強相関トポロジカル相のエッジ状態・低エネルギー励起

• Project Area: Frontiers of materials science spun from topology
• Project/Area Number: 16H00985
• Research Category: Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
• Allocation Type: Single-year Grants
• Review Section: Science and Engineering
• Research Institution: The University of Tokyo
• Principal Investigator: 桂 法称 東京大学, 大学院理学系研究科(理学部), 准教授 (80534594)
• Project Period (FY): 2016-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: トポロジカル相 / エッジ状態 / 超対称性 / 指数定理 / 機械学習 / 強相関電子系

Outline of Annual Research Achievements

トポロジカル相の低エネルギー準粒子励起について、理論的な観点から研究を行った。当該年度の主な結果として、以下の３つに関するものが挙げられる。（１）乱れのあるトポロジカル絶縁体・超伝導体の特徴づけ、（２）超対称性の破れと南部・ゴールドストーンフェルミオン、（３）キタエフ物質における光学応答。

（１）トポロジカル絶縁体・超伝導体を特徴づけるトポロジカル不変量は、通常、波数空間における積分の形で表現されるが、乱れにより系に並進対称性が存在しない場合、その定義は自明ではない。しかし、このような場合にも、任意の次元・対称性クラスにおいてトポロジカル不変量を実空間で定義することが可能であることを、非可換幾何の手法を用いて示した。また、この手法が有限系でも有効であることを、２・３次元のクラスAIIトポロジカル絶縁体を例に、数値的に実証した。また、乱れのあるトポロジカル超伝導体（２次元・クラスDIII）にもこの手法を応用した。さらに、機械学習を用いてこの系の相図を調べ、エッジ状態を画像として学習する手法が有効であることを明らかにした。
（２）前年度は、スピンレスフェルミオンにより超電荷が定義される、N=2超対称性を持つ格子フェルミオン系を調べたが、本年度はマヨラナフェルミオンにより超電荷が定義される、N=1超対称性を持つ系を構成し、その性質を調べた。その結果、i) 自発的超対称性の破れが起こる相が存在すること、ii) 超対称性が破れた相での低エネルギー励起の分散関係は波数の３乗に比例すること、iii) 超対称性が破れない相も存在し、厳密に基底状態が求まる場合があること、などを明らかにした。
（３）キタエフ物質における光学応答
α-RuCl3における最近の実験結果を念頭に、この系における局所的電気分極のスピンを用いた表式を導出し、キタエフ模型を用いて、光学伝導度の低エネルギーの振る舞いを計算した。

