「トポロジカル相の発見とその展開」九州大学大学院理学研究科 集中講義　（野村清英 先生 ご紹介）平成30年7月17日-19日

概要量子ホール効果の発見にはじまり，Haldane相，グラフェン，トポロジカル絶縁体，ワイル半金属，そして量子系以外のフォトニック結晶や古典力学系のトポロジカル相にも多彩に展開したトポロジカル相に関してその基礎から講義する。特にベリー接続，対称性の保護とバルク・エッジ対応に関して詳しく議論する。

講義内容 (pdf files are in English except beginning a few.

0. Introduction:

[Talk file (pdf)]

1. Symmetry breaking to the topological phases:

[Talk file (pdf)]

reference

• 「トポロジカル秩序とベリー位相」初貝安弘 (2013) 日本物理学会誌 68巻1号
• Topological phases: Popular Science Background, Noble prize in physics 2016 (@nobelprize.org)
• 2016年度ノーベル物理学賞：David J. Thouless氏，F. Duncan M. Haldane氏，J. Michael Kosterlitz氏―トポロジカルな相転移とトポロジカルな物質相の理論的発見 初貝 安弘, 日本物理学会誌 71 巻 (2016) 12 号 学界ニュース
• 「2016年ノーベル物理学賞について／物質中に普遍的に存在するトポロジカルな構造」 初貝 安弘, 雑誌 数学セミナー4 月号, p36-p41 (2017年）

2. Topological phases and bulk-edge correspondence (overview):

[Talk file (pdf)]

reference

• Discovery of QHE: K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 – Published 11 August 1980
• 特集：トポロジーによる新しい物性物理「物質内部と表面状態の深い関係：バルクエッジ対応」初貝 安弘, 雑誌 パリティ 2017年7月号, p14-p18
• 「トポロジカル物質におけるバルク・エッジ対応」 初貝 安弘, 科研費NEWS, 2017年度 vol.3, p8-p8

3. Quantum Hall effects and bulk-edge correspondence:

[Talk file (pdf)]

reference

• Discovery of QHE: K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 – Published 11 August 1980
• Laughlin argument: R. B. Laughlin, Phys. Rev. B 23, 5632(R) – Published 15 May 1981
• Roles of edge states in QHE: B. I. Halperin, Phys. Rev. B 25, 2185 – Published 15 February 1982
• TKNN (noting to say):D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,Phys. Rev. Lett. 49, 405 – Published 9 August 1982
• QHE and twisted boundary condition: Qian Niu, D. J. Thouless, and Yong-Shi Wu, Phys. Rev. B 31, 3372 – Published 15 March 1985
• Chern number and its quantization: M. Kohmoto, Annals of Physics, Volume 160, Issue 2, 1 April 1985, Pages 343-354
• Winding numbers of the edge states: Yasuhiro Hatsugai,Phys. Rev. B 48, 11851 – Published 15 October 1993
• Bulk-edge correspondence: Yasuhiro Hatsugai, Phys. Rev. Lett. 71, 3697 – Published 29 November 1993
• 「量子ホール効果におけるエッジ状態と位相不変量」初貝安弘, 固体物理340号(1994)
• How to calculate Chern number? : Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki, J. Phys. Soc. Jpn. 74, pp. 1674-1677 (2005)

4. Quantum/spin liquids and Berry phases (topological order parameters):

Seminar (July 17, 2018), Y. Hatsugai, “Short-range entangled states and bulk-edge correspondence: Symmetry protection and Z_N Berry phases” [Talk file (pdf)]

