Singularities in the Structure Factor of the anisotropic Heisenberg spin-1/2 chain

Transcription

1 Singularities in the Structure Factor of the anisotropic Heisenberg spin-1/ chain Author: P.M. Adamopoulou Supervisor: Dr. Jean-Sébastien Caux Master s Thesis MSc in Physics Track: Theoretical Physics Faculteit der atuurwetenschappen, Wiskunde en Informatica Instituut voor Theoretische Fysica Universiteit van Amsterdam Valckenierstraat XE Amsterdam the etherlands

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3 Abstract The spin-1/ chain is one of the most studied examples of one-dimensional, strongly correlated, systems. Theoretical predictions obtained using the Bethe Ansatz have been successfully linked to experimental observations in several spin chain compounds. In the present thesis we focus on the study of the anisotropic XXZ Heisenberg spin-1/ model in the presence of a magnetic field and show how an exact solution, obtained through the Bethe Ansatz, points to new singularities in the lineshape of the dynamical structure factor of the -particle state [1]. Some developments are described in order to determine the phase space inside which these singularities occur, providing possible predictions for inelastic neutron scattering experiments. 3

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5 Contents 1. Introduction 7. The Spin-1/ Heisenberg Chain 9.1. The XXZ model Eigenstates of the XXZ model - The Bethe Ansatz One downturned spin Two downturned spins M downturned spins - General case Rapidities XXX model XXZ model Quantum umbers Zero field Excitations on-zero field Excitations Thermodynamics General Zero magnetic field XXX model Ground state Excitations XXZ model Ground state Excitations on-zero magnetic field Ground state Excitations Dynamical Structure Factor Linear response theory Experimental measurement Singularities in the DSF umerical form factors Folding condition

6 Contents 6. Conclusion 53 A. Technicalities 55 A.1. Spin operators A.. Trigonometric identities A.3. Scattering phase shift B. TBA formalism 59 C. ABACUS 61 D. XXZ Singularities Program 63 Bibliography 67 6

7 1. Introduction The anisotropic Heisenberg spin-1/ chain with antiferromagnetic interaction is a onedimensional quantum model of spin-1/ interacting particles, widely studied for the past decades. In particular, it was Heisenberg who first introduced the, nowadays know as, Heisenberg model [] in 198, in order to explain ferromagnetism in metals. Soon after that, in 1931, Hans Bethe tried to give an exact solution to this problem, introducing his well-known Bethe Ansatz. In the meantime, antiferromagnetism was introduced by éel. However, it was not before 1958, that the anisotropic extension of the Heisenberg model the XXZ chain was introduced and exact results for the ground state of the antiferromagnetic chain were obtained [3]. Thereafter, the excitation spectrum, as well as the ground state properties, for the infinite chain were extensively studied [4, 5, 6]. Bethe s famous Ansatz opened the door to the study of spin chains, and since then several sophisticated techniques have been developed in order to study them extensively and obtain analytic results for numerous physical quantities related to them. One of such quantities of interest is the Dynamical Structure Factor DSF, defined as the Fourier transform of the spin-spin correlation functions. This quantity is considered of high importance due to the fact that it can give information about the dynamics of the system, as well as it can be measured directly through inelastic neutron scattering experiments In the present thesis we investigate the case of a spin-1/ chain with antiferromagnetic interactions, in the presence of an external magnetic field. In particular, after observing an atypical behavior in the lineshape of the DSF, we attempt to explain it and further investigate it. The outline of this thesis is as follows. In Chapter we introduce the Bethe Ansatz framework and derive all the eigenstates and the corresponding eigenenergies of the XXZ model. Then, in Chapter 3 we demonstrate a pictorial way commonly used in order to classify all the eigenstates of the systems in terms of quantum numbers. Afterwards, in Chapter 4, we focus on the thermodynamic limit approach, that is used to describe the infinite chain, and we show how the ground state and the excitations of the system can be obtained analytically. Finally, the last Chapter is dedicated to the Dynamical Structure Factor, as well as to some recently obtained results about its behaviour when the spin chain is in the presence of a magnetic field. 7

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9 . The Spin-1/ Heisenberg Chain The Heisenberg model [] was first introduced in 198 as an attempt to explain ferromagnetism in metals. Heisenberg considered only spin interactions in nearest neighbouring valence electrons that occupy the s -orbital of atoms in a linear chain. The Hamiltonian he proposed for a chain of sites was H = J S j S j+1, j=1 where J is the exchange coupling between the spins and S j = Sj x,sy j,sz j the spin operator that acts only on site j. For this linear chain of atoms periodic boundary conditions are imposed, such that S j+ S j. This model is nowadays most widely known as the XXX model, or the isotropic Heisenberg model, and can be extended also to explain antiferromagnetic interactions in insulators. By introducing anisotropy in the interaction between the spins [3] we get the anisotropic generalization of the Heisenberg model, also known as the XXZ model..1. The XXZ model The XXZ model is probably the most studied case of the Heisenberg model. The Hamiltonian of the XXZ Heisenberg chain in a magnetic field is H = j=1 J [Sj x Sj+1 x + S y j Sy j+1 + Sz j Sj+1 z 1 ] 4 hsj z..1 Here J is again the exchange coupling that allows one to treat both ferromagnetic J <0 and antiferromagnetic J >0 interactions, is the anisotropy parameter R, and h is the magnetic field. Since we consider spin- 1 particles, the spin operators can be represented by the Pauli matrices, obeying canonical SU commutation relations see appendix A.1. A common strategy is to use spin raising and lowering operators, defined as S ± = S x ± is y, in order to make the calculations more convenient to deal with. This way the Hamiltonian.1 takes the form H = j=1 [ 1 J S+ j S j+1 + S j S+ j+1 + Sz j Sj+1 z 1 ] 4 hsj z.. 9