Research Progress Status

29年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

29年度が最終年度であるため、記入しない。

Research Products

• [Int'l Joint Research] Beijing Institute of Technology(中国)
• [Int'l Joint Research] Utrecht University(オランダ)
• [Int'l Joint Research] Sungkyunkwan University(韓国)
• [Int'l Joint Research] Sungkyunkwan University(韓国)
• [Int'l Joint Research] Universite de Geneve(スイス)
• [Journal Article] The noncommutative index theorem and the periodic table for disordered topological insulators and superconductors (2018)
  • Author(s): Hosho Katsura and Tohru Koma
  • Journal Title: J. Math. Phys. Volume: 59 Issue: 3
  • DOI: 10.1063/1.5026964
  • Peer Reviewed
• [Journal Article] Mechanism for sub-gap optical conductivity in honeycomb Kitaev materials (2018)
  • Author(s): Adrien Bolens, Hosho Katsura, Masao Ogata, Seiji Miyashita
  • Journal Title: Phys. Rev. B Volume: 97 Issue: 16
  • DOI: 10.1103/physrevb.97.161108
  • Peer Reviewed
• [Journal Article] Exact zero modes in twisted Kitaev chains (2017)
  • Author(s): Kohei Kawabata, Ryohei Kobayashi, Ning Wu, and Hosho Katsura
  • Journal Title: Phys. Rev. B Volume: 95 Issue: 19
  • DOI: 10.1103/physrevb.95.195140
  • NAID: 130007510338
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Synergetic effect of spin-orbit coupling and Zeeman splitting on the optical conductivity in the one-dimensional Hubbard model (2017)
  • Author(s): Adrien Bolens, Hosho Katsura, Masao Ogata, and Seiji Miyashita
  • Journal Title: Phys. Rev. B Volume: 95 Issue: 23
  • DOI: 10.1103/physrevb.95.235115
  • Peer Reviewed
• [Journal Article] Finite-size gap, magnetization, and entanglement of deformed Fredkin spin chain (2017)
  • Author(s): Takuma Udagawa and Hosho Katsura
  • Journal Title: J. Phys. A: Math. Theor. Volume: 50 Issue: 40 Pages: 405002-405002
  • DOI: 10.1088/1751-8121/aa85b5
  • Peer Reviewed
• [Journal Article] A new kind of spin-1 chain model with competing dimer and trimer interactions (2017)
  • Author(s): Yun-Tak Oh, Hosho Katsura, Hyunyong Lee, and Jung Hoon Han
  • Journal Title: Phys. Rev. B Volume: 96 Issue: 16
  • DOI: 10.1103/physrevb.96.165126
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Zero-energy states in conformal field theory with sine-square deformation (2017)
  • Author(s): Shota Tamura and Hosho Katsura
  • Journal Title: Prog. Theor. Exp. Phys Volume: 2017 Issue: 11 Pages: 1-17
  • DOI: 10.1093/ptep/ptx147
  • NAID: 130007509147
  • Peer Reviewed
• [Journal Article] A New Numerical Method for Z2 Topological Insulators with Strong Disorder (2017)
  • Author(s): Yutaka Akagi, Hosho Katsura, and Tohru Koma
  • Journal Title: Journal of the Physical Society of Japan Volume: 86 Issue: 12 Pages: 123710-123710
  • DOI: 10.7566/jpsj.86.123710
  • Peer Reviewed / Open Access
• [Journal Article] Resonating Valence Bond States with Trimer Motifs (2017)
  • Author(s): Hyunyong Lee, Yun-Tak Oh, Jung Hoon Han, and Hosho Katsura
  • Journal Title: Phys. Rev. B Volume: 95 Issue: 6
  • DOI: 10.1103/physrevb.95.060413
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion (2017)
  • Author(s): Noriaki Sannomiya, Hosho Katsura, and Yu Nakayama
  • Journal Title: Phys. Rev. D Volume: 95 Issue: 6 Pages: 0650011-13
  • DOI: 10.1103/physrevd.95.065001
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Entanglement and corner Hamiltonian spectra of integrable open spin chains (2016)
  • Author(s): Panjin Kim, Hosho Katsura, Nandini Trivedi, and Jung Hoon Han
