Near Linearity of the Macroscopic Hall Current Response in Infinitely Extended Gapped Fermion Systems
Marius Wesle, Giovann Marcelli, Tadahiro Miyao, Domenico Monaco, Stefan Teufel
Communications in Mathematical Physics, 406, 8, Springer Science and Business Media LLC, 02 Jul. 2025, [Peer-reviewed], [Invited]
Scientific journal, Abstract

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 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 $$\sigma _\textrm{H}$$ 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 $$ \sigma _{\textrm{H } }$$ 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.

Ground state of one-dimensional Fermion–Phonon systems
Tadahiro Miyao, Seigo Okida, Hayato Tominaga
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 28, 01, World Scientific Pub Co Pte Ltd, 15 Oct. 2024, [Peer-reviewed], [Lead author, Last author, Corresponding author]
Scientific journal, 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.

Stability of Charge Density Waves in Electron–Phonon Systems
Tadahiro Miyao
Journal of Statistical Physics, 191, 3, Springer Science and Business Media LLC, 06 Mar. 2024, [Peer-reviewed], [Lead author]
English, Scientific journal

Ground state properties of the periodic Anderson model with electron–phonon interactions
Tadahiro Miyao, Hayato Tominaga
Annals of Physics, 169381, 169381, Elsevier BV, Jun. 2023, [Peer-reviewed], [Corresponding author]
English, Scientific journal

Rigorous analysis of the effects of electron–phonon interactions on magnetic properties in the one-electron Kondo lattice model
Tadahiro Miyao, Kazuhiro Nishimata, Hayato Tominaga
Quantum Studies: Mathematics and Foundations, Springer Science and Business Media LLC, 21 Dec. 2022, [Peer-reviewed], [Lead author, Corresponding author]
Scientific journal

Electron–phonon interaction in Kondo lattice systems
Tadahiro Miyao, Hayato Tominaga
Annals of Physics, 168467, 168467, Elsevier BV, Apr. 2021, [Peer-reviewed], [Corresponding author]
Scientific journal

On Renormalized Hamiltonian Nets
Tadahiro Miyao
Annales Henri Poincaré, Springer Science and Business Media LLC, 15 Feb. 2021, [Peer-reviewed], [Lead author]
English, Scientific journal

Thermal Stability of the Nagaoka–Thouless Theorems
Tadahiro Miyao
Annales Henri Poincaré, 21, 12, 4027, 4072, Springer Science and Business Media LLC, Dec. 2020, [Peer-reviewed], [Lead author]
Scientific journal

Note on the Retarded van der Waals Potential within the Dipole Approximation
Tadahiro Miyao
Symmetry, Integrability and Geometry: Methods and Applications, SIGMA (Symmetry, Integrability and Geometry: Methods and Application), 26 Apr. 2020, [Peer-reviewed], [Lead author]
Scientific journal

Correlation Inequalities for Schrödinger Operators
Tadahiro Miyao
Mathematical Physics, Analysis and Geometry, 23, 1, Springer Science and Business Media LLC, Mar. 2020, [Peer-reviewed], [Lead author]
Scientific journal

Stability of Ferromagnetism in Many-Electron Systems
Tadahiro Miyao
Journal of Statistical Physics, 176, 5, 1211, 1271, Springer Science and Business Media LLC, Sep. 2019, [Peer-reviewed], [Lead author]
Scientific journal

On the semigroup generated by the renormalized Nelson Hamiltonian
Miyao Tadahiro
Journal of Functional Analysis, 276, 6, 1948‐1977, 2019, [Peer-reviewed]
English

Ground State Properties of the Holstein–Hubbard Model
Tadahiro Miyao
Annales Henri Poincaré, 19, 8, 2543, 2555, Springer Science and Business Media LLC, Aug. 2018, [Peer-reviewed], [Lead author]
Scientific journal

Nagaoka's Theorem in the Holstein-Hubbard Model
Tadahiro Miyao
ANNALES HENRI POINCARE, 18, 9, 2849, 2871, Sep. 2017, [Peer-reviewed]
English, Scientific journal

Rigorous Results Concerning the Holstein-Hubbard Model
Tadahiro Miyao
ANNALES HENRI POINCARE, 18, 1, 193, 232, Jan. 2017, [Peer-reviewed]
English, Scientific journal

Long-Range Charge Order in the Extended Holstein-Hubbard Model
Tadahiro Miyao
JOURNAL OF STATISTICAL PHYSICS, 165, 2, 225, 245, Oct. 2016, [Peer-reviewed]
English, Scientific journal

Quantum Griffiths Inequalities
Tadahiro Miyao
JOURNAL OF STATISTICAL PHYSICS, 164, 2, 255, 303, Jul. 2016, [Peer-reviewed]
English, Scientific journal

