Ordering of Energy Levels in the Fröhlich Model

2025/05/11 by Fumio Hiroshima, Hiroshima, Fumio, Akihiro Kobayashi +5
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum and electron transport phenomena #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.07152

openalex publication_date 2025/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Fröhlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass
• The Fröhlich polaron at strong coupling: Part II — Energy-momentum relation and effective mass
• A functional central limit theorem for Polaron path measures
• On the semigroup generated by the renormalized Nelson Hamiltonian
• Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration
• Exact Solution of a Many-Fermion System and Its Associated Boson Field
• Electrons in lattice fields

Related

• Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit
• Energy of a many-electron system in an ensemble ground-state, versus electron number and spin: piecewise-linearity and flat plane condition generalized
• Ferromagnetic Lieb-Mattis Theorem
• The Lieb--Thomas strategy for strongly coupled fermionic multipolarons with general external fields
• On the ground state of a completely filled lowest Landau level in two dimensions