arXiv subjects

Tadahiro Miyao

Publications and source records attributed to Tadahiro Miyao.

At least 19 recordsLinked to original sources

arXiv2026

The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram

We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux $b$, we construct an open region in the three-dimensional parameter space $(b,μ,λ)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane $λ=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2πσ^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

math-ph↗

arXiv2026

Hubbard--Heisenberg Thermodynamic Comparison at Half Filling in a Fixed Staggered Field

We study the repulsive Hubbard model at half filling in the strong-coupling regime, with a staggered magnetic field of strength $h$. The analysis is carried out in the canonical half-filled ensemble, with temperature measured on the Heisenberg scale $J_0(U)=4t^2/U$. Uniformly for $|h|\le h_0$ and $βJ_0(U)\ge \ell_0$, we prove finite-volume Hubbard--Heisenberg pressure estimates with errors uniform in the system size. These estimates pass to thermodynamic limits whenever the limiting pressures exist. The proof uses a strong-coupling unitary transformation which separates the singly occupied spin sector from sectors containing empty or doubly occupied sites. On the singly occupied sector, the effective Hamiltonian is compared with the Heisenberg reference Hamiltonian; the remaining sectors are controlled through a decomposition of the transformed partition function according to the set of empty or doubly occupied sites. For fixed positive staggered-field windows $I\Subset(0,h_0]$, the magnetisation comparison is then derived from the pressure comparison by convexity of finite-volume pressures. We also prove charge-sector suppression estimates, uniformly for $|h|\le h_0$: in the large positive-$U$ Heisenberg-scale regime, the density of empty or doubly occupied sites and the double-occupancy density are small, and the squared staggered charge divided by $|Λ|^2$ is small. Thus the results give a quantitative Gibbs-state formulation of the strong-coupling picture in which the half-filled repulsive Hubbard model is described, at the Heisenberg scale, by effective antiferromagnetic spin degrees of freedom, while charge fluctuations are suppressed.

math-ph↗

arXiv2025

Ordering of Energy Levels in the Fröhlich Model

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.

math-ph↗

arXiv2025

Near linearity of the macroscopic Hall current response in infinitely extended gapped fermion systems

We consider an infinitely extended system of fermions on a $d$-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a locally unique gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $\varepsilon$ is of order $\mathcal{O}(\varepsilon^\infty)$, i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $\varepsilon$ up to terms of order $\mathcal{O}(\varepsilon^\infty)$. The proportionality factor $σ_\mathrm{H}$ is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $d=2$ the relevant observable that "measures" the Hall conductivity in experiments, the Hall conductance, not only agrees with $σ_{\mathrm{H}}$ in expectation up to $\mathcal{O}(\varepsilon^\infty)$, but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.

math-ph↗

arXiv2024

Ground state of one-dimensional fermion-phonon systems

A new framework for the reflection positivity, based on order-preserving operator inequalities, is proposed. This framework is utilized to investigate one-dimensional fermion-phonon systems, with a particular focus on the detailed examination of ground state properties. Our analysis reveals that the reflection positivity provides a consistent description of the charge-density-wave (CDW) order in such systems. Additionally, we establish the existence of CDW long-range order in the ground state when the Coulomb interaction is long range.

math-ph↗

arXiv2024

Stability of charge density waves in electron-phonon systems

We demonstrate that electron-phonon interactions enhance the stability of charge density waves in low-temperature phases of many-electron systems. Our proof method involves an appropriate application of the Pirogov--Sinai theory to electron-phonon systems. Combining our findings with existing results, we obtain rigorous information regarding the low-temperature phase diagram for half-filled electron-phonon systems.

math-ph↗

arXiv2023

A finite temperature version of the Nagaoka--Thouless theorem in the $\mathrm{SU}(n)$ Hubbard model

The Aizenman--Lieb theorem for the $\mathrm{SU}(2)$ Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization $m(β, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(β, b)=\tanh(βb)$. In this manuscript, we present an extension of the Aizenman--Lieb theorem to the $\mathrm{SU}(n)$ Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.

math-ph↗

arXiv2023

Ground state properties of the periodic Anderson model with electron-phonon interactions

The periodic Anderson model (PAM) is a fundamental model describing heavy fermion systems. In this paper, we examine the PAM with electron-phonon interactions. Introducing a new analytical method based on operator inequalities, we prove that the ground state at half-filling is unique and a singlet. We also prove that the ground state exhibits short-range antiferromagnetism.

math-ph↗

arXiv2022

Rigorous analysis of the effects of electron-phonon interactions on magnetic properties in the one-electron Kondo lattice model

