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

2. Zoology of topological quantum phases in solids
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. Zoology of topological quantum phases in solids
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
Different phases are characterized by different symmetries.
Emergent Laudau order parameter
Successfully describes a large set of phenomena in solids
Landau’s Fermi Liquid (metals)
Landau Theory of broken symmetry.

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

6. First topological phases: IQHE and FQHE
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!
2D electron gas in a strong magnetic field
--- Quantized Hall conductance:
von Klitzing, Tsui, Stormer, Laughlin ….

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

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

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

11. In fact, all topological quantum phases can be viewed as:
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:
Generalizations of integer quantum hall phases
Generalizations of fractional quantum hall phases

12. In fact, all topological quantum phases can be viewed as:
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 key features of integer/fractional quantum hall phases
--- Why we call them topological phases?
Generalizations of integer quantum hall phases
Generalizations of fractional quantum hall phases

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

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

15. Integer quantum hall phases: key features
2DEG in a magnetic field E
Landau Levels
Quantum
Mechanics EF
C=1
Landau levels are energy bands with non-trivial topology:
Chern number C = 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
Landau Levels
Quantum
Mechanics EF
C=1
Landau levels are energy bands with non-trivial topology:
Chern number C = 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
g=0
g=1
from Charlie Kane’s website

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

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

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

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

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

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

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

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

25. Generalized “integer phases”
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 ….
Key features:
(1) Band insulator --- ordinary gapped bulk excitations
(2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer)
(3) Characteristic gapless edge modes
(2), (3) protected by time-reversal symmetry
---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities)

26. Symmetry protected topological phases
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 ….
Key features:
(1) Band insulator --- ordinary gapped bulk excitations
(2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer)
(3) Characteristic gapless edge modes
(2), (3) protected by time-reversal symmetry
---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities)

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 ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
Landau phases
Ising
ferromagnet
Ising
paramagnet
+ Topological
Phases ….
“Standard model”
Generalization of FQH phases

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

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

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

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

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

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

36. Fractional quantum hall phases: Key features
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
Fractional statistics
sphere E
Gap E
Gap
torus
V.S.
Wavefunctions locally identical
Local perturbations cannot lift degeneracy

37. Fractional quantum hall phases: Key features
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
Fractional statistics
sphere E
Gap E
Gap
torus
V.S.
Wavefunctions locally identical
Local perturbations cannot lift degeneracy

38. Fractional quantum hall phases: Key features
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
Fractional statistics
sphere E
Gap E
Gap
torus
V.S.
These features can be used to characterize different phases.

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”
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. E
Gap
torus

42. Generalized “Fractional phases”
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 E
Gap
torus

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”
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
anyon excitations
Topological ground state deg.
Fractional statistics

45. Generalized “Fractional phases”
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
anyon excitations
Topological ground state deg.
Fractional statistics

46. entanglement protected topological phases
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
anyon excitations
Topological ground state deg.
Fractional statistics

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

48. The modern view of gapped quantum phases
symmetry protected topological phases
Top. Insulator
Top. superconductor ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
Landau phases
Ising
ferromagnet
Ising
paramagnet
+ 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 ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
Landau phases
Ising
ferromagnet
Ising
paramagnet
+ 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. Zoology of topological quantum phases in solids
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?

51. Zoology of topological quantum phases in solids
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?
Searching for topological phases in transition metal oxide heterostructures
Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011)
Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011)
Wang, YR, PRB Rapid Commun. (2011)

52. Motivation A growing family of topological insulators:
‣ CdHgTe/HgTe/CdHgTe (Bernevig et al, Science 2006, Konig et al, Science
2007)
‣ Bi1-xSbx (Fu and Kane, PRB 2007, Hsieh et al, Nature 2008)
‣ Bi2Se3, Bi2Te3, Sb2Te3 (Zhang et al, Nat Phys 2009, Xia et al, Nat Phys
2009, Chen et al, Science 2009)
‣ TlBiTe2 and TlBiSe2 (Lin et al, PRL 2010, Yan et al, EPL 2010, Sato et al,
PRL 2010, Chen et al, PRL 2011)
‣ Half-heuslers, Chalcopyrites (Lin et al, Nat Mat. 2010, Chadov et al, Nat Mat
2010, Xiao et al, PRL, 2010, Feng et al, PRL 2010)
‣ Many more...

