The open Haldane-Shastry chain: thermodynamics and criticality

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 energyquasimomentum 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.

Thermodynamics and criticality of su($m$) spin chains of Haldane-Shastry type

“…As shown in Ref. [41], similar expressions are valid for the HS-B chain (8). Indeed, from (12) we obtain…”

“…Since all the matrices in the expression for the partition function have again positive entries, it follows from Perron's theorem that the thermodynamic free energy per spin is given by Eq. ( 15) also in this case (see [41] for details). Note that when computing the thermodynamic free energy from Eq.…”

“…In fact, Eq. ( 15) has been shown to hold for the su(m|n) supersymmetric version of the chains ( 4)-( 8) studied in this paper, even with the addition of a general chemical potential term [38,41]. Thus the thermodynamic functions of all of these models can be computed in closed form provided that the Perron eigenvalue λ 1 (x) of the transfer matrix A(x) in Eq.…”

“…( 14) can be explicitly found. So far, this has only been done in the su(2) case (bosonic or fermionic) [37,41] and in the supersymmetric case [42] with 1 m, n 2 [38,41].…”

“…In Section II we recall the formula for the free energy per spin of su(m) chains of HS-type in terms of the Perron eigenvalue of a suitable transfer matrix derived in Refs. [37] and [41]. We then evaluate this eigenvalue, thus obtaining a simple closed-form expression for the free energy of the latter models valid for arbitrary values of m. This expression is used in Section III to derive explicit formulas for the energy, entropy and specific heat per spin.…”