Spiral Phases of Doped Antiferromagnets from a Systematic Low-Energy Effective Field Theory

Transcription

1 Spiral Phases of Doped Antiferromagnets from a Systematic Low-Energy Effective Field Theory Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Christoph Brügger von Biglen ḂE Leiter der Arbeit: Prof. Dr. U.-J. Wiese Institut für theoretische Physik Universität Bern

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3 Spiral Phases of Doped Antiferromagnets from a Systematic Low-Energy Effective Field Theory Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Christoph Brügger von Biglen ḂE Leiter der Arbeit: Prof. Dr. U.-J. Wiese Institut für theoretische Physik Universität Bern Von der Philosophisch-naturwissenschaftlichen Fakultät angenommen. Der Dekan: Bern, den 28. Februar 2008 Prof. Dr. P. Messerli

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5 ABSTRACT We present a low-energy effective theory for magnons and holes or electrons which are doped with homogeneous and finite density in an antiferromagnet. In particular we investigate how doping affects the configuration of the staggered magnetisation, the order parameter of the spontaneously broken spin rotation symmetry in the antiferromagnets. The cuprates showing high temperature superconductivity, a phenomenon for which a suitable theoretical explanation is still missing, are doped antiferromagnets. Therefore the physics of doped antiferromagnets is of particular interest. The cuprates are expected to be well described by the two-dimensional Hubbard or t-j-type models. The presented approach explores the low-energy physics described by these models based on symmetry considerations. We identify the magnons as Goldstone bosons of the spontaneously broken spin rotation symmetry of the Hubbard model. Then we represent the charge carriers. These are holes centred inside pockets at lattice momenta (± π 2a, ± π 2a ) or electrons centred inside pockets at lattice momenta (π a,0) and (0, π a ). The low-energy effective Lagrangians are then constructed in terms of magnon fields and Grassmann valued hole or electron fields by demanding that the Lagrangian inherits the symmetries of the underlying model. In the lightly hole-doped antiferromagnet the various configurations of the staggered magnetisation are accessed by a variational calculation. We show that the order parameter either uniformly points in one direction or forms a helical state known as a spiral. In the effective theory framework one can prove that the homogeneous state and the spiral are the only two classes of configurations that exist at a constant fermion density. For the electron-doped system the low-energy effective theory predicts the absence of spiral phases. The low-energy effective theory for magnons and charge carriers is inspired by baryon chiral perturbation theory (BChPT) the low-energy effective theory for pions and nucleons in QCD. It describes pions as the Goldstone bosons of the spontaneously broken chiral symmetry and baryons which are the analogue of the doped fermions. The low-energy effective theory reproduces results obtained with earlier approaches in a particularly clear manner. In contrast to the latter, it is completely systematic and model-independent. The results presented in this work are also in good agreement with experiments. v

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7 Contents Introduction 1 1 The Underlying Theory Microscopic Models Electrons are different from Holes Symmetries of the Hubbard Model Strong Coupling Limit of the Hubbard Model and the Heisenberg Hamiltonian 14 2 Construction Of The Effective Lagrangians The Magnons Non-linear Realisation of the SU(2) s Symmetry Fermionic Degrees of Freedom Effective Lagrangian for Holes Effective Lagrangian for Electrons Spiral Phases of the Doped Antiferromagnet Spirals with Uniform Composite Vector Fields Hole Doped Antiferromagnet - Spiral Phases Electron Doped Antiferromagnet - Homogeneous Phase QCD and Chiral Perturbation Theory Spontaneous Breakdown of Continuous Global Symmetries and Order Parameters Non-Linear Realisation and Construction of the Lagrangian Nonrelativistic Regime And Universality Conclusion and Outlook 71 Acknowledgments 73 A The Most General Configuration with Uniform Background Fields 75 B Minimum of the Dispersion Relation 79 B.1 Minimum Energy of the Doped Hole B.2 Minimum Energy of the Doped Electron Bibliography 83 vii

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9 Introduction In 1986 Bednorz and Müller discovered the phenomenon which now occupies a large number of experimental and theoretical physicists: High-temperature superconductivity (HTSC) [1]. The phenomenon was first discovered in doped La 2 CuO 4. Doping means that a small amount x of e.g. La (Lanthanum) in the La 2 CuO 4 -crystal is replaced by another atom. As a result one obtains La [2 x] Sr x CuO 4 (LSCO). Nowadays, there is a wide range of doped cuprates which also contain Halogenes (Halogen family, including Ca 2 CuO 2 Cl 2 ), Mercury (Hg family, including HgBa 2 CuO [3+x] ) or the famous YBCO family with e.g. YaBa 2 Cu 3 O 6+x. The doping changes the amount of electrons in the crystal. On the one hand, doping creates an electron vacancy known as a hole and the resulting materials are called hole-doped (e.g. LSCO). On the other hand, it may add an additional electron to the crystal and the resulting materials are called electron-doped (e.g. Nd 2 x Ce x CuO 4 ). Although the mechanism of doping may differ a lot in the various materials, they have in common a two-dimensional CuO 2 plane, depicted in figure 1 with an occupancy of one electron per unit cell. As long as the HTSC materials are undoped (or at least at low doping) this copper oxide plane shows antiferromagnetic order. Due to this antiferromagnetic order, one can conclude, that understanding the physics Figure 1: (A) Crystal structure of the La 2 CuO 4 unit cell, parent compound of the La [2 x] Sr x CuO 4 family of high-temperature superconductors. (B) CuO 2 plane which extends in the a-b direction. The red arrows displayed in Néel order illustrate the antiferromagnetic ordering in the plane. [2] 1

