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On the thermodynamic limit of interacting fermions in the continuum

Oliver Siebert

TL;DR

This work develops a resolvent-algebra–inspired extension of the CAR framework to continuum interacting fermions, enabling a continuous Heisenberg time evolution in each $n$-particle sector and a globally invariant, norm-continuous $C^*$-dynamical system via time averaging. It first analyzes the particle-number–preserving subalgebra, identifying $n$-particle sectors with a coherent sequence of algebras $ rak K_n$ and an inverse-limit structure, proving an isomorphism with the extended CAR algebra and establishing sector coherence. The full extension to the complete CAR algebra is then constructed, with invariance of the extended algebra under dynamics, and the existence of a dense, norm-continuous subalgebra of analytic elements that supports a $C^*$-dynamical system. The framework is designed to support KMS-state construction for continuum interacting fermions by considering regularized models and taking suitable limits, offering a principled path to equilibrium states in the thermodynamic limit.

Abstract

We study the dynamics of non-relativistic fermions in $\mathbb R^d$ interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we identify an extension of the CAR algebra where the dynamics acts as a group of *-automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we identify a suitable dense subalgebra where the time evolution is also strongly continuous. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.

On the thermodynamic limit of interacting fermions in the continuum

TL;DR

This work develops a resolvent-algebra–inspired extension of the CAR framework to continuum interacting fermions, enabling a continuous Heisenberg time evolution in each -particle sector and a globally invariant, norm-continuous -dynamical system via time averaging. It first analyzes the particle-number–preserving subalgebra, identifying -particle sectors with a coherent sequence of algebras and an inverse-limit structure, proving an isomorphism with the extended CAR algebra and establishing sector coherence. The full extension to the complete CAR algebra is then constructed, with invariance of the extended algebra under dynamics, and the existence of a dense, norm-continuous subalgebra of analytic elements that supports a -dynamical system. The framework is designed to support KMS-state construction for continuum interacting fermions by considering regularized models and taking suitable limits, offering a principled path to equilibrium states in the thermodynamic limit.

Abstract

We study the dynamics of non-relativistic fermions in interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we identify an extension of the CAR algebra where the dynamics acts as a group of *-automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we identify a suitable dense subalgebra where the time evolution is also strongly continuous. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.
Paper Structure (5 sections, 21 theorems, 94 equations)

This paper contains 5 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Model and main results
  3. The structure of the particle number preserving CAR algebra
  4. Invariance under the dynamics for particle number preserving observables
  5. Full CAR algebra

Key Result

Theorem 2.1

The dynamics $(\alpha_t)_{t\in{\mathbb R}}$ leaves $\widehat{\mathfrak{A}}$ (resp. $\widehat{\mathfrak{A}}_0$) invariant and therefore forms a group of *-automorphisms on $\widehat{\mathfrak{A}}$ (resp. $\widehat{\mathfrak{A}}_0$). Furthermore, for each $A \in \widehat{\mathfrak{A}}$ (resp. $\wideha

Theorems & Definitions (45)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 35 more