10 . The Spin-1/ Heisenberg Chain Depending on the values of the anisotropy parameter and the magnetic field h, much can be learned about the behaviour of the ground state, the elementary excitations and the low-temperature thermodynamics of the model [7]. Here we will focus on the region where 0 < < 1, in which a gapless excitation spectrum occurs. Special cases that are of high importance are the limits where, =1and =0, which are also known as the Ising model, the XXX model and the XY model, respectively... Eigenstates of the XXZ model - The Bethe Ansatz The XXZ Hamiltonian. commutes with the ẑ- projection of the total spin operator Stot z = j=1 Sz j, [ ] H, Stot z =0, in such a way that the Hilbert space separates into subspaces of fixed magnetization, H M, along the ẑ- axis. These subspaces are characterized by the number of spins pointing downwards in the ẑ direction, M = Sz tot, with M {0, 1,..., }. The dimensionality of the subspaces can be found by counting the number of M-element subsets of an -element set, or in other words by the binomial coefficient, dimh M =. M By summing over all the subspaces H M we get the dimensionality of the Hilbet space, M=0 dimh M=. We introduce the state with all spins pointing up in the ẑ direction M = 0 to be the reference state 0 j,.3 j=1 or the state of zero energy at zero field, as H 0 =0. This way, we can denote an eigenstate as j 1,j,..., j M = S j 1 S j... S j M 0, where j 1,,...M are the positions of the down spins and Sj 1,,...M the spin operators that act on each site.3. We specialize to the case where j 1 <j <... < j M. We should note here that the choice of the particular reference state allows us to consider only the cases with M. Since the Hamiltonian conserves the ẑ-component of the total spin, one can consequently write any eigenfunction of a state with M down spins as Ψ M = {j} f M j 1,j,..., j M j 1,j,..., j M,.4 where the coefficients f M j 1,j,..., j M of an eigenstate are still to be calculated, in order to fully determine a state. From now on we choose to set h =0, without any loss of generality. We will only re-introduce the magnetic field in subsequent chapters. 10

11 .. Eigenstates of the XXZ model - The Bethe Ansatz..1. One downturned spin Following Bethe s original paper [8] in constructing the eigenstates of the Hamiltonian., we first consider the subspace with M = 1, the dimensionality of which is dimh 1 = =. The wavefunction for this subsector can be written as 1 Ψ 1 = f 1 j 1 j 1, j=1 with j 1 = Sj 1 0, as mentioned above. If we act with the Hamiltonian. on this wavefunction { J H Ψ 1 = S j + S j+1 + S j S+ j+1 + J Sj z Sj+1 z 1 } Ψ 1, 4 where summation over all j s is understood, then the first two terms of the Hamiltonian, also called hopping terms, will just interchange the position of the down spin with its neighbouring up spins. The second term will give two terms of non-zero contribution, since there are three neighbouring sites with opposite spins this term is only non-zero if two adjacent sites have opposite spins. Taking now the Schrödinger equation, H Ψ 1 = E 1 Ψ 1, and projecting it onto the state j 1 will give us the condition J {f 1j f 1 j 1 + 1} J f 1 j 1 =E 1 f 1 j 1.5 for the coefficients f 1 j 1, with 1 <j 1 <. By taking the free wave solution, f 1 j 1 = expik 1 j 1, as the solution of the above equation.5, we get the energy of the state E 1 = Jcos k 1..6 Imposing periodic boundary conditions, f 1 j 1 =f 1 j 1 +, we get that e ik 1 =1from which we can determine the allowed values of the momentum k 1, k 1 = πĩ, Ĩ =0, 1, Equation.7 shows that there are linearly independent solutions generated by this Ansatz, in accordance with the dimensionality of the Hilbert space, dimh 1 =.... Two downturned spins For the case of two downturned spins we have to consider two different sub-cases; one in which the two down spins are adjacent and another one where the two spins are well separated. Following the same procedure as before, this time the eigenfunction assumes the form Ψ = j 1 <j f j 1,j j 1,j, 11

12 . The Spin-1/ Heisenberg Chain were j 1,j = S j 1 S j 0. For the case in which the two spins are well separated, the Schrödinger equation for Ψ projected onto the state j 1,j yields J [f j 1 1,j +f j 1 +1,j +f j 1,j 1 + f j 1,j + 1] J f j 1,j =E f j 1,j,.8 with <j 1 +1<j <. On the other hand, for adjacent down spins we get J [f j 1 1,j +f j 1,j + 1] J f j 1,j =E f j 1,j,.9 with <j = j <. This set of equations can be solved by the Ansatz [8] f j 1,j =A 1 e ik 1j 1 +ik j + A 1 e ik j 1 +ik 1 j,.10 i.e. the momenta are permuted among the two down spins for an extended introduction to the Bethe Ansatz one can consult [9], [10], [11]. The coefficients A 1 and A 1 in the above equation are yet to be determined. For the case of well separated down spins, this Ansatz gives for the energy of the state E = Jcos k 1 + cos k..11 The essential part of the solution is to determine the coefficients A 1, A 1 so that equations.8 and.9 become identical when j = j. To this end, we substitute the Ansatz wavefunction.10 into.9, using at the same time.11, finally obtaining see appendix A.3 with A 1 A 1 = 1+eik1+k e ik1 1+e ik 1+k e ik e iφk 1,k, φk 1,k = arctan cos k1 +k sin k1 k cos k1 k,.1 the so-called scattering phase shift function. So, it should be noted that the scattering phase shift is antisymmetric under the interchange of k 1,k, or in other words φk 1,k = φk,k 1. Expressing φ in such a form gives one the opportunity to identify that the phase shift, as well as the wavefunction, vanishes for k 1 = k. Thus, the above Ansatz wavefunction only generates states with non-coinciding momenta, as the opposite would not lead to a meaningful solution. Using the scattering phase shift we can write.10 as unnormalized f j 1,j =e ik 1j 1 +ik j i φk 1,k e ik j 1 +ik 1 j + i φk 1,k..13 Periodic boundary conditions require that f j 1,j =f j,j 1 +, or f j 1,j =f j, j 1, 1