53. Motivation A growing family of topological insulators:
‣ CdHgTe/HgTe/CdHgTe (Bernevig et al, Science 2006, Konig et al, Science
2007)
‣ Bi1-xSbx (Fu and Kane, PRB 2007, Hsieh et al, Nature 2008)
‣ Bi2Se3, Bi2Te3, Sb2Te3 (Zhang et al, Nat Phys 2009, Xia et al, Nat Phys
2009, Chen et al, Science 2009)
‣ TlBiTe2 and TlBiSe2 (Lin et al, PRL 2010, Yan et al, EPL 2010, Sato et al,
PRL 2010, Chen et al, PRL 2011)
‣ Half-heuslers, Chalcopyrites (Lin et al, Nat Mat. 2010, Chadov et al, Nat Mat
2010, Xiao et al, PRL, 2010, Feng et al, PRL 2010)
‣ Many more...
---they are all s/p-orbital electronic systems

54. Motivation
What about d-orbital?

55. Motivation What about d-orbital? Correlation-driven physics:
e.g., various symmetry breaking phases
Superconductivity
Magnetism
Ferroelectricity ….

56. Motivation What about d-orbital? Correlation-driven physics:
e.g., various symmetry breaking phases
Superconductivity
Magnetism
Ferroelectricity ….
+ TI physics ?

57. Motivation What about d-orbital? Correlation-driven physics:
e.g., various symmetry breaking states
Superconductivity
Magnetism
Ferroelectricity ….
+ TI physics ?
Novel applications of TI physics require proximity effects between TIs and symmetry-breaking states.
(e.g., magnetoelectric effects, Majorana fermions)
New regime: interplay between Mott physics and TI physics

58. Motivation What about d-orbital? Correlation-driven physics:
e.g., various symmetry breaking states
Superconductivity
Magnetism
Ferroelectricity ….
+ TI physics ?
Novel applications of TI physics require proximity effects between TIs and symmetry-breaking states.
(e.g., magnetoelectric effects, Majorana fermions)
New regime: interplay between Mott physics and TI physics

59. I will show:
Certain transition metal oxide heterostructures could host:
room-temperature 2D TI phases

60. Certain transition metal oxide heterostructures could host:
I will show:
Certain transition metal oxide heterostructures could host:
room-temperature 2D TI phases
and much more than that:
quantum anomalous hall insulator,
abelian/non-abelian fractional Chern insulators,
Dirac half-semimetal, quantum spin liquids……

61. Certain transition metal oxide heterostructures could host:
I will show:
Certain transition metal oxide heterostructures could host:
room-temperature 2D TI phases
and much more than that:
quantum anomalous hall insulator,
abelian/non-abelian fractional Chern insulators,
Dirac half-semimetal, quantum spin liquids……
Lesson from previous TI materials (HgTe, Bi2Se3…):
semi-metal + spin-orbit interaction: generates (inverts) the band gap. E k EF E k EF
Gap
+Spin-orbit
coupling

62. Heterostructures of transition metal oxides
Layered structure can be prepared with atomic precision
Great flexibility: tunable lattice constant, carrier concentration, spin-orbit interaction, correlation strength...
Correlation physics of d-orbitals: Mott physics, magnetism, superconductivity…

63. Crystal structure Current technology focus on perovskites ABO3.
Experimental efforts are mainly on interface/hetero-structures grown along the (001) direction
For example, superconductivity
is found on STO/LAO interface
Perovskite structure of SrTiO3
Reyren et al, Science 2007

64. This is partially because the current efforts are on (001) direction.
Previously, possible topological phases have not been investigated in TMOH.
This is partially because the current efforts are on (001) direction.
(square lattice---large fermi surface, or large band gap…)
I will show that, heterostructures grown along the (111) direction are particularly interesting for topological phases of matter. Z X Y Y X
Square lattice of transition metal atoms

65. Perovskite (111)-bilayer
Example:
LaAuO3
LaAlO3
substrate
LaAlO3
substrate
(111) direction
d-electrons hopping on a buckled honeycomb lattice

66. Perovskite (111)-bilayer
Example:
LaAuO3
LaAlO3
substrate
LaAlO3
substrate
(111) direction
d-electrons hopping on a buckled honeycomb lattice
Graphene-like band structure?

