Topology and exotic orders in quantum solids

1. Topology and exotic orders in quantum solids Ying Ran Boston College ITP, CAS, June 2013

2. This talk is about: • Zoology of topological quantum phases in solids Introduction and overview. • How to realize them in materials? where to look for them? what kind of new materials? • How to systematically understand them? New theoretical framework?

3. This talk is about: • Zoology of topological quantum phases in solids Introduction and overview. • How to realize them in materials? where to look for them? what kind of new materials? • How to systematically understand them? New theoretical framework?

4. The “Standard Model” of condensed matter Landau Theory of broken symmetry. Landau’s Fermi Liquid (metals) Different phases are characterized by different symmetries. Emergent Laudau order parameter Successfully describes a large set of phenomena in solids

5. First topological phases: IQHE and FQHE 2D electron gas in a strong magnetic field --- Quantized Hall conductance: von Klitzing, Tsui, Stormer, Laughlin …. In 1980’s, integer/fractional quantum hall phases

6. First topological phases: IQHE and FQHE 2D electron gas in a strong magnetic field --- Quantized Hall conductance: von Klitzing, Tsui, Stormer, Laughlin …. In 1980’s, integer/fractional quantum hall phases --- striking counterexamples of the “Standard Model”: All have the same symmetry, yet there are many different phases!

7. Beyond the “Standard Model” in solids? Previously, violations only in “extreme conditions” one dimension, 2DEG in strong magnetic field

8. Beyond the “Standard Model” in solids? • Topological insulators • Quantum spin liquids Bi2Se3 Herbertsmithite dmit organic salts HgTe quantum well Previously, violations only in “extreme conditions” one dimension, 2DEG in strong magnetic field New patterns of emergence in solids e.g.

9. Beyond the “Standard Model” in solids? • Topological insulators • Topological superconductors • Quantum spin liquids • Fractional Chern insulators • --fractional quantum hall states in solids • in the absence of magnetic field • So far • not realized in experiments Bi2Se3 Herbertsmithite dmit organic salts HgTe quantum well ? ? Previously, violations only in “extreme conditions” one dimension, 2DEG in strong magnetic field New patterns of emergence in solids e.g.

10. With a growing list of topological phases, it may be helpful to organize them in a certain way --- a zoology.

11. Generalizations of integer quantum hall phases • Generalizations of fractional quantum hall phases With a growing list of topological phases, it may be helpful to organize them in a certain way --- a zoology. In fact, all topological quantum phases can be viewed as:

12. Generalizations of integer quantum hall phases • Generalizations of fractional quantum hall phases With a growing list of topological phases, it may be helpful to organize them in a certain way --- a zoology. In fact, all topological quantum phases can be viewed as: To perform generalization, helpful to review the keyfeatures of integer/fractional quantum hall phases --- Why we call them topological phases?

13. Integer quantum hall phases 2DEG in a magnetic field E Quantum Mechanics Landau Levels

14. Integer quantum hall phases 2DEG in a magnetic field E Quantum Mechanics EF Landau Levels

15. Integer quantum hall phases: key features 2DEG in a magnetic field E Quantum Mechanics EF C=1 Landau Levels • Landau levels are energy bands with non-trivialtopology: Chern number C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982) Chern number = Integral of Berry’s curvatures of wavefunctions

16. Integer quantum hall phases: key features 2DEG in a magnetic field E Quantum Mechanics EF C=1 Landau Levels g=0 g=1 • Landau levels are energy bands with non-trivialtopology: Chernnumber C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982) Chern number = Integral of Berry’s curvatures of wavefunctions Analogy: genus g (number of handles). Integral of Gaussian curvature: K from Charlie Kane’s website

17. Integer quantum hall phases: key features 2DEG in a magnetic field E Quantum Mechanics EF C=1 Landau Levels Band insulator Landau levels are energy bands with non-trivialtopology: Chernnumber C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982) IQH phases are band insulators: ordinary gapped bulk excitations

