An analytical study is presented of the crossover in the gapless attractive XXZ chain from antiferromagnetic to ferromagnetic behaviour at low to high temperature, respectively. In particular, an analytic formula for the crossover in the long distance asymptotics and explicit results for the nearest-neighbour longitudinal correlation are obtained. We also provide results for the specific heat and magnetic susceptibility for various anisotropies.
An analytical study is presented of the crossover in the gapless attractive XXZ chain from antiferromagnetic to ferromagnetic behaviour at low to high temperature, respectively. In particular, an analytic formula for the crossover in the long distance asymptotics and explicit results for the nearest-neighbour longitudinal correlation are obtained. We also provide results for the specific heat and magnetic susceptibility for various anisotropies.
Abstract We present a systematic analytical study of finite-temperature spin diffusion in the gapped spin-21 XXZ chain with easy-axis anisotropy Δ>1 at zero magnetic field. While the finiteness of the spin-diffusion constant Ds(T,Δ) for all T>0 and Δ>1 has recently been established analytically \cite{R01Carmelo2025a,R02Carmelo2025b}, its precise values away from the asymptotic limits remain an outstanding open problem \cite{R03Carmelo2026BJP}. We provide a unified treatment covering three complementary analytical regimes. First, we formulate the two-sided bound Ds∞(Δ)≤Ds(T,Δ)≤DsLT(T,Δ) between the exact high-temperature Bethe-ansatz value and the leading-order low-temperature activated formula; this bound is a conjecture supported by both asymptotic limits and the available numerical data, and it provides qualitative constraints at every temperature. Second, we exploit the high-temperature approximation of Carmelo and Sacramento \cite{R01Carmelo2025a} to obtain quantitative values of Ds throughout the window βJ∈[0,1], tabulated for Δ∈{1.25,1.5,1.75,2.0}. Third, we identify and compute the crossover temperature T∗(Δ) at which the high-temperature approximation meets the low-temperature activated formula; since both expressions are used outside their strict domains of validity at that point, T∗ is a heuristic indicator of the window in which neither closed-form expression is controlled, rather than a sharp regime boundary. We find that T∗(Δ) increases monotonically with anisotropy, ranging from T∗/J≈0.27 at Δ=1.25 to T∗/J≈0.60 at Δ=2.0, and give an explicit expression for it. Finally, we show that the high-temperature series can be resummed exactly into the form Ds∞=(4J/3π)sinhη∑m≥1m/sinh(mη) (Δ=coshη), from which the near-isotropic divergence Ds∞(Δ)≃C0(Δ−1)−1/2 follows with the exact amplitude C0=πJ/(32). These results provide a quantitative analytical picture of spin diffusion over the temperature range accessible to closed-form expressions. The exact high-temperature floor Ds∞(Δ) and the crossover temperature T∗(Δ) serve as reference values against which future tensor-network and generalized-hydrodynamics calculations may be compared.
We study the zero- and the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. We evaluate the local magnetization on the impurity site at zero temperature analytically and derive the impurity susceptibility exactly from it. In the graphs of the impurity specific heat versus temperature, we show how the impurity spin becomes more liberated from the bulk many-body effect as the exchange coupling between the impurity spin and other spins decreases and that at low temperature it couples strongly to them such as in the Kondo effect. Thus, we observe not only the crossover behavior from the high- to the low-temperature regime, but another from the N -site chain to the ( N − 1)-site chain with a free impurity spin. We also show that the estimate of the Wilson ratio at a given low temperature is independent of the impurity parameter if its absolute value is small enough with respect to the temperature and the universality class is described by the XXZ anisotropy in terms of the dressed charge.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
A crossover phenomenon of the longitudinal spin–spin correlation function is investigated on the spin-1/2 XXZ chain in a magnetic field at finite temperatures. A commensurate–incommensurate transition of the correlation function in the asymptotic behaviour has been pointed out by Klumper et al in the framework of the quantum transfer matrix method. We show the field dependence and the anisotropy-constant dependence of the critical temperature by the quantum transfer matrix method. Using a quantum Monte Carlo method with the continuous time loop algorithm, we directly observe the commensurate–incommensurate transition. Detailed quantum Monte Carlo analyses for the longitudinal spin–spin correlation in both the high and low temperature regions are presented. Especially in the low temperature region, evaluating the precise distance dependence of the correlation function, we prove that the incommensurate oscillation has detectable amplitude, which has not been estimated before. We also estimate the phase factor of the incommensurate oscillation, which accurately coincides with the one from the quantum transfer matrix method.
The thermodynamics of finite open antiferromagnetic XXZ chains is studied using field theory, Bethe ansatz and quantum Monte Carlo methods. For the susceptibility a parameter-free result as a function of the number of sites L and temperature T beyond the scaling limit is derived. The limiting cases ( J being the exchange constant), where the boundary correction shows a logarithmically suppressed Curie behaviour, and , where a crossover to the ground state behaviour takes place, are discussed in detail. On the basis of this analysis we present a simple formula for the averaged susceptibility of a spin-1/2 chain doped with non-magnetic impurities. We show that the effective Curie constant has a highly non-trivial temperature dependence and shows scaling in the low temperature limit. Finally, corrections due to intrachain and interchain couplings and implications for experiments on Sr 2 Cu 1− x Pd x O 3+δ and related compounds are discussed.
The anisotropic XXZ spin-21 Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction sixsjx+siysjy induces an antiferromagnetic correlation in the out-of-plane siz component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temperatures in the ferromagnetic XXZ chain, and, in-between, at all temperatures in the XY chain. We find that the converse effect also occurs in the antiferromagnetic XXZ chain: an antiferromagnetic sizsjz interaction induces a correlation in the sixy component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.