Upper Bounds on the Charge Susceptibility of Many-Electron Systems Coupled to the Quantized Radiation Field
Tadahiro Miyao
LETTERS IN MATHEMATICAL PHYSICS, 105, 8, 1119, 1133, Aug. 2015, [Peer-reviewed]
English, Scientific journal

MONOTONICITY OF THE POLARON ENERGY
Tadahiro Miyao
REPORTS ON MATHEMATICAL PHYSICS, 74, 3, 379, 398, Dec. 2014, [Peer-reviewed]
English, Scientific journal

Monotonicity of the Polaron Energy II: General Theory of Operator Monotonicity
Tadahiro Miyao
JOURNAL OF STATISTICAL PHYSICS, 153, 1, 70, 92, Oct. 2013, [Peer-reviewed]
English, Scientific journal

Ground State Properties of the SSH Model
Tadahiro Miyao
JOURNAL OF STATISTICAL PHYSICS, 149, 3, 519, 550, Nov. 2012, [Peer-reviewed]
English, Scientific journal

Scale dependence of the retarded van der Waals potential
Tadahiro Miyao, Herbert Spohn
JOURNAL OF MATHEMATICAL PHYSICS, 53, 9, 095215-095215-15, Sep. 2012, [Peer-reviewed]
English, Scientific journal

Note on the One-Dimensional Holstein-Hubbard Model
Tadahiro Miyao
JOURNAL OF STATISTICAL PHYSICS, 147, 2, 436, 447, Apr. 2012, [Peer-reviewed]
English, Scientific journal

SELF-DUAL CONE ANALYSIS IN CONDENSED MATTER PHYSICS
Tadahiro Miyao
REVIEWS IN MATHEMATICAL PHYSICS, 23, 7, 749, 822, Aug. 2011, [Peer-reviewed]
English

Nondegeneracy of ground states in the nonrelativistic quantum field theory　　　　　　　　　　　　　　　
Tadahiro Miyao
Journal of Operator Theory, 64, 207, 241, 2010, [Peer-reviewed]

Time reversal symmetries and properties of ground states in nonrelativistic QED
Tadahiro Miyao
RIMS Kokyuroku Bessatsu, B21, 35, 44, Kyoto University, 2010, [Peer-reviewed]
English

Polaron with at most one phonon in the weak coupling limit
Tadahiro Miyao
MONATSHEFTE FUR MATHEMATIK, 157, 4, 365, 378, Aug. 2009, [Peer-reviewed]
English, Scientific journal

The retarded van der Waals potential: Revisited
Tadahiro Miyao, Herbert Spohn
JOURNAL OF MATHEMATICAL PHYSICS, 50, 7, 072103, Jul. 2009, [Peer-reviewed]
English, Scientific journal

Kramers Degeneracy Theorem in Nonrelativistic QED
Michael Loss, Tadahiro Miyao, Herbert Spohn
LETTERS IN MATHEMATICAL PHYSICS, 89, 1, 21, 31, Jul. 2009, [Peer-reviewed]
English, Scientific journal

Spectral analysis of the semi-relativistic Pauli-Fierz hamiltonian
Tadahiro Miyao, Herbert Spohn
JOURNAL OF FUNCTIONAL ANALYSIS, 256, 7, 2123, 2156, Apr. 2009, [Peer-reviewed]
English, Scientific journal

Lowest energy states in nonrelativistic QED: Atoms and ions in motion
Michael Loss, Tadahiro Miyao, Herbert Spohn
JOURNAL OF FUNCTIONAL ANALYSIS, 243, 2, 353, 393, Feb. 2007, [Peer-reviewed]
English, Scientific journal

Nonrelativistic limit of the abstract chiral quark soliton model and confining effects　　　　　　　　　　　　　　　
Tadahiro Miyao
Journal of Operator Theory, 57, 429, 441, 2007, [Peer-reviewed]

The bipolaron in the strong coupling limit
Tadahiro Miyao, Herbert Spohn
ANNALES HENRI POINCARE, 8, 7, 1333, 1370, 2007, [Peer-reviewed]
English, Scientific journal

Momentum operators with a winding gauge potential
Tadahiro Miyao
Hokkaido Mathematical Journal, 34, 1, 159, 184, 2005, [Peer-reviewed]
English, Scientific journal

Dirac-Weyl operators with a winding gauge potential
Tadahiro Miyao
Hokkaido Mathematical Journal, 34, 1, 185, 218, 2005, [Peer-reviewed]
English, Scientific journal

