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Spin chain with 1/r2 interactions

Yoshio Kuramoto, Yusuke S. Kato

Published Aug 6, 2009• 0 citations

Abstract

In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation. There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.

DOI: 10.1017/cbo9780511596827.005

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•Jan 1, 1995•5 citations

In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation. There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.

Recently, the one dimensional model of

N

spins with

S=\frac{1}{2}

on a circle, interacting with an exchange that falls off with the inverse square of the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}π \sin(\frac{i-j}{N}π)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model, has received ample attention. Its special features include: relatively simple eigenfunctions, non-interacting elementary excitations that obey semionic statistics (spinons), and a large ``quantum group'' symmetry algebra called the Yangian. This model is fully integrable, albeit in a slightly different sense than the more traditional nearest neighbor exchange (NNE) Heisenberg chain. This thesis comes in 4 chapters. Chapter 1 introduces the model and presents the construction of a subset of the eigenfunctions. The other eigenfunctions are shown to be generated by the action of the Yangian symmetry algebra of

H_{\rm ISE}

. Chapter 2 presents a method to construct the set of constants of the motion of the ISE-model. The ISE-model is tractable enough to obtain its zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend this to a non-zero magnetic field, where, due to the presence of spinons in the groundstate, more complicated excitations contribute: small numbers of magnons and spinons. discuss the relation to the more complicated NNE-model. Finally chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon propagator in the ISE-model, and verified numerically. We briefly comment on its relevance to stabilizing superconductivity in the layered cuprates.

Starting from a Calogero-Sutherland model with the hyperbolic interaction confined by an external field with Morse potential, we construct a Heisenberg spin chain with exchange interaction varies as 1/sinh 2 x on a lattice given in terms of the zeros of Laguerre polynomials. Varying the strength of the Morse potential, the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with non-uniform nearest-neighbour coupling in a non-uniform magnetic field which allows us to study the thermodynamics in the limit of infinite chains.

The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.

Starting from a Calogero--Sutherland model with hyperbolic interaction confined by an external field with Morse potential we construct a Heisenberg spin chain with exchange interaction

\propto 1/\sinh^2 x

on a lattice given in terms of the zeroes of Laguerre polynomials. Varying the strength of the Morse potential the Haldane--Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with nonuniform nearest neighbour coupling in a nonuniform magnetic field which allows to study the thermodynamics in the limit of infinite chains.

We study the thermodynamics and critical behavior of su(m) spin chains of Haldane-Shastry type at zero chemical potential, both in the A_{N-1} and BC_{N} cases. We evaluate in closed form the free energy per spin for arbitrary values of m, from which we derive explicit formulas for the energy, entropy, and specific heat per spin. In particular, we find that the specific heat features a single Schottky peak, whose temperature is well approximated for m≲10 by the corresponding temperature for an m-level system with uniformly spaced levels. We show that at low temperatures the free energy per spin of the models under study behaves as that of a one-dimensional conformal field theory with central charge c=m-1 (with the only exception of the Frahm-Inozemtsev chain at zero value of its parameter). However, from a detailed study of the ground-state degeneracy and the low-energy excitations, we conclude that these models are only critical in the antiferromagnetic case, with a few exceptions that we fully specify.

The su(1|1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators. Several years ago Haldane and Shastry independently discovered a soluble spin chain with exchange interaction falling off as the inverse square of the distance between the spins [1, 2]. This model can be thought of as the static limit [3] of the Sutherland model with internal degrees of freedom [4, 5, 6]. The idea of putting internal degrees of freedom into the Calogero-Sutherland models has proved to be very fruitful since then. A rich family of related models has been found. These long range interacting systems are under active investigation now. A common algebraic structure underlying their integrability is still waiting for its discovery. In this letter we discuss a linear transformation, that turns the su(m|n) supersymmetric extension of the original Haldane-Shastry Hamiltonian into a more compact form,