The Kondo lattice model (KLM) is a typical model describing heavy fermion systems. In this paper, we consider the interaction of phonons with the system described by the one-electron KLM. Magnetic properties of the ground state of this model are revealed in a rigorous form. Furthermore, we derive the effective Hamiltonian in the strong coupling limit ($J\to \infty$) for the strength of the spin-exchange interaction $J$; we examine the magnetic properties of the ground state of the effective Hamiltonian and prove that the Aizenman--Lieb theorem concerning the magnetization holds for the effective Hamiltonian at finite temperatures. Generalizing the obtained results, we clarify a mechanism for the stability of magnetic properties of the ground state in the one-electron KLM system.

math-ph↗

arXiv2021

An algebraic approach to revealing magnetic structures of ground states in many-electron systems

Mathematical understanding of the origin of ferromagnetism is still incomplete and remains an important research topic in mathematical physics. In this paper, we give a model-independent mathematical framework describing the magnetic features of the ground states in many-electron systems. Within this framework, we also present a new approach to understanding magnetic orders in macroscopic systems. Based on these, we construct a general theory that explains the stability of magnetic orders in the ground states despite the interaction of electrons with the environment. Methodologically, the theory presented in this paper is formulated using von Neumann algebras and their associated standard forms. A benefit of working in such an algebraic setting is that we can define operator inequalities that preserve the ordered structures that naturally follow from the standard forms; by exploiting these operator inequalities, we can develop new descriptions of the magnetic structures of the ground states. As specific applications of the proposed theory, we first analyze the Marshall--Lieb--Mattis theorem, Lieb's theorem, and their stabilities under various perturbations. Next, the Nagaoka--Thouless theorem and its stability are addressed. In addition, we interpret various other examples from the new theory and give a unified perspective on existing results.

math-ph↗

arXiv2021

Electron-phonon interaction in Kondo lattice systems

We study ground state properties of the Kondo lattice model with an electron-phonon interaction. The ground state is proved to be unique; in addition, the total spin of the ground state is determined according to the lattice structure. To prove the assertions, an extension of the method of spin reflection positivity is given in terms of order preserving operator inequalities.

math-ph↗

arXiv2021

On renormalized Hamiltonian nets

We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as follows:(i) in contrast to the probabilistic approach in the Schrödinger representation, our method works well in the Fock representation; (ii) the method covers the massless case.

math-ph↗

arXiv2020

Thermal stability of the Nagaoka-Thouless theorems

We prove that the Aizenman-Lieb theorem on ferromagnetism in the Hubbard model holds true even if the electron-phonon interactions and the electron-photon interactions are taken into account. Our proof is based on path integral representations of the partition functions.

math-ph↗

arXiv2020

Note on the Retarded van der Waals Potential within the Dipole Approximation

We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as $R^{-7}$, provided that the distance between atoms $R$ is sufficiently large. We employ the Feynman's representation of the quantized radiation fields which enables us to diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture by Casimir and Polder.

math-ph↗

arXiv2019

Correlation inequalities for Schrödinger operators

This paper analyzes Schödinger operators from viewpoint of correlation inequalities. We construct Griffiths inequalities for the ground state expectations by applying operator-theoretic correlation inequalities. As an example of such an application, we analyze the momentum distribution, i.e., the Fourier transform of the ground state density.

math-ph↗

arXiv2019

Stability of good quantum numbers in ground states

Let $H$ be a self-adjoint operator, bounded from below and let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Suppose that (i) $E(H)=\inf \mathrm{spec}(H)$ is a simple eigenvalue, and (ii) $H$ strongly commutes with $O$. Let $ψ_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $ψ_H$ is an eigenvector of $O$: $Oψ_H=μ(H)ψ_H$. In the context of quantum mechanics, $μ(H)$ is called a good quantum number. In this note, we examine the stability of $μ(H)$ under perturbations of $H$ from a viewpoint of the order theory.

math.FA↗

arXiv2019

Stability of ferromagnetism in many-electron systems

We construct a model-independent framework describing stabilities of ferromagnetism in strongly correlated electron systems; Our description relies on the operator theoretic correlation inequalities. Within the new framework, we reinterpret the Marshall-Lieb-Mattis theorem and Lieb\rq{}s theorem; in addition, from the new perspective, we prove that Lieb\rq{}s theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. We also examine the Nagaoka-Thouless theorem and its stabilities. These examples verify the effectiveness of our new viewpoint.

math-ph↗

arXiv2018

On the semigroup generated by the renormalized Nelson Hamiltonian

Let us consider the renormalized Nelson model at a fixed total momentum $P$: $H_{\mathrm{ren}}(P)$; The Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{-βH_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in \mathbb{R}^3$ and $β>0$ in the Fock representation.

math-ph↗