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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!