Thermodynamics of the Heisenberg XXX chain with negative spin

TL;DR

This work analyzes the thermodynamics of the isotropic Heisenberg XXX chain with negative spin $s=-1$, showing that it maps to the quantum lattice nonlinear Schrödinger model and serves as an effective description of reggeized gluons in high-energy QCD. Using the algebraic and thermodynamic Bethe Ansatz, the authors determine the ground-state root distribution, classify elementary excitations as particle–hole modes, and obtain finite-temperature thermodynamics via the TBA, revealing a distinct vacuum structure and a quantum phase transition. The low-energy sector realizes a Luttinger-liquid-like phase with velocity $v_s(n)$ and a density-dependent LL parameter, while the finite-temperature regime exhibits universal scaling near a quantum critical point at $h_c=-2$, with $z=2$, $ u=1/2$. Despite formal similarities to Lieb-Liniger and positive-spin XXX models, the negative-spin chain exhibits qualitative differences in thermodynamics and scaling due to its unique vacuum and excitation structure. The results enhance understanding of negative-spin integrable systems and their connections to DIS and high-energy QCD.

Abstract

We study the thermodynamics of the isotropic Heisenberg XXX spin chain with negative spin, focusing on the case $s=-1$. The model is equivalent to the quantum lattice nonlinear Schrödinger (NLS) model and appears as an effective theory in deep inelastic scattering in high-energy quantum chromodynamics. Owing to its integrability, it admits a consistent Bethe Ansatz description and a well-defined thermodynamic limit. Using the thermodynamic Bethe Ansatz, we analyze the ground state, elementary excitations, and finite-temperature properties. In contrast to the conventional positive spin XXX chain, the negative spin model exhibits a distinct vacuum structure and excitation spectrum, leading to modified TBA equations and unconventional low-temperature behavior. Although the integral equations resemble those of the Lieb-Liniger Bose gas, the thermodynamics and scaling properties are qualitatively different and cannot be continuously connected. We derive the free energy, entropy, and specific heat, and identify a quantum phase transition separating different thermodynamic regimes. At zero temperature, the excitation spectrum becomes linear in the continuum limit and can be described by a conformal field theory. The low-temperature regime realizes a Luttinger-liquid like phase with features unique to the negative spin XXX chain.

Figures (10)

• Figure 1: Schematic illustration of particle and hole excitations relative to the Fermi surface. (a) Ground state: all single-particle levels within the Fermi surface are occupied. (b) Hole excitation: one particle is removed from a filled state inside the Fermi surface. (c) Particle excitation: one particle is promoted to a state above the Fermi surface. (d) Particle-hole excitation: a particle is excited from within the Fermi surface to an unoccupied state outside it, leaving a hole inside. In this process, the total number of particles $N$ is conserved.
• Figure 2: Variation of the distribution $g(x)$ as a function of the dimensionless parameter $x$ for various densities $n$. The inset compares the analytical approximation given by Eq. \ref{['eq:g_weak']} (red squares) with the numerical solution (red solid line) of Eqs. \ref{['eq:g']}-\ref{['eq:gamma']} at $n=0.1$.
• Figure 3: Excitation energy $\Delta E$ as a function of excitation momentum $\Delta P$ for various densities $n$. Solid lines represent hole excitations ($\lambda_p\to\lambda_F$), while dash-dotted lines denote particle excitations ($\lambda_h\to\lambda_F$).
• Figure 4: Sound velocity $v_s$ as a function of density $n$. The inset shows the variation of $s/T = \pi/(3v_s)$ with respect to $n$.
• Figure 5: Schematic of excited states at finite temperature: solid circles indicate occupied quantum numbers, while open circles indicate unoccupied ones. Due to the correspondence between quantum numbers and Bethe roots, there are $\Delta I$ allowed quantum numbers within the interval $\Delta \lambda$. Here, $\rho$ denotes the density of occupied states, and $\rho_h$ represents the density of unoccupied hole states.