10 of the antiferromagnet is essential in order to understand high-temperature superconductivity. Soon after the discovery of HTSC, Anderson showed that such cuprates can be described by quasi-two-dimensional models [3]. He argued, that the interlayer couplings between the CuO 2 planes are very weak, the relevant physics is contained in a single CuO 2 plane displaying antiferromagnetic order. The standard antiferromagnetic model is the two-dimensional nearest-neighbour Heisenberg Hamiltonian. This model is not analytically solvable. Substantial progress has been made by Chakravarty, Halperin, and Nelson [4] as well as by Hasenfratz, Leutwyler and Niedermayer [5,6]. Assuming that there is long range order at T = 0 in the Heisenberg antiferromagnet, they showed that it can be described by the (classical) O(3) nonlinear σ-model and that this is a good model to describe undoped La 2 CuO 4. The simplest models to describe interacting and itinerant fermions are the Hubbard or t-j-type models. Often these models are used to describe doped systems. A lot of effects which give rise to the phase diagram displayed in figure 2 are not included in the Hubbard or t-j-type models. The doping for example necessarily gives rise to impurities in the real materials. The doped holes and electrons get localised on the impurities. This effect may prevent the coexistence of antiferromagnetism and superconductivity in the real cuprates. The Hubbard or t-j-type models are clean systems where both phases may coexist. In addition, both models are formulated on a lattice which is put by hand. Therefore, they do not contain phonons and can not describe effects due to the phonons which are present in the cuprates. The statement, that the relevant physics takes place in two dimensions is also an approximation. As shown e.g. in [7], the T c of the cuprates may depend very much on the number of layers in the material. This shows that there is an interlayer interaction which has an effect on the phase diagram of the cuprates. Investigating the cuprates with the ARPES technique (Angular Resolved Photo Emission Spectroscopy) one found out that holes and electrons appear at different lattice momenta. While holes enter the Brillouin zone at ( ± π 2a, ± 2a) π, electrons occur at ( π a,0) and ( 0, π a). A priori it is not obvious, that this experimental fact is contained in the Hubbard and t-j-type models. Only for specific values of U or t and J, respectively, numerical simulations show that the doped holes or electrons appear at the same places in the Brillouin zone as the ARPES experiments have shown. The above comparison between real cuprates and the Hubbard or t-j-type models was not meant to suggest that these models are inappropriate to describe at least the antiferromagnetic phase of the cuprates. Since they are free of impurity effects, these models are useful to focus on the physics purely driven by magnons and doped fermions in a two-dimensional system. In the Hubbard or t-j-type models, systematic numerical and analytical investigations are both very difficult because the fermions in these system are strongly correlated. Away from half-filling numerical simulations are not possible due to a severe fermion sign problem. Due to this problem numerous approaches also investigated the dynamics of the doped holes and electrons with effective models. Starting with Shraiman and Siggia [8], Wen [9] and Shankar [10], these approaches use composite vector fields to couple magnons and fermions. The spin of the fermions then occurs as the charge of an Abelian gauge field. In this approach, the Lagrangian is usually obtained from an underlying system (typically the above mentioned Hubbard or t-j-type models) by integrating out high-energy degrees of freedom [8 10]. However, these 2

11 theories do not agree on the fermion field content. Often it is not even discussed, how various symmetries are realised on those fields. In addition, it has never been demonstrated convincingly, that any existing effective model indeed correctly describes the low-energy physics of the underlying microscopic model quantitatively. The aim of this work is to apply a technique that has been used successfully in particle physics to condensed matter systems. Quantum chromodynamics (QCD) is the theory of the strong interactions between quarks and gluons. Just as the underlying models describing the strongly correlated electron systems, QCD is very difficult to simulate numerically at nonzero baryon density due to a complex action problem. Baryons (e.g. neutron and proton) are the particle physics analogue of the holes or electrons doped into a cuprate. In QCD we have a spontaneously broken continuous global symmetry known as chiral symmetry. According to the Nambu-Goldstone theorem, a spontaneously broken continuous global symmetry gives rise to massless excitations in the spectrum [11]. These excitations are identified as the pions which are the Goldstone bosons of the spontaneously broken chiral symmetry. An effective theory for the low-energy regime of QCD has been proposed by Weinberg [12]. It is based on the idea that, given the Goldstone degrees of freedom, the most general Lagrangian describing the low-energy structure of the strong interactions can be obtained by respecting the symmetries of the underlying QCD together with the general principles of quantum field theory (e.g. analyticity, unitarity, etc). In the mid-eighties Weinbergs idea has been developed into a theory by Gasser and Leutwyler [13]. Chiral Perturbation Theory (ChPT) uses the nonlinear realisation of the spontaneously broken global continuous chiral symmetry of QCD. In analogy to doping a cuprate with holes or electrons, the purely pionic system can be doped with baryons. In this way one obtains baryon chiral perturbation theory (BChPT) [14]. The Lagrangian of this low-energy effective theory for QCD is then constructed using the nonlinear realisation. The transformation properties of the pion and baryon fields to be included in the theory are completely specified. The Lagrangian can be obtained by constructing all terms which are invariant under the symmetries of the underlying QCD. In this thesis we want to apply the above principles to the Hubbard or t-j type models. It is known from numerical simulations, that these systems undergo a spontaneous breakdown of the global continuous SU(2) spin rotation symmetry. This breakdown can be seen in numerical simulations, but a rigorous proof for spin S = 1 2 is still missing. The Goldstone bosons of the spontaneously broken spin symmetry are known as the magnons. Applying the nonlinear realisation technique, we obtain magnons which are represented as composite vector fields. The Lagrangian describing magnons in an antiferromagnet is obtained by constructing all terms which are invariant under the symmetries of the underlying Hubbard or t-j-type models. These terms are organised according to the number of derivatives they contain. The leading terms of this derivative expansion have been obtained in [4, 5]. In order to include fermions into the theory, one may proceed along the lines of BChPT. Once one knows how the fermion fields representing the doped holes or electrons transform, one can construct all terms which are invariant under the symmetries of the Hubbard or t-j-type models to obtain the most general Lagrangian. All the couplings which appear in the Lagrangian are so-called low-energy constants. They cannot be determined by symmetry arguments and must be extracted from numerical simulations or from experiment. We have already argued that in the cuprates hole and electron pockets appear at given 3

