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

▶
phase diagram and quench dynamics of the cluster xy spin
SMART_READER_LITE
LIVE PREVIEW

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

Dec 01, 2023 139 likes •536 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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 Z2 symmetry

(Z-C. Gu, X-G. Wen, arXiv:1201.2648v1)

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 2 / 32

slide-3
SLIDE 3

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 Z2 symmetry

(Z-C. Gu, X-G. Wen, arXiv:1201.2648v1)

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 2 / 32

slide-4
SLIDE 4

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

slide-5
SLIDE 5

Cluster state

Cluster state

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 4 / 32

slide-6
SLIDE 6

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

slide-7
SLIDE 7

Cluster state

Preparing a cluster state

|ψ = |↑⊗N , σz |↑ = |↑

  • 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

slide-8
SLIDE 8

Cluster state

Preparing a cluster state

U = exp (iπ |+ +| ⊗ |+ +|) , σx |+ = |+

  • 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

slide-9
SLIDE 9

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

slide-10
SLIDE 10

Cluster state

Stabilizers

We can also obtain the cluster state as the ground state of a particular stabilizer Hamiltonian: Kµ = σz

µ

  • ν∼µ

σx

ν

HC = −

  • µ

Kµ

Z X X X X X

  • 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

slide-11
SLIDE 11

Cluster-XY model

Cluster-XY model

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 7 / 32

slide-12
SLIDE 12

Cluster-XY model

Cluster-XY model

Cluster-XY Hamiltonian

H(λx, λy, h) := −

N

  • i=1

σx

i−1σz i σx i+1 − h N

  • i=1

σz

i

+ λy

N

  • i=1

σy

i σy i+1 + λx N

  • i=1

σx

i σx i+1

For periodic boundary conditions Q =

N

  • i=1

σz

i ,

[H, Q] = 0, Q = (−1)q

  • 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

slide-13
SLIDE 13

Cluster-XY model

Cluster-XY model

Jordan-Wigner transformation

c†

l =

l−1

  • m=1

σz

m

  • σ+

l

{cn, cm} = 0 {cn, c†

m} = δnm

Fourier transform

ck = 1 √ N

N

  • n=1

eikncn, k = π N (2m + 1 − q), m = 0, · · · , N − 1

  • S. Sachdev. Quantum phase transitions. Cambridge University Press, 1999
  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 9 / 32

slide-14
SLIDE 14

Cluster-XY model

Cluster-XY model

H = 2

  • k>0
  • ǫk
  • c†

kck + c† −kc−k

  • + iδk
  • c†

kc† −k + ckc−k

  • ǫk = cos(2k) − (λx + λy) cos(k) − h,

δk = sin(2k) − (λx − λy) sin(k)

Bogoliubov transformation

γk = cos(θk/2)ck − i sin(θk/2)c†

−k

θk = − arctan δk ǫk

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 10 / 32

slide-15
SLIDE 15

Cluster-XY model

Cluster-XY model

Diagonal Hamiltonian

H = 2

  • k>0

∆k

  • γ†

kγk + γ† −kγ−k − 1

  • ∆k =
  • ǫ2

k + δ2 k

Ground state

|Ω =

  • k>0
  • cos(θk/2) + i sin(θk/2)c†

kc† −k

  • |0c
  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 11 / 32

slide-16
SLIDE 16

Cluster-XY model

Gapless regions

∆k = 0, for some k

Ising planes

h = ±(λx + λy) + 1

Cluster transitions

h = λ2

y − λxλy − 1,

−2 ≤ λx − λy ≤ 2 (Video)

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 12 / 32

slide-17
SLIDE 17

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(λx, λy, h; λ′

x, λ′ y, h′) = | Ω(λx, λy, h)|Ω(λ′ x, λ′ y, h′) |

=

  • k>0
  • cos

θk(λx, λy, h) − θk(λ′

x, λ′ y, h′)

2

  • 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

slide-18
SLIDE 18

Cluster-XY model

Fidelity

F(λy, λy + δλy), δλy = 0.05, N = 500

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 14 / 32

slide-19
SLIDE 19

Cluster-XY model

“Ghost” phases

h = 0

H = −

N

  • i=1

σx

i−1σz i σx i+1 + λy N

  • i=1

σy

i σy i+1 + λx N

  • i=1

σx

i σx i+1

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 15 / 32

slide-20
SLIDE 20

Cluster-XY model

“Ghost” phases

λx = 0

H = −

N

  • i=1

σx

i−1σz i σx i+1 − h N

  • i=1

σz

i + λy N

  • i=1

σy

i σy i+1

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 16 / 32

slide-21
SLIDE 21

Quench dynamics

Quench dynamics

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 17 / 32

slide-22
SLIDE 22

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

  • f 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

slide-23
SLIDE 23

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) = |ψ0|U(t)|ψ0|2 , with U(t) = exp(−itHQ). 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¨

  • l¨

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

slide-24
SLIDE 24

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 vLR Trev ≈ N 2vLR .

