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Quantum effects in rotationally invariant spin glass models

Yoshinori Hara, Yoshiyuki Kabashima

TL;DR

This work addresses quantum effects in mean-field spin-glass models with rotationally invariant interactions and tests the validity of the quasi-static approximation (qSA) beyond the static approximation. It combines the Suzuki--Trotter decomposition with the replica method to obtain a replica-symmetric (RS) framework for time-dependent order parameters and derives an Almeida--Thouless–like stability condition. The authors compute zero-temperature transverse-field thresholds $\Gamma_c$ for the Sherrington--Kirkpatrick and Hopfield models, and show that quantum fluctuations qualitatively modify the random first-order transition scenario in the random orthogonal model (ROM). The results support using a quasi-static RS treatment in quantum spin glasses and offer a practical route to inform quantum optimization strategies, with potential validation via quantum Monte Carlo comparisons.

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 quasi-static approximation 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, $Γ_\mathrm{c}$, which agrees with previous Monte Carlo-based estimations. For the Hopfield model, it provides an estimate of $Γ_\mathrm{c}$, 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 quasi-static treatment for analyzing quantum spin glasses and may offer useful insights into the analysis of quantum optimization algorithms.

Quantum effects in rotationally invariant spin glass models

TL;DR

This work addresses quantum effects in mean-field spin-glass models with rotationally invariant interactions and tests the validity of the quasi-static approximation (qSA) beyond the static approximation. It combines the Suzuki--Trotter decomposition with the replica method to obtain a replica-symmetric (RS) framework for time-dependent order parameters and derives an Almeida--Thouless–like stability condition. The authors compute zero-temperature transverse-field thresholds for the Sherrington--Kirkpatrick and Hopfield models, and show that quantum fluctuations qualitatively modify the random first-order transition scenario in the random orthogonal model (ROM). The results support using a quasi-static RS treatment in quantum spin glasses and offer a practical route to inform quantum optimization strategies, with potential validation via quantum Monte Carlo comparisons.

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 quasi-static approximation 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, , which agrees with previous Monte Carlo-based estimations. For the Hopfield model, it provides an estimate of , 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 quasi-static treatment for analyzing quantum spin glasses and may offer useful insights into the analysis of quantum optimization algorithms.

Paper Structure

This paper contains 15 sections, 53 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model setup
  3. Analytical treatment
  4. Suzuki--Trotter decomposition
  5. Replica method
  6. Assessment of $\left [Z_M^n \right ]_J$ for $n \in \mathbb{N}$
  7. Replica symmetry (RS) and analytic continuation
  8. Quasi-static solution and its stability
  9. Quasi-static solution
  10. Stability analysis
  11. Numerical evaluation
  12. SK model
  13. Hopfield model
  14. Random Orthogonal Model
  15. Summary and discussion

Figures (8)

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