Thermodynamics of the $q$-deformed Kittel--Shore model

TL;DR

This work analyzes the thermodynamics of the $q$-deformed Kittel–Shore Hamiltonian for spin-$\tfrac{1}{2}$ particles, implementing a $U_q(\mathfrak{su}(2))$ symmetry that dilates energy spacings and shifts thermodynamic features to higher temperatures. The undeformed spectrum is extended to $E^{q}_{N J m}$ with $[n]_{q}$, and the partition function $Z_N^{q}$ is built from the same degeneracies, recovering the $q\to1$ limit. In the ferromagnetic case, deformation raises the Curie temperature and narrows the specific-heat peak, while in the antiferromagnetic case a two-level approximation captures the main thermodynamic behavior and phase transitions, with parity playing a significant role. In the thermodynamic limit, deformation manifests as larger energy gaps and a more pronounced separation of ground and excited states, with $q=e^{\eta/N}$ producing explicit $q$-dependent forms for $C_V$, $\chi$, and $M$ and shifting convergence between even and odd particle numbers to higher temperatures. Overall, the deformation introduces non-identical spin couplings and provides analytic insight into the thermodynamics of long-range quantum spin models, with potential applications to ultrasmall spin clusters and engineered non-uniform couplings.

Abstract

The Kittel--Shore Hamiltonian characterizes $N$ spins with identical long-range interactions, and the $\mathfrak{su}(2)$ coalgebra has been proven to be a symmetry of this model, which can be exactly solved. By using quantum groups and, in particular, $\mathfrak{su}_{q}(2)$, this Hamiltonian was deformed. In this work, we study the thermodynamic properties of this deformed model for spin-$1/2$ particles. In particular, we discuss how this deformation affects the specific heat, magnetic susceptibility, magnetisation, and phase transitions as a function of the parameter $q$ of the deformation and compare them with those of the undeformed model. Deformation was found to shift the thermodynamic behaviours to higher temperatures and alter the phase transitions. The potential applications of this $q$-deformed model for describing few-spin quantum systems with non-identical couplings are discussed.

Figures (50)

• Figure 1: Specific heat as a function of temperature for values of the number of particles $N=2$ (red), $3$ (orange), $4$ (yellow), $5$ (green), $6$ (cyan), $7$ (blue), $8$ (purple), and $9$ (black) for the ferromagnetic case.
• Figure 2: Magnetic susceptibility as a function of temperature for values of the number of particles $N=2$ (red), $3$ (orange), $4$ (yellow), $5$ (green), $6$ (cyan), $7$ (blue), $8$ (purple), and $9$ (black) for the ferromagnetic case.
• Figure 3: Magnetisation as a function of temperature for values of the number of particles $N=2$ (red), $3$ (orange), $4$ (yellow), $5$ (green), $6$ (cyan), $7$ (blue), $8$ (purple), and $9$ (black), with $h=\gamma=1$ for the ferromagnetic case.
• Figure 4: Magnetisation as a function of magnetic field for values of the number of particles $N=2$ (red), $3$ (orange), $4$ (yellow), $5$ (green), $6$ (cyan), $7$ (blue), $8$ (purple), and $9$ (black) for temperature $T=0^{+}$, for the ferromagnetic case.
• Figure 5: Specific heat as a function of temperature for values of the number of particles $N=100$ (red), $500$ (orange), and $1000$ (yellow), for the interaction constant $I=1/N$ ferromagnetic case.