (6.12) j In consequence, the evaluation of the DSF can be done with the form factor expression in (5.51). 6.2 Sum Rules In this section, we calculate the analytical results for the integrated intensity and the first frequency moment (f-sum rule) which are used as independent checks for the numerical evaluation of the dynamical structure factor. We give more details about the use of the sum rules in section

84 Integrated intensity The sum over all the momenta and frequencies of the dynamical structure factors is 2π 1 N 1 N N q=0 1 dωs exch. (q, ω) = 1 2π N 2 q N j,j =1 e iq(j j ) GS (S j 1 S j + S j S j+1 )(S j 1 S j + S j S j +1) GS = 1 4 ln(2) + 9 ζ(3) (6.13) 8 where we used the results in [92] and the following intermediate results valid in the isotropic zero field case: 1 N j GS (S j S j+1 ) (S j S j+1 ) GS = N GS Sj z Sj+1 z GS (6.14) j 1 N j GS (S j 1 S j ) (S j S j+1 ) GS = 3 4N GS Sj 1S z j+1 z GS. (6.15) j First frequency moment We express the f-sum rule as a double commutator: 1 dωωs exch. (q, ω) 2π = 4 cos 2 (q/2) 1 (E 0 E µ ) GS e iqj S j S j+1 µ 2 µ N j = 2 cos2 (q/2) [ [ ) ] GS S i S i+1, S j S j+1 ], S j S j N +1 GS. j,j e iq(j j Using the intermediate results i (6.16) [S j 1 S j, S j S j+1 ] = i α,β,γ ε αβγ S α j 1S β j Sγ j+1 e iqj[ ] S i S i+1, S j S j+1 = i(e iq 1) e iqj ε αβγ Sj 1S α β j Sγ j+1 j i j α,β,γ [ ] Sj α S β j+1, Sγ j Sγ j+1 = 0, for α γ, β γ (6.17) 82

85 6.3 with ε αβγ the Levi-Civita symbol. The first frequency moment reads then 1 dωωs exch. (q, ω) 2π [ (1 = 6 sin 2 (q) 4 cos 2 (q/2) ) ( 1 8 ζ(3) 1 ) 6 ln ] ln 2 ζ(3) 8 2 [ = 6 sin 2 (q) ( 1 4 cos 2 (q/2) ) ] (6.18) We used during the calculation isotropy and rotational invariance of the correlators, the result here is therefore valid only for isotropic spin chain in zero magnetic field. 6.3 Numerical results of the spin-exchange DSF and comparison between RIXS and INS We evaluate here numerically the spin-exchange DSF at zero magnetic field. The process of computation being the same for all the DSFs in this thesis, the details for the procedure can be found in section The resulting map of the spin-exchange DSF as defined in (6.12) is presented in figure 6.1(a). RIXS and INS are two different setups which can probe the dynamics of 1D XXX spin chains in e.g. the compound Sr 2 CuO 3 (see chapter 4). As both response signal are determined by DSFs that we can numerically evaluate via the non perturbative methods explained in this work, we then naturally compare the two results and analyze the specificities of each response. In addition to the computation for spinexchange DSF (4.13) in figure 6.1(a), we therefore also compute the single spin DSF (4.4) in figure 6.1(b). Whereas the INS response which includes one spin operator associated which a S = 1, is dominated by 2-spinon excitations (see table 5.3), we might expect the RIXS response function that includes two spin operators, to be mainly carried by 4-spinon states. However, we notice from the computation that the RIXS excitations splits almost completely into two spinons. The contributions to the sum rule shows that the 2-spinon states cover all but 10 4 % of the spin-exchange DSF for a chain of 400 sites. Therefore as both response functions are dominated by two spinons, it is not surprising to see that both shapes are very similar and correspond to the 2-spinon spectrum. Although the excitation continuum measured by RIXS coincides with the one probed by INS, the signal has a noticeably different distribution of weight and this difference is partly caused by cos 2 (q/2). This static prefactor originates from the perturbation of two adjacent exchange couplings in the x-ray scattering process. Excitations are then produced with a typical length 2a (with a the lattice spacing). Equivalently, in momentum space, they mainly carry ± π 2 (mod 2π) momentum. This interpretation is very well illustrated by the two figures: for the INS (figure 6.1(b)) the signal is at its highest at the antiferromagnetic wavevector q = π at ω close to zero, although the RIXS amplitude 83

86 6.3 vanishes there but is concentrated above the continuum threshold at q = π/2. The profiles at fixed-momentum in figure 6.3 show further differences between RIXS and INS by removing the cos 2 (q/2) pre-factor. We notice that the signal distribution between these two normalized responses are clearly distinct: the RIXS response has a broader distribution in energy and this difference is important enough to be measurable experimentally. The asymptotic behavior of correlations along the spin chain are well described in the vicinity of q = π and ω = 0 by low energy effective theories such as the Luttinger liquid theory. Indeed, in the long distance or time asymptotic regime, the leading and subleading terms of a correlation functions are directly calculated.we find, for example, from bosonization in [96], a formula for the adjacent spin operator correlation functions. The behavior of a DSF singularity at ω 0 (e.g. for the single spin DSF in figure 6.1(b)) could then be compared, after Fourier transform, to the predicted effective field theory form. Either the exponent of the polynomial decay can be confirmed or undefined parameters can be pinned down. However, such comparison is not possible with the RIXS intensity since the signal vanishes precisely at low energy. The spin-exchange operator mainly probes high-energy spectrum and the non perturbative method we have used here is therefore the only way of describing RIXS so far. Nevertheless, it is important to mention that the non-linear Luttinger liquid method is able to describe the singularity of the single spin DSF at such high-energies (lower boundary of the spectrum, around q = π/2) [97, 98] and one might expect an equivalent effective field theory result for the spin-exchange DSF. We have evaluated in this chapter the spin-exchange dynamical structure factor which is proportional to the K-edge RIXS intensity. Using the intermediate expression developed in the chapter 5, we have given a non-perturbative expression for the RIXS cross section (4.3). 84

87 6.3 (a) (b) Figure 6.1: (a) Spin-exchange DSF S exch (q, ω) and (b) single spin DSF S zz (q, ω) for N = 400 sites in h = 0. The intensity around q = π is markedly different in the two cases and the signal of the spin-exchange DSF around q = π 2, 3π 2 is enhanced. 85