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Existence of de Almeida-Thouless-type instability in the transverse field Sherrington-Kirkpatrick model

C. Itoi, K. Fujiwara, Y. Sakamoto

Abstract

The interpolation method for mean field spin glass models developed by Guerra and Talagrand is extended to a quantum mean field spin glass model. This extension enables us to obtain both replica-symmetric (RS) and one step replica-symmetry breaking (1RSB) solutions of the free energy density in the transverse field Sherrington-Kirkpatrick model. It is shown that the RS solution is exact in the paramagnetic phase. We provide a sufficient condition on coupling constants where the 1RSB solution gives better bound than the RS one. This condition reduced to physical quantities in disordered single spin systems allows a simple computer-assisted proof for the existence of the de Almeida-Thouless-type instability.

Existence of de Almeida-Thouless-type instability in the transverse field Sherrington-Kirkpatrick model

Abstract

The interpolation method for mean field spin glass models developed by Guerra and Talagrand is extended to a quantum mean field spin glass model. This extension enables us to obtain both replica-symmetric (RS) and one step replica-symmetry breaking (1RSB) solutions of the free energy density in the transverse field Sherrington-Kirkpatrick model. It is shown that the RS solution is exact in the paramagnetic phase. We provide a sufficient condition on coupling constants where the 1RSB solution gives better bound than the RS one. This condition reduced to physical quantities in disordered single spin systems allows a simple computer-assisted proof for the existence of the de Almeida-Thouless-type instability.
Paper Structure (6 sections, 7 theorems, 81 equations)

This paper contains 6 sections, 7 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Definitions of the model
  3. RS bound on the free energy density
  4. 1RSB bound on the free energy density
  5. AT-type instability
  6. Discussions

Key Result

Lemma 3.1

(Extended Guerra's identity for RS bound) Define a function by where the above random variable is defined by For arbitrary $(\beta,b,q) \in [0,\infty)^2 \times [0,1]$, the following identity is valid Proof. Integration of the identity (phi'2) over $s\in[0,1]$ gives The model at $s=0$ becomes independent spin model, and therefore $\varphi_N(0)$ is represented in terms of the partition function

Theorems & Definitions (7)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 5.1