12 momenta in the Brillouin zone. While holes are doped into pockets centred at ( ± π 2a, ± 2a) π, electrons appear in pockets centred at ( π a,0) and ( 0, π a) respectively. By applying the symmetry analysis one does not automatically obtain this property of the cuprates in the effective theory. Instead, one has to put it by hand. Figure 2: Schematic phase diagram of high T c superconductors showing electron doping on the left-hand side and hole doping on the right-hand side [15] Many publications [8, 16 33] investigate the influence of arbitrarily small doping on the antiferromagnetic ordering. In particular, it was suggested that the Néel phase of the undoped antiferromagnet is replaced by a spiral phase, a helical structure in the staggered magnetisation configuration already at arbitrarily small doping. The idea of a spiraling spin state goes back to Nagaoka [34]. In terms of t-j-like models it has been investigated by Shraiman and Siggia [8]. Although there is no doubt about the existence of spiral phases in hole-doped systems (neither theoretically nor experimentally), there are different opinions about the stability of the phases obtained. The magnetic properties of an antiferromagnet are often measured with neutron scattering experiments. The antiferromagnet has a period given by the wavevector Q = (π,π) in the Brillouin zone. Neutron scattering experiments measure periodic structures and therefore the antiferromagnetic wavevector. If the wavevector is a simple rational multiple of Q then the state of the system is called commensurate else it is called incommensurate. For the hole-doped cuprates people measure incommensurate order [35, 36] even at low doping. There exist two different interpretations for the incommensurate order. Either the incommensurate peaks stem from a so-called stripe ordered phase or from the helical spin ordered phase described in this thesis. In the stripe phase, the magnetic structure alternates in a given direction between an antiferromagnetic phase without holes and a phase where the doped holes accumulate. Perpendicular to this alternation the magnetic structure does not change. There are two main differences between stripes and the spiral. The spiral is 4

13 homogeneously doped, while the stripe is charge ordered besides the magnetic ordering. While incommensurate order is a generic feature in the hole-doped cuprates its interpretation may differ a lot. For low-doped antiferromagnets the incommensurability is doping independent [36, 37]. Currently there is no agreement on what picture, the spiral or stripes, is favourable to explain the physics of the antiferromagnet at low-doping. Recent experimental data [35,38,39] is likely to be interpreted as a consequence of a spiraling staggered magnetisation [40]. In the electron doped materials no incommensurate order, i.e. no spiral can be found [41]. In this thesis we want to investigate these spiral phases at homogeneous and low hole or electron doping using the low-energy effective theory. We access the spiral configuration through a variational calculation. The composite vector fields are treated as constant to supply a homogeneous magnetic background for the homogeneously doped fermions. The effective theory reproduces some results obtained by other authors for hole-doped antiferromagnets [8, 18, 27] in a particularly clear way. We have proved that the only two possible configurations leading to a homogeneous fermion background field are the Néel phase and the spiral phase. For electron-doped cuprates, the low-energy effective theory predicts the absence of a spiral phase in agreement with experiments [42 45]. It should be noted that the spiral phase has an analogue in QCD - the pion condensate in nuclear matter [46 50]. The thesis is organised as follows. Chapter 1 deals with the underlying microscopic models. We perform a symmetry analysis in the Hubbard model since symmetries are one key point in the formulation of a low-energy effective theory. The strong coupling limit and spontaneous symmetry breaking are discussed. In chapter 2 we describe in detail how to construct a nonlinear realisation for the magnons. Using symmetry considerations, the fermion content of the theory is identified by relating the Hubbard model lattice operator to continuum Grassmann numbers. ARPES results are used to include the doped holes and electrons in the right positions in the Brillouin zone. As stated above, the Lagrangian can be constructed by respecting the symmetries of the underlying models. We give both the Lagrangian for holes and electrons to second order in the derivative expansion and discuss its symmetries. A short comment on power-counting is made. In chapter 3 we investigate fermions doped into a homogeneous magnetic background. Performing a U(1) gauge transformation we obtain constant background fields. We investigate the various phases of the staggered magnetisation. The influence of the four-fermi contact interactions in the Lagrangian is addressed and we show that a spiral phase exists and may be stable. The stability depends on the low-energy constants in the Lagrangian. For doped electrons, it can be shown that a spiral is energetically less favourable than a homogeneous phase. The reduction of the staggered magnetisation upon doping is also considered for both hole and electron doped systems. In chapter 4, we review the basic properties of QCD. Spontaneous symmetry breaking and its consequences are discussed. We review the technique of realising a symmetry non-linearly both in ChPT and BChPT. Some special features of nonrelativistic theories as well as the universality of the low-energy effective theory approach are considered. Chapter 5 contains the conclusions and outlook on future applications of the effective theory approach. In Appendix A we prove that a homogeneous background field only allows the staggered magnetisation to be either homogeneous in space or form a spiral. Appendix B investigates the correct dispersion relation 5