67. Perovskite (111)-bilayer
Example:
LaAuO3
LaAlO3
substrate
LaAlO3
substrate
(111) direction
d-electrons hopping on a buckled honeycomb lattice
Naturally give semi-metallic band structure
--Similar physics to graphene?
(“correlated versions” of graphene ? )

68. d-electrons in a crystal
d-orbitals
5x2=10 states
Octahedral
Crystal field eg
t2g

69. d-electrons in a crystal
d-orbitals
5x2=10 states
Octahedral
Crystal field
Example
Au 3+:
8 electrons in 5d orbitals
 eg orbitals half-filled eg
t2g

70. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Tight-binding band structure without Spin-Orbital coupling EF
Interestingly, similar to graphene + 2 flat bands.
The exact flatness of these bands are consequence of the nearest neighbor model.
Further neighbor hoppings destroy the exact flatness.
Xiao,Zhu,YR et.al, (2011)

71. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Tight-binding band structure without Spin-Orbital coupling EF
Interestingly, similar to graphene + 2 flat bands.
Can S-O coupling generate topological gap?
similar to graphene (Kane&Mele 2005)…
Xiao,Zhu,YR et.al, (2011)

72. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Tight-binding band structure with Spin-Orbital coupling EF
Xiao,Zhu,YR et.al, (2011)

73. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Tight-binding band structure with Spin-Orbital coupling
with S-O coupling:
Gapless edge states EF EF
LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a 2D TI.
Xiao,Zhu,YR et.al, (2011)

74. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Tight-binding band structure with Spin-Orbital coupling EF EF
LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a 2D TI.
Topological band is nearly flat!
Xiao,Zhu,YR et.al, (2011)

75. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Comparing with first principle calculation:
Tight-binding analysis
First-principle calculation
LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a room-temp. 2D TI.
Xiao,Zhu,YR et.al, (2011)

76. Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
Comparing with first principle calculation:
Tight-binding analysis
First-principle calculation
LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a room-temp. 2D TI.
Flat band is slightly dispersive (further neighbor hopping)
Xiao,Zhu,YR et.al, (2011)

77. What if the nearly flat band is partially filled?

78. What if the nearly flat band is partially filled?
Lesson from FQHE:
Partially filled topological flat band (Laudau level) + Correlation:
Fractional topological phases E
fractional plateaus
Landau Levels EF
C=1

79. What if the nearly flat band is partially filled?
Fractional topological phases?
---Realizable: e.g., electron-doped SrTiO3/SrPtO3/SrTiO3 EF
+ Correlation
Xiao,Zhu,YR et.al, (2011)

80. What if the nearly flat band is partially filled?
Fractional topological phases?
---Realizable: e.g., electron-doped SrTiO3/SrPtO3/SrTiO3
Our calculations show signature of fractional quantum hall effects!
--- FQHE in the absence of magnetic field,
has been called “fractional Chern insulator”
(Mudry, Chamon,Tang, Wen, Sun, Sheng, Gu,Bernevig, Fiete, 2011….) EF
+ Correlation
Xiao,Zhu,YR et.al, (2011)

81. What if the nearly flat band is partially filled?
Our calculations: EF
+ Correlation
Xiao,Zhu,YR et.al, (2011)

82. What if the nearly flat band is partially filled?
Our calculations: EF
Correlation
Ferromagnetism EF
nearly flat band with C=1
--- analogy of Landau level
In the absence of mag. field.
Xiao,Zhu,YR et.al, (2011)

83. What if the nearly flat band is partially filled?
Our calculations:
With realistic interactions… EF
Correlation
Ferromagnetism EF
And choose band-filling º=1/3
Xiao,Zhu,YR et.al, (2011)

84. What if the nearly flat band is partially filled?
Our calculations:
Exact diagonalization simulations show: EF
Correlation
Ferromagnetism EF
(1) 3-fold ground state degeneracy on torus
(2) Quantized hall conductance:
(many-body Chern number =1/3)
Xiao,Zhu,YR et.al, (2011)

85. What if the nearly flat band is partially filled?
Our calculations:
Numerical signatures of 1/3-Laughlin fractional Chern insulator: EF
Correlation
Ferromagnetism EF
(1) 3-fold ground state degeneracy on torus
(2) Quantized hall conductance:
(many-body Chern number =1/3)
Xiao,Zhu,YR et.al, (2011)

86. Why fractional Chern insulators are interesting?

87. Why fractional Chern insulators are interesting?
For practical purpose:
High-temperature FQHE without magnetic field

88. Why fractional Chern insulators are interesting?
For practical purpose:
High-temperature FQHE without magnetic field
For fundamental science’s purpose:
Are there intrinsically new regime/phases?