  • Journal Title: Phys. Rev. B Volume: 94 Issue: 19 Pages: 1951101-13
  • DOI: 10.1103/physrevb.94.195110
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Hofstadter's Butterfly in Quantum Geometry (2016)
  • Author(s): Yasuyuki Hatsuda, Hosho Katsura, and Yuji Tachikawa
  • Journal Title: New. J. Phys. Volume: 18 Issue: 10 Pages: 1030231-16
  • DOI: 10.1088/1367-2630/18/10/103023
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model (2016)
  • Author(s): Noriaki Sannomiya, Hosho Katsura, and Yu Nakayama
  • Journal Title: Phys. Rev. D Volume: 94 Issue: 4 Pages: 0450141-13
  • DOI: 10.1103/physrevd.94.045014
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Spontaneous SUSY Breaking and Nambu-Goldstone fermions in extended Nicolai models (2018)
  • Author(s): Noriaki Sannomiya
  • Organizer: APS March Meeting (Los Angeles)
  • Int'l Joint Research
• [Presentation] Machine Learning Disordered Topological Phases by Statistical Recovery of Symmetry (2018)
  • Author(s): Nobuyuki Yoshioka
  • Organizer: APS March Meeting (Los Angeles)
  • Int'l Joint Research
• [Presentation] 散逸のあるXYスピン鎖のマヨラナフェルミオンを用いた解法 (2018)
  • Author(s): 柴田直幸, 桂法称
  • Organizer: 日本物理学会第73回年次大会 (東京理科大)
• [Presentation] S=1 ダイマー・トライマー模型の基底状態相図 (2018)
  • Author(s): 桂法称, Yun-tak Oh, Hyunyong Lee, Jung Hoon Han
  • Organizer: 日本物理学会第73回年次大会 (東京理科大)
• [Presentation] 流入・散逸のあるキタエフ鎖におけるパリティ・時間対称性によって誘起されたエッジ状態 (2018)
  • Author(s): 川畑幸平, 蘆田祐人, 桂法称, 上田正仁
  • Organizer: 日本物理学会第73回年次大会 (東京理科大)
• [Presentation] U(1)対称性のない拡張Nicolai模型における超対称性の破れと南部-Goldstoneフェルミオン (2017)
  • Author(s): 三ノ宮典昭, 桂法称, 中山優
  • Organizer: 日本物理学会 第72回年次大会
  • Place of Presentation: 大阪大学（大阪府豊中市）
  • Year and Date: 2017-03-20
• [Presentation] 乱れのある3次元トポロジカル絶縁体の非可換Z2指数 (2017)
  • Author(s): 赤城 裕, 桂 法称, 高麗 徹
  • Organizer: 日本物理学会 第72回年次大会
  • Place of Presentation: 大阪大学（大阪府豊中市）
  • Year and Date: 2017-03-18
• [Presentation] Majorana zero mode without edges (2017)
  • Author(s): Ryohei Kobayashi, Kohei Kawabata, Ning Wu and Hosho Katsura
  • Organizer: Theory of Correlated Topological Materials (TCTM2017)
  • Place of Presentation: 物性研究所（千葉県柏市）
  • Year and Date: 2017-03-02
• [Presentation] Noncommutative Z2 index of 3D topological insulators with disorder (2017)
  • Author(s): Yutaka Akagi
  • Organizer: International Conference on Topological Materials Science 2017 (TopoMat2017)
  • Int'l Joint Research
• [Presentation] SUSY breaking and NG fermions in \mathbb{Z}_2 Nicolai model (2017)
  • Author(s): Noriaki Sannomiya
  • Organizer: International Conference on Topological Materials Science 2017 (TopoMat2017)
  • Int'l Joint Research
• [Presentation] Anomalous Thermal Hall Effect in Nodal Chiral Superconductors (2017)
  • Author(s): Nobuyuki Yoshioka
  • Organizer: International Conference on Topological Materials Science 2017 (TopoMat2017)
  • Int'l Joint Research
• [Presentation] Spontaneous Supersymmetry Breaking and Nambu-Goldstone Fermions in Z_2 Nicolai Model (2017)
  • Author(s): Noriaki Sannomiya
  • Organizer: Croucher Summer Course "Quantum Entanglement and Topological Order
  • Int'l Joint Research
• [Presentation] Learning Disordered Topological Phases by Statistical Recovery of Symmetry (2017)
  • Author(s): Nobuyuki Yoshioka
  • Organizer: Novel Quantum States in Condensed Matter 2017 (NQS2017)
  • Int'l Joint Research
• [Presentation] Quantum Hangul (2017)
  • Author(s): Hosho Katsura
  • Organizer: Novel Quantum States in Condensed Matter 2017 (NQS2017)
