Dec. 10, 2008 at KITP (3,1) (1,1) Bulk-Edge correspondence and Fractionalization As a topological (spin) insulator with strong interaction � Y. Hatsugai i γ C ( A ψ ) = A ψ C Institute of Physics Univ. of Tsukuba JAPAN
Plan With time reversal invariance Berry phase for a topological order parameter Z 2 Fractionalization for the Bulk in 1D & 2D Entanglement Entropy to detect edge states (effective) Description by the Edges : Fractionalization at the Edges in 1D deconfined spinons in 2D & 3D ?? Time Reversal operators with interaction Θ 2 = 1 , or Global to Local : super-selection rule − 1 Let us consider Gapped spin liquid as a topological insulator with strong interaction
Quantum Liquids without Symmetry Breaking Quantum Liquids in Low Dimensional Quantum Systems Low Dimensionality, Quantum Fluctuations No Symmetry Breaking Topological Order X.G.Wen No Local Order Parameter Various Phases & Quantum Phase Transitions Gapped Quantum Liquids in Condensed Matter Integer & Fractional Quantum Hall States Dimer Models of Fermions and Spins Integer spin chains Valence bond solid (VBS) states Half filled Kondo Lattice How to understand gapped quantum liquids ?
How to understand gapped quantum liquids ?
How to understand gapped quantum liquids ? Bulk classically featureless : need geometrical phase
How to understand gapped quantum liquids ? Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN
How to understand gapped quantum liquids ? Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN low energy localized modes in the gap Edge
How to understand gapped quantum liquids ? Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN low energy localized modes in the gap Edge edge states for QHE Laughlin, Halperin, YH
How to understand gapped quantum liquids ? Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN low energy localized modes in the gap Edge edge states for QHE Laughlin, Halperin, YH
How to understand gapped quantum liquids ? Bulk-Edge correspondence Common property of topological ordered states Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN low energy localized modes in the gap Edge edge states for QHE Laughlin, Halperin, YH
How to understand gapped quantum liquids ? Bulk-Edge correspondence Common property of topological ordered states Bulk classically featureless : need geometrical phase 1-st Chern number for QHE TKNN low energy localized modes in the gap Edge edge states for QHE Laughlin, Halperin, YH As for quantum spins Z 2 Berry Phase as a Topological Order Parameter of bulk Entanglement Entropy to detect edge states (generic Kennedy triplets)
Quantum Liquid (Example 1) The RVB state by Anderson 1 √ | Singlet Pair 12 � = 2( | ↑ 1 ↓ 2 � − | ↓ 1 ↑ 2 � ) � | G � = c J ⊗ ij | Singlet Pair ij � J =Dimer Covering Local Singlet Pairs : (Basic Objects) Purely Quantum Objects are basic Purely Quantum Objects are basic
Quantum Liquid (Example 1) The RVB state by Anderson 1 √ | Singlet Pair 12 � = 2( | ↑ 1 ↓ 2 � − | ↓ 1 ↑ 2 � ) � | G � = c J ⊗ ij | Singlet Pair ij � J =Dimer Covering Spins disappear Local Singlet Pairs : as a Singlet pair (Basic Objects) Purely Quantum Objects are basic Purely Quantum Objects are basic
Quantum Liquid (Example 2) The RVB state by Pauling 1 1 2( c † 1 + c † √ √ | Bond 12 � = 2( | 1 � + | 2 � ) = 2 ) | 0 � � | G � = c J ⊗ ij | Bond ij � J =Dimer Covering Do Not use the Fermi Sea Local Covalent Bonds : (Basic Objects) Purely Quantum Objects are basic Purely Quantum Objects are basic
Quantum Liquid (Example 2) The RVB state by Pauling 1 1 2( c † 1 + c † √ √ | Bond 12 � = 2( | 1 � + | 2 � ) = 2 ) | 0 � � | G � = c J ⊗ ij | Bond ij � J =Dimer Covering Do Not use the Fermi Sea Delocalized charge as a covalent bond Local Covalent Bonds : (Basic Objects) Purely Quantum Objects are basic Purely Quantum Objects are basic
Quantum Interference for the Classification “Classical” Observables Charge density, Spin density,... O = n ↑ ± n ↓ , · · · � O � G = � G |O| G � = � G ′ |O| G ′ � = � O � G ′ | G ′ � = | G � e i φ “Quantum” Observables ! Quantum Interferences: 2 � e i ( φ 1 − φ 2 ) � G 1 | G 2 � = � G ′ 1 | G ′ Probability Ampliture (overlap) i � e i φ i | G i � = | G ′ Aharonov-Bohm Effects Phase (Gauge) dependent :Berry Connection A = � G | dG � � G | G + dG � = 1 + � G | dG � � :Berry Phase i γ = A Use Quantum Interferences To Classify Quantum Liquids