14 of one single doped electron or hole. 6

15 Chapter 1 The Underlying Theory In the introduction, we have already argued that the Hubbard model is a good starting point to describe the antiferromagnetic phase of the cuprates. It is not analytically solveable and numerical simulations suffer from a severe fermion sign problem. Therefore an effective theory approach to the low-energy physics of the Hubbard model is natural. In contrast to the earlier attempts [8, 18, 27] the theory to be constructed is systematic. In the earlier works, phenomenological models are used to describe the low energy physics of the Hubbard model. In the low-energy effective theory approach considered here we will proceed along the lines of BChPT. BChPT is based on symmetry considerations. The Lagrangian to be constructed inherits all the symmetries of the underlying model. In the following chapter we discuss all the symmetries of the Hubbard and the t-j-type models. The most important symmetry is the spin rotation symmetry SU(2) s, where the index s stands for spin. We will argue later in this chapter that this symmetry is spontaneously broken down to a U(1) s symmetry. This allows us to apply the BChPT method to the Hubbard model. The Hubbard model is formulated on a lattice. The continuous Poincaréinvariance which plays an imporant role in BChPT is replaced by several discrete symmetries. In addition, for a vanishing chemical potential at half-filling (one electron per lattice site on average) the Hubbard model possesses a non-abelian extension of the fermion number conservation symmetry. This symmetry will play an important role in the construction of the effective theory. Since symmetries are the essential ingredient of the low-energy effective theory approach to the Hubbard model, the following chapter concentrates on the symmetry analysis. A detailed discussion of the model can be found e.g. in [51,52]. 1.1 Microscopic Models As pointed out in the introduction, we concentrate on the Hubbard and t-j type models. They are supposed to contain the relevant physics of the cuprates. 7

16 B A B A A B A B B A B A A B A B Figure 1.1: Sublattice structure of the bipartite square lattice The Hubbard Model The Hubbard model was introduced simultaneously by Hubbard, Gutzwiler, and Kanamori [53] to give a description of the Mott transitions from metallic to insulating behaviour. This transition was clearly identified to be correlation-driven. This means that the Coulomb force between the electrons on the lattice sites is responsible for the phase transition between insulator and conductor. The Hubbard model has the simplest Hamiltonian which describes mobile interacting electrons on a lattice. In this thesis we describe the Hubbard model on a special class of lattices, the so-called bipartite lattices. Bipartite means that the lattice can be divided into two sublattices A and B with all lattice sites of A having nearest neighbours on sublattice B and vice versa. The bipartite nature of the square lattice is illustrated in figure 1.1. The Hubbard model is based on the following second quantised Hamiltonian H = t x,i (c x c x+î + c x+î c x + c x c x+î + c x+î c x ) + U x c x c x c x c x µ x (c x c x + c x c x ). (1.1) Here, the fermions at a site x are represented by creation and annihilation operators c x,c x and c x,c x, which obey the standard anticommutation relations {c xs,c ys } = δ xyδ ss, {c xs,c ys } = {c xs,c ys } = 0. (1.2) Here î represents a unit-vector in the i-th direction of the lattice. The Hubbard model describes the hopping of electrons between (nearest neighbour) lattice sites (if not forbidden by the Pauli principle). Each hop costs a certain amount of energy which is determined by the hopping parameter t. The correlations between the electrons are simply described by the Coulomb repulsion of two fermions sitting on the same lattice site. The strength (energy cost) of this repulsion is fixed by the parameter U > 0. In addition, one can switch on a chemical potential µ for fermion number relative to an empty lattice. The interaction in the Hubbard model is due to the charge of the fermions. The fermions also carry a spin. The total spin of the system is given by the operator S = x S x = x 8 c x σ 2 c x. (1.3)

17 where σ i are the Pauli matrices and c x is an SU(2) s Pauli spinor ( ) cx c x =. (1.4) The Hubbard model describes two opposing processes: The electrons can hop to other lattice sites. The hopping term alone leads to an itinerant state and the system is a metal which one can describe with band theory. The repulsive on-site coulomb interaction U localises the electrons to on a lattice site and leads to the formation of an insulator The t-j-type models Another model which is supposed to describe the physics of the antiferromagnets is the socalled t-j model. It is described by the Hamiltonian (in terms of SU(2) s Pauli spinors) c x H = P t c xc y + J S xsy + µ x,y x,y x c xc x P. (1.5) P is an operators which projects doubly occupied states out of the Hilbert space. The bracket x, y allows for nearest neighbour interactions only. The t-j-model is a limiting case of the Hubbard model. It is obtained by second order perturbation theory in the coupling t2 U = J 2. As mentioned in the introduction, t-j-type models gained popularity when Anderson [3] suggested that these models can be applied to high-t c compounds. It has been pointed out by many authors (see e.g. [54 56]), that the inclusion of an additional next-to-nearest neighbour coupling t in the t-j model is favourable. In particular some ARPES measurements on the high-t c materials can only be reproduced with the inclusion of a coupling t. In addition, the t-t -J model allows us to access electron doped systems as well [57]. It s Hamiltonian is given by H = P t c xc y t x,y x,y (c xc y + c yc x ) + J S xsy + µ c xc x P. (1.6) x,y x The bracket x, y allows for next to nearest neighbour hopping only. Again, doubly occupied sites are projected out. As pointed out by Tohyama and Maekawa [58] the sign of t depends on the type of the charge carriers doped into the system. Near half-filling we have t > 0 for holes and t < 0 for electrons. 1.2 Electrons are different from Holes The fact that the sign of t is different depending on the type of fermions doped into the system has consequences for the dispersion relation of holes and electrons. Angle resolved photoemission spectroscopy studies as well as numerical simulations show that holes and electrons appear at different lattice momenta in the Brillouin zone. Hole-doped cuprates have hole pockets centred at lattice momenta (± π 2a, ± π 2a ). This can be seen in numerical simulations of the t-j-model as displayed in figure 1.2. In electron-doped cuprates the fermions reside in momentum space pockets centred at ( π a,0) or (0, π a ) [59 65]. We have computed the single-electron dispersion relation in the t-t -J model 9

