[Paper Review] Competition of Ferromagnetic and Antiferromagnetic Order in the Spin-1/2 XXZ Chain at Finite Temperature

K. Fabricius, Andreas Klümper+1 more|arXiv (Cornell University)|Oct 21, 1998

Theoretical and Computational Physics8 references3 citations

TL;DR

This paper analytically investigates the competition between ferromagnetic and antiferromagnetic order in the spin-1/2 XXZ chain at finite temperature, using exact solutions via the thermodynamic Bethe ansatz. It derives an analytic formula for the crossover in long-distance asymptotics and provides explicit results for longitudinal spin correlations, specific heat, and magnetic susceptibility across varying anisotropies, revealing a transition from antiferromagnetic to ferromagnetic dominance with increasing temperature.

ABSTRACT

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.

Motivation & Objective

• To understand the interplay between ferromagnetic and antiferromagnetic order in quantum spin chains under thermal fluctuations.
• To analyze the crossover from antiferromagnetic to ferromagnetic behavior in the attractive XXZ chain as temperature increases.
• To derive exact analytical expressions for long-distance asymptotics of spin correlations in the finite-temperature regime.
• To compute thermodynamic quantities such as specific heat and magnetic susceptibility for various anisotropy parameters.
• To establish a quantitative description of the transition between dominant magnetic orders in the gapless regime.

Proposed method

• Employing the thermodynamic Bethe ansatz (TBA) to solve the integrable XXZ spin chain at finite temperature.
• Using exact solutions of the TBA equations to analyze the long-distance asymptotics of the longitudinal spin correlation function.
• Deriving an analytic formula for the crossover behavior between antiferromagnetic and ferromagnetic correlation decay.
• Computing the specific heat and magnetic susceptibility from the free energy obtained via TBA.
• Performing analytical continuation and asymptotic analysis to extract temperature-dependent correlation lengths and decay exponents.
• Focusing on the attractive (negative exchange) XXZ model where both ferromagnetic and antiferromagnetic tendencies coexist.

Experimental results

Research questions

• RQ1 How does the long-distance asymptotic behavior of the longitudinal spin correlation function evolve from antiferromagnetic to ferromagnetic decay with increasing temperature?
• RQ2 What is the analytic form of the crossover region between antiferromagnetic and ferromagnetic correlation regimes in the finite-temperature XXZ chain?
• RQ3 How do the specific heat and magnetic susceptibility depend on the anisotropy parameter Δ and temperature in the gapless attractive regime?
• RQ4 What is the role of the anisotropy Δ in determining the dominance of ferromagnetic or antiferromagnetic order at finite T?
• RQ5 Can exact analytical expressions be derived for the nearest-neighbor longitudinal correlation function at finite temperature?

Key findings

• An exact analytic formula is derived for the crossover in the long-distance asymptotics of the longitudinal spin correlation function in the attractive XXZ chain at finite temperature.
• Explicit results are obtained for the nearest-neighbor longitudinal correlation function, showing a transition from antiferromagnetic to ferromagnetic character with increasing temperature.
• The specific heat exhibits a non-monotonic dependence on temperature, with features sensitive to the anisotropy parameter Δ.
• Magnetic susceptibility shows a strong temperature and anisotropy dependence, reflecting the competition between ferromagnetic and antiferromagnetic fluctuations.
• The crossover from antiferromagnetic to ferromagnetic dominance is analytically characterized by a change in the sign and magnitude of the correlation function's asymptotic decay exponent.
• The results are exact and derived from the thermodynamic Bethe ansatz, providing a rigorous finite-temperature description of competing magnetic orders in the integrable XXZ chain.