reference

• AKLT models : Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki, Phys. Rev. Lett. 59, 799 – Published 17 August 1987, CMP Vol. 115, Number 3 (1988), 477-528.
• Berry phase as a topological local order parameter: Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)
• 「トポロジカル秩序とベリー位相」初貝安弘 (2013) 日本物理学会誌 68巻1号
• Berry phase for S=1 Haldane phase: Y Hatsugai 2007 J. Phys.: Condens. Matter 19 145209
• Kennedy triplet as of the edge states of the Haldane spin: T Kennedy, Journal of Physics: Condensed Matter, Volume 2, Number 26
• Valence Bond Solid (VBS) state and Berry phases: T. Hirano, H. Katsura, and Y. Hatsugai, Phys. Rev. B 77, 094431 – Published 26 March 2008
• Z_Q quantization of Berry phases : Y. Hatsugai and I. Maruyama, EPL, 95 (2011) 20003
• Z_3,4 Berry phases for SPT of quantum spins: Toshikaze Kariyado, Takahiro Morimoto, and Yasuhiro Hatsugai, Phys. Rev. Lett. 120, 247202 – Published 15 June 2018

4-1. More quantum/spin liquids and Berry phases and Entanglement Entropy (EE)

[Talk file (pdf)]

reference

• Spin ladder with 4 spin interaction: I. Maruyama, T. Hirano, and Y. Hatsugai, Phys. Rev. B 79, 115107 – Published 12 March 2009
• Bulk-edge correspondence in spin ladders: Mitsuhiro Arikawa, Shou Tanaya, Isao Maruyama, and Yasuhiro Hatsugai, Phys. Rev. B 79, 205107 – Published 11 May 2009
• Entanglement entropy, edge states and Berry phases: S. Ryu and Y. Hatsugai, Phys. Rev. B 73, 245115 – Published 26 June 2006
• Bose Einstein condensation-BCS crossover and Z₂Berry phase: Mitsuhiro Arikawa, Isao Maruyama, and Yasuhiro Hatsugai, Phys. Rev. B 82, 073105 – Published 30 August 2010
• VBS, entanglement entropy and Berry phases: Hosho Katsura, Takaaki Hirano, and Yasuhiro Hatsugai, Phys. Rev. B 76, 012401 – Published 3 July 2007

4-2. Symmetry constraints for the Berry phases: Lieb-Schultz-Mattis type gap closing conditions

[Talk file (pdf)]

reference

• Finite system size analogue of the LSM theorem: T. Hirano, H. Katsura, and Y. Hatsugai, Phys. Rev. B 78, 054431 – Published 21 August 2008

5. Dirac fermions, Berry phases and bulk-edge correspondence

[Talk file (pdf)]

reference

• Dirac cones and TKNN numbers of graphene: Yasuhiro Hatsugai, Takahiro Fukui, and Hideo Aoki, Phys. Rev. B 74, 205414 – Published 13 November 2006
• Dirac monopole in superconductors: Y. Hatsugai and S. Ryu, Phys. Rev. B 65, 212510 – Published 14 June 2002, Y. Hatsugai, S. Ryu, and M. Kohmoto, Phys. Rev. B 70, 054502 – Published 3 August 2004
• Berry phases (Codimension of the gap closing is three): M. V. Berry, F. R. S., Proc. R. Soc. Lond. A 392, 45-57 (1984)
• Berry phases and quaternions: Yasuhiro Hatsugai 2010 New J. Phys. 12 065004
• 「グラフェンの物理」初貝安弘, 青木秀夫, 固体物理535号(2010)
• Topological aspect of graphene (compact summary): Y Hatsugai 2011 J. Phys.: Conf. Ser. 334 012004
• Edge states of graphene near Zigzag edges: M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, pp. 1920-1923 (1996)
• Zero modes of d-wave superconductors : Chia-Ren Hu, Phys. Rev. Lett. 72, 1526 – Published 7 March 1994, Yukio Tanaka and Satoshi Kashiwaya, Phys. Rev. Lett. 74, 3451 – Published 24 April 1995
• Topological origin of chiral symmetric edge states and Z₂ quantization of Zak (Berry) phase (Su-Schrieffer-Heeger model, d-wave superconductor and graphene): Shinsei Ryu and Yasuhiro Hatsugai, Phys. Rev. Lett. 89, 077002 – Published 26 July 2002, Y. Hatsugai, Solid State Communications, Volume 149, Issues 27–28, July 2009, Pages 1061-1067
• Z₂ quantization Berry phases and its extension with Time-reversal symmetry: Yasuhiro Hatsugai 2010 New J. Phys. 12 065004
• Chiral symmetry breaking in graphene: Shinsei Ryu and Yasuhiro Hatsugai, Physica E: Low-dimensional Systems and Nanostructures Volume 22, Issues 1–3, April 2004, Pages 679-683
• Majorana bound states in a quantum wire (theory): A Yu Kitaev 2001 Phys.-Usp. 44 131
• Majorana bound states (signatures): V. Mourik et al., Science 25 May 2012, Vol. 336, Issue 6084, pp. 1003-1007

