Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Parisi Formula for Quantum Spin Glasses

Chokri Manai, Simone Warzel

Abstract

We establish three equivalent versions of a Parisi formula for the free energy of mean-field spin glasses in a transversal magnetic field. These results are derived from available results for classical vector spin glasses by an approximation method using the functional integral representation of the partition function. In this approach, the order parameter is a non-decreasing function with values in the non-negative, real hermitian Hilbert-Schmidt operators. For the quantum Sherrington-Kirkpatrick model, we also show that under the assumption of self-averaging of the self-overlap, the optimising Parisi order parameter is found within a two-dimensional subspace spanned by the self-overlap and the fully stationary overlap.

A Parisi Formula for Quantum Spin Glasses

Abstract

We establish three equivalent versions of a Parisi formula for the free energy of mean-field spin glasses in a transversal magnetic field. These results are derived from available results for classical vector spin glasses by an approximation method using the functional integral representation of the partition function. In this approach, the order parameter is a non-decreasing function with values in the non-negative, real hermitian Hilbert-Schmidt operators. For the quantum Sherrington-Kirkpatrick model, we also show that under the assumption of self-averaging of the self-overlap, the optimising Parisi order parameter is found within a two-dimensional subspace spanned by the self-overlap and the fully stationary overlap.
Paper Structure (25 sections, 26 theorems, 194 equations)

This paper contains 25 sections, 26 theorems, 194 equations.

Table of Contents

  1. Setting the stage
  2. Models and assumptions
  3. Functional integral
  4. Space of asymptotic overlap operators
  5. Correlation functional
  6. Quantum Parisi functional
  7. Main results
  8. Quantum Parisi formula
  9. Self-overlap corrected partition function
  10. Structure of the order parameter and conjectures
  11. Approximation results
  12. Paths space and functional integral
  13. Partition functions
  14. Correlation functional
  15. Quantum Parisi functional
  16. ...and 10 more sections

Key Result

Lemma 1.2

Theorems & Definitions (52)

  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • ...and 42 more