Berry Phases with Time Reversal Invariance Characterization of - PowerPoint PPT Presentation

June 26, 2008 YITP, TASSP (3,1) (1,1) Berry Phases with Time Reversal Invariance Characterization of Gapped Quantum Liquids & LSM theorem Quantum interferences in Many body states � Y. Hatsugai γ C � = A ψ = � ψ | C d Institute of Physics ψ � C Univ. of Tsukuba with H. Katsura, I. Maruyama, T. Hirano

� � � � � � � � June 26, 2008 YITP, TASSP (3,1) (1,1) Berry Phases with Time Reversal Invariance Characterization of Gapped Quantum Liquids & LSM theorem Quantum interferences in Many body states � Y. Hatsugai γ C � = A ψ = � ψ | C d Institute of Physics ψ � C Univ. of Tsukuba with H. Katsura, I. Maruyama, T. Hirano

Plan Motivation: Topological Quantum Phase Transition Quantum liquids & Spin liquids Berry Phase as a Quantum Order Parameter Berry phases & Time reversal invariance Use of the Berry phases as a topological order parameter Another look at of the Berry phases Gapped or Gapless LSM & Berry phases

Plan Motivation: Topological Quantum Phase Transition Quantum liquids & Spin liquids Berry Phase as a Quantum Order Parameter Berry phases & Time reversal invariance Use of the Berry phases as a topological order parameter Another look at of the Berry phases � 1 Θ 2 = Gapped or Gapless − 1 LSM & Berry phases

Plan Motivation: Topological Quantum Phase Transition Quantum liquids & Spin liquids Berry Phase as a Quantum Order Parameter Berry phases & Time reversal invariance Use of the Berry phases as a topological order parameter Another look at of the Berry phases � 1 Θ 2 = Gapped or Gapless − 1 LSM & Berry phases

Plan Motivation: Topological Quantum Phase Transition Quantum liquids & Spin liquids Berry Phase as a Quantum Order Parameter Berry phases & Time reversal invariance Use of the Berry phases as a topological order parameter Another look at of the Berry phases � 1 Θ 2 = Gapped or Gapless − 1 LSM & Berry phases next time

Plan Motivation: Topological Quantum Phase Transition Quantum liquids & Spin liquids Berry Phase as a Quantum Order Parameter Berry phases & Time reversal invariance Use of the Berry phases as a topological order parameter Another look at of the Berry phases � 1 Θ 2 = Gapped or Gapless − 1 LSM & Berry phases

June 26, 2008 YITP, TASSP Topological Quantum Phase Transition & Symmetry (Classical) Phase Transition Finite temperature Phases between different symmetries Order parameter & Symmetry breaking Quantum Phase Transition Zero temperature, Ground state Phases between different symmetries Classical Order parameter & Symmetry breaking Topological Quantum Phase Transition Quantum Phases among the same symmetry No Classical Order parameter Quantum Liquids!

June 26, 2008 YITP, TASSP Quantum Liquids without Symmetry Breaking Quantum Liquids in Low Dimensional Quantum Systems Low Dimensionality, Quantum Fluctuations No Symmetry Breaking Topological Order No Local Order Parameter X.G.Wen Various Phases & Quantum Phase Transitions Gapped Quantum Liquids in Condensed Matter Integer & Fractional Quantum Hall States RVB states in (Doped) Heisenberg Models Dimer Models of Fermions and Spins Integer spin chains Correlated Electrons & Spins with Frustrations Half filled Kondo Lattice How to Classify the Quantum Liquid Phases Quantized Berry Phase as a Topological Order Parameter Entanglement Entropy to pick up global entanglement

��฀฀฀� June 26, 2008 YITP, TASSP Conventional (Classical) Order Parameters (Local) Order Parameters: To characterize the phase Charge Density Wave (CDW) as a Local Order parameter Local Charges : Basic Objects x x Spin Density Wave, Ferromagnets, Antiferromagnets Local Spins : Basic Objects m(x) x Classical Objects as local charges and small magnets for Local Order Parameters

June 26, 2008 YITP, TASSP 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 fundamental

June 26, 2008 YITP, TASSP 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 fundamental

June 26, 2008 YITP, TASSP 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 fundamental PS04, T. Kawarabayashi, TASSP

June 26, 2008 YITP, TASSP 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 fundamental PS04, T. Kawarabayashi, TASSP

June 26, 2008 YITP, TASSP Quantum Liquids are Featureless !! How to characterize the phase Without Symmetry Breaking Use Quantum Interference!

June 26, 2008 YITP, TASSP Quantum Interferences for the Classification “Classical” Observables O = n ↑ ± n ↓ , · · · Charge density, Spin density,... � O � G = � G |O| G � = � G ′ |O| G ′ � = � O � G ′ | G ′ � = | G � e i φ “Quantum” Observables ! 2 � e i ( φ 1 − φ 2 ) � G 1 | G 2 � = � G ′ 1 | G ′ Quantum Interferences: i � e i φ i | G i � = | G ′ Probability Ampliture (overlap) Aharonov-Bohm Effects :Berry Connection A = � G | dG � Phase (Gauge) dependent � :Berry Phase � G | G + dG � = 1 + � G | dG � 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 H = S i · S j √ | 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

June 26, 2008 YITP, TASSP Quantized Berry Phases for Spin Liquids Generic Heisenberg Models (with frustrations) Time Reversal Invariant � J ij S i · S j H = ∀ j, S j → Θ − 1 T S j Θ T = − S j ij U(1) twist as a Local Probe to define Berry Phases 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> H ( x = e i θ ) Parameter dependent Hamiltonian C = { x = e i θ | θ : 0 → 2 π } Define Berry Phases by the Entire Many Spin Wavefunction Require excitation Gap! Quantization � � � 0 � ψ | d ψ � = Z2 A ψ = γ C = : mod 2 π π C C Time Reversal ( Anti-Unitary ) Invariance

� � � June 26, 2008 YITP, TASSP 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 (Also Non-Abelian extension)

June 26, 2008 YITP, TASSP 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 )

June 26, 2008 YITP, TASSP Anti-Unitary Invariant State and � 1 Quantized Berry Phases Θ 2 = − 1 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 ) = mod 2 π π γ C ( A Ψ ) = − γ C ( A ΘΨ ) ≡ − γ C ( A Ψ ) , mod2 π