Stability of discrete ground state
Tadahiro Miyao, Itaru Sasaki
Hokkaido Mathematical Journal, 34, 3, 689, 717, 2005, [Peer-reviewed]
English, Scientific journal

Strongly superconnnuting self-adjoint operators
T Miyao
INTEGRAL EQUATIONS AND OPERATOR THEORY, 50, 4, 505, 535, Dec. 2004, [Peer-reviewed]
English, Scientific journal

Reflection positivity and operator theoretic correlaion inequalities　　　　　　　　　　　　　　　
Tadahiro Miyao, Oberwolfach report, 55, 42, 45, 2018, [Invited]
English

Frohlich模型における相関不等式 (量子場の数理とその周辺)
宮尾 忠宏, 数理解析研究所講究録, 2010, 127, 131, Dec. 2016
京都大学数理解析研究所, Japanese

Correlation inequalities for the Fro¨hlich model　　　　　　　　　　　　　　　
Tadahiro Miyao, RIMS Kokyuroku, 2010, 127, 131, 2016, [Invited]

電子-格子相互作用系における基底状態の性質について (量子場の数理とその周辺)
宮尾 忠宏, 数理解析研究所講究録, 1961, 12, 16, Aug. 2015, [Invited]
京都大学, Japanese

Monotonicity of the Polaron Energy (Applications of Renormalization Group Methods in Mathematical Sciences)
Miyao Tadahiro, RIMS Kokyuroku, 1904, 46, 53, Jul. 2014, [Invited]
Kyoto University, English

フレーリッヒ模型の数学的側面 (量子場の数理とその周辺)
宮尾 忠宏, 数理解析研究所講究録, 1859, 41, 49, Oct. 2013, [Invited]
京都大学, Japanese

Stability and instability of moving atoms and ions in nonrelativistic QED(Spectral and Scattering Theory and Related Topics)
Miyao Tadahiro, RIMS Kokyuroku, 1563, 109, 123, Jun. 2007, [Invited]
Kyoto University, Japanese

A comprehensive study on the mathematical aspects of metallic ferromagnetism
Grants-in-Aid for Scientific Research
01 Apr. 2023 - 31 Mar. 2027
宮尾 忠宏, 坂井 哲, 桂 法称, 松澤 泰道, 松井 卓
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, 23K25783

Comprehensive study on mathematical aspects of metallic ferromagnetism
Grants-in-Aid for Scientific Research
01 Apr. 2023 - 31 Mar. 2027
宮尾 忠宏, 松井 卓, 坂井 哲, 桂 法称, 松澤 泰道
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, 23H01086

Rigorous construction of linear response theory for many-fermion systems interacting with environment and its applications
Grants-in-Aid for Scientific Research
2022 - 2024
宮尾 忠宏
Japan Society for the Promotion of Science, Fund for the Promotion of Joint International Research (Fostering Joint International Research (A)), Hokkaido University, 20KK0304

Hubbard模型における磁性の安定性の厳密解析：作用素不等式によるアプローチ
科学研究費助成事業
01 Apr. 2018 - 31 Mar. 2023
宮尾 忠宏
近藤格子模型で記述される電子系とフォノンが相互作用する系の基底状態の磁気特性を解析した．その結果，近藤格子系の基底状態の磁気特性は，フォノンとの相互作用がある程度小さければ安定であることを証明した．さらに，基底状態におけるスピン2点相関関数の性質も解析した．これまでに近藤格子系とフォノンの相互作用に関する厳密な解析は為されておらず，ここで得られた結果は，筆者が提唱する作用素論不等式理論が多電子系の厳密解析において有効であることを例証したと見做せる．


この研究で得られた解析と，筆者の過去の解析結果を総合することにより，多電子系の基底状態の磁気特性の安定性を記述する一般論を，富田-竹崎理論を用いて構成した．このような統一的な視点はこれまでになかったものであり，今後の発展が十分に期待できる．実際，この理論から派生した，新しいプロジェクトが現在進行中である．


上述の結果とは独立に，SU(n) Hubbard模型に関する解析を行った．通常のSU(2) Hubbard模型で良く知られている，Aizenman-Liebの定理とは，Nagaoka-Thoulessの定理の有限温度版であると言われている．筆者はAizenman-Liebの定理をSU(n) Hubbard模型に拡張した．ここで用いられた解析手法は，分配関数の経路積分表示を基にしたランダムループ展開であり，通常のスピン系におけるランダムループ展開に比較してその取扱いが難しい．SU(n)ハバード模型に関する厳密な結果は少なく，ここで得られた結果を基に，この模型に対する新しいプロジェクトを準備している最中である．
日本学術振興会, 基盤研究(C), 北海道大学, 18K03315