18 π π /2 0 -π /2 -π -π -π /2 0 π /2 π Figure 1.2: The dispersion relation E( p) of a single hole in the t-j model (on a lattice for J = 2t) with hole pockets centred at (± π 2a, ± π 2a ). shown in figure 1.3. The location of the pockets has important effects on the dynamics of the fermions doped into the system. As pointed out in [58], the sign of the next-to-nearest neighbour coupling t depends of the type of charge carriers doped into the system. The value of t also governs the location of the fermion-pockets in the Brillouin zone. Therefore, the location of the fermionpockets is not a general property of the model. Depending on the parameter values t,t,j,u of the model the electrons and holes appear at different lattice momenta in the Brillouin zone. 1.3 Symmetries of the Hubbard Model The effective Lagrangian to be constructed is obtained by constructing all terms that are invariant under the symmetries of the Hubbard model. These symmetries will be investigated in the following section. In addition, we will represent the Hubbard Hamilton in a form which is manifestly invariant under all the symmetries of the model. This analysis has been performed in great detail in [66] SU(2) s Spin Symmetry In order to examine the Hamiltonian, it is useful to rewrite it in the spinor notation of eq.(1.4). Up to an irrelevant constant, the Hamiltonian can be expressed in terms of c x and c x. It takes the form H = t x,i (c xc x+î + c x+î c x) + U 2 (c xc x 1) 2 µ x x (c xc x 1). (1.7) 10

19 π π/2 0 π/2 π π π/2 0 π/2 π Figure 1.3: The dispersion relation E( p) of a single electron in the t-t -J model (on a lattice for J = 0.4t and t = 0.3t) with electron pockets centred at ( π a,0) and (0, π a ) in the 2-d Brillouin zone with p = (p 1,p 2 ) [ π a, π a ]2. Here µ = µ U/2 is the chemical potential for fermion number but now relative to half-filling. The spin symmetry is implemented on the Hilbert space by the operator The three components of S are given by V = exp(i η S) (1.8) S 1 = 1 (c 2 x c x + c x c x ), S 2 = i (c 2 x c x c x c x ), S 3 = 1 (c 2 x c x c x c x ). (1.9) x x Recalling the definition of the spin operator in eq.(1.3) it is straightforward to show that under the SU(2) s spin rotation the fermion operators transform as x c x = V c x V = exp(i η σ 2 )c x = gc x. (1.10) With the above transformation behaviour it is easy to see that the Hubbard Hamiltonian eq.(1.7) is invariant under spin rotations. SU(2) s is indeed a symmetry of the Hubbard model. The spin operator commutes with the Hamiltonian, [H, S] = Charge Symmetry In the Hubbard model, the electric charge of the fermions is a conserved quantity. The charge is given by the operator Q = x Q x = x (c x c x 1) = x (c x c x + c x c x 1). (1.11) 11

20 This operator defines a U(1) Q symmetry of eq.(1.7). It is defined to count the fermion number with respect to a half filled system. In the Hilbert space, fermion number conservation is implemented as W = exp(iωq), (1.12) and the fermion operators transform as Q c x = W c x W = exp(iω)c x. (1.13) With this transformation behaviour the Hubbard model is obviously invariant under U(1) Q, [H, Q] = Shift Symmetry Since the Hubbard Hamiltonian is formulated on a lattice, there is no continuous translation invariance. However there is an invariance under a shift by one lattice spacing in the î-th direction of the lattice. We call this symmetry the shift symmetry. A shift in the î-th direction of the lattice is generated by the unitary operator D i which acts as D i c x = D i c xd i = c x+î. (1.14) By redefining the sum over lattice points x, it is easy to show that [H,D i ] = 0 and that the U(1) Q as well as the SU(2) s symmetry commute with the translation, i.e. [Q,D i ] = [ S,D i ] = 0. It will turn out to be very useful (see section 2.2.1) to consider a combined symmetry D. D i cx = D i cx D i = (iσ 2) D i c x = (iσ 2 )c. (1.15) x+î Apparently D combines a global spin rotation with a shift. Since D i is a combination of a spin rotation and a shift it is again a symmetry: [H,D i ] = 0. It commutes with the charge conservation and spin rotation symmetry: [Q,D i ] = [ S,D i ] = 0. D i is similar to the charge conjugation symmetry in particle physics. It flips the spin-charge of the fermion operators Discrete Symmetries In relativistic systems the effective Lagrangians are restricted by Lorentz invariance. Our underlying model is a lattice model, rotation invariance is reduced to an invariance under rotations by a multiple of 90 degrees. A spatial rotation O by 90 degrees turns a lattice point x = (x 1,x 2 ) into Ox = ( x 2,x 1 ). In nonrelativistic systems, the spin is an internal quantum number and therefore not affected by this rotation as global SU(2) s rotations can be applied independently of O. Under O the fermion operators transform as O c x = O c x O = c Ox. (1.16) The Hubbard model is also invariant under a spatial reflection R at the x 1 -axis. R acts on a lattice point x as Rx = (x 1, x 2 ). Since the hopping parameter is isotropic, it is straightforward to see that these reflections must be a symmetry of the Hubbard model as well. The fermion operators transform as R c x = R c x R = c Rx. (1.17) 12

21 By combining R and O, one can generate reflections also at the x 2 -axis. The parity transformation is not an additional symmetry. It can be obtained by a applying a spatial rotation O twice. We assume the Hubbard and t-j-type models also to contain a time revearsal symmetry. How time revearsal acts on the level of fermion creation and annihilation operators on the lattice can be found in [67] An additional SU(2) Symmetry of the Hubbard Model The Abelian U(1) Q fermion number conservation can be extended to a non-abelian SU(2) Q charge symmetry. This charge symmetry is, however, only present in a half-filled system defined on a bipartite lattice. The Hubbard model at half-filling corresponds to µ = 0. The non-abelian symmetry is generated by the operators Q + = x ( 1) x c x c x, Q = x ( 1) x c x c x, Q 3 = x 1 2 (c x c x +c x c x 1) = 1 Q. (1.18) 2 These operators indeed form indeed an SU(2) algebra with the characteristic commutation relations [Q +,Q ] = 2Q 3 [Q ±,Q 3 ] = 0. (1.19) First discovered by Yang and Zhang [68], this symmetry is often called pseudospin symmetry. It is straightforward to prove that the above operators define a new symmetry i.e. [H,Q ± ] = [H,Q 3 ] = 0, (1.20) which is discussed in [66]. Using the above commutation relations together with eq.(1.19) one can interpret the non-abelian charge symmetry as follows. It relates all charge sectors away from half filling to eachother Q. The Hubbard Hamiltonian of eq.(1.1) is not manifestly invariant under the pseudospin symmetry. However, a s shown in detail in [66], the construction of a manifestly SU(2) s SU(2) Q invariant formulation is still possible. The new fermion representation is a 2 2 matrix valued operator ( ) c x ( 1) x c x C x = c x ( 1) x c. (1.21) x Under SU(2) Q it transforms as Q C x = C x Ω T. (1.22) The SU(2) s and the SU(2) Q symmetries commute since the spin rotation acts on the left and the pseudospin symmetry acts on the right of the fermion operator matrix as displayed in eq.(1.10), i.e. Q C x = gc xω T. (1.23) 13

