ESQPT in systems with long-range interactions Lea F. Santos Yeshiva - - PowerPoint PPT Presentation

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lea F. Santos

Yeshiva University, New York, NY, USA

ESQPT in systems with long-range interactions

Francisco Pérez-Bernal

Universidad de Huelva, Spain PRA 94, 012113 (2016)

arXiv:1604.06851 (Fort. Physik) PRA 92, 050101R (2015)

Ø Consequences of the presence of an ESQPT (static and dynamics). Ø Lipkin model [U(2)]: experiments with ion traps, BEC, NMR but valid also for U(n+1)

• G. Luca Celardo and Fausto Borgonovi

Universita Cattolica del Sacro Cuore, Italy

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Trapped ions: long-range interaction

H = B ! n

z n

!

+ J | n " m |" ! n

x! m x n<m

!

• P. Richerme et al, Nature 511, 198 (2014)
• P. Jurcevi et al, Nature 511, 202 (2014)

0 ~ ! ! 3

Ion Traps

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Trapped ions: long-range interaction

H = B ! n

z n

!

+ J | n " m |" ! n

x! m x n<m

!

• P. Richerme et al, Nature 511, 198 (2014)
• P. Jurcevi et al, Nature 511, 202 (2014)
• P. Hauke and L. Tagliacozzo, PRL 111, 207202 (2013)

N=100, excitation on 50

!!..!!"!!..!!Z ! = 3

Magnetization in z of each site Faster than Lieb-Robinson bound

! = 0.7

0 ~ ! ! 3

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: infinite-range interaction

H = B ! n

z n

!

+ J ! n

x! m x n<m

!

! = 0

H = B ! n

z n

!

+ J | n " m |" ! n

x! m x n<m

!

dimension = 2N to dimension = N 2 +1

S z = ! n

z n

!

S x = ! n

x n

!

N = number of sites

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: infinite-range interaction

H = B ! n

z n

!

+ J ! n

x! m x n<m

!

! = 0

H = B ! n

z n

!

+ J | n " m |" ! n

x! m x n<m

!

dimension = 2N to dimension = N 2 +1

S z = ! n

z n

!

S x = ! n

x n

!

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

N = number of sites

Control parameter

Lipkin-Meshkov-Glick model

!c = 0.2

Ground state quantum phase transition

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Lipkin Model: U(2) algebraic structure

Schwinger representation:

S + = 2t+s S z = 1 2 (t+t ! s+s)

H = (1!!)nt + ! N (t+s+ s+t)2

nt = t+t

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

U(2) U(1) SO(2) Two species of scalar bosons Ground state QPT U(2) U(1) SO(2)

HU(n+1) = (1!!)HU(n) + ! N HSO(n+1)

In general:

PRA 92, 050101R (2015)

!c = 0.2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Excited State Quantum Phase Transition

!c = 0.2 ! = 0.4 ! = 0.6 ! = 0.8

HU(n+1) = (1!!)HU(n) + ! N HSO(n+1)

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Separatrix

that marks the ESQPT

QPT ESQPT ESQPT ESQPT

EESQPT = (1! 5!)2 16!

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

ESQPT: participation ratio in U(1) basis

!U(2)

(k)

= Csz

(k) S mz sz=!N/2 N/2

"

Participation Ratio

PR(k) ! 1 |Csz

(k) |4 sz="N/2 N/2

#

LFS & Pérez-Bernal PRA92, 050101R (2015).

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

U(1)-basis

Large PR: delocalized state Small PR: localized state

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

!U(2)

(k)

= Csz

(k) S mz sz=!N/2 N/2

"

Participation Ratio

PR(k) ! 1 |Csz

(k) |4 sz="N/2 N/2

#

PRA92, 050101R (2015).

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

U(1)-basis

N=600, 2000

! = 0.4

EESQPT

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2

N=600

! = 0.6

ez '/ N ez '/ N ez '/ N ez '/ N

Csz

(k) 2

!U(2)

(k)

= Csz

(k) s mz sz=!N/2 N/2

"

EESQPT

U(1) basis s mz

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

ez = s mz HU(2) s mz

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2

N=600

! = 0.6

ez '/ N ez '/ N ez '/ N ez '/ N

Csz

(k) 2

Energy of the U(1) basis vectors

ez = s mz HU(2) s mz

!!..!!!!..!!Z

mz = !N / 2

!U(2)

(k)

= Csz

(k) s mz sz=!N/2 N/2

"

EESQPT

U(1) basis s mz

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

ez = s mz HU(2) s mz

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Eigenstate at ESQPT localized at U(1) basis with mz=-N/2

N=600

! = 0.6

ez '/ N ez '/ N ez '/ N ez '/ N

Csz

(k) 2

Energy of the U(1) basis vectors

ez = s mz HU(2) s mz

!!..!!!!..!!Z

mz = !N / 2

U(1): z SO(2): x

!U(2)

(k)

= Csz

(k) s mz sz=!N/2 N/2

"

EESQPT

U(1) basis s mz

!c

Ground state QPT H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

!(0) = S mz

Initial state U(1)-basis

Quench from U(n) to U(n+1)

F(t) = !(0) | !(t)"

2

Survival Probability

!!..!!!!..!!Z

U(1) ground state

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

!(0) = s mz = Csz

(k) !U(2) (k) k

"

Initial state

Initial state: U(1)-basis vector Slow decay

Survival Probability

10 0.2 0.4 0.6 0.8 1

Time

10

• 4

10

• 2

10

F(t) !!..!!!!..!!Z

!!..!""!..!!Z

U(1) ground state

N=1000

! = 0.6

PRA 94, 012113 (2016)

F(t) = !(0) | !(t)"

