Classification of Symmetry Classification of Symmetry Classification of Symmetry Classification of Symmetry Protected Topological Phases Protected Topological Phases Protected Topological Phases Protected Topological Phases in Interacting Systems in Interacting Systems in Interacting Systems in Interacting Systems Zhengcheng Gu (PI) Zhengcheng Gu (PI) Zhengcheng Gu (PI) Zhengcheng Gu (PI) Collaborators: Collaborators: Collaborators: Collaborators: Prof. Xiao- G ang Wen ( PI/ MIT) Prof. M. Levin (U. of Chicago ) Prof. Xiao- Prof. Xiao- Prof. Xiao-G Gang Wen ( G ang Wen (PI/ ang Wen ( PI/ PI/MIT) MIT) MIT) Prof. Prof. Prof. M. Levin M. Levin M. Levin (U. of (U. of (U. of Chicago Chicago Chicago) ) ) Dr. Xie Chen Dr. Xie Chen (UC Berkeley) (UC Berkeley) Dr. Dr. Zheng-Xin Liu Zheng-Xin Liu ( ( Tsinghua U. Tsinghua U. ) ) Dr. Xie Chen Dr. Xie Chen(UC Berkeley) (UC Berkeley) Dr. Dr. Zheng-Xin Liu Zheng-Xin Liu( (Tsinghua U. Tsinghua U.) ) Vienna. Aug Vienna. Aug . 201 . 201 4 4 Vienna. Aug Vienna. Aug. 201 . 2014 4
Outline Outline Outline Outline � Intrinsic topological(IT) order and symmetry Intrinsic topological(IT) order and symmetry Intrinsic topological(IT) order and symmetry Intrinsic topological(IT) order and symmetry protected topological(SPT) order. protected topological(SPT) order. protected topological(SPT) order. protected topological(SPT) order. 1D SPT phases in interacting bosonic systems. 1D SPT phases in interacting bosonic systems. � 1D SPT phases in interacting bosonic systems. 1D SPT phases in interacting bosonic systems. � 2D and 3D SPT phases in interacting bosonic 2D and 3D SPT phases in interacting bosonic 2D and 3D SPT phases in interacting bosonic 2D and 3D SPT phases in interacting bosonic systems. systems. systems. systems. SPT phases in interacting fermionic systems. � SPT phases in interacting fermionic systems. SPT phases in interacting fermionic systems. SPT phases in interacting fermionic systems. Summ Summ a a ry and outlook ry and outlook . . � Summ Summa ary and outlook ry and outlook. .
Topological phenomena in strongly Topological phenomena in strongly Topological phenomena in strongly Topological phenomena in strongly correlated systems correlated systems correlated systems correlated systems Fractional Quantum Hall Fractional Quantum Hall Effect Effect Fractional Quantum Hall Fractional Quantum Hall Effect Effect D C Tsui, et al 1982 P W Anderson, 1987 Hong Ding, et al ,1996 N P Ong's group, 2000 Spin liquid Spin liquid Spin liquid Spin liquid � Frustrated magnets SL? SL? SL? SL? � High-Tc cuprates AF SC AF AF AF SC SC SC S Yan, D Huse and S White Science, 2011
New phases of matter: intrinsic New phases of matter: intrinsic New phases of matter: intrinsic New phases of matter: intrinsic t t opological opological o o rder rder t topological opological o order rder X.-G. Wen,1989 � Can have the same symmetry as disordered systems. � Gapped ground state without long range correlations. � Ground state degeneracy depends on the topology of the manifold. � Ground state degeneracy is robust against any local perturbations. � Excitations carry fractional statistics. � Protected chiral edge states(chiral topological order, e.g. FQHE).
Topological terms for intrinsic Topological terms for intrinsic Topological terms for intrinsic Topological terms for intrinsic t t opological opological o o rder rder t topological opological o order rder FQHE FQHE FQHE FQHE R B Laughlin 1983 E Witten, 1989 S C Zhang, et al 1989 X G Wen, et al 1989 Z Z spin liquid spin liquid Z Z 2 2 spin liquid spin liquid R. Moessner and S. L. Sondhi 2001 2 2
Symmetry protected topological(SPT) Symmetry protected topological(SPT) Symmetry protected topological(SPT) Symmetry protected topological(SPT) phenomena phenomena phenomena phenomena Topological insulator in 2D/3D Topological insulator in 2D/3D Topological insulator in 2D/3D Topological insulator in 2D/3D C L Kane, et al , 2005 B A Bernevig, et al 2006 W Molenkamp's group 2007 M Zahid Hasan, et al , 2008 from Wikipedia
New phases of matter: symmetry New phases of matter: symmetry New phases of matter: symmetry New phases of matter: symmetry protected t protected t opological opological o o rder rder protected t protected topological opological o order rder Z C Gu and X G Wen, 2009 � Can have the same symmetry as trivial disordered systems. � Gapped ground state without long range correlations. � Excitations do not carry fractional statistics. � Indistinguishable from trivial disordered systems if symmetry is broken in bulk. � Stable against any local perturbations preserving symmetry. � Protected gapless edge states if symmetry is not (spontaneously or explicitly)broken on the edge.
