Search

The Perron--Frobenius theorem in infinite-dimensional Hilbert spaces can be breifly stated as follows: Given a Hilbert cone in a real Hilbert space, a bounded positive self-adjoint operator $A$ is ergodic with respect to this cone if and only if the maximum eigenvalue $\|A\|$ of $A$ is simple, and the corresponding eigenvector is strictly positive with respect to this cone. This paper addresses the inverse problem of the Perron--Frobenius theorem: Does there exist a Hilbert cone such that a given bounded positive self-adjoint operator $A$ becomes ergodic when its maximum eigenvalue $\|A\|$ is simple? We provide an affirmative answer to this question in this paper. Furthermore, we conduct a detailed analysis of a specialized Hilbert cone introduced to obtain this result. Additionally, we provide an illustrative example of an application of the obtained results to the heat semigroup generated by the magnetic Schr\"odinger operator

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., Comment: This version corrects the definition of the order parameter and the assertion in Theorem 1.1

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., Comment: Minor revision

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., Comment: v2: minor corrections

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 $m(\beta, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(\beta, b)=\tanh(\beta b)$. 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., Comment: Fixed errors in previous version and improved presentation

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., Comment: Minor revision

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.

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., Comment: Minor revision: 34 pages, 2 figures

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., Comment: Improved version, 44 pages

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 $\psi_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $\psi_H$ is an eigenvector of $O$: $O\psi_H=\mu(H)\psi_H$. In the context of quantum mechanics, $\mu(H)$ is called a good quantum number. In this note, we examine the stability of $\mu(H)$ under perturbations of $H$ from a viewpoint of the order theory., Comment: Improved version

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.

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 Schr\"odinger representation, our method works well in the Fock representation; (ii) the method covers the massless case., Comment: 34 pages, minor revision

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 $e^{-\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in \mathbb{R}^3$ and $\beta >0$ in the Fock representation., Comment: V2: Revised version

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., Comment: Ver3: Title changed. Improved version

Nagaoka's theorem on ferromagnetism in the Hubbard model is extended to the Holstein-Hubbard model. This shows that Nagaoka's ferromagnetism is stable even if the electron-phonon interaction is taken into account. We also prove that Nagaoka's ferromagnetism is stable under the influence of the quantized radiation field., Comment: 23 pages. V3: Improved version, to appear in Annales Henri Poincare

We study the ground state properties of the Holstein-Hubbard model on some bipartite lattices at half-filling; The ground state is proved to exhibit ferrimagnetism whenever the electron-phonon interaction is not so strong. In addition, the antiferromagnetic long range order is shown to exist in the ground state. In contrast to this, we prove the absence of the long range charge order., Comment: V3: Improved version, to appear in Annales Henri Poincare

This paper analyzes Sch\"odinger 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., Comment: Improved version, to appear in Mathematical Physics, Analysis and Geometry

We rigorously investigated the charge-charge correlation function of the ionic Hubbard model in two dimensions by reflection positivity. We prove the existence of charge-density waves for large staggered potential $\Delta$ (i.e., $\frac{\Delta}{2}+2V>U$) at low temperatures, where $U$ and $V$ are the on-site and nearest-neighbor Coulomb repulsions, respectively. The results are consistent with previous numerical simulation results. We argue that the absence of charge-density waves for $\Delta=0$ and $U$ are large enough (i.e., $U>\frac{\Delta}{2}$+2V)., Comment: 32 pages. v2: typos corrected

This study investigated the extended Holstein--Hubbard model at half-filling as a model for describing interplay of the electron-electron and electron-phonon couplings. When the electron-phonon and nearest-neighbor electron-electron interactions are strong, we prove the existence of long-range charge order at sufficiently low temperature in three or more dimensions. As a result, we rigorously justify the phase competition between the antiferromagnetic and charge orders., Comment: 23 pages (Revised and improved version), to appear in J. Stat. Phys

