Topology and exotic orders in quantum solids
Ying Ran
Boston College
ITP, CAS, June 2013
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?
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?
The “Standard Model” of condensed matter
Landau’s Fermi
Liquid (metals)
Landau Theory of
broken symmetry.
• Different phases are characterized by different symmetries.
• Emergent Laudau order parameter
• Successfully describes a large set of phenomena in solids
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 ….
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 ….
Beyond the “Standard Model” in solids?
• Previously, violations only in “extreme conditions”
one dimension, 2DEG in strong magnetic field
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
HgTe quantum well
• Quantum spin liquids
Bi2Se3
dmit organic salts
Herbertsmithite
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
HgTe quantum well
• Quantum spin liquids
Bi2Se3
• Topological superconductors
dmit organic salts
Herbertsmithite
• Fractional Chern insulators
--fractional quantum hall states in solids
in the absence of magnetic field
?
?
So far
not realized in experiments
• With a growing list of topological phases, it may be helpful to
organize them in a certain way --- a zoology.
• 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
• 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
• To perform generalization, helpful to review the key features of
integer/fractional quantum hall phases
--- Why we call them topological phases?
Integer quantum hall phases
2DEG in a magnetic field
E
Quantum
Mechanics
Landau Levels
Integer quantum hall phases
2DEG in a magnetic field
E
Quantum
Mechanics
EF
Landau Levels
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)
Chern number = Integral of Berry’s curvatures of wavefunctions
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)
Chern number = Integral of Berry’s curvatures of wavefunctions
Analogy:
genus g (number of handles).
Integral of Gaussian curvature: K
 K dS  4 (1  g )
g=0
g=1
from Charlie Kane’s website
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
Band insulator
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
• Characteristic gapless edge modes
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
• Characteristic gapless edge modes
--- Similar features in generalized phases
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,….
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
Gap
A schematic band structure
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)
Gap
A schematic band structure
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
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)
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)
• Other examples: topological superconductors, bosonic analogs ….
Symmetry protected topological 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)
• Other examples: topological superconductors, bosonic analogs ….
The modern view of gapped quantum phases
Landau phases
Ising
ferromagnet
Ising
paramagnet
+ Topological
Phases
….
“Standard model”
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
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
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?
Fractional quantum hall phases
E
Integer plateaus
Quantum
Mechanics
EF
C=1
Landau Levels
Fractional quantum hall phases
E
Quantum
Mechanics
EF
C=1
Landau Levels
• A partially filled Laudau level: C=1 flat band
?
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
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
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
• Topological ground state degeneracy
E
E
Gap
Gap
sphere
V.S.
torus
Wen,Niu 1990
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:
Fractional statistics
• Topological ground state degeneracy
E
E
Gap
Gap
sphere
V.S.
torus
• Wavefunctions locally identical
• Local perturbations cannot lift degeneracy
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:
Fractional statistics
• Topological ground state degeneracy
E
E
Gap
Gap
sphere
V.S.
torus
• Wavefunctions locally identical
• Local perturbations cannot lift degeneracy
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:
Fractional statistics
• Topological ground state degeneracy
E
E
Gap
Gap
sphere
V.S.
torus
These features can be used to characterize different phases.
Generalized “Fractional phases”
Generalized “Fractional phases”
Examples:
• Gapped quantum spin liquids
Candidate material
Herbertsmithite:
Gapless or a small gap?
(Helton,Lee,McQueen,Nocera,Broholm,….)
-- Mott insulators without any symmetry breaking
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
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
Generalized “Fractional phases”
Examples:
• Gapped quantum spin liquids
• Fractional Chern insulators
--fractional quantum hall states in solids
in the absence of magnetic field
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
Fractional statistics
• Topological ground state deg.
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
Fractional statistics
• Topological ground state deg.
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
Fractional statistics
• Topological ground state deg.
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
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
“Standard model”
•
•
•
Fractional
Chern
insulator
….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
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
“Standard model”
Will come back to this later
•
•
•
Fractional
Chern
insulator
….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
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?
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)
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...
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
Motivation
• What about d-orbital?
Motivation
• What about d-orbital?
Correlation-driven physics:
e.g., various symmetry breaking phases
• Superconductivity
• Magnetism
• Ferroelectricity
….
Motivation
• What about d-orbital?
Correlation-driven physics:
e.g., various symmetry breaking phases
• Superconductivity
• Magnetism
• Ferroelectricity
….
+ TI physics ?
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
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
I will show:
Certain transition metal oxide heterostructures could host:
• room-temperature 2D TI phases
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
E
Gap
EF
+Spin-orbit
coupling
k
EF
k
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…
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
• 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…)
Z
Y
Y
X
X
Square lattice of transition metal atoms
• I will show that, heterostructures grown along the (111) direction
are particularly interesting for topological phases of matter.
Perovskite (111)-bilayer
Example:
LaAlO3
substrate
• d-electrons hopping on a buckled honeycomb lattice
LaAuO3
LaAlO3
substrate
(111) direction
Perovskite (111)-bilayer
Example:
LaAlO3
substrate
• d-electrons hopping on a buckled honeycomb lattice
Graphene-like band structure?
