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arXiv:2504.18034 · Physical review. E

Quantum effects in rotationally invariant spin glass models

Yoshinori Hara, Yoshiyuki Kabashima

Published Oct 7, 2025• 0 citations
arXivHTMLPDF

Abstract

This study investigates the quantum effects in transverse-field Ising spin glass models with rotationally invariant random interactions. The primary aim is to evaluate the validity of a quasistatic approach that captures the imaginary-time dependence of the order parameters beyond the conventional static approximation. Using the replica method combined with the Suzuki-Trotter decomposition, we established a stability condition for the replica symmetric solution, which is analogous to the de Almeida-Thouless criterion. Numerical analysis of the Sherrington-Kirkpatrick model estimates a value of the critical transverse field, Γ_{c}, which is consistent with previous Monte Carlo-based estimations. For the Hopfield model, it provides an estimate of Γ_{c}, which has not been previously evaluated. For the random orthogonal model, our analysis suggests that quantum effects alter the random first-order transition scenario in the low-temperature limit. This study supports a quasistatic treatment for analyzing quantum spin glasses and may offer useful insights into the analysis of quantum optimization algorithms.

DOI
10.1103/tk5w-dr5j

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This study investigates the quantum effects in transverse-field Ising spin glass models with rotationally invariant random interactions. The primary aim is to evaluate the validity of a quasistatic approach that captures the imaginary-time dependence of the order parameters beyond the conventional static approximation. Using the replica method combined with the Suzuki-Trotter decomposition, we established a stability condition for the replica symmetric solution, which is analogous to the de Almeida-Thouless criterion. Numerical analysis of the Sherrington-Kirkpatrick model estimates a value of the critical transverse field, Γ_{c}, which is consistent with previous Monte Carlo-based estimations. For the Hopfield model, it provides an estimate of Γ_{c}, which has not been previously evaluated. For the random orthogonal model, our analysis suggests that quantum effects alter the random first-order transition scenario in the low-temperature limit. This study supports a quasistatic treatment for analyzing quantum spin glasses and may offer useful insights into the analysis of quantum optimization algorithms.
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I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
The effect of a transverse field on the spin-glass freezing transition in the Sherrington-Kirkpatrick version of the quantum Ising model is studied using the thermofield dynamics (the real-time, finite-temperature quantum field theory) due to Takahashi and Umezawa (1975) as a substitute for the 'n-replica trick'. The field-dependent transition line to the spin-glass phase is found and the averaged local static metastable susceptibility in the transverse direction is evaluated in the simple mean-field approximation for the effective dynamic Lagrangian.
In the present paper we present a detailed analysis of quantum effects in the spin-glass transition as described by a quantum Heisenberg analogue of the Sherrington-Kirkpatrick model. The spin operators are represented in terms of two fermion fields and the problem is reduced to that of n fermion levels at one site in a random time-dependent field and with an interaction delocalized in time. It is shown that within a Hartree-Fock approximation in a replica-symmetric theory one obtains a mean-field description of the transition with satisfactory zero-temperature properties. The transition is described by two order parameters: the static magnetic susceptibility and the spin-glass order parameter. The saddle-point equations obtained in this work are analogous to those of Sommers for a theory of classical spins.