SPT phase in strongly interacting 1D model SPT phase in strongly interacting 1D model SPT phase in strongly interacting 1D model SPT phase in strongly interacting 1D model Spin one Haldane chain realizes 1D topological order(even with strong interaction) � stable up to U~1 But Haldane phase requires symmetry protection! But Haldane phase requires symmetry protection! But Haldane phase requires symmetry protection! But Haldane phase requires symmetry protection! � Haldane phase can be protected by many kinds of symmetries: time reversal, spin rotation, etc... Z C Gu and X G Wen, 2009, F Pollmann, et al , 2010 Fixed point wavefunction: spin-(1/2,1/2) dimer model Fixed point wavefunction: spin-(1/2,1/2) dimer model Fixed point wavefunction: spin-(1/2,1/2) dimer model Fixed point wavefunction: spin-(1/2,1/2) dimer model Z C Gu and X G Wen, 2009 The key observation: edge states form projective representation of the symmetry group!
A revisit of transverse Ising model: A revisit of transverse A revisit of transverse A revisit of transverse Ising Ising Ising model: model: model:
A n example of Ising SPT phase in 2D A An A n example of n example of example of Ising Ising Ising SPT phase SPT phase in 2D SPT phase in 2D in 2D How many different paramagnetic phases? Two! Two! Two! Two! (M. Levin and Z.-C. Gu, Phys. Rev. B 86, 115109 (2012)) Domain deformation rule Domain deformation rule Domain deformation rule Domain deformation rule But why not? But why not? But why not? But why not?
Topologically consistent condition for fixed Topologically consistent condition for fixed Topologically consistent condition for fixed Topologically consistent condition for fixed point wavefunction point wavefunction point wavefunction point wavefunction
Duality between Ising model and Duality between Ising model and Duality between Ising model and Duality between Ising model and Z Z gauge model gauge model Z Z 2 2 gauge model gauge model 2 2 Duality map requires Z Duality map requires Z Duality map requires Z Duality map requires Z 2 2 2 2 symmetry to be symmetry to be symmetry to be symmetry to be preserved! preserved! preserved! preserved! � String condensation corresponds to domain wall condensation
Z Z gauge model(toric code model) gauge model(toric code model) Z Z 2 2 gauge model(toric code model) gauge model(toric code model) 2 2 Kitaev 2003, M. Levin and X.G. Wen 2005 Ground state Ground state Ground state Ground state
Topological properties Topological properties Topological properties Topological properties The same topological order as Z The same topological order as Z spin liquid spin liquid The same topological order as Z The same topological order as Z 2 2 spin liquid spin liquid 2 2 Four-fold ground Four-fold ground Four-fold ground Four-fold ground e e e e state degeneracy state degeneracy state degeneracy state degeneracy on a torus on a torus on a torus on a torus m m m m Quasi-particle in toric code model: Quasi-particle in toric code model: Quasi-particle in toric code model: Quasi-particle in toric code model: 1, e, m, f=em 1, e, m, f=em 1, e, m, f=em 1, e, m, f=em
Dehn twist and T matrix Dehn twist and T matrix Dehn twist and T matrix Dehn twist and T matrix Dehn twist: Dehn twist: Dehn twist: Dehn twist: from Wikipedia T matrix: T matrix: T matrix: T matrix:
The twisted toric code: double semion The twisted toric code: double semion The twisted toric code: double semion The twisted toric code: double semion model model model model M. Levin and X.G. Wen 2005 T matrix: T matrix: T matrix: T matrix: � End of string is a semion or anti-semion. Quasi-particle types in double semion model: Quasi-particle types in double semion model: Quasi-particle types in double semion model: Quasi-particle types in double semion model: 1, s, 1, s, s s , b=s , b=s s s 1, s, 1, s, s s, b=s , b=ss s
Dual theory of double semion model Dual theory of double semion model Dual theory of double semion model Dual theory of double semion model The dual theory of double semion model The dual theory of double semion model The dual theory of double semion model The dual theory of double semion model is an SPT ordered phase! is an SPT ordered phase! is an SPT ordered phase! is an SPT ordered phase! Different SPT orders Different SPT orders Different SPT orders Different SPT orders Different (intrinsic) Different (intrinsic) Different (intrinsic) Different (intrinsic) M Levin and Z.-C. Gu (Phys. Rev. B 86, 115109 (2012)) topological orders topological orders topological orders topological orders
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