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IEEE Transactions on Magnetics

Thermal Phase Transitions in a Deformable Quantum Spin-1/2 XX Chain in a Transverse Magnetic Field

Dávid Sivý, Jozef Strečka

Published Nov 3, 2025• 0 citations
Publisher

Abstract

We investigate the deformable quantum spin-1/2 XX chain in a transverse magnetic field, which is exactly solvable via the Jordan-Wigner transformation under the assumption of a linear dependence of the exchange interaction on uniform lattice distortion. By calculating the magnetization, magnetic susceptibility, distortion parameter, and inverse compressibility, we explore the coupled magnetic and elastic properties of the deformable quantum spin chain. It is demonstrated that the deformable spin-1/2 XX chain in a transverse magnetic field exhibits a line of discontinuous phase transitions emerging at low but finite temperatures, which terminates at a critical point corresponding to a continuous phase transition. The discontinuous thermal phase transitions are accompanied by magnetic hysteresis due to metastable states, which gradually vanishes as the temperature increases.

DOI
10.1109/tmag.2025.3628061

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We investigate the deformable quantum spin-1/2 XX chain in a transverse magnetic field, which is exactly solvable via the Jordan-Wigner transformation under the assumption of a linear dependence of the exchange interaction on uniform lattice distortion. By calculating the magnetization, magnetic susceptibility, distortion parameter, and inverse compressibility, we explore the coupled magnetic and elastic properties of the deformable quantum spin chain. It is demonstrated that the deformable spin-1/2 XX chain in a transverse magnetic field exhibits a line of discontinuous phase transitions emerging at low but finite temperatures, which terminates at a critical point corresponding to a continuous phase transition. The discontinuous thermal phase transitions are accompanied by magnetic hysteresis due to metastable states, which gradually vanishes as the temperature increases.
We investigate a deformable spin-1/2 Ising chain subjected to either a longitudinal or a transverse magnetic field, which incorporates a magnetoelastic coupling linearly dependent on a lattice distortion parameter. Within the harmonic and static adiabatic approximations, the variational Gibbs free energy is evaluated exactly using transfer-matrix and Jordan-Wigner fermionization techniques and then minimized self-consistently with respect to the lattice distortion parameter. This approach enables a unified description of magnetic and elastic properties including the magnetization, magnetic susceptibility, lattice distortion, inverse compressibility, and relative change in the sound velocity. In a longitudinal magnetic field, the deformable Ising chain displays a line of discontinuous thermal phase transitions terminating at a critical point. The discontinuous transitions are accompanied by metastable states, which give rise to a hysteresis loop at low temperatures. In contrast, the deformable Ising chain in a transverse field undergoes exclusively a continuous quantum phase transition at zero temperature with no indication of thermal phase transitions. The magnetic susceptibility and inverse compressibility exhibit cusp- and dip-like anomalies at discontinuous phase transitions, while a diverging susceptibility and vanishing inverse compressibility characterize the continuous phase transitions. An elastic softening of the deformable chain near thermal and quantum phase transitions manifests itself also through a significant sound attenuation.
The quantum phase transition in the spin-12\frac{1}{2}21​ XX Heisenberg chain model with three-spin interaction is studied. Using the Jordan-Wigner transformation, several thermodynamic, as well as thermal and spin transport quantities of the system are calculated exactly. It is shown that the three-spin interaction influences the calculated quantities and leads to characteristic features of the quantum phase transition. The effects of a finite magnetic field on the magnetic moment and magnetic susceptibility are also discussed.
The thermal pairwise entanglement (TE) of the S=1/2S=1/2S=1/2 XYXYXY chain in a transverse magnetic field is exactly resolved by means of the Jordan-Wigner transformation in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that the TE vanishes at a fixed point with temperature ${T}_{c}\ensuremath{\simeq}0.484\text{ }3J$, which is independent of the magnetic field. A thermal quantity is proposed to witness the entangled state. Furthermore, the TE of the S=1/2S=1/2S=1/2 antiferromagnetic-ferromagnetic (AF-F) Heisenberg chain is studied by the transfer-matrix renormalization-group method. The TEs of the spins coupled by AF and F interactions are found to behave distinctively. The vanishing temperature of the field-induced TE of the spins coupled by F interactions is observed dependent on the magnetic field. The results are further confirmed and analyzed within a mean-field framework.
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the ``$\ensuremath{\tau}$ order metric'' of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the xxx-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the ``$\ensuremath{\tau}$ order metric,'' even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.
Using the Jordan-Wigner transformation and continued fractions we calculate rigorously the thermodynamic quantities for the spin-1/2 transverse Ising chain with periodically varying intersite interactions and/or on-site fields. We consider in detail the properties of the chains having a period of the transverse field modulation equal to 3. The regularly alternating transverse Ising chain exhibits several quantum phase transition points, where the number of transition points for a given period of alternation strongly depends on the specific set of the Hamiltonian parameters. The critical behavior in most cases is the same as for the uniform chain. However, for certain sets of the Hamiltonian parameters the critical behavior may be changed and weak singularities in the ground-state quantities appear. Due to the regular alternation of the Hamiltonian parameters the transverse Ising chain may exhibit plateaulike steps in the zero-temperature dependence of the transverse magnetization vs transverse field and many-peak temperature profiles of the specific heat. We compare the ground-state properties of regularly alternating transverse Ising and transverse XX chains and of regularly alternating quantum and classical chains. Making use of the corresponding unitary transformations we extend the elaborated approach to the study of thermodynamics of regularly alternating spin-1/2 anisotropic XY chains without field. We use the exact expression for the ground-state energy of such a chain of period 2 to discuss how the exchange interaction anisotropy destroys the spin-Peierls dimerized phase.
Quantum phase transitional behavior of a finite periodic XX spin-1/2 chain with nearest neighbor interaction in a uniform transverse field is studied based on the simple exact solutions. It is found that there are [N/2] level-crossing points in the ground state, where N is the periodic number of the system and [x] stands for the integer part of x, when the interaction strength and magnitude of the magnetic field satisfy certain conditions. The quantum phase transitions are of the first order due to the level-crossing. The ground state in the thermodynamic limit will be divided into three distinguishable quantum phases.