Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subcriticality at High Temperatures in Spin Lattice Systems

Nicolò Drago, Lorenzo Pettinari, Christiaan J. F. van de Ven

TL;DR

This work establishes dimension-uniform sufficient conditions for subcriticality of quantum and classical spin lattice systems by proving a novel quantum KMS state uniqueness result. The approach blends a non-commutative Kirkwood-Salzburg framework with an eta-free decomposition of local observables, and handles non-commuting single-site and multilocal potentials via a Dyson-series-based dynamics and robust norm bounds. The main theorem delivers a subcritical inverse temperature $\beta_u$ independent of the single-site dimension, and the analysis extends to classical KMS states with a parallel bound, yielding a common subcritical region under strict deformation quantization. Collectively, the results broaden the class of admissible interactions, avoid growth-derivative assumptions, and bridge quantum-classical behavior at high temperatures with practical implications for the absence of phase transitions in a broad, physically relevant setting.

Abstract

We provide new sufficient conditions for subcriticality of classical and quantum spin lattice systems, formulated in terms of the uniqueness of Kubo-Martin-Schwinger (KMS) states. This is achieved by exploiting a non-commutative analog of the Kirkwood-Salzburg equations together with a novel decomposition of local observables. In contrast to standard approaches \cite{Bratteli_Robinson_97,Frohlich_Ueltschi_2015}, our condition is uniform with respect to the dimension of the single-site Hilbert space. Moreover, unlike the results of \cite{Drago_Pettinari_Van_de_Ven_2025}, which required control over the growth of the derivatives of the interaction potentials, our result only involves estimating the natural $C^*$-norm of these potentials. This substantially enlarges the class of interactions for which the theorems apply and provides better lower bounds on the subcritical inverse temperature. Finally, our results are flexible enough to cover situations where no assumptions are imposed on the single-site potentials.

Subcriticality at High Temperatures in Spin Lattice Systems

TL;DR

This work establishes dimension-uniform sufficient conditions for subcriticality of quantum and classical spin lattice systems by proving a novel quantum KMS state uniqueness result. The approach blends a non-commutative Kirkwood-Salzburg framework with an eta-free decomposition of local observables, and handles non-commuting single-site and multilocal potentials via a Dyson-series-based dynamics and robust norm bounds. The main theorem delivers a subcritical inverse temperature independent of the single-site dimension, and the analysis extends to classical KMS states with a parallel bound, yielding a common subcritical region under strict deformation quantization. Collectively, the results broaden the class of admissible interactions, avoid growth-derivative assumptions, and bridge quantum-classical behavior at high temperatures with practical implications for the absence of phase transitions in a broad, physically relevant setting.

Abstract

We provide new sufficient conditions for subcriticality of classical and quantum spin lattice systems, formulated in terms of the uniqueness of Kubo-Martin-Schwinger (KMS) states. This is achieved by exploiting a non-commutative analog of the Kirkwood-Salzburg equations together with a novel decomposition of local observables. In contrast to standard approaches \cite{Bratteli_Robinson_97,Frohlich_Ueltschi_2015}, our condition is uniform with respect to the dimension of the single-site Hilbert space. Moreover, unlike the results of \cite{Drago_Pettinari_Van_de_Ven_2025}, which required control over the growth of the derivatives of the interaction potentials, our result only involves estimating the natural -norm of these potentials. This substantially enlarges the class of interactions for which the theorems apply and provides better lower bounds on the subcritical inverse temperature. Finally, our results are flexible enough to cover situations where no assumptions are imposed on the single-site potentials.

Paper Structure

This paper contains 13 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Data availability statement.
  4. Conflict of interest statement.
  5. Uniqueness result for quantum KMS states
  6. Quantum quasi-local algebra
  7. Dynamics for non-commuting $\Psi,\bar{\Phi}$
  8. Decomposition in $\eta$-free elements
  9. Proof of Theorem \ref{['Thm: uniqueness for KMS quantum states']}
  10. Consequences of Theorem \ref{['Thm: uniqueness for KMS quantum states']}
  11. Comparison of the subcritical inverse temperatures
  12. Uniqueness result for classical KMS states
  13. Common subcritical region for quantum and classical systems

Key Result

Theorem 1.1

Let $\beta>0$ be such that Then there exists a unique $(\beta,\tau^\Gamma)$-KMS state on $\mathcal{A}_\Gamma$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 8 more