LaAuO3
LaAlO3
substrate
(111) direction
Perovskite (111)-bilayer
Example:
LaAlO3
substrate
• d-electrons hopping on a buckled honeycomb lattice
• Naturally give semi-metallic band structure
--Similar physics to graphene?
(“correlated versions” of graphene ? )
LaAuO3
LaAlO3
substrate
(111) direction
d-electrons in a crystal
eg
Octahedral
Crystal field
d-orbitals
5x2=10 states
t2g
d-electrons in a crystal
Example
eg
Au 3+:
8 electrons in 5d orbitals
Octahedral
Crystal field
 eg orbitals half-filled
d-orbitals
5x2=10 states
t2g
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)
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)
Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer
• Tight-binding band structure with Spin-Orbital coupling
EF
Xiao,Zhu,YR et.al, (2011)
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)
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)
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)
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)
• Flat band is slightly dispersive (further neighbor hopping)
What if the nearly flat band is partially filled?
What if the nearly flat band is partially filled?
• Lesson from FQHE:
E
fractional plateaus
EF
C=1
Landau Levels
Partially filled topological flat band (Laudau level) + Correlation:
Fractional topological phases
What if the nearly flat band is partially filled?
• Fractional topological phases?
+ Correlation
EF
---Realizable: e.g., electron-doped SrTiO3/SrPtO3/SrTiO3
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Fractional topological phases?
+ Correlation
EF
---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….)
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Our calculations:
+ Correlation
EF
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Our calculations:
Correlation
Ferromagnetism
EF
EF
nearly flat band with C=1
--- analogy of Landau level
In the absence of mag. field.
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Our calculations:
Correlation
Ferromagnetism
EF
EF
With realistic interactions…
And choose band-filling º=1/3
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Our calculations:
Correlation
Ferromagnetism
EF
EF
Exact diagonalization simulations show:
(1) 3-fold ground state degeneracy on torus
(2) Quantized hall conductance:
(many-body Chern number =1/3)
Xiao,Zhu,YR et.al, (2011)
What if the nearly flat band is partially filled?
• Our calculations:
Correlation
Ferromagnetism
EF
EF
Numerical signatures of 1/3-Laughlin fractional Chern insulator:
(1) 3-fold ground state degeneracy on torus
(2) Quantized hall conductance:
(many-body Chern number =1/3)
Xiao,Zhu,YR et.al, (2011)
Why fractional Chern insulators are interesting?
Why fractional Chern insulators are interesting?
• For practical purpose:
High-temperature FQHE without magnetic field
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?
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)
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)
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
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)
• Is it possible to realize nearly-flat C>1 bands in materials?
Nearly flat C=2 bands: SrIrO3 (111) trilayer
• Trilayer: transition metal atoms form a Dice-lattice.
Wang,YR (2011)
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
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
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
+ correlation
Without S-O
With S-O
What about experiments?
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)
Experiment progress!
• Motivated by our theoretical investigations on possible
correlation-driven topological phases in LaNiO3 (111) bilayer:
Yang,YR,et.al (2011)
Also Fiete et.al, (2011)
• The successful synthesis of (111) bilayer LaAlO3/LaNiO3/LaAlO3
heterostructure was reported recently:
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)
Motivation
• Crystals: lessons from the “standard model”
Cool down
Crystal
Liquid
Crystals = spontaneous breaking of translational symmetry
• symmetry group theory allows systematic understandings:
230 space groups
All realized in nature!
Motivation
• Crystals: lessons from the “standard model”
Cool down
Crystal
Liquid
Crystals = spontaneous breaking of translational symmetry
• symmetry group theory allows systematic understandings:
230 space groups
All realized in nature!
• What is the “group theory” for topological phases?
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
“Standard model”
•
•
•
Fractional
Chern
insulator
….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
The modern view of gapped quantum phases
New mathematics introduced
for systematic understandings
symmetry protected topological phases
Group
cohomology
K-theory
….
Landau phases
•
•
Group theory
Ordinary bulk excitation
Symmetry-protected gapless edge modes
+ Topological
Phases
….
“Standard model”
•
(Wen, Kitaev, Levin, Senthil, Turner, Pollmann,Chen, •
•
Gu, Vishwanath, Lu, Ryu, Schnyder, Ludwig….)
entanglement protected topological phases
Gauge
theory
Tensor
Category
….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
The modern view of gapped quantum phases
symmetry protected topological phases
Group
cohomology
• What about topological phases protected
by both symmetry AND entanglement?
•
•
K-theory
….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
entanglement protected topological phases
Gauge
theory
•
•
•
Tensor
Category
….
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
The modern view of gapped quantum phases
symmetry protected topological phases
Group
cohomology
• What about topological phases protected
by both symmetry AND entanglement?
• Directly relevant to physical systems
•
•
K-theory
….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
sym. and entanglement show up together:
entanglement protected topological phases
e.g.:
Quantum spin liquid --- spin rotation sym.
Gauge
theory
Fractional Chern insulator --- lattice sym.
Tensor
Category
….