18. Integer quantum hall phases: key features 2DEG in a magnetic field E Quantum Mechanics EF C=1 Landau Levels • Landau levels are energy bands with non-trivial topology: Chern number C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982) • IQH phases are band insulators: ordinary gapped bulk excitations • Characteristicgapless edge modes

19. Integer quantum hall phases: key features 2DEG in a magnetic field E Quantum Mechanics EF C=1 Landau Levels --- Similar features in generalized phases • Landau levels are energy bands with non-trivial topology: Chern number C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982) • IQH phases are band insulators: ordinary gapped bulk excitations • Characteristicgapless edge modes

20. Generalized “integer phases” 2D TI: HgTe quantum well 3D TI: Bi2Se3, Bi2Te3,…. Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

21. Generalized “integer phases” • Key features: • (1) Band insulator --- ordinary gapped bulk excitations Gap A schematic band structure Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

22. Generalized “integer phases” • Key features: • (1) Band insulator --- ordinary gapped bulk excitations • (2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer) Gap A schematic band structure Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

23. Generalized “integer phases” • Key features: • (1) Band insulator --- ordinary gapped bulk excitations • (2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer) • (3) Characteristicgapless edge modes Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

24. Generalized “integer phases” • Key features: • (1) Band insulator --- ordinary gapped bulk excitations • (2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer) • (3) Characteristicgapless edge modes • (2), (3) protected by time-reversal symmetry • ---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities) Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

25. Generalized “integer phases” • Key features: • (1) Band insulator --- ordinary gapped bulk excitations • (2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer) • (3) Characteristicgapless edge modes • (2), (3) protected by time-reversal symmetry • ---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities) Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… ) Other examples: topological superconductors, bosonic analogs ….

26. Symmetry protected topological phases • Key features: • (1) Band insulator --- ordinary gapped bulk excitations • (2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer) • (3) Characteristicgapless edge modes • (2), (3) protected by time-reversal symmetry • ---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities) Examples: Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… ) Other examples: topological superconductors, bosonic analogs ….

27. The modern view of gapped quantum phases Landau phases Ising ferromagnet Ising paramagnet + Topological Phases …. “Standard model”

28. The modern view of gapped quantum phases Generalization of IQH phases Landau phases Ising ferromagnet Ising paramagnet + Topological Phases …. “Standard model” Generalization of FQH phases

29. The modern view of gapped quantum phases symmetry protected topological phases Top. Insulator Top. superconductor …. Landau phases Ising ferromagnet Ising paramagnet • Ordinary bulk excitation • Symmetry-protected gapless edge modes + Topological Phases …. “Standard model” Generalization of FQH phases

30. The modern view of gapped quantum phases symmetry protected topological phases Top. Insulator Top. superconductor …. Landau phases Ising ferromagnet Ising paramagnet • Ordinary bulk excitation • Symmetry-protected gapless edge modes + Topological Phases …. “Standard model” Generalization of FQH phases Key features?

31. Fractional quantum hall phases E Integer plateaus Quantum Mechanics EF C=1 Landau Levels

32. Fractional quantum hall phases E Quantum Mechanics ? EF C=1 Landau Levels A partially filled Laudau level: C=1 flat band

33. Fractional quantum hall phases E fractional plateaus Quantum Mechanics EF C=1 Landau Levels A partially filled Laudau level: C=1 flat band Electron-electron Coulomb interactions lift degeneracy  Fractional quantum hall phases

34. Fractional quantum hall phases: Key features Fractional statistics Apart from the quantized hall conductance NOT a band insulator in the bulk anyon excitations with a finite gap:

35. Fractional quantum hall phases: Key features Fractional statistics V.S. E E torus sphere Gap Gap Wen,Niu 1990 Apart from the quantized hall conductance NOT a band insulator in the bulk anyon excitations with a finite gap: Topological ground state degeneracy

36. Fractional quantum hall phases: Key features Fractional statistics V.S. E E torus sphere • Wavefunctions locally identical • Local perturbations cannot lift degeneracy Gap Gap Robust towards any local perturbations! DO NOT require symmetry NOT a band insulator in the bulk anyon excitations with a finite gap: Topological ground state degeneracy

