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Large deviations in mean-field quantum spin systems

Matthias Keller, Christiaan J. F. van de Ven

Abstract

Continuous fields (or bundles) of $C^*$-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^*$-bundle generated by increasing symmetric tensor powers of the complex $\ell\times\ell$ matrices $M_\ell(\mathbb{C})$, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.

Large deviations in mean-field quantum spin systems

Abstract

Continuous fields (or bundles) of -algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous -bundle generated by increasing symmetric tensor powers of the complex matrices , which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.

Paper Structure

This paper contains 12 sections, 6 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Continuous $C^*$-bundles and $C^*$-algebraic deformation quantization
  3. Principle of large deviations
  4. Large deviations in statistical mechanics
  5. Mean-field quantum spin systems in algebraic quantum theory
  6. Mean-field theories
  7. Strict deformation quantization of Poisson manifolds
  8. Logarithmic moment generating function and local Gibbs states
  9. A Baker-Campbell-Hausdorff approach
  10. Method of majorants
  11. A "classical" counterpart of the BCH formula
  12. Large deviation upper bounds

Key Result

Theorem 3

Let $\delta>0$ be given as above. The series $Z(1)=\sum_{n=1}^\infty z(n)$ converges absolutely for all $a,b \in \mathfrak{L}:=\{c \in M_\ell(\mathbb{C})\ | \ \|c\|< \delta/4\}$, its sum defines an analytic map from $\mathfrak{L}\times\mathfrak{L}$ into $M_\ell(\mathbb{C})$, and $e^{a}e^{b}=e^{Z}$.

Theorems & Definitions (19)

  • Definition 1: LDP
  • Remark
  • Remark
  • Example 2
  • Remark
  • Remark
  • Theorem 3: Theorem 2.15.4 in Varadarajan
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 9 more