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Quantum Mean-Fields Spin Systems in a Random External Field

Chokri Manai

TL;DR

The paper develops a non-commutative large-deviation framework to analyze disordered quantum mean-field spin systems with a random external field, deriving a explicit variational formula for the limiting quenched free energy $p(V,\mathfrak{b}) = \sup_{\mathbf{m} \in B_{\mathbb{R}^3}} ( V(\mathbf{m}) - \Lambda_{\mathfrak{b}}^{*}(\mathbf{m}) )$. The main methodological pillars are Bloch coherent states and Berezin–Lieb bounds to connect quantum Hamiltonians with classical symbols, a Gibbs variational principle for sharp lower bounds, and a microcanonical analysis complemented by Gaussian perturbations to control thermal fluctuations and obtain sharp upper bounds. The authors demonstrate the result for polynomial mean-field Hamiltonians, with a Toland–Singer duality enabling a clean variational form, and extend the argument to general continuous symbols $V$ via Weierstrass approximation. The approach not only resolves the disordered quantum Curie–Weiss-type models but also provides a flexible framework potentially applicable to multi-species mean-field Hamiltonians and other disordered quantum systems, linking non-commutative analysis with large-deviation techniques.

Abstract

In this work, we consider general exchangeable quantum mean-field Hamiltonian such as the prominent quantum Curie-Weiss model under the influence of a random external field. Despite being arguably the simplest class of disordered quantum systems, the random external field breaks the symmetry of the mean-field Hamiltonian and hence standard quantum de Finetti type or semiclassical arguments are not directly applicable. We introduce a novel strategy in this context, which can be seen as non-commutative large deviation analysis, allowing us to characterize the limiting free energy in terms of a simple and explicit variational formula. The proposed method is general enough to be used for other classes of mean-field models such as multi species Hamiltonians.

Quantum Mean-Fields Spin Systems in a Random External Field

TL;DR

The paper develops a non-commutative large-deviation framework to analyze disordered quantum mean-field spin systems with a random external field, deriving a explicit variational formula for the limiting quenched free energy . The main methodological pillars are Bloch coherent states and Berezin–Lieb bounds to connect quantum Hamiltonians with classical symbols, a Gibbs variational principle for sharp lower bounds, and a microcanonical analysis complemented by Gaussian perturbations to control thermal fluctuations and obtain sharp upper bounds. The authors demonstrate the result for polynomial mean-field Hamiltonians, with a Toland–Singer duality enabling a clean variational form, and extend the argument to general continuous symbols via Weierstrass approximation. The approach not only resolves the disordered quantum Curie–Weiss-type models but also provides a flexible framework potentially applicable to multi-species mean-field Hamiltonians and other disordered quantum systems, linking non-commutative analysis with large-deviation techniques.

Abstract

In this work, we consider general exchangeable quantum mean-field Hamiltonian such as the prominent quantum Curie-Weiss model under the influence of a random external field. Despite being arguably the simplest class of disordered quantum systems, the random external field breaks the symmetry of the mean-field Hamiltonian and hence standard quantum de Finetti type or semiclassical arguments are not directly applicable. We introduce a novel strategy in this context, which can be seen as non-commutative large deviation analysis, allowing us to characterize the limiting free energy in terms of a simple and explicit variational formula. The proposed method is general enough to be used for other classes of mean-field models such as multi species Hamiltonians.
Paper Structure (9 sections, 15 theorems, 83 equations)

This paper contains 9 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Coherent States and the Hamiltonian for general $V$
  3. Proof of Theorem \ref{['thm:main']} via a Non-Commutative Microcanonical Approach
  4. Sharp Lower Bound by Gibbs Variational Principle
  5. Control of Thermal Fluctuations by Random Perturbations
  6. Negative Definite Quadratic Models via Linearization
  7. Upper Bound via a Microcanonical Analysis
  8. Conclusion
  9. Supplementary Proofs

Key Result

Theorem 1.1

Let $V : B_{{\mathbb{R}}^3} \to {\mathbb{R}}$ be a continuous function on the three-dimensional unit ball, $\mathfrak{b}$ an integrable (i.e. an $L^1$-)vector-valued random variable and $H_{V, \mathfrak{b}}$ the corresponding family of self-adjoint random Hamiltonians defined on the $N$-particle Hil In fact, if the random variables $p_N(V,\mathfrak{b})$ are defined on a common probability space, t

Theorems & Definitions (29)

  • Theorem 1.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop:lower']}
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 19 more