  • Int'l Joint Research
• [Presentation] Supersymmetry Breaking in an Interacting Majorana Chain (2017)
  • Author(s): Noriaki Sannomiya
  • Organizer: Novel Quantum States in Condensed Matter 2017 (NQS2017)
  • Int'l Joint Research
• [Presentation] Topological order and zero modes in interacting Kitaev/Majorana chains (2017)
  • Author(s): Hosho Katsura
  • Organizer: Quantum Devices (Natal Brazil)
  • Int'l Joint Research / Invited
• [Presentation] 量子トライマー模型のRVB状態とZ3トポロジカル秩序 (2017)
  • Author(s): 桂法称, Hyunyong Lee, Yun-tak Oh, Jung Hoon Han
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] 乱れたトポロジカル超伝導体の非可換指数と量子相図 (2017)
  • Author(s): 吉岡信行, 赤城裕, 桂法称
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] 捻り境界条件を課したKitaev模型におけるマヨラナ・ゼロモードの出現 (2017)
  • Author(s): 川畑幸平, 小林良平, Ning Wu, 桂法称
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] Charge fluctuation and optical response in two-dimensional Mott insulators (2017)
  • Author(s): Adrien Bolens, Hosho Katsura, Masao Ogata, Seiji Miyashita
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] Majorana Nicolai模型における超対称性の破れと南部-Goldstoneフェルミオン (2017)
  • Author(s): 三ノ宮典昭, 桂法称
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] 機械学習による乱れたトポロジカル超伝導体の相判定 (2017)
  • Author(s): 吉岡信行, 赤城裕, 桂法称
  • Organizer: 日本物理学会2017年秋季大会 (岩手大学)
• [Presentation] 対称性の統計的回復による乱れたトポロジカル相の分類 (2017)
  • Author(s): 吉岡信行
  • Organizer: 第11回 物性科学領域横断研究会 (東大物性研)
• [Presentation] Z2 指数の新しい計算法: 乱れたトポロジカル絶縁体 (2017)
  • Author(s): 赤城裕
  • Organizer: 第11回 物性科学領域横断研究会 (東大物性研)
• [Presentation] A supersymmetric Kitaev honeycomb model (2016)
  • Author(s): 桂 法称
  • Organizer: 第２回トポ物質科学領域研究会
  • Place of Presentation: 東北大学（宮城県仙台市）
  • Year and Date: 2016-12-17
• [Presentation] Disordered topological insulators with time-reversal symmetry: Z2 index (2016)
  • Author(s): Hosho Katsura
  • Organizer: Topological Materials Science: Intensive-Interactive Meeting
  • Place of Presentation: 慶応大学（神奈川県横浜市）
  • Year and Date: 2016-11-17
  • Int'l Joint Research
• [Presentation] Topological Excitations in Quantum Spin-nematics (2016)
  • Author(s): Yutaka Akagi
  • Organizer: Highly Frustrated Magnetism 2016
  • Place of Presentation: 台北（台湾）
  • Year and Date: 2016-09-07
  • Int'l Joint Research
• [Presentation] Topological Excitations in Frustrated Magnets (2016)
  • Author(s): Yutaka Akagi
  • Organizer: International Symposium on New Horizons in Condensed Matter Physics
  • Place of Presentation: 東京大学（東京都文京区）
  • Year and Date: 2016-06-18
  • Int'l Joint Research
• [Presentation] Supersymmetry breaking and Nambu-Goldstone fermions in lattice models (2016)
  • Author(s): Hosho Katsura
  • Organizer: Quantum Matter, Spacetime and Information
  • Place of Presentation: 京都大学（京都府京都市）
  • Year and Date: 2016-06-15
  • Int'l Joint Research
• [Presentation] Topological defects in quantum spin-nematics (2016)
  • Author(s): Yutaka Akagi
  • Organizer: International Conference on Strongly Correlated Electron Systems
  • Place of Presentation: 杭州（中国）
  • Year and Date: 2016-05-08
  • Int'l Joint Research
• [Presentation] Microscopic mechanisms of spin-driven ferroelectricity and the thermal Hall effect in insulating magnets (2016)
  • Author(s): Hosho Katsura
  • Organizer: International Conference on Strongly Correlated Electron Systems
  • Place of Presentation: 杭州（中国）
  • Year and Date: 2016-05-08
  • Int'l Joint Research
• [Remarks] Katsura group webpage
  • URL: http://park.itc.u-tokyo.ac.jp/hkatsura-lab/