Phase diagram and quench dynamics of the Cluster-XY spin chain - PowerPoint PPT Presentation

Phase diagram and quench dynamics of the Cluster-XY spin chain Sebasti´ an Montes arXiv:1112.4414 (with Alioscia Hamma (PI)) Centro de ciencias de Benasque Pedro Pascual Networking Tensor Networks May 16th, 2012 S. Montes (PI) Cluster-XY chain May 16th, 2012 1 / 32

Motivation Motivation Dinamics of closed quantum systems ◮ Recent experiments with cold atoms, quantum dots, nanowires ◮ Foundations of statistical mechanics and thermodynamics: Equilibration, thermalization, closed quantum systems out of equilibrium ◮ Universal features? (e.g. Kibble-Zurek scaling) Effective boundary Hamiltonians Effective behavior of the edge in a non-trivial 2D fermionic symmetry-protected topological state with Z 2 symmetry (Z-C. Gu, X-G. Wen, arXiv:1201.2648v1) S. Montes (PI) Cluster-XY chain May 16th, 2012 2 / 32

Motivation Motivation Dinamics of closed quantum systems ◮ Recent experiments with cold atoms, quantum dots, nanowires ◮ Foundations of statistical mechanics and thermodynamics: Equilibration, thermalization, closed quantum systems out of equilibrium ◮ Universal features? (e.g. Kibble-Zurek scaling) Effective boundary Hamiltonians Effective behavior of the edge in a non-trivial 2D fermionic symmetry-protected topological state with Z 2 symmetry (Z-C. Gu, X-G. Wen, arXiv:1201.2648v1) S. Montes (PI) Cluster-XY chain May 16th, 2012 2 / 32

Outline Outline ◮ Motivation ◮ Cluster state ◮ Model and exact solution ◮ Phase diagram ◮ Quench dynamics ◮ Summary S. Montes (PI) Cluster-XY chain May 16th, 2012 3 / 32

Cluster state Cluster state S. Montes (PI) Cluster-XY chain May 16th, 2012 4 / 32

Cluster state Preparing a cluster state A. Doherty and S. Bartlett. Phys. Rev. Lett. 103 , 020506 (2009); S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009) S. Montes (PI) Cluster-XY chain May 16th, 2012 5 / 32

Cluster state Preparing a cluster state | ψ � = |↑� ⊗ N , σ z |↑� = |↑� A. Doherty and S. Bartlett. Phys. Rev. Lett. 103 , 020506 (2009); S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009) S. Montes (PI) Cluster-XY chain May 16th, 2012 5 / 32

Cluster state Preparing a cluster state σ x | + � = | + � U = exp ( i π | + � � + | ⊗ | + � � + | ) , A. Doherty and S. Bartlett. Phys. Rev. Lett. 103 , 020506 (2009); S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009) S. Montes (PI) Cluster-XY chain May 16th, 2012 5 / 32

Cluster state Preparing a cluster state C | ψ � = | C � A. Doherty and S. Bartlett. Phys. Rev. Lett. 103 , 020506 (2009); S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009) S. Montes (PI) Cluster-XY chain May 16th, 2012 5 / 32

Cluster state Stabilizers We can also obtain the cluster state as the ground state of a particular stabilizer Hamiltonian: X X � K µ = σ z σ x µ ν ν ∼ µ Z X � H C = − K µ X X µ A. Doherty and S. Bartlett. Phys. Rev. Lett. 103 , 020506 (2009); S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009) S. Montes (PI) Cluster-XY chain May 16th, 2012 6 / 32

Cluster-XY model Cluster-XY model S. Montes (PI) Cluster-XY chain May 16th, 2012 7 / 32

