Ergodicity-nonergodicity transitions in driven many-body systems - PowerPoint PPT Presentation

Ergodicity-nonergodicity transitions in driven many-body systems Tomaž Prosen Department of Physics, FMF, University of Ljubljana, SLOVENIA Superbagneres de Luchon, 20 March 2015 Tomaž Prosen Ergodicity breaking transitions

subtitle: MANY-BODY QUANTUM CHAOS WITHOUT � Tomaž Prosen Ergodicity breaking transitions

Outline Quantum ergodicity, decay of correlations and fidelity decay Kicked Ising chain Integrability breaking ergodicity/non-ergodicity transition Heisenberg XXZ chain Integrable ergodicity/non-ergodicity transition Ergodicity/non-ergodicity transition in a completely integrable classical-mechanical model (Lattice-Laudau-Lifshitz) Kicked Ising spin system on a 2D lattice – dynamical phase transitions Tomaž Prosen Ergodicity breaking transitions

My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 L − 1 � � � J σ z j σ z j + 1 + ( h x σ x j + h z σ z � H ( t ) = j ) δ ( t − m ) j = 0 m ∈ Z � 1 + � � d t ′ H ( t ′ ) � − i ( h x σ x j + h z σ z − i J σ z j σ z U Floquet = T exp − i = exp � j ) � exp � � j + 1 0 + j where [ σ α j , σ β k ] = 2 i ε αβγ σ γ j δ jk . Tomaž Prosen Ergodicity breaking transitions

My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 L − 1 � � � J σ z j σ z j + 1 + ( h x σ x j + h z σ z � H ( t ) = j ) δ ( t − m ) j = 0 m ∈ Z � 1 + � � d t ′ H ( t ′ ) � − i ( h x σ x j + h z σ z − i J σ z j σ z U Floquet = T exp − i = exp � j ) � exp � � j + 1 0 + j where [ σ α j , σ β k ] = 2 i ε αβγ σ γ j δ jk . The model is completely integrable in terms of Jordan-Wigner transformation if h x = 0 (longitudinal field) h z = 0 (transverse field) Tomaž Prosen Ergodicity breaking transitions

My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 L − 1 � � � J σ z j σ z j + 1 + ( h x σ x j + h z σ z � H ( t ) = j ) δ ( t − m ) j = 0 m ∈ Z � 1 + � � d t ′ H ( t ′ ) � − i ( h x σ x j + h z σ z − i J σ z j σ z U Floquet = T exp − i = exp � j ) � exp � � j + 1 0 + j where [ σ α j , σ β k ] = 2 i ε αβγ σ γ j δ jk . The model is completely integrable in terms of Jordan-Wigner transformation if h x = 0 (longitudinal field) h z = 0 (transverse field) Tomaž Prosen Ergodicity breaking transitions

My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 L − 1 � � � J σ z j σ z j + 1 + ( h x σ x j + h z σ z � H ( t ) = j ) δ ( t − m ) j = 0 m ∈ Z � 1 + � � d t ′ H ( t ′ ) � − i ( h x σ x j + h z σ z − i J σ z j σ z U Floquet = T exp − i = exp � j ) � exp � � j + 1 0 + j where [ σ α j , σ β k ] = 2 i ε αβγ σ γ j δ jk . The model is completely integrable in terms of Jordan-Wigner transformation if h x = 0 (longitudinal field) h z = 0 (transverse field) Tomaž Prosen Ergodicity breaking transitions

My favourite toy model of many-body quantum chaos: Kicked Ising Chain Prosen PTPS 2000, Prosen PRE 2002 L − 1 � � � J σ z j σ z j + 1 + ( h x σ x j + h z σ z � H ( t ) = j ) δ ( t − m ) j = 0 m ∈ Z � 1 + � � d t ′ H ( t ′ ) � − i ( h x σ x j + h z σ z − i J σ z j σ z − i � � � � U Floquet = T exp = exp j ) exp j + 1 0 + j where [ σ α j , σ β k ] = 2 i ε αβγ σ γ j δ jk . The model is completely integrable in terms of Jordan-Wigner transformation if h x = 0 (longitudinal field) h z = 0 (transverse field) Time-evolution of local observables is quasi-exact, e.g. for computing U − t Floquet σ α j U t Floquet only 2 t + 1 sites in the range [ j − t , j + t ] are needed!. Quantum cellular automaton in the sense of Schumacher and Werner (2004). Tomaž Prosen Ergodicity breaking transitions

Quasi-energy level statistics of KI [C. Pineda, TP, PRE 2007] Fix J = 0 . 7 , h x = 0 . 9 , h z = 0 . 9, s.t. KI is (strongly) non-integrable. Diagonalize U Floquet | n � = exp ( − i ϕ n ) | n � . For each conserved total momentum K quantum number, we find N ∼ 2 L / L levels, normalized to mean level spacing as s n = ( N / 2 π ) ϕ n . Tomaž Prosen Ergodicity breaking transitions

