Quantum group deformation of the Kittel--Shore model

TL;DR

The paper addresses constructing and solving a $U_q(\mathfrak{su}(2))$-invariant deformation of the Kittel–Shore model to preserve integrability for arbitrary spins. It develops the q-KS Hamiltonian by enforcing quantum group coalgebra symmetry, derives exact spectra and eigenvectors via $q$-Clebsch–Gordan coefficients, and analyzes spin-\tfrac{1}{2}$ cases at $N=2$ and $N=3$ with higher-spin extensions sketched. A thorough numerical study of the Curie temperature reveals an exponential rise with deformation parameter $\eta$, and the work outlines how to extend these results to arbitrary spin and potential connections to thermodynamics and quantum information. The results establish a new, exactly solvable long-range spin model with nonlocal, $q$-dependent couplings, enriching the landscape of integrable quantum spin systems and offering a framework for exploring deformation effects in finite-size and thermodynamic regimes.

Abstract

The Kittel--Shore (KS) Hamiltonian describes $N$ spins with long-range interactions that are identically coupled; therefore, this (mean-field) model is also known as the Heisenberg XXX model on the complete graph. In this paper, the underlying $U(\mathfrak{su}(2))$ coalgebra symmetry of the KS model is demonstrated for arbitrary spins, and the quantum deformation of the KS Hamiltonian ($q$-KS model) is obtained using the corresponding $U_q(\mathfrak{su}(2))$ quantum group. By construction, the existence of such a symmetry guarantees that all integrability properties of the KS model are preserved under $q$-deformation. In particular, the $q$-KS model for spin-$1/2$ particles is analysed, the cases with $N=2$ and $3$ spins are studied in detail, and higher-spin $q$-KS models are sketched. As a first excursion into the thermodynamic properties of the spin-$1/2$ $q$-KS model, the dependence of the Curie temperature on the deformation parameter is studied through numerical analysis.

Figures (19)

• Figure 1: Energy levels (in the units $|I|/4$) and their degeneracies for $N=2,3,4,5$ spins with $j=1/2$ coupled antiferromagnetically (a) without an external magnetic field $(h=0)$ and (b) with an external magnetic field $(h=0.5)$. Adapted from al1998exact. Note that $(1/2)^{\otimes 2} = 1 \oplus 0$, $(1/2)^{\otimes 3} = 3/2 \oplus (1/2)^{\oplus 2}$, $(1/2)^{\otimes 4} = 2 \oplus 1^{\oplus 3} \oplus 0^{2}$ and $(1/2)^{\otimes 5} = 5/2 \oplus (3/2)^{\oplus 4} \oplus (1/2)^{\oplus 4}$.
• Figure 2: Density of energy levels (with zero energy ground state) for $N=20$ in the antiferromagnetic case.
• Figure 3: Density of energy levels for $N=100$, $I\rightarrow\frac{I}{N}$ in the antiferromagnetic case.
• Figure 4: Density of energy levels for $N=1000$, $I\rightarrow\frac{I}{N}$ in the antiferromagnetic case.
• Figure 5: Density of energy levels for $N=20$ as a function of the parameter $\eta=0$ (red) and $\eta=0.2$ (dashed blue) in the antiferromagnetic case.