Infinite dimensional analysis by stochastic methods and their application to quantum field theory
Grants-in-Aid for Scientific Research
01 Apr. 2016 - 31 Mar. 2020
Hiroshima Fumio
A. Construction and application of Gibbs measure by rough path theory, B. Ground state study, C. Stochastic UV renormalization, D. Study of quantum field theory on manifolds, E. Study of SDE and semi-classical limit in quantum field theory. In addition, every other year, an international workshop on spectral analysis of quantum field theory was held. Great progress has been made on A and C, and they are summarized in a paper. The results of B were also summarized and completed as a paper. Regarding E, joint research has begun and research is ongoing. Every year, I was able to support travel expenses for the holding of international research meetings.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Kyushu University, 16H03942

Studies of condensed matter physics in terms of operator inequalities
Grants-in-Aid for Scientific Research
01 Apr. 2013 - 31 Mar. 2016
Miyao Tadahiro
(1) I studied some operator inequalities associated with self-dual cones. By applying these operator inequalities, I constructed a novel method of analyzing removal of the ultraviolet cutoff for the Froehlich model. By the method, I solved a longstanding problem, that is, I proved uniqueness of ground states for the Froehlich model. These results are published in international journals.
(2) I investigated a many-electron system coupled to the quantized radiation field. I obtained a useful upper bound of the charge susceptibility of the system. As a result, I proved the absence of long-range charge order. This result is published in an international journal.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 25400123

Applications of Renormalization Group Methods in Mathematical Sciences
Grants-in-Aid for Scientific Research
2011 - 2013
ITO Keiichi, TERAMOTO Toshiaki, TAMURA Hiroshi, HIROKAWA Masao, HIROSHIMA Funio
We applied the renormalization group methods to the longstanding problems (the so-called Millennium problems) in mathematical sciences which go back to the last century. We apply this method to solve the 2D O(N) symmetric spin model which is a prototype model of the gauge theory model. In this model, one needs to integrate over infinite variables which form a functional space. We find a set of subspaces (called flow) which dominate the integral. This enables us to solve the recursive integrals and obtain the final result.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Setsunan University, 23540257

Applications of operator inequalities induced by self-dual cones to the solid state physics
Grants-in-Aid for Scientific Research
2011 - 2012
MIYAO Tadahiro
Self-dual cones in the Hilbert space naturally induce useful operator inequalities. We studied mathematical aspects of the inequalities and applied these to the condensed matter physics. For instance, the Holstein-Hubbard, SSH and Frohlich models are investigated from the viewpoint of our operator inequalities. We showed that ground states are unique in the Holstein-Hubbard and SSH models. This reveals interesting magnetic structures in the ground states.
Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), 23740112

ポーラロンモデルの作用素論的側面
科学研究費助成事業
2010 - 2010
宮尾 忠宏
日本学術振興会, 若手研究(B), 摂南大学, 22740096

粒子と量子場の相互作用系における束縛の強化の解析的研究
科学研究費助成事業
2005 - 2007
宮尾 忠宏
イオン結晶中の2つの電子を考える。この2電子の間には、次の2つの力が働く。(1)各電子が同じ電荷を持つために働くクーロン斥力。(2)格子-電子間相互作用により生じる電子間引力。2電子間の引力がクーロン斥力より強いならば2つの電子はベアを形成する。このイオン結晶中の2電子ベアをバイポーラロンと呼ぶ。本年度の主な研究業績は、この直感的な話を数学的に厳密に証明したことである。より詳しく述べると、このような系は一般にFroehlich Hamiltonianと呼ばれる、フォック空間上の自己共役作用素で記述される。バイポーラロンの存在は、このHamiltonianが基底状態を持つことに対応している。(ここで基底状態とはHamiltonianのスペクトルの下限が固有値である場合の、対応する固有ベクトルのことである。)ここで考察しているFroehlich Hamiltonianとは、非常に複雑であり、そのスペクトル解析は困難である。執筆者とミュンヘン工科大学のH. Spohn教授は実際にFroehlich Hamiltonianが基底状態を持つことを証明した。これが業績欄の論文である。先に述べたようにFroehlich Hamiltonianの解析は困難だが、我々は以前の共同研究を通じて発展させてきた手法をさらに発展、応用することにより、このHamiltonianが強結合領域において実際に基底状態を持つことを証明した。さらに、この証明の中で自然に現れてくるPekar-Tomasevich汎関数と呼ばれるエネルギー汎関数の解析が、今後の執筆者とH. Spohn氏の共同研究に繋がっている。また、バイポーラロンに関連する論文を2本現在投稿中である。
日本学術振興会, 特別研究員奨励費, 岡山大学, 05J03551