37. Fractional quantum hall phases: Key features Fractional statistics V.S. E E torus sphere • Wavefunctions locally identical • Local perturbations cannot lift degeneracy Gap Gap Robust towards any local perturbations! DO NOT require symmetry Protected by long-range quantum entanglement NOT a band insulator in the bulk anyon excitations with a finite gap: Topological ground state degeneracy

38. Fractional quantum hall phases: Key features Fractional statistics V.S. E E torus sphere Gap Gap These features can be used to characterize different phases. Robust towards any local perturbations! DO NOT require symmetry Protected by long-range quantum entanglement NOT a band insulator in the bulk anyon excitations with a finite gap: Topological ground state degeneracy

39. Generalized “Fractional phases”

40. Generalized “Fractional phases” Candidate material Herbertsmithite: Gapless or a small gap? (Helton,Lee,McQueen,Nocera,Broholm,….) Examples: Gapped quantum spin liquids -- Mott insulators without any symmetry breaking

41. Generalized “Fractional phases” E torus Gap Examples: Gapped quantum spin liquids -- Mott insulators without any symmetry breaking Hastings’ Theorem (2004): A gapped quantum spin liquid has ground state deg. on torus.

42. Generalized “Fractional phases” E torus Gap Examples: Gapped quantum spin liquids -- Mott insulators without any symmetry breaking Hastings’ Theorem (2004): A gapped quantum spin liquid has ground state deg. on torus. But by definition of QSL, not due to symmetry breaking due to long-range entanglement

43. Generalized “Fractional phases” Examples: • Gapped quantum spin liquids • Fractional Chern insulators --fractional quantum hall states in solids in the absence of magnetic field

44. Generalized “Fractional phases” Fractional statistics • anyon excitations • Topological ground state deg. Examples: • Gapped quantum spin liquids • Fractional Chern insulators --fractional quantum hall states in solids in the absence of magnetic field shared key features: protected by long-range quantum entanglement, do not require any symmetry

45. Generalized “Fractional phases” Fractional statistics • anyon excitations • Topological ground state deg. Examples: • Gapped quantum spin liquids • Fractional Chern insulators --fractional quantum hall states in solids in the absence of magnetic field shared key features: can be used to characterize different phases protected by long-range quantum entanglement, do not require any symmetry

46. entanglement protected topological phases Fractional statistics • anyon excitations • Topological ground state deg. Examples: • Gapped quantum spin liquids • Fractional Chern insulators --fractional quantum hall states in solids in the absence of magnetic field shared key features: can be used to characterize different phases protected by long-range quantum entanglement, do not require any symmetry

47. The modern view of gapped quantum phases symmetry protected topological phases Top. Insulator Top. superconductor …. Landau phases Ising ferromagnet Ising paramagnet • Ordinary bulk excitation • Symmetry-protected gapless edge modes + Topological Phases …. • Generalization of FQH phases “Standard model”

48. The modern view of gapped quantum phases symmetry protected topological phases Top. Insulator Top. superconductor …. Landau phases Ising ferromagnet Ising paramagnet • Ordinary bulk excitation • Symmetry-protected gapless edge modes + Topological Phases …. entanglement protected topological phases Gapped spin liquid Fractional Chern insulator …. “Standard model” • Anyon bulk excitation • Topological ground state degeneracy • Robust even without any symmetry

49. The modern view of gapped quantum phases symmetry protected topological phases Top. Insulator Top. superconductor …. Landau phases Ising ferromagnet Ising paramagnet • Ordinary bulk excitation • Symmetry-protected gapless edge modes + Topological Phases …. entanglement protected topological phases Gapped spin liquid Fractional Chern insulator …. “Standard model” • Anyon bulk excitation • Topological ground state degeneracy • Robust even without any symmetry Will come back to this later

50. This talk is about: • Zoology of topological quantum phases in solids Introduction and overview. • How to realize them in materials? where to look for them? what kind of new materials? • How to systematically understand them? New theoretical framework?