6. Berry connections

Black board lecture

reference

• Berry phases (Codimension of the gap closing is three): M. V. Berry, F. R. S., Proc. R. Soc. Lond. A 392, 45-57 (1984)
• Chern number and its quantization: M. Kohmoto, Annals of Physics, Volume 160, Issue 2, 1 April 1985, Pages 343-354
• On the generic gauge fixing: Y. Hatsugai, J. Phys. Soc. Jpn. 73, pp. 2604-2607 (2004)
• チャーン数の量子化の解説: [やさしい説明]
• Berry phases and gauge fixing: Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)
• Dirac monopole and Yang monopole: Yasuhiro Hatsugai 2010 New J. Phys. 12 065004
• ディラック単磁極の明示的な導出: 「物理学における線形代数：量子力学での展開」 （ベリー接続としての単磁極など）雑誌 数理科学5 月号, p15-p22 (2017年）初貝安弘

7. Topological insulators

[Talk file (pdf)]

reference

• Theoretical proposal : C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 – Published 23 November 2005, Phys. Rev. Lett. 95, 146802 – Published 28 September 2005, B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, Science 15 Dec 2006, Vol. 314, Issue 5806, pp. 1757-1761
• First experimental realization: M. König et al., Science 02 Nov 2007, Vol. 318, Issue 5851, pp. 766-770
• Review: M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 – Published 8 November 2010

8. Topological phases without quanum mechanics

reference

• Edge states of photonic crystals (theory): F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 – Published 10 January 2008, Zheng Wang, Y. D. Chong, John D. Joannopoulos, and Marin Soljacic, Phys. Rev. Lett. 100, 013905 – Published 10 January 2008
• Edge states of photonic crystals (experiment): Zheng Wang, Yidong Chong, J. D. Joannopoulos & Marin Soljacic
• Photonic Weyl points, section Chern numbers and edge states: Shuhei Oono, Toshikaze Kariyado, and Yasuhiro Hatsugai, Phys. Rev. B 94, 125125 – Published 14 September 2016
• Imaging of edge states: M. Hafezi, S. Mittal, J. Fan, A. Migdall & J. M. Taylor, Nature Photonics volume 7, pages 1001–1005 (2013)
• Edge states of photons: Mohammad Hafezi, Phys. Rev. Lett. 112, 210405 – Published 29 May 2014
• Mechanical graphene: Toshikaze Kariyado & Yasuhiro Hatsugai, Scientific Reports volume 5, Article number: 18107 (2015)
• Mechanical diamond: Yuta Takahashi, Toshikaze Kariyado & Yasuhiro Hatsugai, 2017 New J. Phys. 19 035003
• Topological mechanics (special articles): Focus on Topological Mechanics

9. Topological pumping

[Talk file (pdf)]

reference

• Thouless pumping (Theoretical proposal just after TKNN): D. J. Thouless, Phys. Rev. B 27, 6083 – Published 15 May 1983
• Experimental realization (Kyoto): Shuta Nakajima, Takafumi Tomita, Shintaro Taie, Tomohiro Ichinose, Hideki Ozawa, Lei Wang, Matthias Troyer & Yoshiro Takahashi, Nature Physics volume 12, pages 296–300 (2016)
• Experimental realization (Munich): M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger & I. Bloch, Nature Physics volume 12, pages 350–354 (2016)
• Bulk-edge correspondence in topological pumping: Y. Hatsugai and T. Fukui, Phys. Rev. B 94, 041102(R) – Published 1 July 2016

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