22 The 2 2 matrix-valued fermion operator has the following transformation properties under the symmetries of the Hubbard model C x = gc x (1.24) Q C x = C x Ω T (1.25) D C x = C x+i σ 3 (1.26) D C x = (iσ 2 )C x+i σ 3 (1.27) T C x = (iσ 2 )C x (1.28) We can now write down the Hubbard model in an SU(2) s SU(2) Q invariant form H = t Tr[C 2 xc + x+î C C x+î x] + U Tr[C 12 xc x C xc x ] µ x x x,i Tr[C xc x σ 3 ]. (1.29) With eq.(1.25) it is obvious, that the pseudospin symmetry is only present at half-filling, i.e. at µ = 0. The effective theory to be constructed below needs to be only U(1) Q invariant. We want to describe systems doped with either holes or electrons. Since the SU(2) Q symmetry connects the hole-doped with the electron-doped phase, it is not present in real materials. Nevertheless, the pseudospin symmetry plays an important role in identifying the fermion fields of the effective theory Symmetries of t-j-type models The t-j-model as well as the t-t -J-model share all symmetries with the Hubbard model except for the non-abelian extension of fermion number symmetry. The reason why the t-j-model can t have the additional SU(2) Q comes from the projection that moves out the double occupied sites (if one wants to describe electron-doping, one projects out the unoccupied sites [57]). We argued in the previous section, that the pseudospin relates different particle sectors relative to half-filling. Such a relation can t exist in t-j-type models since they describe either the sector Q 0 or Q 0, in contrast to the Hubbard model which describes both sectors. 1.4 Strong Coupling Limit of the Hubbard Model and the Heisenberg Hamiltonian The physics of the cuprates is dominated by a single CuO 2 plane which shows antiferromagnetic order as mentioned in the introduction. The Hubbard model is suitable to describe the physics of the cuprates if it shows long range antiferromagnetic order. We will now argue, that the Hubbard model turns into the Heisenberg antiferromagnet in the strong coupling limit U t. We explained above that there are two competing processes in the Hubbard Hamiltonian: the hopping of electrons and the localisation of the electrons on a lattice site by the strong Coulomb interaction. For U t, one hopping step becomes energetically unfavourable since it causes one lattice site to be doubly occupied. This costs a large energy U. Instead of hopping once, the particle will choose to hop twice. In the strong coupling limit, we can therefore 14

23 replace the hopping term H t with H 2 t because the doubly occupied states are only virtual. Instead of the above heuristic argument one can perform a second order perturbation theory as it can be found in [51,52]. There one finds, that the leading contribution to the Hamiltonian in the strong coupling limit is H 2 t = t2 U y,i (c x c x+î + c c x+î x + c x c x+î + c c x+î x ) (1.30) x,i (c y c y+î + c y+î c y + c y c y+î + c y+î c y ). Due to the large U we project out all energetically unfavourable doubly occupied sites. Eq.(1.30) then turns into H 2 t = J x,i (c x c xc c 2t2 x+î x+î ), J = U (1.31) Using the definition eq.(1.3) it is straightforward to see that Ht 2 = J S x S y, (1.32) <x,y> in the strong coupling limit at half-filling. For U t the Hubbard model is replaced by the antiferromagnetic Heisenberg model. At half-filling, also the t-j-type models turn into the Heisenberg antiferromagnet. Due to the projectors P in eq.(1.5) one can find exactly one electron per lattice site. At exact halffilling, we have µ = 0. The only contribution stems from the term proportional to J. At exact half-filling the t-j-type models are therefore equivalent to the Heisenberg antiferromagnet The Heisenberg Antiferromagnet The Heisenberg model has the simplest Hamiltonian describing spin interactions on a square lattice. It is given by H = J S x S y. (1.33) x,y The bracket x, y indicate a nearest neighbour summation only. For J < 0 this model is ferromagnetic and the ground state is known analytically. We are interested in the case J > 0. This is the antiferromagnetic Heisenberg model. A natural ground state candidate for the Hamiltonian of eq.(1.33) is the classical antiferromagnetic Néel state in which neighbouring spins point in antiparrallel directions. The characteristic antiparallel alignment of the spins stems from the bipartite structure of the lattice. On a non-bipartite lattice as e.g. the triangular lattice, the spins can not show perfectly antiparallel alignement. The classical Néel order gives rise to an order parameter - the staggered magnetisation vector. Although the total magnetisation vanishes in the antiferromagnet, we can define M s = x ( 1) x Sx = x ( 1) x 1 +x 2 a Sx. (1.34) 15

