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arXiv:1008.5285 · Symmetry Integrability and Geometry Methods and Applications

One-Dimensional Vertex Models Associated with a Class of Yangian Invariant Haldane-Shastry Like Spin Chains

B. Basu-Mallick

Published Dec 10, 2010• 10 citations
arXivHTMLPDF

Abstract

We define a class of Y (sl (m|n) ) Yangian invariant Haldane-Shastry (HS) like spin chains, by assuming that their partition functions can be written in a particular form in terms of the super Schur polynomials.Using some properties of the super Schur polynomials, we show that the partition functions of this class of spin chains are equivalent to the partition functions of a class of one-dimensional vertex models with appropriately defined energy functions.We also establish a boson-fermion duality relation for the partition functions of this class of supersymmetric HS like spin chains by using their correspondence with onedimensional vertex models.

DOI
10.3842/sigma.2010.091

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We define a class of Y (sl (m|n) ) Yangian invariant Haldane-Shastry (HS) like spin chains, by assuming that their partition functions can be written in a particular form in terms of the super Schur polynomials.Using some properties of the super Schur polynomials, we show that the partition functions of this class of spin chains are equivalent to the partition functions of a class of one-dimensional vertex models with appropriately defined energy functions.We also establish a boson-fermion duality relation for the partition functions of this class of supersymmetric HS like spin chains by using their correspondence with onedimensional vertex models.
Abstract We introduce a novel family of translationally invariant su ( m | n ) supersymmetric spin chains with long-range interaction not directly associated with a root system. We study the symmetries of this model, establishing in particular the existence of a boson–fermion duality characteristic of this type of system. Taking advantage of the relation of the new chains with an associated many-body supersymmetric spin dynamical model, we are able to compute their partition function in closed form for all values of m and n and for an arbitrary number of spins. When both m and n are even, we show that the partition function factorizes as the product of the partition functions of two supersymmetric Haldane–Shastry spin chains, which in turn leads to a simple expression for the thermodynamic free energy per spin in terms of the Perron eigenvalue of a suitable transfer matrix. We use this expression to study the thermodynamics of a large class of these chains, showing in particular that the specific heat presents a single Schottky peak at approximately the same temperature as a suitable k-level model. We also analyze the critical behavior of the new chains, and in particular, the ground-state degeneracy and the existence of low-energy excitations with a linear energy–momentum dispersion relation. In this way, we show that the only possible critical chains are those with m = 0 , 1 , 2 . In addition, using the explicit formula for the partition function we are able to establish the criticality of the su ( 0 | n ) and su ( 2 | n ) chains with even n, and to evaluate the central charge of their associated conformal field theory.
The energy level density distributions of some Haldane-Shastry like spin chains associated with the AN − 1 root system have been computed recently by Enciso et al., exploiting the connection of these spin systems with inhomogeneous one-dimensional vertex models whose energy functions depend on the vertices through specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions depend on the vertices through arbitrary polynomials of any possible degree. We estimate the order of mean and variance for such energy functions and show that the level density distribution of all vertex models belonging to this class asymptotically follow the Gaussian pattern for large number of vertices. We also present some numerical evidence in support of this analytical result.
Level density distributions for some Haldane-Shastry like spin chains associated with A N −1 root system have been computed recently by Enciso et al. , using the connection of these spin chains with inhomogeneous one-dimensional vertex models. The energy functions of such vertex models are given by specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions are given by arbitrary polynomials of any possible degree and show that the level densities for this class of vertex models asymptotically follow the Gaussian distribution for large number of vertices.
In this paper, we study an su(m)\mathrm{su}(m)su(m)-invariant open version of the Haldane-Shastry spin chain whose ground state can be obtained from the chiral correlator of the $c=m\ensuremath{-}1$ free boson boundary conformal field theory. We show that this model is integrable for a suitable choice of the chain sites depending on the roots of the Jacobi polynomial ${P}_{N}^{\ensuremath{\beta}\ensuremath{-}1,{\ensuremath{\beta}}^{\ensuremath{'}}\ensuremath{-}1}$, where NNN is the number of sites and $\ensuremath{\beta},{\ensuremath{\beta}}^{\ensuremath{'}}$ are two positive parameters. We also compute in closed form the first few nontrivial conserved charges arising from the twisted Yangian invariance of the model. We evaluate the chain's partition function, determine the ground-state energy, and deduce a complete description of the spectrum in terms of Haldane's motifs and a related classical vertex model. In particular, this description entails that the chain's level density is normally distributed in the thermodynamic limit. We also analyze the spectrum's degeneracy, proving that it is much higher than for a typical Yangian-invariant model.
We analyze the thermodynamics and criticality properties of four families of su(m|n) supersymmetric spin chains of Haldane-Shastry (HS) type, related to both the A_{N-1} and the BC_{N} classical root systems. Using a known formula expressing the thermodynamic free energy per spin of these models in terms of the Perron (largest in modulus) eigenvalue of a suitable inhomogeneous transfer matrix, we prove a general result relating the su(kp|kq) free energy with arbitrary k=1,2,⋯, to the su(p|q) free energy. In this way we are able to evaluate the thermodynamic free energy per spin of several infinite families of supersymmetric HS-type chains, and study their thermodynamics. In particular, we show that in all cases the specific heat at constant volume features a single marked Schottky peak, which in some cases can be heuristically explained by approximating the model with a suitable multilevel system with equally spaced energies. We also study the critical behavior of the models under consideration, showing that the low-temperature behavior of their thermodynamic free energy per spin is the same as that of a (1+1)-dimensional conformal field theory with central charge c=m+n/2-1. However, using a motif-based description of the spectrum we prove that only the three families of su(1|n) chains of type A_{N-1} and the su(m|n) HS chain of BC_{N} type with m=1,2,3 (when the sign ɛ_{B} in the Hamiltonian takes the value -1 in the latter case) are truly critical.
Abstract We study the thermodynamics and critical behavior of s u ( m | n ) supersymmetric spin chains of Haldane–Shastry type with a chemical potential term. We obtain a closed-form expression for the partition function and deduce a description of the spectrum in terms of the supersymmetric version of Haldane’s motifs, which we apply to obtain an analytic expression for the free energy per site in the thermodynamic limit. By studying the low-temperature behavior of the free energy, we characterize the critical behavior of the chains with 1 ⩽ m , n ⩽ 2 , determining the critical regions and the corresponding central charge. We also show that in the s u ( 2 | 1 ) , s u ( 1 | 2 ) and s u ( 2 | 2 ) chains the bosonic or fermionic densities can undergo first-order (discontinuous) phase transitions at T = 0, in contrast with the previously studied s u ( 2 ) case.