13 .. Eigenstates of the XXZ model - The Bethe Ansatz and it is these conditions that will determine the allowed values for the momenta k 1,k. Substituting one of these conditions into.13 we have that e ik 1j 1 +ik j i φk 1,k e ik j 1 +ik 1 j + i φk 1,k = e ik 1j +ik j 1 ik 1 i φk 1,k e ik j +ik 1 j 1 ik + i φk 1,k. In order for the above equation to hold, the first term of the LHS must be equal to the second term of the RHS, since they have the same dependence on j 1,j. The same holds for the two other terms. These restrictions yield the so-called Bethe equations for the allowed values of k 1,k e ik 1 = e iφk 1,k, e ik = e iφk 1,k..14 It is more convenient to write the Bethe equations in the logarithmic form k 1 + φk 1,k = πĩ1, k φk 1,k =πĩ.15 where { Ĩ1, Ĩ are half-odd integers called quantum numbers, taking values from 1,..., 1 }. As Bethe already pointed out in his original paper [8] the above Ansatz does not generate all the possible solutions that we should get, that is. For a fixed value of Ĩ, Ĩ1 has Ĩ 1 solutions with Ĩ Ĩ1 + Ĩ1 =0,..., Ĩ and Ĩ =,..., 1 [1]. Thus, there is a number of solutions equal to 1 Ĩ = 1 Ĩ 1 = That is obviously less than the number of solutions expected, a fact that led Bethe to the conclusion that there must also exist complex solutions to the Bethe equations.15. Here and in the following we will only restrict ourselves to real solutions of the Bethe equations, which will turn out to be very important for certain configurations of states...3. M downturned spins - General case Since the previous Ansatz worked for two downturned spins we can try to generalize it for an arbitrary number M of down spins, that obeys the condition M. Again, we have to consider two different cases according to the relative position of the down spins. If none of the M down spins are adjacent j a +1<j a+1 then we get a condition of the form J. M [f M j 1,..., j a 1,..., j M +f M j 1,... j a +1,..., j M ] a=1 J M f M j 1,..., j M =E M f M j 1,..., j M,.16 a=1 13

14 . The Spin-1/ Heisenberg Chain whereas if two of the M down spins are adjacent, say j k + 1 = j k+1, then the condition reads M a k,k+1 { J [f Mj 1,..., j a 1,..., j M +f M j 1,... j a +1,..., j M ] } J f M j 1,..., j a,..., j M + J [f Mj 1,... j k 1,j k +1,..., j M +f M j 1,..., j k,j k +,..., j M ] =E M + J f M j 1,..., j M..17 Bethe proposed [8] that the solution to the above system of equations is a wavefunction of the form [ ] M! M f M j 1,..., j M = A P exp i k Pa j a,.18 P=1 where A P are unknown coefficients, the summation over P denotes a summation over all permutations M! among the down spins and P a is the image of a under the permutation P. The condition for well separated down spins.16 gives us the energy of the state E M = J a=1 M cos k a..19 a=1 Following the same procedure as before, we finally obtain the scattering phase shift A P A P = 1+eik P k +k P k e ik Pk 1+e ik e iφkpk,k P k P k +k P ik k e P,.0 k where P denotes a permutation which differs from P by the exchange of two adjacent elements, such that P k = P k+1, as well as P k+1 = P k. Since we can express every permutation as a sequence of permutations of two adjacent elements, we can write the coefficients A P as a product of scattering phases, one for every pair of two adjacent elements. Luckily, this product is independent of the path of the permutations see e.g. [13]. This way we can write the Ansatz wavefunction for M down spins as M! f M j 1,..., j M = 1 [P] e i P M a=1 k Pa ja i P M a>b 1 φk Pa,k P b P=1 with [P] the parity of the permutation. Periodic boundary conditions impose.1 f M j 1,..., j M =f M j,..., j 1 + and by substituting.1 in this equation we get that 1 [P] A P e i P M a=1 k Pa ja = 1 [P ] A P e i P M P P a=1 k P a ja+ik P 1. 14

15 .3. Rapidities Using.0 and taking into account that since P is a permutation that differs from P only by the exchange of two adjacent spins we can obtain it by shifting the elements in P as P =P M, P 1,..., P M 1, we end up with the Bethe equations e ika = 1 M 1 e i P b φka,k b.. The factor in front of the exponential comes from the shifting of elements of the permutation, otherwise interpreted as M 1 exchange phases that a particle down spin will pick up if it travels once around the chain to its initial position. The logarithmic form of the Bethe equations gives us k a + b φk a,k b = πĩa,.3 with a =1,,..., M and Ĩa integers when M is odd, or half-odd integers when M is even. Thus, the Bethe equations are M coupled non-linear equations for the momenta of the M down spins, the solution of which characterizes the energy and momentum of an eigenstate of the XXZ Hamiltonian. M M E M = J cos k a hstot, z P = k a..4 a=1 In the equations above we have re-introduced the magnetic field h and the total magnetization of the system in the ẑ-axis Stot z = σ = M, with σ the magnetization per site. According to the discussion made above, we have managed to express the scattering of particles down spins as a sequence of a two-particle scattering, reducing, therefore, the problem of finding a complete wavefunction that satisfies our Hamiltonian to just determining all the two-particle scattering phases. This assumption is an intrinsic property of integrable systems, commonly found in one dimension. When two particles scatter in one dimension their mometa are either exchanged or conserved. As a result the particle wavefunctions just pick up a scattering phase. Thus, there is a fundamental difference between -particle scattering in one- and higher-dimensional systems. a=1.3. Rapidities A common and convenient way to study the Bethe equations is by parametrizing the bare momenta k a in terms of rapidities λ a, in such a way that the two-particle scattering phase shift φk a,k b becomes a new function that depends only on the rapidity difference [3]. This parametrization varies depending on the value of the anisotropy parameter. In the following we will only consider the cases where =1and < 1, or in other words the XXX and the gapless XXZ model respectively. 15

16 . The Spin-1/ Heisenberg Chain.3.1. XXX model For the isotropic Heisenberg antiferromagnet =1 the momentum k is written in terms of rapidities λ as λ + i/ e ik =..5 λ i/ Taking the logarithm of.5 yields λ + i/ k = i ln λ i/ 1 = arctan λ = π arctanλ,.6 or λ = 1 cot k so that k = π when λ =0. Using.5 the LHS of the Bethe equations. becomes whereas for the RHS we have that e ika = λa + i/, λ a i/ e iφka,k b = 1+eik P k +k P k e ik Pk 1+e ik P k +k P k e ik P k = λ a λ b + i λ a λ b i,.7 finally leading to a new form of the Bethe equations, this time in terms of the rapidities, λa + i/ = λ a i/ M b a λ a λ b + i, a =1,..., M..8 λ a λ b i Taking the logarithm of the Bethe equations.8 and by using some of the trigonometric identities found in A. we can rewrite them as or in a more compact form arctanλ a 1 M b=1 arctanλ a λ b =π I a, φ 1 λ a 1 M b=1 φ λ a λ b =π I a, a =1,..., M,.9 where φ n λ arctan λ n and the quantum numbers I a are integers for M odd and half-odd integers for M even. The scattering phase shift φk a,k b is obtained as a 16

