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arXiv:2609.31094

Thermodynamics of the $q$-deformed Kittel--Shore model for spin-1 particles

Víctor Mariscal, Jose Javier Relancio

Published Sep 25, 2026• 0 citations
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Abstract

Quantum algebras ($q$-algebras) have been used in the literature to deform spin-chain models. In particular, in previous works by the present authors, the Kittel--Shore Hamiltonian was deformed using $\mathfrak{su}_q(2)$ algebra. The deformed Hamiltonian was presented for a general spin, and the thermodynamic properties were studied for spin-1/2 particles. In this paper, we focus on the $q$-deformation of the Kittel--Shore model for spin-1 systems. We analyse the impact of this deformation on the thermodynamic properties and phase transitions of the system as a function of the deformation parameter $q$. Specifically, we study the specific heat, magnetic susceptibility, and magnetisation, including a detailed analysis of the Curie temperature characterising the ferromagnetic phase. Furthermore, a finite-size scaling analysis is conducted for both the ferromagnetic and antiferromagnetic cases. Results are systematically compared with those of the undeformed model to reveal the physical effects of the $q$-deformation.

DOI
10.48550/arxiv.2609.31094

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Thermodynamics of the qqq-deformed Kittel--Shore model for spin-1 particles
Quantum algebras (qqq-algebras) have been used in the literature to deform spin-chain models. In particular, in previous works by the present authors, the Kittel--Shore Hamiltonian was deformed using suq(2)\mathfrak{su}_q(2)suq​(2) algebra. The deformed Hamiltonian was presented for a general spin, and the thermodynamic properties were studied for spin-1/2 particles. In this paper, we focus on the qqq-deformation of the Kittel--Shore model for spin-1 systems. We analyse the impact of this deformation on the thermodynamic properties and phase transitions of the system as a function of the deformation parameter qqq. Specifically, we study the specific heat, magnetic susceptibility, and magnetisation, including a detailed analysis of the Curie temperature characterising the ferromagnetic phase. Furthermore, a finite-size scaling analysis is conducted for both the ferromagnetic and antiferromagnetic cases. Results are systematically compared with those of the undeformed model to reveal the physical effects of the qqq-deformation.
Thermodynamics of the qqq-deformed Kittel--Shore model
The Kittel--Shore Hamiltonian characterizes NNN spins with identical long-range interactions, and the su(2)\mathfrak{su}(2)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, suq(2)\mathfrak{su}_{q}(2)suq​(2), this Hamiltonian was deformed. In this work, we study the thermodynamic properties of this deformed model for spin-1/21/21/2 particles. In particular, we discuss how this deformation affects the specific heat, magnetic susceptibility, magnetisation, and phase transitions as a function of the parameter qqq 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 qqq-deformed model for describing few-spin quantum systems with non-identical couplings are discussed.
The Kittel–Shore Hamiltonian characterizes N spins with identical long-range interactions, and the [Formula: see text] coalgebra has been proven to be a symmetry of this model, which can be exactly solved. By using quantum groups and, in particular, [Formula: see text], this Hamiltonian was deformed. In this work, we study the thermodynamic properties of this deformed model for spin-[Formula: see text] particles. In particular, we discuss how this deformation affects the specific heat, magnetic susceptibility, magnetization 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 behaviors to higher temperatures and alter the phase transitions. Additionally, we extend the framework to complex deformations on the unit circle, [Formula: see text], for q not being a root of unity. For an [Formula: see text] tetrahedral cluster, this phase establishes a thermodynamic equivalence between a second-neighbor spin-Hamiltonian and the q-deformed quantum model. At least, we have found experimental compounds well reproduced by the Kittel–Shore model for [Formula: see text]. Validating our effective algebraic model against exact diagonalization data for the Cu 2 Te 2 O 5 X 2 (X [Formula: see text] Cl, Br) compound family demonstrates that [Formula: see text] accurately reproduces the structural anisotropy.
Quantum deformations of U(sl(2,R))\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))U(sl(2,R)). Part I: Fidelity and experimental benchmarking
This work explores the effects of both the standard quantum qqq-deformation and the non-standard hhh-deformation of the Hopf algebra U(sl(2,R))\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))U(sl(2,R)) on multi-qubit systems. By constructing the states of a Hilbert space of NNN qubits through the Clebsch-Gordan coefficients associated with the deformed algebras, we show that these states naturally coincide with the eigenstates of the Hamiltonian of the qqq- and hhh-deformed Kittel-Shore models. We compare the resulting deformed states with those typically targeted in quantum information experiments, providing a bridge between algebraic constructions and experimentally relevant quantum resources. Fidelities with respect to the undeformed states are computed to establish how the quantum correlations are affected, both for few-qubit systems (including Dicke and non-Dicke states), and in the macroscopic limit (N→∞N \to \inftyN→∞) through closed-form formulas derived for arbitrary Dicke states. The results reveal different behaviors between the two deformations. The qqq-deformation smoothly modifies the states and maintains a residual overlap with the original configurations, while the hhh-deformation rapidly makes the states orthogonal to their undeformed counterparts. Both models demand a standard N−1N^{-1}N−1 rescaling to preserve fidelity stability in the macroscopic limit.
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 ( 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 ( 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.
The magnetization process, the susceptibility, and the specific heat of the spin-1∕21∕21∕2 antiferromagnet (AF)-AF-ferromagnet (F) and F-F-AF trimerized quantum Heisenberg chains have been investigated by means of the transfer matrix renormalization group (TMRG) technique as well as the modified spin-wave (MSW) theory. A magnetization plateau at m=1∕6m=1∕6m=1∕6 for both trimerized chains is observed at low temperature. The susceptibility and the specific heat show various behaviors for different ferromagnetic and antiferromagnetic interactions and in different magnetic fields. The TMRG results of susceptibility and the specific heat can be nicely fitted by a linear superposition of double two-level systems, where two fitting equations are proposed. Three branch excitations, one gapless excitation and two gapful excitations, for both systems are found within the MSW theory. It is observed that the MSW theory captures the main characteristics of the thermodynamic behaviors at low temperatures. The TMRG results are also compared with the possible experimental data.
Assuming q-deformed commutation relations for the fermions, an extension of the standard Lipkin Hamiltonian is presented. The usual quasi-spin representation of the standard Lipkin model is also obtained in this q-deformed framework. A variationally obtained energy functional is used to analyse the phase transition associated with the spherical symmetry breaking. The only phase transitions in this q-deformed model are of second order. As an outcome of this analysis a critical parameter is obtained which is dependent on the deformation of the algebra and on the number of particles.