Lu, YR, PRB (2012)
•
•
•
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
The modern view of gapped quantum phases
symmetry protected topological phases
Group
cohomology
• What about topological phases protected
by both symmetry AND entanglement?
• Directly relevant to physical systems
•
•
K-theory
….
Ordinary bulk excitation
Symmetry-protected gapless edge modes
sym. and entanglement show up together:
entanglement protected topological phases
e.g.:
Quantum spin liquid --- spin rotation sym.
Gauge
theory
Fractional Chern insulator --- lattice sym.
Tensor
Category
….
Lu, YR, PRB (2012)
• How to glue the two pieces together?
•
•
•
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
Roughly speaking, my picture is like:
• The space of topological phases:
entanglement protected topological phases
Gauge
theory
•
•
•
Tensor
category
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
….
Roughly speaking, my picture is like:
• The space of topological phases:
entanglement but no sym.
entanglement protected topological phases
Gauge
theory
•
•
•
Tensor
category
Anyon bulk excitation
Topological ground state degeneracy
Robust even without any symmetry
….
Roughly speaking, my picture is like:
• The space of topological phases:
entanglement but no sym.
entanglement protected topological phases
Roughly speaking, my picture is like:
• The space of topological phases:
Systematic understanding in the “bulk”
--- Mission impossible?
entanglement protected topological phases
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
A classification of topological phases
protected by both symmetry and entanglement
Mesaros, YR (2012)
• Assumptions:
(1) bosonic gapped quantum phases (e.g. quantum spin systems)
(2) A local symmetry group SG. (e.g. spin rotations)
(3) Entanglement described by a gauge group GG
A classification of topological phases
protected by both symmetry and entanglement
Mesaros, YR (2012)
• Assumptions:
(1) bosonic gapped quantum phases (e.g. quantum spin systems)
(2) 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))
A classification of topological phases
protected by both symmetry and entanglement
Mesaros, YR (2012)
• Assumptions:
(1) bosonic gapped quantum phases (e.g. quantum spin systems)
(2) 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.
A classification of topological phases
protected by both symmetry and entanglement
Mesaros, YR (2012)
• Assumptions:
(1) bosonic gapped quantum phases (e.g. quantum spin systems)
(2) 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.
A math theorem
Künneth formula:
= H3(SG, U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1)) £ H2(SG, GG) £ ….
A math theorem
Künneth formula:
= H3(SG, U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1)) £ H2(SG, GG) £ ….
Now has full physical meaning
SG: symmetry
GG: entanglement
New understanding in the “bulk”
=
H3(SG,
U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1)) £ H2(SG, GG) £ ….
Mesaros, YR (2012)
entanglement protected topological phases
New understanding in the “bulk”
=
H3(SG,
U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1))
Mesaros, YR (2012)
entanglement protected topological phases
New understanding in the “bulk”
=
H3(SG,
U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1))
Mesaros, YR (2012)
entanglement protected topological phases
New understanding in the “bulk”
=
H3(SG,
U(1)) £
H3(SG£GG, U(1))
H3(GG, U(1))
Mesaros, YR (2012)
entanglement protected topological phases
Dijkgraaf-Witten, 1990
New understanding in the “bulk”
=
H3(SG,
U(1)) £
H3(SG£GG, 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
Dijkgraaf-Witten, 1990
Example: new phases
• 2-spatial dimension:
Mesaros, YR (2012)
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z27
But H3(SG,U(1))=Z2
---128 phases.
H3(GG,U(1))=Z23  There are Z23 new indices in the “bulk”
Example: new phases
• 2-spatial dimension:
Mesaros, YR (2012)
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z27
---128 phases.
• Among them, we identify phases hosting new kinds of interplay
between symmetry and entanglement.
Consequence: new phenomena
For excited state with two quasiparticles
E
Ising sym. protect
two-fold degeneracy!
E
degeneracy lifted
if sym. is broken
Example: new phases
• 2-spatial dimension:
Mesaros, YR (2012)
SG=Z2 (Ising symmetry), GG=Z2 £ Z2
H3(SG£ GG, U(1) ) = Z27
---128 phases.
• Among them, we identify phases hosting new kinds of interplay
between symmetry and entanglement.
Consequence: new phenomena
For excited state with two quasiparticles
In order to obtain results like this ….
E
Ising sym. protect
two-fold degeneracy!
E
degeneracy lifted
if sym. is broken
new kinds of calculations …
• Solving models in a geometric fashion:
Models:
Solution of models:
Mesaros, YR (2012)
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
Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011)
Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011)
Wang, YR, PRB Rapid Commun. (2011)
Theoretically:
• introducing new framework, new methods
--- e.g. systematic understanding of topological phases protected by both sym. and entanglement
Lu, YR, PRB (2012)
Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)
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
Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011)
Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011)
Wang, YR, PRB Rapid Commun. (2011)
Theoretically:
• introducing new framework, new methods
--- e.g. systematic understanding of topological phases protected by both sym. and entanglement
Lu, YR, PRB (2012)
Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)
In addition, finding new detectable signatures and developing new experimental
probes are also very important.
Acknowledgement
• 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!