17 .3. Rapidities function of the rapidities if we take the logarithm of.7 φk a,k b = i ln λ a λ b + i λ a λ b i [ ] 1 = i iπ +iarctan λ a λ b = arctanλ a λ b,.30 so indeed we see that it has become a function of the rapidity difference. The energy and momentum of a state.4 can also be described in terms of rapidities as and E = J a=1 = J M cos k a 1 hstot z a=1 = πj a=1 M [cos arctanλ a + 1] hstot z a=1 M a 1 λ a hstot z.31 a=1 M M P = k a = π φ 1 λ a = πm π respectively, where a n λ = 1 π d dλ φ nλ = 1 π n λ +n /4. M I a modπ.3 Yang and Yang [5] proved that the ground state can be obtained from the following choice of quantum numbers considering even a=1 Ia GS = M +1 + a, a =1,..., M.33 and all the excited states can be found from different choices of sets of quantum numbers {I a }..3.. XXZ model Similarly, for the XXZ case the parametrization reads e ik = sinhλ + iζ/, = cos ζ..34 sinhλ iζ/ This time the Bethe equations are written in terms of rapidities as sinhλa + iζ/ = sinhλ a iζ/ M b a sinh λ a λ b + iζ sinh λ a λ b iζ

18 . The Spin-1/ Heisenberg Chain In logarithmic form, and with the use of A.,.35 are written as or ln siniλa ζ/ = siniλ a + ζ/ M ln b=1 [ ] tanζ/ + i tanhλa iπ ln = tanζ/ i tanhλ a M b=1 sin iλa λ b ζ, sin iλ a λ b +ζ [ ] sin ζ + iλa λ b iπ ln. sin ζ iλ a λ b After a few manipulations the Bethe equations can be transformed into or tanh λa arctan 1 tanζ/ φ 1 λ a 1 M b=1 M tanhλa λ b arctan tan ζ b=1 with φ n this time defined as φ n λ arctan tanh λ tannζ/ =π I a, φ λ a λ b = π I a, a =1,..., M.36 and I a being integers for M odd and half-odd integers for M even, as before. Energy and momentum are again transformed into functions of the rapidities in which a=1 E = πj sin ζ a=1 M a 1 λ a hstot z,.37 a=1 M M P = k a = πm φ 1 λ a =πm π a n = 1 π d dλ φ nλ = 1 π M I a modπ.38 a=1 sinnζ coshλ cosnζ..39 It should be noted that the parametrization for the isotropic case can be obtained by λ taking the limit of the anisotropic one λ =1 = lim <1 ζ 0 ζ. As for the XXX model, the ground state can be constructed from quantum numbers that satisfy.33. As it was mentioned above, there exist complex solutions to the Bethe equations, a fact introduced by Bethe [8] and further developed by Takahashi [14]; it is known as the string hypothesis. These solutions have complex rapidities, which means complex momenta, and they represent bound states of magnons group of down spins. For the purpose of this thesis we will only focus on the real solutions of the Bethe equations. To conclude, after a brief introduction to the XXZ spin-1/ chain, the Bethe Ansatz for this model was presented. Using the Bethe Ansatz one can obtain the exact eigenstates and eigenenergies of the XXZ chain in terms of rapidities. These rapidities are closely related to the so-called quantum numbers. Thus, a set of quantum numbers and rapidities can uniquely label an eigenstate of the system and in what follows we will see how one can determine these quantum numbers. 18

19 3. Quantum umbers As it emerged from previous discussions, it is the choice of quantum numbers that can fully characterize a state and indicate whether that state actually represents an eigenstate of the system. In the following, it will become clear that not all quantum numbers can be chosen for a system with given, M. Thus, it turns out useful, as well as common, to picture the chosen occupied quantum numbers as particles and the vacant ones as holes. As it was already mentioned in the previous chapter, we will only restrict ourselves to real solutions of the Bethe equations Zero field In order to derive the bounds for the quantum numbers in the case of zero magnetic field we will consider the isotropic model XXX chain, for which we will search for a set of real, finite rapidities that satisfy the Bethe equations.9. The strategy is to first send one rapidity to infinity, and afterwards take the limit λ a of the Bethe equations.9. The maximum bound for the quantum numbers IM is then related to the infinite rapidity with the rest M 1 rapidities being finite according to lim λ a φ 1 λ a 1 M φ λ a λ b b=1 The LHS of the equation above is written as lim λ a thus 3.1 yields arctan λ a 1 M arctan λ a λ b b=1 I M = M +1 which for M even odd is half-odd even integer. number we can define the maximun quantum number π I M. 3.1 = π 1 M 1,, 3. From the infinite quantum IM max = IM 1= M 1, 3.3 such that all quantum numbers that correspond to real, finite rapidities obey the restriction I a <I M. 19

20 3. Quantum umbers ow that we found the boundaries of the quantum numbers that yield finite, real solutions to the Bethe equations, we can actually count the number of these solutions. The number of vacancies for the M non-coinciding quantum numbers is equal to IM max+1, taking also into account the I =0case. Thus, putting M particles into the IM max +1 available positions yields a number of solutions equal to I max M +1 M = M M which is obviously less than the number of required solutions M. Therefore, the rest of the solutions are expected to either have coinciding quantum numbers, or lie on the complex plane. It is only the ground state in the case of zero magnetic field that consists of only real, finite rapidities. Since the number of down spins is M = the number of solutions can be found to be equal to I max / +1 = M, / =1, / which leads to the conclusion that we only have one non-degenerate ground state with real, finite rapidities, represented by the following set of quantum numbers according to.33 { } M 1,..., M 1 = { 4 + 1,..., 4 1 }. 3.4 Thus, all the available quantum numbers are occupied in the case of the ground state at zero field. The ground state can be visualized as a fully packed distribution of quantum numbers, shown in Fig Excitations Starting from the ground state configuration with M = down spins, we can create excitations to our system by upturning down spins. These upturned spins correspond to unoccupied positions holes in the quantum number distribution, otherwise known as spinons. These excitations can be interpreted as domain walls topological excitations in an antiferromagnetic order parameter between sites with parallel spins [15]. After they are created they separate regions of antiferromagnetic ordering. The simplest excited state to construct is by upturning one of the down spins, providing our rapidities are real and finite. The number of down spins is now M = 1 leading, according to 3.3, to a number of available positions for the quantum numbers equal to IM max + 1 = +1. Since we have M = 1 particles to put in +1 available positions the number of different states we can construct is I max M +1 /+1 = = M / also known as -spinon states. The number of spinons is IM max +1 M = M and so for even the number of spinons is also even, reassuring that spinons always appear in pairs. A visualization of -spinon excitations is illustrated below in Fig. 3.1., 0

