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Ordering of Energy Levels in the Fröhlich Model

Fumio Hiroshima, Akihiro Kobayashi, Tadahiro Miyao, Shunsuke Tomioka

TL;DR

This work extends the Mattis–Lieb ordering of energy levels from one-dimensional Schrödinger systems to the Fröhlich model describing electron–phonon interactions in a continuous 1D setting without an ultraviolet cutoff. By developing a Feynman–Kac-type representation for the Fröhlich Hamiltonian and exploiting the Schrödinger representation, the authors establish existence and uniqueness of ground states in each total-spin sector, alongside a strict energy ordering E(|M|) with E(N/2) > E(N/2-1) > ... and the corresponding ground-state non-ferromagnetic tendency. They show that electron–phonon coupling lowers the ground-state energy across all spin sectors and that the ordering persists in the UV-cutoff-free limit, with rigorous path-integral control via positivity-improving semigroups and hypercontractivity. The results provide a robust mathematical foundation for non-magnetic tendencies in 1D many-electron systems with phonons and offer a framework for further exploration of ground-state properties in polarons and related models, including potential finite-temperature extensions and quantum-field interactions.

Abstract

Consider a one-dimensional system of \( N \) electrons subject to an external potential \( U \). Let \( E_{\rm el}(S) \) denote the ground state energy of the system with total spin \( S \). The Mattis--Lieb theorem asserts that, for a broad class of potentials \( U \), the inequality \( E_{\rm el}(S) < E_{\rm el}(S') \) holds whenever \( S < S' \). This result implies that the ground state of a one-dimensional many-electron system is non-ferromagnetic. In the present work, we demonstrate that the Mattis--Lieb theorem can be extended to electron-phonon interacting systems governed by the Fröhlich model. Our analysis is carried out in the setting without an ultraviolet cutoff. The cornerstone of our approach is the construction of a Feynman--Kac-type formula for the heat semigroup generated by the Fröhlich Hamiltonian.

Ordering of Energy Levels in the Fröhlich Model

TL;DR

This work extends the Mattis–Lieb ordering of energy levels from one-dimensional Schrödinger systems to the Fröhlich model describing electron–phonon interactions in a continuous 1D setting without an ultraviolet cutoff. By developing a Feynman–Kac-type representation for the Fröhlich Hamiltonian and exploiting the Schrödinger representation, the authors establish existence and uniqueness of ground states in each total-spin sector, alongside a strict energy ordering E(|M|) with E(N/2) > E(N/2-1) > ... and the corresponding ground-state non-ferromagnetic tendency. They show that electron–phonon coupling lowers the ground-state energy across all spin sectors and that the ordering persists in the UV-cutoff-free limit, with rigorous path-integral control via positivity-improving semigroups and hypercontractivity. The results provide a robust mathematical foundation for non-magnetic tendencies in 1D many-electron systems with phonons and offer a framework for further exploration of ground-state properties in polarons and related models, including potential finite-temperature extensions and quantum-field interactions.

Abstract

Consider a one-dimensional system of electrons subject to an external potential . Let \( E_{\rm el}(S) \) denote the ground state energy of the system with total spin . The Mattis--Lieb theorem asserts that, for a broad class of potentials , the inequality \( E_{\rm el}(S) < E_{\rm el}(S') \) holds whenever . This result implies that the ground state of a one-dimensional many-electron system is non-ferromagnetic. In the present work, we demonstrate that the Mattis--Lieb theorem can be extended to electron-phonon interacting systems governed by the Fröhlich model. Our analysis is carried out in the setting without an ultraviolet cutoff. The cornerstone of our approach is the construction of a Feynman--Kac-type formula for the heat semigroup generated by the Fröhlich Hamiltonian.
Paper Structure (27 sections, 34 theorems, 190 equations)

This paper contains 27 sections, 34 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Previous Studies
  4. Ordering of Energy Levels in the Fröhlich Model
  5. Structure of $M$-subspaces
  6. Preliminaries
  7. Definition of $\mathfrak{K}_{{\rm as}, p}$
  8. Representation of Vectors in the $M$-subspace
  9. Structure of $\mathfrak{K}_{{\rm as}, p}$
  10. Characteristics of the Ground State
  11. Proof of Theorem \ref{['EnOrd']}
  12. Outline of the Proof
  13. Proof of Theorem \ref{['AbstEO']}
  14. Path Integral Representation of $\mathrm{e}^{-\beta H_{\rm F} \restriction \mathfrak{K}_{D^{N, (p)}}}$
  15. Main Theorem of Section \ref{['FKIF']}
  16. ...and 12 more sections

Key Result

Theorem 2.2

For each $S \in \mathbb{S}_N$, define $E_{\rm el}(S) \stackrel{\mathrm{def}}{=} \inf \mathrm{spec}(H_{\rm el} \restriction \mathfrak{E}^S).$ Then, the following holds: Here, the minimum value in the above inequality corresponds to $E_{\rm el}(0)$ when $N$ is even, and $E_{\rm el}(1/2)$ when $N$ is odd.

Theorems & Definitions (59)

  • Remark 2.1
  • Theorem 2.2: Mattis--Lieb Lieb1962
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 3.1
  • Remark 3.2
  • proof
  • Corollary 3.3
  • ...and 49 more