89. Why fractional Chern insulators are interesting?
For practical purpose:
High-temperature FQHE without magnetic field
For fundamental science’s purpose:
Are there intrinsically new regime/phases?
Yes:
A band structure can have Chern number C >1 bands
--- no analog in Landau Level (C=1)
Intrinsically new regime:
Partially filled nearly flat C>1 bands + correlation
Lu,YR(2011)

90. Why fractional Chern insulators are interesting?
For practical purpose:
High-temperature FQHE without magnetic field
For fundamental science’s purpose:
Are there intrinsically new regime/phases?
Yes:
A band structure can have Chern number C >1 bands
--- no analog in Landau Level (C=1)
Intrinsically new regime:
Partially filled nearly flat C>1 bands + correlation
e.g., the natural counterpart of 1/3-Laughlin state in C=2 band is non-Abelian
--- SU(3)1 Abelian Chern-Simons theory SU(3)2 non-Abelian Chern-Simons theory
Lu,YR(2011)

91. Why fractional Chern insulators are interesting?
For practical purpose:
High-temperature FQHE without magnetic field
For fundamental science’s purpose:
Are there intrinsically new regime/phases?
Yes:
A band structure can have Chern number C >1 bands
--- no analog in Landau Level (C=1)
Intrinsically new regime:
Partially filled nearly flat C>1 bands + correlation
Is it possible to realize nearly-flat C>1 bands in materials?
Lu,YR(2011)

92. Nearly flat C=2 bands: SrIrO3 (111) trilayer
Trilayer: transition metal atoms form a Dice-lattice.
Wang,YR (2011)

93. Nearly flat C=2 bands: SrIrO3 (111) trilayer
Trilayer: transition metal atoms form a Dice-lattice.
Wang,YR (2011)
Dice lattice is known to support flat band. (one of the many Lieb’s theorems)
Without S-O
With S-O

94. Nearly flat C=2 bands: SrIrO3 (111) trilayer
Trilayer: transition metal atoms form a Dice-lattice.
Wang,YR (2011)
Dice lattice is known to support flat band. (one of the many Lieb’s theorems)
The spin degenerate flat band is half-filled.
--- correlation-driven ferromagnetism
Without S-O
With S-O

95. Nearly flat C=2 bands: SrIrO3 (111) trilayer
Trilayer: transition metal atoms form a Dice-lattice.
Wang,YR (2011)
Dice lattice is known to support flat band. (one of the many Lieb’s theorems)
The spin degenerate flat band is half-filled.
--- correlation-driven ferromagnetism
Our calculations show:
ferromagnetism  nearly-flat C=2 bands
Without S-O
With S-O
+ correlation

96. What about experiments?

97. What about experiments?
Motivated by our theoretical investigations on possible correlation-driven topological phases in LaNiO3 (111) bilayer:
---Strongly correlated 3d electrons on honeycomb lattice
Within realistic regime, we identified:
Dirac half-semimetal (spinless graphene)
Quantum anomalous hall insulator
Yang,YR,et.al (2011)
Also Fiete et.al, (2011)

98. Experiment progress!
Motivated by our theoretical investigations on possible correlation-driven topological phases in LaNiO3 (111) bilayer:
The successful synthesis of (111) bilayer LaAlO3/LaNiO3/LaAlO3 heterostructure was reported recently:
Yang,YR,et.al (2011)
Also Fiete et.al, (2011)

99. Zoology of topological quantum phases in solids
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?
Lu,YR, PRB (2012)
Mesaros,YR (arXiv. Dec. 2012, to appear on PRB 2013)

100. Motivation Crystals: lessons from the “standard model”
Crystals = spontaneous breaking of translational symmetry
symmetry group theory allows systematic understandings:
Liquid
Cool down
Crystal
230 space groups
All realized in nature!

101. Motivation Crystals: lessons from the “standard model”
Crystals = spontaneous breaking of translational symmetry
symmetry group theory allows systematic understandings:
What is the “group theory” for topological phases?
Liquid
Cool down
Crystal
230 space groups
All realized in nature!