21 3.. on-zero field Fig. 3.1.: Representation of different eigenstates in the case of zero field for = 0. The black circles represent the occupied quantum numbers particles, while the empty ones represent the unoccupied ones holes. The thick lines set the boundaries of the quantum number distribution. The first set corresponds to the ground state configuration. The second and third one correspond to the state with M =9and two holes. The holes can be placed everywhere in the quantum number distribution. It is, therefore, in the case of zero magnetic field that all available positions for the quantum numbers are filled both for the ground state and the excited ones. In addition, one can generalize the spinon picture explained above to the anisotropic chains as well. 3.. on-zero field The finite field case is fundamentaly different from the zero field one, since even in the ground state a certain number of available quantum numbers remains unoccupied. More precisely, in order to construct the ground state we have the following choice of quantum numbers, in accordance with.33, Ia GS = M +1 + a, a =1,..., M. Focusing on the XXZ model, this time we require λ a < and from the Bethe equations.36 we have I M = M ζ M π, and thus, the quantum numbers again have to obey the restriction I a <IM since we deal with real, finite rapidities. Due to the presence of the magnetic field though, the boundary IM this time is larger, leaving space for unoccupied positions in the quantum number distribution, even in the case of the ground state. A way to represent the ground state is shown in Fig Excitations In order to create excitations in the presence of a magnetic field we can take one of the particles, that belongs to the ground state configuration, and move it outside the ground state interval, creating this way a hole in the ground state quantum number distribution. Thus, we can construct particle-hole-like excitations over a Fermi sea, which in our case These particles and holes are sometimes called psinons and antipsinons, respectively [16], but we will not follow this convention throughout this thesis. 1

22 3. Quantum umbers is represented by the ground state quantum number distribution. An illustrative picture of the excitations above the ground state is shown in Fig. 3.. We can determine the momentum of an excited particle-hole state which is also called two-particle state by calculating the difference between the quantum numbers of the ground state and those of the excited one δi = a I GS a I a. Hence, the momentum is equal to q = π δi. So, by shifting the particle and hole quantum numbers, always leaving their difference fixed, we can create q π two-particle states with the same momentum. A two-particle continuum is thus generated, since the energy of these states is non-degenerate. In subsequent chapters, we will come across this two-particle continuum again. Following the same procedure we can construct excitations with more than one particlehole pairs, such as four-particle excitations. The latter are obtained by creating two particle-hole pairs in the quantum number distribution. Fig. 3..: Representation of different eigenstates in terms of quantum numbers in the case of non-zero magnetic field. The black circles correspond to occupied quantum numbers particles, whereas the empty ones to unoccupied ones holes. The thick lines represent the quantum number boundaries, while the dotted lines represent the limits of the ground state interval. The first set stands for the ground state configuration. The second and third ones correspond to two-particle excitations at different momenta. So far, we have managed to express all the eigenstates of the XXZ model and their corresponding eigenenergies in terms of sets of rapidities and quanutm numbers using the Bethe equations. We have also demonstrated a way to represent the eigenstates of the model according to these quantum numbers. In this formulation the length of the chain was considered to be finite with a known number of sites. However, in order to obtain exact analytic expressions for the eigenenergies and the momenta of these states we will pass to the so-called thermodynamic limit, where the number of sites is sent to infinity.

23 4. Thermodynamics The Bethe equations.36 is a system of coupled non-linear equations for the rapidities and quantum numbers, thus finding solutions to this system seems impossible. However, going to the thermodynamic limit, that is let the system size approach infinity while keeping the magnetization fixed, can lead to several analytic results concerning the ground state and the excitations of the spin chain General Consider sets of rapidities {λ a } and quantum numbers {I a } that satisfy the Bethe equations.36. We begin by defining the real parameter x = Ia for the quantum numbers. This way a function λx of the parameter x can be introduced, such that takes the value of the real rapidity λ a when evaluated at argument Ia Ia λx =λ = λ a, a =1,..., M. ext, we introduce a density function also depending on the quantum numbers ρx = 1 M a=1 δ x I a. 4.1 Therefore, the Bethe equations.36 can be written as φ 1 λx ˆx + x dy φ λx λy ρy πx x= Ia =0, with x,x + real numbers. What we want is for the function λx to take values for all x R, generalizing this way the above equation to the whole real axis φ 1 λx x ˆ + x dy φ λx λy ρy = πx. 4. Thus, equation 4. is also defined on the set of points that represents the unoccupied quantum numbers, {Ī}. These quantum numbers together with the occupied ones {I}, satisfy the condition {I}+{Ī} = {T }. Here {T } = Z+1/ for M even and {T } = Z 3

24 4. Thermodynamics for M odd, as for.36. Consequently, we can define particle and hole densities in x-space as ρx = 1 δ x l, ρ h x = 1 δ x m l {I} m/ {I} respectively, such that the total density is ρ tot x =ρx+ρ h x. In the thermodynamic limit ρ tot becomes ρ tot x dy δx y =1. These densities can also be written in rapidity space using the transformation rules for δ functions as ρλ =ρxλ dxλ dλ, ρh λ =ρ h xλ dxλ dλ, ρ totλ = dxλ dλ. 4.3 This way the Bethe equations 4. become φ 1 λ ˆB + B dλ φ λ λ ρλ =πxλ, 4.4 with B ± = λx ±. Differentiating with respect to λ and using 4.3 leads to a 1 λ ˆB + B dλ a λ λ ρλ =ρλ+ρ h λ 4.5 which is the continuous version of the Bethe equations.36 for a given eigenstate with rapidity distribution ρλ. The limits of integration are equal to a magnetizationdependent constant B and always depend on the limits of the quantum numbers that characterize an eigenstate according to Chapter 3. As will be discussed in what follows, this realization is crucial and actually determines the solutions to the Bethe equations 4.5 for the XXZ chain in the presence or absence of a magnetic field. The energy.37 and momentum.38 of an eigenstate in terms of the rapidity distribution ρλ become and E = πj sin ζ M a 1 λ a hstot z = πj sin ζ a=1 P = πm M φ 1 λ a = a=1 ˆB B ˆB B dλ a 1 λρλ hs z tot 4.6 dλ π φ 1 λ ρλ, 4.7 We suppose that the mapping λx defined in 4. is invertible, so that xλ exists. 4

