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Asymptotic behavior of the ground state energy of a Fermionic Fröhlich multipolaron in the strong coupling limit

Ioannis Anapolitanos, Michael Hott

TL;DR

This work establishes the leading strong-coupling behavior of the ground-state energy for a fermionic Fröhlich multipolaron by showing it is governed by the Pekar–Tomasevich functional with fermionic statistics, i.e. $E^{(N,\alpha)} \approx E_{PT}^{(N,\alpha)}$ with $E_{PT}^{(N,\alpha)} = \alpha^2 E_{PT}^{(N,1)}$. The authors develop a localization-based framework to incorporate fermionic exchange and to handle general external fields, avoiding a previously used verifiable energy assumption. The analysis proceeds via a sequence of reductions: localizing electrons into well-separated clusters, localizing phonons, applying an ultraviolet cutoff and partitioning phonons into block modes, and then invoking Pekar’s variational argument on the block-model to bound the full energy from below by the PT energy plus a rigorously controlled error of order $N^{82/30}\alpha^{42/23}$. The results thus confirm that in the strong-coupling limit the multipolaron energy is captured by an effective Pekar–Tomasevich energy, with the leading term scaling as $\alpha^2$, and provide explicit control of the subleading error terms and the dependence on external fields.

Abstract

In this article, we investigate the asymptotic behavior of the ground state energy of the Fröhlich Hamiltonian for a fermionic multipolaron in the strong coupling limit. We prove that it is given, to leading order, by the ground state energy of the Pekar-Tomasevich functional with fermionic statistics, a much simpler model. Our analysis builds upon \cite{wellig}, which itself used and generalized methods developed in \cite{liebthomas}, \cite{frank-lieb-seiringer-thomas-stability-absence} and \cite{griesemer-wellig-strong-polaron-static-fields}. Our main new contribution is two-fold. First, we take into account the fermionic statistics of the multipolaron, by employing a localization method used in \cite{liebloss}. Second, we relax an assumption on the external electric and magnetic fields, which is not easily verifiable, unless the fields are periodic. Instead, we allow for general fields that only ensure self-adjointness of the Fröhlich Hamiltonian.

Asymptotic behavior of the ground state energy of a Fermionic Fröhlich multipolaron in the strong coupling limit

TL;DR

This work establishes the leading strong-coupling behavior of the ground-state energy for a fermionic Fröhlich multipolaron by showing it is governed by the Pekar–Tomasevich functional with fermionic statistics, i.e. with . The authors develop a localization-based framework to incorporate fermionic exchange and to handle general external fields, avoiding a previously used verifiable energy assumption. The analysis proceeds via a sequence of reductions: localizing electrons into well-separated clusters, localizing phonons, applying an ultraviolet cutoff and partitioning phonons into block modes, and then invoking Pekar’s variational argument on the block-model to bound the full energy from below by the PT energy plus a rigorously controlled error of order . The results thus confirm that in the strong-coupling limit the multipolaron energy is captured by an effective Pekar–Tomasevich energy, with the leading term scaling as , and provide explicit control of the subleading error terms and the dependence on external fields.

Abstract

In this article, we investigate the asymptotic behavior of the ground state energy of the Fröhlich Hamiltonian for a fermionic multipolaron in the strong coupling limit. We prove that it is given, to leading order, by the ground state energy of the Pekar-Tomasevich functional with fermionic statistics, a much simpler model. Our analysis builds upon \cite{wellig}, which itself used and generalized methods developed in \cite{liebthomas}, \cite{frank-lieb-seiringer-thomas-stability-absence} and \cite{griesemer-wellig-strong-polaron-static-fields}. Our main new contribution is two-fold. First, we take into account the fermionic statistics of the multipolaron, by employing a localization method used in \cite{liebloss}. Second, we relax an assumption on the external electric and magnetic fields, which is not easily verifiable, unless the fields are periodic. Instead, we allow for general fields that only ensure self-adjointness of the Fröhlich Hamiltonian.

Paper Structure

This paper contains 15 sections, 11 theorems, 126 equations, 2 figures.

Table of Contents

  1. Statements and Declarations
  2. Data availablity
  3. Acknowledgments
  4. Introduction
  5. Background
  6. Further directions
  7. Polarons in the continuum limit: The Fröhlich model
  8. Statement of main result: Strong-coupling limit
  9. Sketch of the proof of Theorem \ref{['thm:mainthm']}
  10. Localization of the Fröhlich multipolaron
  11. UV cut-off and partition into block modes
  12. Proof of main result
  13. Derivation of the Pekar-Tomasevich functional
  14. Localization results
  15. Phonon integration

Key Result

Theorem 1.1

Let $V$ be relatively $-\Delta$-form-bounded with relative form bound $0$, $A_k\in L^2_{\mathrm{loc}}(\mathbb{R}^3)$, $k=1,2,3$, and $\nu\in\mathbb{R}$. Then there exists a constant $C\in\mathbb{R}$ such that for any $N\in\mathbb{N}$, and $\alpha\geq1$ we have that

Figures (2)

  • Figure 1: Ionic crystal with and without traveling test charge
  • Figure 2: Definition of phonon modes

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4: Dependence of constant $C$
  • Remark 1.5: $N$-dependence without external fields
  • Remark 1.6: Binding
  • Remark 1.7: Vanishing of non-rescaled fields
  • Lemma 2.1: Covering lemma
  • proof
  • Lemma 2.2
  • ...and 21 more