Examples: RVB state by Anderson 1 √ | Singlet Pair 12 � = 2( | ↑ 1 ↓ 2 � − | ↓ 1 ↑ 2 � ) � | G � = c J ⊗ ij | Singlet Pair ij � J =Dimer Covering Spins disappear No Long Range Order as a Singlet pair in Spin-Spin Correlation Local Singlet Pair is a Basic Object How to Characterize the Local Singlet Pair ? 1 √ | G � = 2( | ↑ i ↓ j � − | ↓ i ↑ j � ) Use Berry Phase to characterize the Singlet! Singlet does not carries spin but does Berry phase γ singlet pair = π mod 2 π
Examples: RVB state by Anderson 1 √ | Singlet Pair 12 � = 2( | ↑ 1 ↓ 2 � − | ↓ 1 ↑ 2 � ) � | G � = c J ⊗ ij | Singlet Pair ij � J =Dimer Covering Spins disappear No Long Range Order as a Singlet pair in Spin-Spin Correlation Local Singlet Pair is a Basic Object How to Characterize the Local Singlet Pair ? 1 √ | G � = 2( | ↑ i ↓ j � − | ↓ i ↑ j � ) Use Berry Phase to characterize the Singlet! Singlet does not carries spin but does Berry phase γ singlet pair = π mod 2 π
Z 2 Berry phases for gapped quantum spins generic Heisenberg Models (with frustration) � J ij S i · S j H = ij Time Reversal Invariant Θ N S i Θ − 1 N = − S i [ H, Θ N ] = 0 y ) · · · ( i σ N Θ N = ( i σ 1 y ) ⊗ ( i σ 2 y ) K Θ 2 N = ( − ) N Mostly N: even Θ 2 N = 1 (probability 1/2 in HgTe)
Z 2 Berry phases for gapped quantum spins Define a many body hamiltonian by local twist as a parameter H ( x = e i θ ) C = { x = e i θ | θ : 0 → 2 π } U(1) S i · S j → 1 2( e − i θ S i + S j − + e + i θ S i − S j + ) + S iz S jz Only link <ij> numerically Calculate the Berry Phases using the Entire Many Spin Wavefunction Require excitation Gap! Z 2 quantization � � � 0 Z 2 � ψ | d ψ � = A ψ = γ C = : mod 2 π π C C Time Reversal ( Anti-Unitary ) Invariance
� � � Berry Connection and Gauge Transformation Parameter Dependent Hamiltonian H(x) (x) (x) (x) = H ( x ) | ψ ( x ) � = E ( x ) | ψ ( x ) � , � ψ ( x ) | ψ ( x ) � = 1. Berry Connections A ψ = � ψ | d ψ � = � ψ | d dx ψ � dx . � Berry Phases i γ C ( A ψ ) = A ψ (Abelian) Phase Ambiguity of the eigen state C | ψ ′ ( x ) � e i Ω ( x ) | ψ ( x ) � = Gauge Transformation ψ + id Ω A ′ ψ + id Ω = A ′ = A ψ dx dx Berry phases are not well-defined without � specifying the gauge γ C ( A ψ ) = γ C ( A ψ ′ ) + d Ω C 2 π × (integer) if e i Ω is single valued Well Defined up to mod 2 π γ C ( A ψ ) ≡ γ C ( A ψ ′ ) mod 2 π
Anti-Unitary Operator and Berry Phases Anti-Unitary Operator (Time Reversal, Particle-Hole) K : Complex conjugate Θ = KU Θ , U Θ : Unitary (parameter independent) � � | Ψ � = C J | J � J C J = � Ψ | Ψ � = 1 C ∗ J J � | Ψ Θ � = Θ | Ψ � = J | J Θ � , | J Θ � = Θ | J � C ∗ J Berry Phases and Anti-Unitary Operation � A Ψ = � Ψ | d Ψ � = C ∗ J dC J � � dC ∗ C ∗ J C J + J dC J =0 J J J � A ΘΨ = � Ψ Θ | d Ψ Θ � = J = − A Ψ C J dC ∗ J γ C ( A ΘΨ ) = − γ C ( A Ψ )
Anti-Unitary Invariant State and Z 2 Berry Phase Anti-Unitary Symmetry [ H ( x ) , Θ ] = 0 Invariant State ∃ ϕ , | Ψ Θ � = Θ | Ψ � = | Ψ � e i ϕ ex. Unique Eigen State ≃ | Ψ � Gauge Equivalent(Different To be compatible with the ambiguity, Gauge) the Berry Phases have to be quantized as � 0 γ C ( A Ψ ) = Z 2 Berry phase mod 2 π π γ C ( A Ψ ) = − γ C ( A ΘΨ ) ≡ − γ C ( A Ψ ) , mod2 π
Numerical Evaluation of the Berry Phases (incl. non-Abelian) (1) Discretize the periodic parameter space θ 0 = 0 , θ N = 2 π x 0 , x 1 , · · · , x N = x 0 ∀ ∆ θ n → 0 x n = e i θ n θ n +1 = θ n + ∆ θ n (2) Obtain eigen vectors H ( x n ) | ψ i n � = E i ( x n ) | ψ i n � (3) Define Berry connection in a discretized form A n = Im log � ψ n | ψ n +1 � n | ψ j A n = Im log det D n , { D n } ij = � ψ i non-Abelian n +1 � (4) Evaluate the Berry phase non-Abelian N − 1 ( ) � = Im log � ψ 0 | ψ 1 �� ψ 1 | ψ � · · · = Im log det D 1 D 2 · · · D n A n γ = n =0 | ψ n � → | ψ n � ′ e i Ω n Independent of the choice of the phase Gauge invariant Luscher ’82 (Lattice Gauge Theory) King-Smith & Vanderbilt ’93 (polarization in solids) after the discretization Convenient for Numerics T. Fukui, H. Suzuki & YH ’05 (Chern numbers)
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