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The Euclidean $φ^4_2$ theory as a limit of an inhomogeneous Bose gas

Cristina Caraci, Antti Knowles, Alessio Ranallo, Pedro Torres Giesteira

Abstract

We prove that the grand canonical Gibbs state of an interacting two-dimensional quantum Bose gas confined by a trapping potential converges to the complex Euclidean field theory with local quartic self-interaction, when the density of the gas becomes large and the range of the interaction becomes small. We obtain convergence of the relative partition function and convergence in $L^1 \cap L^\infty$ of the renormalised reduced density matrices. The field theory is supported on distributions of negative regularity, which requires a renormalisation by divergent mass and energy counterterms. Unlike previous results in the homogeneous setting of the torus without a trapping potential, the counterterms are not given by a finite collection of scalars but by diverging counterterm functions. This leads to significant new mathematical challenges. For our proof, we also derive quantitative bounds on the Green function of Schrödinger operators and of its gradient, which might be of independent interest.

The Euclidean $φ^4_2$ theory as a limit of an inhomogeneous Bose gas

Abstract

We prove that the grand canonical Gibbs state of an interacting two-dimensional quantum Bose gas confined by a trapping potential converges to the complex Euclidean field theory with local quartic self-interaction, when the density of the gas becomes large and the range of the interaction becomes small. We obtain convergence of the relative partition function and convergence in of the renormalised reduced density matrices. The field theory is supported on distributions of negative regularity, which requires a renormalisation by divergent mass and energy counterterms. Unlike previous results in the homogeneous setting of the torus without a trapping potential, the counterterms are not given by a finite collection of scalars but by diverging counterterm functions. This leads to significant new mathematical challenges. For our proof, we also derive quantitative bounds on the Green function of Schrödinger operators and of its gradient, which might be of independent interest.
Paper Structure (23 sections, 51 theorems, 170 equations)

This paper contains 23 sections, 51 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. The one-particle Hamiltonian
  4. Classical field theory
  5. Quantum many-body system
  6. Results
  7. Strategy of the proof
  8. Proof of Proposition \ref{['prop:convergence classical field theories']}
  9. Partition function
  10. Correlation functions
  11. Functional integral representations
  12. Proof of Proposition \ref{['Prop:regularisation of euclidean field theory']}
  13. Brownian paths
  14. Proof of Proposition \ref{['Prop:regularisation of phi42 quantum rel part funct']}
  15. Proof of Proposition \ref{['prop:convergence quantum many body system and Weps']}
  16. ...and 8 more sections

Key Result

Theorem 2.6

Let $d =2$ and $\theta >10$. Let $U$ satisfy Assumption assumption:P and D for U, and $H$ be the Hamiltonian eq:WickHamiltonian. Suppose that Assumption assumptions: non local interaction holds and $\varepsilon\equiv \varepsilon(\nu)$ satisfies for some $c>0$. Then as $\varepsilon, \nu \to 0$ we have the convergence of the partition function and of the Wick-ordered correlation functions for all

Theorems & Definitions (69)

  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Remark 2.9
  • Remark 2.10
  • Remark 2.12
  • Remark 2.13
  • ...and 59 more