102. The modern view of gapped quantum phases
symmetry protected topological phases
Top. Insulator
Top. superconductor ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
Landau phases
Ising
ferromagnet
Ising
paramagnet
+ 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

103. The modern view of gapped quantum phases
New mathematics introduced
for systematic understandings
symmetry protected topological phases
Group cohomology
K-theory ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
Landau phases
Group theory
+ Topological
Phases ….
entanglement protected topological phases
Gauge theory
Tensor Category ….
“Standard model”
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
(Wen, Kitaev, Levin, Senthil, Turner, Pollmann,Chen,
Gu, Vishwanath, Lu, Ryu, Schnyder, Ludwig….)

104. The modern view of gapped quantum phases
symmetry protected topological phases
Group cohomology
K-theory ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
What about topological phases protected
by both symmetry AND entanglement?
entanglement protected topological phases
Gauge theory
Tensor Category ….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry

105. The modern view of gapped quantum phases
symmetry protected topological phases
Group cohomology
K-theory ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
What about topological phases protected
by both symmetry AND entanglement?
Directly relevant to physical systems
sym. and entanglement show up together:
e.g.:
Quantum spin liquid --- spin rotation sym.
Fractional Chern insulator --- lattice sym.
entanglement protected topological phases
Gauge theory
Tensor Category ….
Lu, YR, PRB (2012)
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry

106. The modern view of gapped quantum phases
symmetry protected topological phases
Group cohomology
K-theory ….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
What about topological phases protected
by both symmetry AND entanglement?
Directly relevant to physical systems
sym. and entanglement show up together:
e.g.:
Quantum spin liquid --- spin rotation sym.
Fractional Chern insulator --- lattice sym.
How to glue the two pieces together?
entanglement protected topological phases
Gauge theory
Tensor Category ….
Lu, YR, PRB (2012)
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry

107. Roughly speaking, my picture is like:
The space of topological phases:
Group cohomology
K-theory ….
Symmetry-protected gapless edge modes
Ordinary bulk excitation
symmetry protected topological phases
Gauge theory
Tensor category
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
entanglement protected topological phases

108. Roughly speaking, my picture is like:
The space of topological phases:
entanglement but no sym.
Group cohomology
K-theory ….
Symmetry-protected gapless edge modes
Ordinary bulk excitation
symmetry protected topological phases
Gauge theory
Tensor category
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
entanglement protected topological phases
sym. but no entanglement

109. Roughly speaking, my picture is like:
The space of topological phases:
entanglement but no sym.
entanglement protected topological phases
sym. but no entanglement
symmetry protected topological phases ?
topological Phases protected
By both symmetry and entanglement

110. Roughly speaking, my picture is like:
The space of topological phases:
Systematic understanding in the “bulk”
--- Mission impossible?
entanglement protected topological phases
symmetry protected topological phases ?
topological Phases protected
By both symmetry and entanglement

111. Roughly speaking, my picture is like:
The space of topological phases:
Systematic understanding in the “bulk”
--- Mission impossible?
--- at least we can provide answers to a certain level
entanglement protected topological phases
symmetry protected topological phases ?
topological Phases protected
By both symmetry and entanglement

112. A classification of topological phases protected by both symmetry and entanglement
Assumptions:
bosonic gapped quantum phases (e.g. quantum spin systems)
A local symmetry group SG. (e.g. spin rotations)
(3) Entanglement described by a gauge group GG
Mesaros, YR (2012)

113. A classification of topological phases protected by both symmetry and entanglement
Assumptions:
bosonic gapped quantum phases (e.g. quantum spin systems)
A local symmetry group SG. (e.g. spin rotations)
(3) Entanglement described by a gauge group GG
In d-spatial dimension, topological phases protected by sym. SG and entanglement GG are classified by the (d+1)th cohomology group:
Hd+1(SG£GG, U(1))
Mesaros, YR (2012)

114. A classification of topological phases protected by both symmetry and entanglement
Assumptions:
bosonic gapped quantum phases (e.g. quantum spin systems)
A local symmetry group SG. (e.g. spin rotations)
(3) Entanglement described by a gauge group GG
In d-spatial dimension, topological phases protected by sym. SG and entanglement GG are classified by the (d+1)th cohomology group:
Hd+1(SG£GG, U(1))
For example, when SG=Z2 (Ising symmetry) and GG=Z2,
H3(Z2£ Z2, U(1))=Z23
---in 2-spatial dimension,
8 Ising paramagnetic phases whose entanglement described by Z2 gauge group.
Mesaros, YR (2012)