25 4.. Zero magnetic field respectively. So, in order to calculate the energy spectrum of a given eigenstate of the XXZ Hamiltonian. we always have to solve the integral equation 4.5 that determines the density ρλ of the state. As will become more clear in the following sections, this task is highly non-trivial and analytic results are not always available neither for the ground state nor for the excitations of an XXZ chain. 4.. Zero magnetic field It was Hulthén who first studied the thermodynamic limit of the isotropic Heisenberg antiferromagnet and managed to derive its ground state energy [17]. After that, des Cloizeaux and Pearson [4] derived the antiferromagnetic spin-wave spectrum. In this section, we will derive the ground state and excitations first for the isotropic Heisenberg antiferromagnet and then for the anisotropic gapless case XXX model The isotropic case XXX chain on zero magnetic field is the easiest to tackle analytically, so here we will shown how one can obtain its ground state and the spinon excitation energies Ground state The ground state can be characterized by a symmetric around λ =0distribution of quantum numbers. In the interval [ B, B] the hole density vanishes, whereas the particle density vanishes for λ >B, so 4.5 becomes ˆB ρ GS λ+ dλ a λ λ ρ GS λ =a 1 λ 4.8 B According to a previous discussion see Chapter 3, in the case of zero field the magnetization in the ground state is zero and the set of quantum numbers that characterizes the state occupy all the available values. Thus, the integration limits B can be taken to infinity B, allowing us this way to define Fourier transforms as ρ GS λ = dω π e iωλ ρ GS ω, ρ GS ω = dλ e iωλ ρ GS λ

26 4. Thermodynamics The Fourier transform of the kernels a n λ can be calculated using contour integration a n ω = = dλ e iωλ a n λ dλ e iωλ 1 π n λ + n /4 = e ω n We can now solve for the ground state rapidity distribution ρ GS λ by first Fourier transforming both sides of 4.8 while using the convolution theorem which then substituting 4.10 gives, ρ GS ω+a ωρ GS ω =a 1 ω, ρ GS ω = a 1ω 1+a ω = 1 cosh ω. The distribution ρ GS λ can now be obtained if we take the inverse Fourier transform of ρ GS ω ρ GS λ = dω π e iωλ cosh ω = dω cosωλ i sinωλ 4π cosh ω. Using some formulas from [18], as well as some useful relations from A. the integral takes the form finally leading to ρ GS λ = 1 4π ˆ0 dω cos ωλ cosh ω + 1 dω cosωλ 4π cosh ω, 0 ρ GS λ = 1 π π secλπ = 1 coshλπ Knowing the ground state rapidity distribution it is easy to calculate the ground state energy of the infinite antiferromagnet by substituting 4.11 in 4.6, for h =0, using again some useful formulas from [18] If f and g are two functions with convolution f gλ = dλ fλ λ gλ and F is the Fourier transform operator then, F{f gλ} = F{f} F{g} = fωgω, with fω and gω the Fourier transforms of f and g. 6

27 4.. Zero magnetic field E GS = πj = J 4 = J 0 dλ a 1 λρ GS λ 1 1 dλ λ +1cosh πλ 1 dλ λ +1 1 cosh πλ = J ln. 4.1 The total momentum is P GS = πm π M I j = πm = π j=1 mod π, since for the ground state the summation over the quantum numbers is zero and for the case of zero magnetic field M = Excitations The simplest type of excited states we can construct analytically in zero magnetic field are the ones created by upturning one of the M downturned spins. By doing so, we can construct + 8 excited states, also known as -spinon states, always considering real, finite rapidities see discussion in Section The spinons correspond to unoccupied quantum numbers, otherwise called holes, in the quantum number distribution, represented by I1 h, Ih. This way the x-space hole density can be written as ρ h x = 1 δ x m In rapidity space the above becomes m/ {I} = 1 δ x Ih 1 ρ h λ = 1 δ λ λ h δ x Ih. + 1 δ λ λ h, if we define λ h I h i = λ i. Consequently, the continuous version of the Bethe equations 4.5 is now transformed into a 1 λ dλ a λ λ ρλ =ρλ+ 1 δ λ λ h δ λ λ h, 7

28 4. Thermodynamics which, by Fourier transforming both sides, leads to a 1 ω a ωρω =ρω+ 1 eiωλh eiωλh The rapidity distribution ρλ of the -spinon state can be obtain is we first solve 4.13 for ρω ρω = a 1 ω 1+a ω 1 = ρ GS ω 1 ρ spω and then take its inverse Fourier transform 1 1+a ω e iωλh 1 + e iωλ h e iωλh 1 + e iωλ h ρλ =ρ GS λ 1 ρ spλ λ h 1 1 ρ spλ λ h The spinon density function is denoted by ρ sp ω = 1 1+e ω and ρ GS λ is the ground state rapidity distribution In order to calculate the spinon rapidity distribution we should take the following Fourier transform ρ sp λ = = = 0 dω e iωλ π 1+e ω dω π ˆ0 1 e ω 1+e ω e iωλ + dω π e iωλ 0 dω π e ω ˆ0 1+e ω e iωλ dω 1 eω π 1+e ω e iωλ dω π e ω 1+e ω e iωλ. It turns out that the spinon rapidity distribution ρ sp λ can be written as a function of the Dirac delta distribution and real parts of digamma functions ψ 0, always considering λ to be real, as ρ sp λ =δλ 1 [ψ π R 0 1+i λ ] 1 ψ 0 + iλ The energy of the excited -spinon state over that of the ground state can be calculated as E E GS = πj = πj dλ a 1 λρλ ρ GS λ dλ a 1 λ ρ sp λ λ h 1+ρ sp λ λ h = ε sp λ h 1+ε sp λ h

