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.
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.
Abstract We study the thermodynamics and criticality of the su( m | n ) Haldane–Shastry chain of BC N type with a general chemical potential term. We first derive a complete description of the spectrum of this model in terms of BC N -type motifs, from which we deduce a representation for the partition function as the trace of a product of site-dependent transfer matrices. In the thermodynamic limit, this formula yields a simple expression for the free energy per spin in terms of the Perron–Frobenius eigenvalue of the continuum limit of the transfer matrix. Evaluating this eigenvalue we obtain closed-form expressions for the thermodynamic functions of the chains with m , n ⩽ 2. Using the motif-based description of the spectrum derived here, we study in detail the ground state of these models and their low energy excitations. In this way we identify the critical intervals in chemical potential space and compute their corresponding Fermi velocities. By contrast with previously studied models of this type, we find in some cases two types of low energy excitations with linear energy-quasimomentum relation. Finally, we determine the central charge of all the critical phases by analyzing the low-temperature behavior of the expression for the free energy per spin.
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Abstract We analyze the thermodynamics and the critical behavior of the supersymmetric t - J model with long-range interactions. Using the transfer matrix formalism, we obtain a closed-form expression for the free energy per site both for a finite number of sites and in the thermodynamic limit. Our approach, which is different from the usual ones based on the asymptotic Bethe ansatz and generalized exclusion statistics, can in fact be applied to a large class of models whose spectrum is described in terms of supersymmetric Young tableaux and their associated Haldane motifs. In the simplest and most interesting case, we identify the five ground state phases of the model and derive the complete low-temperature asymptotic series of the free energy per site, the magnetization and charge densities, and their susceptibilities. We verify the model’s characteristic spin-charge separation at low temperatures, and show that it holds to all orders in the asymptotic expansion. Using the low-temperature asymptotic expansions of the free energy, we also analyze the critical behavior of the model in each of its ground state phases. While the standard phase is described by two independent conformal field theories (CFTs) with central charge c = 1 in correspondence with the spin and charge sectors, we find that the low-energy behavior of the and phases is that of a single c = 1 CFT. We show that the model exhibits an even richer behavior on the boundary between zero-temperature phases, where it can be non-critical but gapless, critical in the spin sector but not in the charge one, or critical with central charge c = 3/2.
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 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.
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.