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

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?

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?

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.

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 Quantum spin liquids HgTe quantum well 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 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 ? ?

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: 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

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

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

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

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 =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 Landau Levels Quantum Mechanics EF C=1 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 g=0 g=1 from Charlie Kane’s website

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 =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 Landau Levels Quantum Mechanics EF C=1 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 Landau Levels Quantum Mechanics EF C=1 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 A schematic band structure Gap

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

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…… ) 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)

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)

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

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

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

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

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: Topological ground state degeneracy Fractional statistics sphere E Gap E Gap torus V.S. 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: Topological ground state degeneracy Fractional statistics sphere E Gap E Gap torus V.S. 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: Topological ground state degeneracy Fractional statistics sphere E Gap E Gap torus V.S. 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: Topological ground state degeneracy Fractional statistics sphere E Gap E Gap torus V.S. These features can be used to characterize different phases.

Generalized “Fractional phases”

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

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 Topological ground state deg. Fractional statistics

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

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

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”

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

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

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?

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)

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

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……

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

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

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

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

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

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 ? )

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

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

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. Flat band is slightly dispersive (further neighbor hopping) Xiao,Zhu,YR et.al, (2011)

What if the nearly flat band is partially filled?

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

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)

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)

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

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)

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)

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)

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)

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 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)

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)

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 Without S-O With S-O + correlation

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: The successful synthesis of (111) bilayer LaAlO3/LaNiO3/LaAlO3 heterostructure was reported recently: Yang,YR,et.al (2011) Also Fiete et.al, (2011)

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)

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!

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!

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

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….)

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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) £ ….

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

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

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

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

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

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

Example: new phases 2-spatial dimension: SG=Z2 (Ising symmetry), GG=Z2 £ Z2 H3(SG£ GG, U(1) ) = Z27 ---128 phases. Mesaros, YR (2012) But H3(SG,U(1))=Z2 H3(GG,U(1))=Z23  There are Z23 new indices in the “bulk”

Example: new phases 2-spatial dimension: 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 Mesaros, YR (2012) E E Ising sym. protect two-fold degeneracy! degeneracy lifted if sym. is broken

Example: new phases 2-spatial dimension: 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 Mesaros, YR (2012) In order to obtain results like this …. E E Ising sym. protect two-fold degeneracy! 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 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)

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)

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!