Cluster-XY model Cluster-XY model Cluster-XY Hamiltonian N N � σ x i − 1 σ z i σ x � σ z H ( λ x , λ y , h ) := − i + 1 − h i i = 1 i = 1 N N � σ y i σ y � σ x i σ x + λ y i + 1 + λ x i + 1 i = 1 i = 1 For periodic boundary conditions N � σ z Q = ( − 1 ) q Q = i , [ H , Q ] = 0 , i = 1 S. Skrøvseth and S. Bartlett. Phys. Rev. A 80 , 022316 (2009); W. Son, et. al . Europhys. Lett. 95 , 50001 (2011).; P. Smacchia, et. al. Phys. Rev. A 84 , 022304 (2011).. S. Montes (PI) Cluster-XY chain May 16th, 2012 8 / 32

Cluster-XY model Cluster-XY model Jordan-Wigner transformation � l − 1 � c † � σ + σ z l = m l m = 1 { c n , c † { c n , c m } = 0 m } = δ nm Fourier transform N k = π 1 � e ikn c n , c k = √ N ( 2 m + 1 − q ) , m = 0 , · · · , N − 1 N n = 1 S. Sachdev. Quantum phase transitions . Cambridge University Press, 1999 S. Montes (PI) Cluster-XY chain May 16th, 2012 9 / 32

Cluster-XY model Cluster-XY model � � � � �� � c † k c k + c † c † k c † H = 2 ǫ k + i δ k − k + c k c − k − k c − k k > 0 ǫ k = cos ( 2 k ) − ( λ x + λ y ) cos ( k ) − h , δ k = sin ( 2 k ) − ( λ x − λ y ) sin ( k ) Bogoliubov transformation γ k = cos ( θ k / 2 ) c k − i sin ( θ k / 2 ) c † − k � δ k � θ k = − arctan ǫ k S. Montes (PI) Cluster-XY chain May 16th, 2012 10 / 32

Cluster-XY model Cluster-XY model Diagonal Hamiltonian � � γ † k γ k + γ † � H = 2 ∆ k − k γ − k − 1 k > 0 � ǫ 2 k + δ 2 ∆ k = k Ground state � � � cos ( θ k / 2 ) + i sin ( θ k / 2 ) c † k c † | Ω � = | 0 � c − k k > 0 S. Montes (PI) Cluster-XY chain May 16th, 2012 11 / 32

Cluster-XY model Gapless regions ∆ k = 0 , for some k Ising planes h = ± ( λ x + λ y ) + 1 Cluster transitions h = λ 2 y − λ x λ y − 1 , − 2 ≤ λ x − λ y ≤ 2 (Video) S.M., A. Hamma, arXiv:1112.4414 S. Montes (PI) Cluster-XY chain May 16th, 2012 12 / 32

Cluster-XY model Detecting critical lines Fidelity Different phases must be distinguishable from the point of view of quantum mechanics. We can use the fidelity F ( λ x , λ y , h ; λ ′ x , λ ′ y , h ′ ) = | � Ω( λ x , λ y , h ) | Ω( λ ′ x , λ ′ y , h ′ ) � | � θ k ( λ x , λ y , h ) − θ k ( λ ′ x , λ ′ y , h ′ ) � �� � � � = � cos � � 2 � k > 0 P. Zanardi and N. Paunkovi´ c. Phys. Rev. E, 74 , 031123 (2006); L. Campos Venuti and P. Zanardi. Phys. Rev. Lett. 99 , 095701 (2007) S. Montes (PI) Cluster-XY chain May 16th, 2012 13 / 32

Cluster-XY model Fidelity F ( λ y , λ y + δλ y ) , δλ y = 0 . 05, N = 500 S.M., A. Hamma, arXiv:1112.4414 S. Montes (PI) Cluster-XY chain May 16th, 2012 14 / 32

Cluster-XY model “Ghost” phases h = 0 N N N � � σ y i σ y � σ x i − 1 σ z i σ x σ x i σ x H = − i + 1 + λ y i + 1 + λ x i + 1 i = 1 i = 1 i = 1 S. Montes (PI) Cluster-XY chain May 16th, 2012 15 / 32

