Focus lecture at Kyushu Univ. (2018)

“Discovery of topological phases and development “: Focused lectures at the Kyushu University, organized by Prof. K. Nomura; July17-19, (2018)

Outline: Basic parts of topological phases are discussed in this course. It includes discussion of quantum Hall effects, Haldane phases, graphene, topological insulators, Weyl semimetals and topological phases of photonic crystals and classical mechanics. The focuses are put on the Berry connection, symmetry protection and bulk-edge correspondence.

Content of the lectures (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

Topological phases: Popular Science Background, Noble prize in physics 2016 (@nobelprize.org)

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

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
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)
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
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)
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

7. Topological insulators

[Talk file (pdf)]

reference

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

Q. Problems for the credit ： [problem]