  • J. H¨

app¨

  • l¨

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

slide-25
SLIDE 25

Quench dynamics

Loschmidt echo for the cluster-Ising model

We start with |ψ(t = 0) = |Ω(λx, λy, h) . We can now compare the initial state with the time evolution of the quenched Hamiltonian H = H(λ′

x, λ′ y, h′)

L(t) =

  • k>0
  • cos2(χk/2) + e−i4t∆k sin2(χk/2)
  • 2

=

  • k>0
  • 1 − sin2(χk) sin2(2t∆k)
  • where

χk = θk(λx, λy, h) − θk(λ′

x, λ′ y, h′),

∆k = ∆k(λ′

x, λ′ y, h′).

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 21 / 32

slide-26
SLIDE 26

Quench dynamics

First critical point

h = 0, λx = 0, λy = 1

The critical Hamiltonian is H = −

N

  • i=1

σx

i−1σz i σx i+1 + N

  • i=1

σy

i σy i+1.

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 22 / 32

slide-27
SLIDE 27

Quench dynamics

First critical point

h = 0, λx = 0, λy = 1

The critical Hamiltonian is H = −

N

  • i=1

σx

i−1σz i σx i+1 + N

  • i=1

σy

i σy i+1.

vLR ≃ 3.2e/ √ 2 = 6.15

3 2 1 1 2 3 k 1 2 3 4 2k

  • 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

slide-28
SLIDE 28

Quench dynamics

Starting from the cluster state λy = 0.8, λx = 0, h = 0, N = 400 Starting from the AFM state λy = 1.2, λx = 0, h = 0,

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 23 / 32

slide-29
SLIDE 29

Quench dynamics

Other interactions

Starting from λx = 0.2, λy = 1, h = 0 (N = 400, q = 1)

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 24 / 32

slide-30
SLIDE 30

Quench dynamics

Along the critical line

Revival times

λ λy

x

Loschmidt echo

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 25 / 32

slide-31
SLIDE 31

Quench dynamics

Second critical point

h = 0, λx = − 3

2, λy = 1 2

The critical Hamiltonian is H = −

N

  • i=1

σx

i−1σz i σx i+1 − 3

2

N

  • i=1

σx

i σx i+1 + 1

2

N

  • i=1

σy

i σy i+1.

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 26 / 32

slide-32
SLIDE 32

Quench dynamics

Second critical point

h = 0, λx = − 3

2, λy = 1 2

The critical Hamiltonian is H = −

N

  • i=1

σx

i−1σz i σx i+1 − 3

2

N

  • i=1

σx

i σx i+1 + 1

2

N

  • i=1

σy

i σy i+1.

3 2 1 1 2 3 k 1 2 3 4 5 2k

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 26 / 32

slide-33
SLIDE 33

Quench dynamics

Starting from the cluster state λy = 1

2, λx = −1.3 (N = 400)

Starting from the cluster state λy = 1

2, λx = −1 (N = 400)

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 27 / 32

slide-34
SLIDE 34

Quench dynamics

Starting from λy = 0.7, λx = − 3

2 (N = 400) - z polarized

Starting from λy = 1

2, λx = −1.7 (N = 400) - Ferromagnetic

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 28 / 32

slide-35
SLIDE 35

Quench dynamics

Overlaps

Ground state and one-particle states

F1(λ′

i) =

  • 0≤k≤π
  • Ω(λ′

i)|γ† kγ† −k|Ω(λ(c) i )

  • 2

λ λ y

x

λ λ

x y

S.M., A. Hamma, arXiv:1112.4414

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 29 / 32

slide-36
SLIDE 36

Summary

Summary

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 30 / 32

slide-37
SLIDE 37

Summary

Summary

◮ The cluster-XY model provides a simple benchmark with a rich phase

diagram.

◮ This model may be useful to test new proposals on the dynamics of

composite quantum systems out of equilibrium. In particular, we showed that different critical points have different effects on the Loschmidt echo.

◮ It would be interesting to extend these ideas even further and

characterize the effect of the universality class of a critical point on the behavior of the quench dynamics.

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 31 / 32

slide-38
SLIDE 38

Thank you

Thank you.

  • S. Montes (PI)

Cluster-XY chain May 16th, 2012 32 / 32