2 =

Cmz

(k) 2 e#iEkt k

$

2

= !ini(E)e#iEt dE

#% %

&

2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in z: slow dynamics

PRA 94, 012113 (2016)

mz

(k) / N = !k S z !k / N

U(1): z SO(2): x

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

!(0) =""..""""..""Z !(0) ="".."##"..""Z

Same initial states studied in ion traps

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in z: dip

PRA 94, 012113 (2016)

mz

(k) / N = !k S z !k / N

U(1): z SO(2): x

!!..!!!!..!!Z

E'k / N

At the separatrix is localized at mz = !N / 2

!k

EESQPT

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation

PRA 94, 012113 (2016)

mz

(k) / N = !k S z !k / N

U(1): z SO(2): x

E'k / N

EESQPT

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

EESQPT

mx

(k) / N = !k S x !k / N

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

QPT with parity-symmetry breaking

H = ! 2 S z

2 "S x # !

2 z2 " 1" z2 cos!

Trenkwalder … Inguscio, Fattori arXiv: 1603.02979

Tuning g to large negative values, the ground state of the system goes from a gapped symmetric state (z = 0) to two degenerate asymmetric states (|z| > 0). The system undergoes a second-order QPT where the spatial parity symmetry is broken. Imbalance

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation

mx

(k) / N = !k S x !k / N

U(1): z SO(2): x

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Bifurcation of mx for the ground state as increases

!

! < !c

! > !c

!c = 0.2

Ground state QPT

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation

0.2 0.4 0.6 0.8 1

ξ

0.2 0.4 0.6 0.8 1

E’

k/N

U(1): z SO(2): x

HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Magnetization in x: bifurcation

mx

(k) / N = !k S x !k / N

U(1): z SO(2): x HU(2) = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

Bifurcation of mx for the ground state as increases

!

Bifurcation of mx at the ESQPT EESQPT EESQPT

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Classical bifurcation

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

H = ! 2 J z

2 "J x # !

2 z2 " 1" z2 cos!

Oberthaler’s group PRL 105 (2010) BEC Oliveira’s group PRA 87 (2013) NMR

z = (Na ! Nb) / N

V(!) !

z = Temporal mean magnetization

V(!) = 1! z2 cos!

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Self-trapping: depending on

H = (1!!) N 2 +S z " # $ % & '! 4! N S x

2

H = ! 2 J z

2 "J x # !

2 z2 " 1" z2 cos!

! <1 ! >1 ! = 0.15 < !c ! = 0.6 > !c

!(0) = s mx

!

SO(2) basis

++..+!++..!+X

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Self-trapping: depending on energy

mx

(k) / N = !k S x !k / N

Bifurcation of mx at the ESQPT

Eini < EESQPT Eini > EESQPT

!(0)

Superposition of eigenstates

• nly below or only above the

separatrix

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Conclusions

Ø Different ways to capture an ESQPT: (1) Structure of the Hamiltonian matrix. (2) Level of delocalization of the eigenstates. (3) Magnetization in x and z. (4) Dynamics. Ø The dynamics of the system depends on the interplay between the initial state and the final Hamiltonian.

!!..!!!!..!!Z

!!..!""!..!!Z

Evolution of as done in experiments with ion traps. PRA 94, 012113 (2016) arXiv:1604.06851 (Fort. Physik) PRA 92, 050101R (2015)

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

ESQPT: participation ratio

In the SO(2)-basis

Participation Ratio

PR(k) ! 1 |Cn

(k) |4 n=L N

"

N=500, 2000

!SO(2)

(k)

= Cn

(k) s mx mx=!N/2 N/2

"

LFS, Távora, Bernal arXiv:1604.04289

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Classical bifurcation

H = ! 2 J z

2 "J x # !

2 z2 " 1" z2 cos!

! <1 ! >1

U(1): z SO(2): x

!c = 0.2

Ground state QPT

! < !c

! > !c

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Structure of the eigenstates

QPT

Caprio, Cejnar, and Iachello

• Ann. Phys. 323, 1106 (2008).

Structure of the eigenstates above and below the separatrix.

U(3) model Eigenstates in the U(2)-basis

!U(3)

(k)

= Cn

(k) [N] n L k n=L N

!

HU(3) = (1!!)HU(2) + ! N HSO(3)

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Classical bifurcation

H = ! 2 J z

2 "J x # !

2 z2 " 1" z2 cos!

! <1 ! >1

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

!(0) = s mz = Cmz

(k) !U(2) (k) k

"

Initial state

Initial state: U(1)-basis vector Slow decay

F(t) = !(0) | !(t)"

2 =

Cmz

(k) 2 e#iEkt k

$

2

= !ini(E)e#iEt dE

#% %

&

2

Survival Probability

10 0.2 0.4 0.6 0.8 1

Time

10

• 4

10

• 2

10

F(t) !!..!!!!..!!Z

!!..!""!..!!Z

U(1) ground state

N=1000

! = 0.6

arXiv:1604.04289

!ini(E) = Cmz

(k) 2 k

!

"(E " Ek)

LDOS (local density of states)

Lea F. Santos, Yeshiva University Long-range, Trieste, 2016

Initial state

Initial state: U(1)-basis vector Slow decay

F(t) = !(0) | !(t)"

2 =

Csz

(k) 2 e#iEkt k

$

2

Survival Probability

10 0.2 0.4 0.6 0.8 1

Time

10

• 4

10

• 2

10

F(t) !!..!!!!..!!Z

!!..!""!..!!Z

U(1) ground state

N=1000

! = 0.6

!(0) = S mz = Csz

(k) !U(2) (k) k

"

arXiv:1604.04289