We present a general framework of Griffiths inequalities for quantum systems. Our approach is based on operator inequalities associated with self-dual cones and provides a consistent viewpoint of the Griffiths inequality. As examples, we discuss the quantum Ising model, quantum rotor model, Bose--Hubbard model, Hubbard model, and spin-1/2 quantum Heisenberg model. We present a model-independent structure that governs the correlation inequalities., Comment: Revised and improved version, to appear in J. Stat. Phys

We propose a new method of constructing the quantum Griffiths inequality. From a viewpoint of operator inequalities, we first study the quantum rotor model. This viewpoint clarifies important connections between the reflection positivity and the Griffiths inequality. Next we apply our method to the spin-$1/2$ quantum Heisenberg model. Finally we point out a model-independent structure which governs the correlation inequalities., Comment: This paper has been merged with arXiv:1507.05355

We extend the Kubo-Kishi theorem concerning the charge susceptibility of the Hubbard model in the following way: (i) The electron-photon interaction is taken into account. (ii) Not only on-site but also general Coulomb repulsion is considered., Comment: 15 pages, V2:Minor corrections

The Holstein model has been widely accepted as a model comprising electrons interacting with phonons; analysis of this model's ground states was accomplished two decades ago. However, the results were obtained without completely taking repulsive Coulomb interactions into account. Recent progress has made it possible to treat such interactions rigorously; in this paper, we study the Holstein--Hubbard model with repulsive Coulomb interactions. The ground state properties of the model are investigated; in particular, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on a bipartite connected lattice. In addition, we provide a rigorous upper bound on charge susceptibility., Comment: 42 pages, Title changed, Major improvements, to appear, Annales Henri Poincare

We construct a general theory of operator monotonicity and apply it to the Fr\"ohlich polaron hamiltonian. This general theory provides a consistent viewpoint of the Fr\"ohlich model., Comment: 24 pages

In condensed matter physics, the polaron has been fascinating subject. It is described by the Hamiltonian of H. Fr\"ohlich. In this paper, the Fr\"ohlich Hamiltonian is investigated from a viewpoint of the self-dual cone analysis proposed in Miyao (2011). This point of view clarifies the monotonicity of polaron energy, i.e., denoting the lowest energy of the Fr\"ohlich Hamiltonian with the ultraviolet cutoff $\Lambda$ by $E_{\Lambda}$, we prove $E_{\Lambda}>E_{\Lambda'}$ for $\Lambda<\Lambda'$., Comment: 20 pages, Sec.1 rewritten

We study the ground state energy for a system of two hydrogen atoms coupled to the quantized Maxwell field in the limit $\alpha \to 0$ together with the relative distance between the atoms increasing as $\alpha^{-\gamma} R$, $\gamma > 0$. In particular we determine explicitly the crossover function from the $R^{-6}$ van der Waals potential to the $R^{-7}$ retarded van der Waals potential, which takes place at scale $\alpha^{-2} R$., Comment: 19 pages

The retarded van-der-Waals potential, as first obtained by Casimir and Polder, is usually computed on the basis of nonrelativistic QED. The hamiltonian describes two infinitely heavy nuclei, charge $e$, separated by a distance $R$ and two spinless electrons, charge $-e$, nonrelativistically coupled to the quantized radiation field. Casimir and Polder use the dipole approximation and small coupling to the Maxwell field. We employ here the full hamiltonian and determine the asymptotic strength of the leading $-R^{-7}$ potential, which is valid for all $e$. Our computation is based on a path integral representation and expands in $1/R$, rather than in $e$., Comment: 26 pages

Degeneracy of the eigenvalues of the Pauli-Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli-Fierz Hamiltonian at fixed total momentum is also investigated., Comment: LaTex, 11 pages

We consider a charged particle, spin 1/2, with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as $H(P)$, $P\in \BbbR^3$. We study the spectrum of $H(P)$. In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in $P$, from the rest of the spectrum., Comment: 32 pages

Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and $N$ electrons coupled to the radiation field. Since the total momentum $P$ is conserved, the Hamiltonian $H$ admits a fiber decomposition with respect to $P$ with fiber Hamiltonian $H(P)$. A stable atom, resp. ion, means that the fiber Hamiltonian $H(P)$ has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for $H(P)$ under (i) an explicit bound on $P$, (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons., Comment: 46 pages