24 Figure 1.4: Alignement of the spins in the classical Néel state We define the factor ( 1) x to be 1 for all,x A and 1 for,x B. Therefore we call A the even and B the odd sublattice. The staggered magnetisation is then defined as the difference between the net magnetisation on sublattice A and the net magnetisation on sublattice B. The existence of a non-vanishing order parameter indicates a spontaneously broken symmetry. This would allow us to apply the BChPT technique to the Hubbard and t-j model if there is indeed a spontaneously broken symmetry in the Heisenberg model. If the classical Néel state were the ground state then the question raised above would easy to answer. The Heisenberg Hamiltonian has an SU(2) s spin rotation symmetry which is no longer present in the classical Néel state, so the spin symmetry is spontaneously broken and the staggered magnetisation is the appropriate order parameter. However, it is not difficult to see that the classical Néel state is not the true ground state of the Heisenberg antiferromagnet. Rewriting eq.(1.33), we obtain H = J x,y S 3 x S3 y + J 2 (S x + S+ y + S x S+ y ). (1.35) x,y where S ± are spin raising and lowering operators S + = S 1 + is 2, S 1 = S 1 is 2 (1.36) The classical Neél ordered state cannot be the ground state. First of all, strict antiparallel alignment of the spins is not an eigenstate of the Heisenberg Hamiltonian. In addition, it can be proved that the ground state of eqs.(1.33, 1.35) is a singlet [69] which the Neél state is certainly not. The spontaneous breakdown of the spin rotation symmetry is now no longer obvious because we do not know the ground state analytically. In fact in the isotropic Heisenberg model for spin S = 1 2 in two dimensions, it has never been proved that the system shows long range antiferromagnetic order. There is, however, strong evidence from numerical calculations [70,71] for a spontaneous breakdown of the spin symmetry in the two dimensional 16

25 Heisenberg model which is therefore generally accepted. With the above considerations we have justified that the low-energy regime of the Hubbard model can be described by an effective theory along the lines of BChPT. The Hubbard model undergoes the spontaneous breakdown of a continuous global symmetry because it turns into the Heisenberg model in the strong coupling limit U t. Its order parameter is the staggered magnetisation vector M s. 17

26

27 Chapter 2 Construction Of The Effective Lagrangians The idea to construct a systematic effective theory analogous to chiral perturbation theory for QCD for the Hubbard model has not been realised before. While the effective action for an undoped system is well established since the pioneering work of Chakravarty, Halperin and Nelson as well as Hasenfratz, Leutwyler and Niedermayer respectively [4 6, 72], there is no agreement on how to include fermions into the theory. Starting with Shraiman and Siggia [8], Wen [9], and Shankar [10], numerous approaches [73 75] used composite vector fields to couple magnons and holes. As a starting point, the existing approaches use phenomenological models to describe the low-energy physics of the Hubbard model. However in these models, there is no agreement, how various symmetries are realised on the fermions. In particular, it has not been demonstrated that the models constructed so far correctly describe the low-energy physics of the underlying Hubbard or t-j-type Hamiltonians. 2.1 The Magnons The low-energy effective theory for magnons is well established [4 6]. In [5, 6] the ChPT approach has already been used. The magnons are identified as Goldstone bosons of the spontaneously broken spin rotation symmetry in the Heisenberg antiferromagnet. The Goldstone bosons are described by fields in the coset space of the broken and unbroken symmetry groups. In the case of the spontaneously broken spin symmetry, the coset space is given by the two-sphere G/H = SU(2) 2 /U(1) s = S 2. (2.1) The elements of the two-sphere S 2 are described by a unit vector field e(x) = (e 1 (x),e 2 (x),e 3 (x)) S 2, e(x) 2 = 1, (2.2) where x = (x 1,x 2,t) denotes a point in space-time. The field e(x) represents the local staggered magnetisation vector. The corresponding Euclidean action has the form S[ e] = d 2 x dt ρ s 2 ( i e i e + 1 c 2 t e t e). (2.3) One can check that it inherits the spin rotation, as well as all the other symmetries of the underlying system. The above action contains two low-energy constants, the spin stiffness ρ s 19

28 and the spin-wave velocity c. These constants can t be obtained from symmetry principles and can t be calculated within ChPT. They have to be extracted from numerical simulations of the underlying models or from experiments. It is interesting to consider the behaviour of e(x) under the shift symmetry. Since the staggered magnetisation vector changes sign under D i, we also have D i e(x) = e(x). (2.4) Note that under the shift, unlike in the Hubbard model, the argument of the fields does not change from x to x + î because the fields now live in the continuum. The parametrisation as a 3-dimensional vector in the spin space is not appropriate when one introduces fermions. It is more convenient to use P CP(1) to represent e(x) as a 2 2 matrix P(x) = 1 (½ + e(x) σ). (2.5) 2 Here σ are the Pauli matrices. P(x) has the properties of a Hermitean projector: P(x) = P(x), TrP(x) = 1, P(x) 2 = P(x). (2.6) The space CP(1) = S 2 and therefore is a valid realisation of the low-energy degrees of freedom. In terms of the projection matrices the lowest order Lagrangian is given by S[P] = d 2 x dt ρ s Tr[ i P i P + 1 c 2 tp t P]. (2.7) Under a global spinrotation g SU(2) s the local staggered magnetisation P(x) transforms as P(x) = gp(x)g. (2.8) Obviously the magnon action S[P] is invariant under global SU(2) s spin rotations. Under the shift symmetry D i, the staggered magnetisation changes sign, which means that D i e(x) = e(x) and therefore D i P(x) = 1 (½ e(x) σ) = ½ P(x). (2.9) 2 In order to be a suitable representation for the staggered magnetisation e(x), ½ P(x) has to be an element of P(1) as well. This is obvious for the idempotency and Hermiteicity properties of P(1). However, the trace condition only holds in a SU(2)-model, where Tr½ = 2, so that Tr[ D i P(x)] = Tr[½ P(x)] = 2 1 = 1. Under the charge conjugation D i, which is actually a translation D i combined with a global spin rotation g = iσ 2, the magnon field transforms as D i P(x) = (iσ2 ) D P(x)(iσ 2 ) = (iσ 2 )[½ P(x)](iσ 2 ) = P(x). (2.10) Note that the spin is an internal quantum number. This means that symmetries such as charge conservation, or the 90 degrees rotation as well as the reflection will affect P(x), only through the space-time argument x. Therefore, P(x) transforms as Q P(x) = P(x) (2.11) O P(x) = P(Ox), R P(x) = P(Rx), O (x 1,x 2,t) = ( x 2,x 1,t) (2.12) R (x 1,x 2,t) = (x 1, x 2,t). (2.13) 20

