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$Φ^4_2$ theory limit of a many-body bosonic free energy

Lucas Jougla, Nicolas Rougerie

TL;DR

This work proves that the relative free energy of a 2D interacting Bose gas, in a regime where classical field theory governs the leading behavior, converges to the free energy of the $\Phi^4_2$ measure after Wick renormalization. By a variational approach that couples quantum Gibbs states to classical Hartree-Gibbs measures and then to the $\Phi^4_2$ theory, the authors handle a polynomially shrinking interaction width $\varepsilon$ and establish convergence under precise scaling constraints $\varepsilon=\lambda^\eta$ with $\eta\in(0,1/24)$. They derive strong a priori estimates, a high-momentum correlation bound via a Deift–Knowles–Nam–Zhu–Sohinger–type inequality, and perform a careful finite-dimensional semiclassical reduction using lower-symbol de Finetti measures and Berezin-Lieb inequalities. The result extends prior logarithmic-rate findings to a polynomial-rate regime, bridging quantum many-body Gibbs states and the renormalized $\Phi^4_2$ classical measure on $\mathbb{T}^2$, with potential implications for deriving effective classical field theories from quantum many-body dynamics in low dimensions.

Abstract

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the $Φ^4_2$ non-linear Schr{ö}dinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the $Φ^4_2$ free energy then follows from arguments of Fr{ö}hlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.

$Φ^4_2$ theory limit of a many-body bosonic free energy

TL;DR

This work proves that the relative free energy of a 2D interacting Bose gas, in a regime where classical field theory governs the leading behavior, converges to the free energy of the measure after Wick renormalization. By a variational approach that couples quantum Gibbs states to classical Hartree-Gibbs measures and then to the theory, the authors handle a polynomially shrinking interaction width and establish convergence under precise scaling constraints with . They derive strong a priori estimates, a high-momentum correlation bound via a Deift–Knowles–Nam–Zhu–Sohinger–type inequality, and perform a careful finite-dimensional semiclassical reduction using lower-symbol de Finetti measures and Berezin-Lieb inequalities. The result extends prior logarithmic-rate findings to a polynomial-rate regime, bridging quantum many-body Gibbs states and the renormalized classical measure on , with potential implications for deriving effective classical field theories from quantum many-body dynamics in low dimensions.

Abstract

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the non-linear Schr{ö}dinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the free energy then follows from arguments of Fr{ö}hlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.

Paper Structure

This paper contains 15 sections, 15 theorems, 354 equations.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Quantum and Classical models
  4. Quantum / classical correspondence
  5. Main result
  6. Variational proof strategy
  7. A priori estimates for the relative free energy
  8. Correlation estimate for high momenta
  9. Convergence of the relative free energy
  10. Free energy lower bound
  11. Free energy upper bound
  12. Proof of the main result
  13. Lower symbol for the free Gibbs state
  14. Berezin-Lieb inequalities on the Fock space
  15. Regularization of the $\Phi^4_2$ measure by convolution

Key Result

Theorem 2.1

Let $\lambda > 0$ and $\eta \in (0,1/24)$. Set $\varepsilon = \lambda^\eta$. Consider the renormalized interaction of the $\Phi^4_2$ theory which is well-defined as a $L^2(\textnormal{d}\mu_0)$-limit of its truncated counterpart. Define the classical partition function Then, setting the parameters of eqhamiltonian with fourier as in eq:renorm param, we have as $\lambda \rightarrow 0^+$,

Theorems & Definitions (29)

  • Theorem 2.1: Convergence of the relative free-energy
  • Lemma 3.1: A-priori estimates for the relative free energy
  • proof
  • Theorem 4.1: Correlation estimate for high momenta
  • proof
  • Definition 5.1: $P$-localization of a state
  • Definition 5.2: Bosonic coherent states
  • Definition 5.3: Lower symbol/Husimi function
  • Theorem 5.4: Lower symbols as de Finetti measures, LewNamRou-15
  • Theorem 5.5: Relative entropyquantum to classical, LewNamRou-15
  • ...and 19 more