29 4.. Zero magnetic field where ε sp λ h is the energy of a single spinon, defined as ε sp λ h = πj = πj = dλ a 1 λρ sp λ λ h dω π a 1ωρ sp ωe iωλh πj coshπλ h In a similar manner, the momentum of the excited -spinon state over the momentum of the ground state can be obtained P P GS = = dλ π φ 1 λ ρλ ρ GS λ dλ π φ 1 λ ρ sp λ λ h 1+ρ sp λ λ h = p sp λ h 1+p sp λ h mod π, 4.18 expressed in terms of the single-spinon momentum, p sp λ h = dλρ sp λ λ h π φ 1 λ, which can be calculated explicitly if we consider that lim λ h p sp λ h =0, as well as dp sp dλ h =π dλ a 1 λρ sp λ = ε spλ h J Consequently, we have for the momentum of a spinon p sp λ h = λ h dλ π coshπλ = arctan e πλh, 4.19 which is defined in the interval [ π, 0], since the rapidities take values in,. Furthermore, expressing λ h as a function of p sp and substituting it into the single-spinon energy 4.17, also using some useful formulas from A., finally yields the spinon dispersion relation ε sp p sp = πj sin p sp

30 4. Thermodynamics The complete -spinon continuum Fig. 4.1 is obtained by using the spinon dispersion relation together with 4.16 E E GS = πj sin p sp1 + sin p sp, where p sp1,p sp are the spinon momenta p sp λ h 1,p spλ h, respectively. Putting one of the spinons at zero energy p sp =0or π we obtain the lowest energy state, also known as des Cloizeaux-Pearson mode [4] ω q Fig. 4.1.: The -spinon continuum for the isotropic chain. The y-axis is the energy of the excited state above the ground state ω = E E GS and the x-axis is the momentum q of the excited state. The momentum takes values from 0 to π. The lower line represents the des Cloizeaux-Pearson mode. Therefore, the dispersion relation obtained above is actually the combination of two elementary spinon excitations, whose energies and momenta add up to give the full spectrum that is illustrated in Fig Since the spin of each individual spinon is 1/, the total spin of the excitation is equal to 0 or 1. Also, as we consider a chain of even number of sites, the total spin is always an integer and thus, the spinons always appear in pairs also see discussion in Chapter XXZ model For completeness, in the following we give the derivation for the ground state and the spinon excitations of the XXZ model, although it is completely analogous to the one done for the isotropic model. 30

31 4.. Zero magnetic field Ground state The Bethe equations 4.5 are ρ GS λ+ dλ a λ λ ρ GS λ =a 1 λ, 4.1 where the limits of integration are again taken to infinity and the kernels a n are of the form.39. Therefore, we can define Fourier transforms for the ground state distributions according to 4.9. This time the Fourier transform of the kernels a n take the form a n = = = dλ e iωλ a n λ dλ eiωλ sin nζ π cosh λ cos nζ sinh ωπ 1 nζ π sinh ωπ. 4. Taking the Fourier transform of both sides of the Bethe equations 4.1 and using the convolution theorem, leads to the following distribution for the ground state ρ GS ω = a 1ω 1+a ω = 1 cosh ωζ whose Fourier transform finally yields the ground state distribution in terms of rapidities, ρ GS λ = dω π e iωλ ωζ cosh = 1 cosh λπ ζ. 4.3 As one could expect, the ground state density distribution is similar to the one obtained for the XXX model, since the isotropic case can be regarded as a limit of the anisotropic one. The only difference comes from the factor ζ, which is related to the anisotropy according to Excitations As for the case of the isotropic model, we can similarly construct the simplest type of excitations by upturning one of the M down spins. We can equivalently create -spinon states, with each spinon having a density distribution of the form [19] ρ sp ω = sinh ωπ sinh ωπ ζ π 31

32 4. Thermodynamics The energy and momentum of the excited -spinon state satisfy 4.16 and 4.18 respectively, but the energy and the momentum of a single spinon this time are given by and ε sp λ h πj sin ζ = ζ cosh p sp λ h = arctan πλ h ζ e πλh ζ. The momentum of a single spinon is again defined in the interval [ π, 0]. The spinon dispersion relation assumes the form [19] ε sp p sp = πj sin ζ ζ sin p sp, which is obviously similar to the one obtained for the isotropic case on-zero magnetic field It is the non-zero magnetic field case of the anisotropic Heisenberg chain that is of main interest for the present work. Following the same line as with the zero field case, we will demonstrate how the excitations can be obtained, after we first present a formal solution for the ground state. All the necessary quantities, such as the rapidity distribution as well as the ground state and excitation energies, will stem from the thermodynamic Bethe equations a 1 λ Ground state dλ a λ λ ρλ =ρλ+ρ h λ, λ R. 4.5 Starting from 4.5 we can solve for the ground state occupation density ρ GS λ, which is non-vanishing only inside the interval [ B, B] contrary to the hole density ρ h GS λ which is non-vanishing ouside this interval. Thus, the rapidities λ = ±B are equivalent to the two Fermi points ±λ F in the rapidity distribution. The occupation density distribution for the ground state is therefore obtained if we solve ρ GS λ+ ˆλ F λ F dλ a λ λ ρ GS λ =a 1 λ, λ [ B, B]. 4.6 In the case of zero magnetic field we saw that the boundaries B, since the magnetization for the ground state is zero. However, this is not the case in the presence of a magnetic field; the boundaries B are now finite, making the calculations somehow 3