Quasi-energy level statistics of KI [C. Pineda, TP, PRE 2007] Fix J = 0 . 7 , h x = 0 . 9 , h z = 0 . 9, s.t. KI is (strongly) non-integrable. Diagonalize U Floquet | n � = exp ( − i ϕ n ) | n � . For each conserved total momentum K quantum number, we find N ∼ 2 L / L levels, normalized to mean level spacing as s n = ( N / 2 π ) ϕ n . N ( s ) = # { s n < s } = N smooth ( s ) + N fluct ( s ) Tomaž Prosen Ergodicity breaking transitions

Quasi-energy level statistics of KI [C. Pineda, TP, PRE 2007] Fix J = 0 . 7 , h x = 0 . 9 , h z = 0 . 9, s.t. KI is (strongly) non-integrable. Diagonalize U Floquet | n � = exp ( − i ϕ n ) | n � . For each conserved total momentum K quantum number, we find N ∼ 2 L / L levels, normalized to mean level spacing as s n = ( N / 2 π ) ϕ n . N ( s ) = # { s n < s } = N smooth ( s ) + N fluct ( s ) For kicked quantum quantum systems spectra are expected to be statistically uniformly dense N smooth ( s ) = s Tomaž Prosen Ergodicity breaking transitions

Short-range statistics: Nearest neighbor level spacings We plot cumulative level spacing distribution � s W ( s ) = 0 d sP ( s ) = Prob { s n + 1 − s n < s } . 0.008 0.008 0.01 0.006 0.006 0 W − W Wigner W − W Wigner 0.004 0.004 0.002 0.002 1 0 2 3 0 0 -0.002 -0.002 -0.004 -0.004 0 0 1 1 2 2 3 3 4 4 s s The noisy curve shows the difference between the numerical data for 18 qubits, averaged over the different momentum sectors, and the Wigner RMT surmise. The smooth (red) curve is the difference between infinitely dimensional COE solution and the Wigner surmise. In the inset we present a similar figure with the results for each of quasi-moemtnum sector K . Tomaž Prosen Ergodicity breaking transitions

Long-range statistics: spectral form factor Spectral form factor K 2 ( τ ) is for nonzero integer t defined as 2 � � K 2 ( t / N ) = 1 � 2 = 1 � � � tr U t � � e − i ϕ n t � . � � N N � � � � n Tomaž Prosen Ergodicity breaking transitions

Long-range statistics: spectral form factor Spectral form factor K 2 ( τ ) is for nonzero integer t defined as 2 � � K 2 ( t / N ) = 1 � 2 = 1 � � � tr U t � � e − i ϕ n t � . � � N N � � � � n In non-integrable systems with a chaotic classical lomit, form factor has two regimes: universal described by RMT, non-universal described by short classical periodic orbits. Tomaž Prosen Ergodicity breaking transitions

Long-range statistics: spectral form factor Note that for kicked systems, Heisenberg integer time τ H = N 1 0.8 K 2 0.6 0.02 0.4 0.2 � 0.02 0.5 1 1.5 0.25 0.5 0.75 1 1.25 1.5 1.75 2 t/τ H We show the behavior of the form factor for L = 18 qubits. We perform averaging over short ranges of time ( τ H / 25). The results for each of the K -spaces are shown in colors. The average over the different spaces as well as the theoretical COE(N) curve is plotted as a black and red curve, respectively. Tomaž Prosen Ergodicity breaking transitions

Surprise!? Deviaton from universality at short times Similarly as for semi-classical systems, we find notable statistically significant deviations from universal COE/GOE predictions for short times of few kicks . 2 1 2 -2 -2 3 4 1 -3 -3 log 10 K 2 (1 /τ H ) log 10 K 2 (1 /τ H ) 0 -4 -4 n σ -1 -5 -5 -2 -6 -6 10 10 12 12 14 14 16 16 18 18 20 20 10 12 14 16 18 20 L L L But there is no underlying classical structure! Dynamical explanation of this phenomenon needed! Tomaž Prosen Ergodicity breaking transitions

Quantum ergodicity and its "order parameter" Temporal correlation of an extensive traceless observable A ( tr A = 0 , tr A 2 ∝ L ): 1 L 2 L tr AU − t AU t C A ( t ) = lim L →∞ Average correlator T − 1 1 � D A = lim C A ( t ) T T →∞ t = 0 signals quantum ergodicity if D A = 0. Quantum chaos regime in KI chain seems compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D � = 0. Tomaž Prosen Ergodicity breaking transitions

Quantum ergodicity and its "order parameter" Temporal correlation of an extensive traceless observable A ( tr A = 0 , tr A 2 ∝ L ): 1 L 2 L tr AU − t AU t C A ( t ) = lim L →∞ Average correlator T − 1 1 � D A = lim C A ( t ) T T →∞ t = 0 signals quantum ergodicity if D A = 0. Quantum chaos regime in KI chain seems compatible with exponential decay of correlations. For integrable, and weakly non-integrable cases, though, we find saturation of temporal correlations D � = 0. Tomaž Prosen Ergodicity breaking transitions