115. A classification of topological phases protected by both symmetry and entanglement
Assumptions:
bosonic gapped quantum phases (e.g. quantum spin systems)
A local symmetry group SG. (e.g. spin rotations)
(3) Entanglement described by a gauge group GG
In d-spatial dimension, topological phases protected by sym. SG and entanglement GG are classified by the (d+1)th cohomology group:
Hd+1(SG£GG, U(1))
And we provide exactly solvable models realizing each phase in the classification.
Mesaros, YR (2012)

116. A math theorem H3(SG£GG, U(1))
Künneth formula:
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….

117. Now has full physical meaning
A math theorem
Künneth formula:
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Now has full physical meaning
SG: symmetry
GG: entanglement

118. New understanding in the “bulk”
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
entanglement protected topological phases
symmetry protected topological phases
topological Phases protected
By both symmetry and entanglement

119. New understanding in the “bulk”
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
entanglement protected topological phases
symmetry protected topological phases
topological Phases protected
By both symmetry and entanglement

120. New understanding in the “bulk”
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
entanglement protected topological phases
symmetry protected topological phases
Chen, et.al, 2011
topological Phases protected
By both symmetry and entanglement

121. New understanding in the “bulk”
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
entanglement protected topological phases
symmetry protected topological phases
Dijkgraaf-Witten, 1990
Chen, et.al, 2011
topological Phases protected
By both symmetry and entanglement

122. New understanding in the “bulk”
H3(SG£GG, U(1))
= H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
New results: Characterizing different interplays between symmetry and entanglement
entanglement protected topological phases
symmetry protected topological phases
Dijkgraaf-Witten, 1990
Chen, et.al, 2011
topological Phases protected
By both symmetry and entanglement

123. Example: new phases 2-spatial dimension:
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z phases.
Mesaros, YR (2012)
But H3(SG,U(1))=Z2 H3(GG,U(1))=Z23  There are Z23 new indices in the “bulk”

124. Example: new phases 2-spatial dimension:
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z phases.
Among them, we identify phases hosting new kinds of interplay between symmetry and entanglement.
Consequence: new phenomena
For excited state with two quasiparticles
Mesaros, YR (2012) E E
Ising sym. protect
two-fold degeneracy!
degeneracy lifted
if sym. is broken

125. Example: new phases 2-spatial dimension:
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z phases.
Among them, we identify phases hosting new kinds of interplay between symmetry and entanglement.
Consequence: new phenomena
For excited state with two quasiparticles
Mesaros, YR (2012)
In order to obtain results like this …. E E
Ising sym. protect
two-fold degeneracy!
degeneracy lifted
if sym. is broken

126. new kinds of calculations …
Solving models in a geometric fashion:
Models:
Solution of models:
Mesaros, YR (2012)

127. Summary Topological quantum phases are beyond the “standard model”.
There are many different kinds of them. Some have been realized.
Experimentally:
Progress on new materials would be essential.
--- e.g. searching for topological phases in transition metal oxide heterostructures
Theoretically:
introducing new framework, new methods
--- e.g. systematic understanding of topological phases protected by both sym. and entanglement
Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011)
Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011)
Wang, YR, PRB Rapid Commun. (2011)
Lu, YR, PRB (2012)
Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)

128. Summary Topological quantum phases are beyond the “standard model”.
There are many different kinds of them. Some have been realized.
Experimentally:
Progress on new materials would be essential.
--- e.g. searching for topological phases in transition metal oxide heterostructures
Theoretically:
introducing new framework, new methods
--- e.g. systematic understanding of topological phases protected by both sym. and entanglement
In addition, finding new detectable signatures and developing new experimental probes are also very important.
Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011)
Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011)
Wang, YR, PRB Rapid Commun. (2011)
Lu, YR, PRB (2012)
Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)

129. Acknowledgement Thank you!
Oak Ridge National Lab: Di Xiao(CMU), Satoshi Okamoto, Wenguang Zhu
MIT: Fa Wang ( PKU)
Tokyo Univ.: Naoto Nagaosa
Boston College: Yuan-Ming Lu (UC Berkeley), Bing Ye, Kaiyu Yang, Andrej Mesaros, Ziqiang Wang
Thank you!