Cluster-XY model “Ghost” phases λ x = 0 N N N � � � σ y i σ y σ x i − 1 σ z i σ x σ z H = − i + 1 − h i + λ y i + 1 i = 1 i = 1 i = 1 S. Montes (PI) Cluster-XY chain May 16th, 2012 16 / 32

Quench dynamics Quench dynamics S. Montes (PI) Cluster-XY chain May 16th, 2012 17 / 32

Quench dynamics Nonequilibrum dynamics of closed quantum systems Quantum quenches Local or global change of the parameters of the system. We would like to study the dynamics and characterize the universal features of a system after a quantum quench. Here we are interested in instantaneous critical global quenches. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore. Rev. Mod. Phys. 83 , 863 (2011). S. Montes (PI) Cluster-XY chain May 16th, 2012 18 / 32

Quench dynamics Loschmidt echo If we perform a quantum quench, we can compute the fidelity between the initial state and the time-evolved state L ( t ) = |� ψ 0 | U ( t ) | ψ 0 �| 2 , with U ( t ) = exp ( − itH Q ) . This is known as the Loschmidt echo. It is related to the study of reversibility in statistical mechanics. T. Gorin, T. Prosen, T.H. Seligman, and M. Znidaric. Phys. Rep. 435 , 33 (2006); L. Campos Venuti and P. Zanardi. Phys. Rev. A 81 , 022113 (2010); J. H¨ app¨ ol¨ a, G.B. Hal´ asz, and A. Hamma. Phys. Rev. A 85 , 032114 (2012). S. Montes (PI) Cluster-XY chain May 16th, 2012 19 / 32

Quench dynamics Proposed lower bound for revival time Quasiperiodic systems will have revivals after long enough times. These can be detected using the Loschmidt echo. A proposed lower bound for the revival time in spin chains with (anti)periodic boundary conditions is given by the Lieb-Robinson speed v LR N T rev ≈ . 2 v LR J. H¨ app¨ ol¨ a, G.B. Hal´ asz, and A. Hamma. Phys. Rev. A 85 , 032114 (2012) S. Montes (PI) Cluster-XY chain May 16th, 2012 20 / 32

Quench dynamics Loschmidt echo for the cluster-Ising model We start with | ψ ( t = 0 ) � = | Ω( λ x , λ y , h ) � . We can now compare the initial state with the time evolution of the quenched Hamiltonian H = H ( λ ′ x , λ ′ y , h ′ ) � 2 � � cos 2 ( χ k / 2 ) + e − i 4 t ∆ k sin 2 ( χ k / 2 ) � � L ( t ) = k > 0 � � 1 − sin 2 ( χ k ) sin 2 ( 2 t ∆ k ) � = k > 0 where χ k = θ k ( λ x , λ y , h ) − θ k ( λ ′ x , λ ′ y , h ′ ) , ∆ k = ∆ k ( λ ′ x , λ ′ y , h ′ ) . S. Montes (PI) Cluster-XY chain May 16th, 2012 21 / 32

Quench dynamics First critical point h = 0 , λ x = 0 , λ y = 1 The critical Hamiltonian is N N � � σ y i σ y σ x i − 1 σ z i σ x H = − i + 1 + i + 1 . i = 1 i = 1 S. Montes (PI) Cluster-XY chain May 16th, 2012 22 / 32

Quench dynamics First critical point h = 0 , λ x = 0 , λ y = 1 The critical Hamiltonian is N N � � σ y i σ y σ x i − 1 σ z i σ x H = − i + 1 + i + 1 . i = 1 i = 1 √ v LR ≃ 3 . 2 e / 2 = 6 . 15 2 � k 4 3 2 1 k � 3 � 2 � 1 1 2 3 P. Smacchia, L. Amico, P. Facchi, R. Fazio, G. Florio, S. Pascazio, and V. Vedral, Phys. Rev. A 84 , 022304 (2011). S. Montes (PI) Cluster-XY chain May 16th, 2012 22 / 32