29 We have pointed out that time-reversal is another important symmetry displayed by the Hubbard model. Time-reversal T turns a point x = (x 1,x 2,t) in space-time into Tx = (x 1,x 2, t). The spin is a form of angular momentum and therefore changes sign under timereversal symmetry. Since we identify e(x) with the local staggered magnetisation it originates from the spin S as displayed in eq.(1.34). Therefore e(x) also changes sign under time-reversal. Hence one has T e(x) = e(tx). (2.14) Time-reversal manifests itself as shift T P(x) = ½ P(Tx) = D i P(Tx). (2.15) Similar to the combined symmetry operation D i, it will turn out to be very useful to define a combined time-reversal operation T. That is an ordinary time-reversal operation T combined with a global SU(2) s spin rotation g = iσ 2. Under T the magnon field transforms as T P(x) = (iσ 2 ) T P(x)(iσ 2 ) = (iσ 2 ) D i P(Tx)(iσ 2 ) = D i P(Tx). (2.16) It is easy to show that the action of eq.(2.7) is invariant under all those symmetry transformations. 2.2 Non-linear Realisation of the SU(2) s Symmetry In order to couple magnons to electrons or holes, we construct a non-linear realisation of the spontaneously broken SU(2) s symmetry [76]. We will closely follow the construction of the non-linear realisation of QCD as suggested in [13]. The concept of a non-linear realisation can be generalised to arbitrary groups [77,78]. Since the spin of the holes and electrons transforms under SU(2), we want to represent the magnon field with a SU(2) field. Following closely BChPT [14,79] we uniquely connect the CP(1) matrices with a SU(2) matrix. u(x)p(x)u(x) = 1 ( ) (½ σ 3) =, u (x) 0. (2.17) Using eq.(2.5) one can see that u(x) rotates the local staggered magnetisation into the z- direction. The above procedure is performed in order to uniquely connect the CP(1) representation with the SU(2) representation of the theory. In order to have u(x) completely well-defined, we need to demand that the matrix element u 11 (x) is real and non-negative. If this condition was not imposed, the matrix u(x) would be defined only up to a U(1) phase. This will play an important role. We present the diagonalising field as derived in [66,76] u(x) = = 1 2(1 + e3 (x)) ( sin( θ(x) 2 ( 1 + e3 (x) ) e 1 (x) ie 2 (x) e 1 (x) ie 2 (x) 1 + e 3 (x) ) cos( θ(x) 2 ) sin(θ(x) 2 )exp( iϕ(x)) )exp(iϕ(x)) cos(θ(x) 2 ). (2.18) Up to now we have constructed the local matrix-valued field u(x). Under SU(2) s it transforms as u(x) P(x) u(x) = u(x)p(x)u(x) = 1 2 (½ + σ 3). (2.19) 21

30 The field u(x) was introduced in order to get a new representation for the magnon fields. Since it represents the unique local staggered magnetisation e(x) we demand that u(x) is unique also. To obtain the correct transformation behaviour of u(x) under SU(2) s we have to demand u 11 (x) 0 as in eq.(2.17) to have u(x) uniquely defined. After the spin rotation this is not necessarily true anymore since it can pick up a complex U(1) s phase i.e. u 11 (x) C. To remove the phase, we define h(x) = exp(iα(x)σ 3 ) = ( ) exp(iα(x)) 0 U(1) 0 exp( iα(x)) s. (2.20) Under SU(2) s spin rotations the diagonalising field then transforms as u(x) = h(x)u(x)g, u 11 (x) 0. (2.21) By this last equation the transformation h(x) is uniquely defined. What has happened here? The coset space of the broken symmetry SU(2) s and the invariant subgroup U(1) s is the two-sphere S 2. The elements of S 2 are unitvectors in a three-dimensional space. In order to obtain a two-dimensional representation we constructed a CP(1) model. In this model the global spin rotation was still globally realised. So we connected a uniquely defined SU(2) diagonalising matrix u(x) to the P(x) CP(1). To assure that u(x) is also unique after a spin rotation we needed to adjust its SU(2) s transformation properties. This can be done by using a unique U(1) s transformation h(x). This h(x) however depends on u(x) itself. In eq.(2.21), the spin rotation is realised non-linearly on u(x). In addition, the spin rotation is realised locally. This does not mean that the SU(2) s transformations are now local truly. If the global spin rotation is in the unbroken subgroup U(1) s that is g = diag(exp(iβ),exp( iβ)), it turns out that the transformation h(x) reduces to h(x) = h = g and therefore becomes globally and linearly realised. This behaviour is identical to ChPT as we will show in chapter 4. It is easy to show that the SU(2) s group structure of the global symmetry group g is properly inherited by the non-linearly realised symmetry in the unbroken subgroup U(1) s. One therefore demands that a composite transformation g = g 2 g 1 SU(2) s leads to a composite transformation h(x) = h 2 (x)h 1 (x) U(1) s. First, we perform the global SU(2) s transformation g 1 that is P(x) = g 1 P(x)g 1, u(x) = h 1 (x)u(x)g 1, (2.22) which defines the non-linear transformation h 1 (x). We then perform the second global transformation g 2, which defines the non-linear transformation h 2 (x) that is P(x) = g 2 P(x) g 2 = g 2g 1 P(x)(g 2 g 1 ) = gp(x)g, u(x) = h 2 (x)u(x) g 2 = h 2(x)h 1 (x)u(x)(g 2 g 1 ) = h(x)u(x)g. (2.23) We can indeed identify h(x) = h 2 (x)h 1 (x) and thus conclude that the group structure is properly inherited by the non-linear realisation. 22