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Gauge theory for quantum XYZ spin glasses

C. Itoi, Y. Sakamoto

Abstract

Nishimori's gauge theory is extended to the quantum XYZ $p$-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the $\mathbb Z_2$-symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards-Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.

Gauge theory for quantum XYZ spin glasses

Abstract

Nishimori's gauge theory is extended to the quantum XYZ -spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the -symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards-Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.
Paper Structure (8 sections, 3 theorems, 62 equations, 1 figure)

This paper contains 8 sections, 3 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions of the quantum XYZ spin glass model
  3. Nishimori's gauge theory
  4. $\mathbb Z_2$-symmetry breaking in the spin glass transition
  5. Assumptions
  6. Absence of spontaneous magnetization
  7. Bound on the susceptibility
  8. Summary and discussions

Key Result

Lemma 3.1

Let $u$ be one of $x,y,z$ and $w (\neq u)$ be another one. The one point function for any $X \subset \Lambda_L$ satisfies and two point functions for any $X,Y \subset \Lambda_L$ satisfy Also Duhamel function and truncated Duhamel function satisfy Multiple point functions satisfy corresponding extended formulae. Proof. Here, we prove these relations for a specific case $u=x$ and $w=z$ or $w=y$ f

Figures (1)

  • Figure 1: The gray region depicts $S^x \cup S^y \cup S^z$, which is predicted to be the spin glass region of the quantum Heisenberg XYZ spin glass model in sufficiently low temperature. All $S^u \ (u=x,y,z)$ are congruent square-based pyramids in this coordinate system of the coupling constant space, then the boundary of $S^x \cup S^y \cup S^z$ consists of triangles and squares. Solid and dashed straight lines represent convex and concave edges, respectively.

Theorems & Definitions (3)

  • Lemma 3.1
  • Theorem 4.1
  • Theorem 4.2