33 4.3. on-zero magnetic field more complicated. The integration boundaries and thus, also the two Fermi points can be worked out from the constraint ˆλ F λ F dλρ GS λ = 1 σ, 4.7 where σ is the average magnetization per site, related to the total magnetization through Stot z = σ. Therefore, by calculating the ground state occupation density ρ GS λ we can simultaneously determine not only the boundaries B, but the hole density ρ h GS as well. In order to accomplish that, we will follow a standard procedure [6] that introduces a formal solution for the unknowns ρ GS, ρ h GS and B. First, in order to assure that the ground state occupation density is only non-zero inside the interval [ λ F, λ F ], we define the kernels a λ F 1 λ = θλ F λ a 1 λ, a λ F λ, λ = θλ F λ θλ F λ a λ λ. 4.8 Then we introduce the inverse operators b λ F n, inverse of the ground state kernels 4.8, as dλ [ ][ δλ λ +b λ F n λ, λ δλ λ+a λ F n ] λ λ = δλ λ, λ, λ R 4.9 We should note here that from now on we can send the limits of integration to infinity, since the cutoff of the Fermi points is now encoded in the ground state kernels 4.8. The inverse operator has the properties of being unique, analytic, as well as symmetric, b λ F n λ, λ =b λ F n λ, λ. Although we do not have the explicit expression of this operator, we have the opportunity to write the ground state occupation density in terms of it, expressing in turn the ground state hole density as a function of the occupation density. More precisely, by rewriting 4.6 as [ ] dλ δλ λ +a λ F λ λ ρ GS λ = and applying the inverse integral operator 1+b λ F n manipulations finally yields ρ GS λ = dλ δλ λ a λ F 1 λ on both sides of 4.6, after a few [ ] dλ δλ λ +b λ F n λ, λ a λ F 1 λ λ, λ R Knowing an explicit expression for ρ GS λ gives us the opportunity to solve for ρ h GS λ using the Bethe equations 4.5 ρ h GSλ =a λ F 1 λ dλ a λ F λ λ ρ GS λ ρ GS λ, λ, λ R

34 4. Thermodynamics Another way to determine the ground state is by calculating the free energy f = E TS of the infinite system, which should be minimized at the thermodynamic equilibrium see Appendix B. First, for convenience, we will rewrite the energy of the system 4.6 as E = πj ˆλ F λ F = h ˆλF + dλ a 1 λρλ h M λ F = h ˆλF + where we have defined the bare energy ε 0 λ as dλ h πj sin ζa 1 λ ρλ λ F dλε 0 λρλ, 4.3 ε 0 λ =h πj sin ζa 1 λ So, using 4.3 for the energy E of the system, as well as for the entropy, and given distributions ρλ and ρ h λ we have for the free energy to leading order in f = E TS = h ˆλF + λ F dλε 0 λρλ T [ dλ ρλ+ρ h λ ln ρλ+ρ h λ 4.34 ] ρλ ln ρλ ρ h λ ln ρ h λ. In addition, we define the quasi-energy for the ground state as ελ =T ln ρh λ ρλ. The condition of thermodynamic equilibrium δf =0see Appendix B together with the Bethe equations 4.5 lead to with ε GS λ+ ˆλ F λ F dλ a λ λ ε GS λ =ε 0 λ, λ R, 4.35 ε GS ±λ F =0, ε GS λ 0 for λ [ λ F, λ F ], ε GS λ > 0 for λ / [ λ F, λ F ].4.36 As with the density distributions ρ GS λ and ρ h GS λ, we can equivalently define ε+ λ 0 and ε λ 0 such that ε GS λ =θ λ λ F ε + λ+θλ F λ ε λ

35 4.3. on-zero magnetic field By using the inverse operators b λ F n λ, λ we can write a formal solution for ε λ and then use it to define ε + λ, in a way similar to 4.30, In particular, we obtain and ε λ = ˆλ F λ F dλ ε + λ =ε 0 λ [ ] δλ λ +b λ F λ, λ ε 0 λ, λ R, 4.38 ˆλ F λ F The free energy now assumes the form Excitations f = h ˆλF + dλ λ a F λ λ ε λ, λ R λ F dλ a 1 λε GS λ We will concentrate on the simplest, yet the most important [0], type of excitations that can be created above the ground state, that is, the single particle-hole excitations. In order to create a single particle-hole pair in the infinite chain we choose a quantum number outside of the ground state set {I GS }, related to the particle I p / {I GS }, and another one related to the hole inside the set, I h {I GS }. As we have seen in Section 3..1, this is the analogue of the finite-chain particle-hole excitations, where removing one quantum number from the ground state interval particle and putting in outside of it, creates one unoccupied quantum number inside that interval hole. The particle and hole densities in x-space are respectively, ρx =ρ GS x+ 1 δ x I p ρ h x =ρ h GSx 1 δ x I p Going to rapidity space the density can be written as 1 δ x I h, + 1 δ x I h. ρλ = dxλ dx GS λ ρ GSx+ 1 δλ λ p 1 δλ λ h ρ h λ = dxλ dx GS λ ρh GSx 1 δλ λ p+ 1 δλ λ h with λ p λ F and λ h λ F. Thus, due to the introduction of the particle-hole pair, the density will slightly shift O1/. This shift is encoded to the particle and hole backflow functions, Kλ; λ p, λ h and K h λ; λ p, λ h respectively, defined as ρλ =ρ GS λ+ 1 [ ] Kλ; λ p, λ h +δλ λ p δλ λ h 4.41 ρ h λ =ρ h GSλ+ 1 [ ] K h λ; λ p, λ h δλ λ p +δλ λ h

36 4. Thermodynamics In what follows we will try to give a formal solution by expressing all desired quantities, such as the energy and momentum of the excited state, in terms of the backflow functions. More precisely, the energy of the excited particle-hole state can be written in terms of the backflow function, using 4.3, as ω E E GS = = dλε 0 λρλ ρ GS λ dλε 0 λ[kλ; λ p, λ h +δλ λ p δλ λ h ] = ε 0 λ p ε 0 λ h + dλε 0 λkλ;, λ p, λ h The momentum, on the other hand, satisfies the following equation using 4.7 q P P GS = dλ π φ 1 λ ρλ ρ GS λ = φ 1 λ p +φ 1 λ h dλφ 1 λkλ; λ p, λ h However, according to the Bethe equations.36, as well as.38, the momentum can also be written as q P P GS = M a=1 = π which, by using 4.4, is transformed into q = φ 1 λ p +φ 1 λ h + [ π φ1 λ a π φ GS 1 λ a ] M a=1 IGS a Ia = πx p x h, 4.45 dλ φ λ p λ φ λ h λ ρ GS λ, with an error of O1/, since we have subsituted ρλ with ρ GS λ. We should now try to find a solution to the backflow functions in order to be able to fully determine the excited state. eedless to mention that ρλ obeys the same equations as ρ GS λ, that is, the thermodynamic Bethe equations 4.5. So, subtracting