lea f. santos - crm. · pdf filelea f. santos, yeshiva university beyond integrability 2015...

Lea F. Santos, Yeshiva University Beyond Integrability 2015

Lea F. Santos

Department of Physics, Yeshiva University, New York, NY, USA

Dynamics of Interacting Quantum Systems: Effects of Symmetries, Perturbation Strength,

and Initial States

E. Jonathan Torres-Herrera (U. Puebla, Mexico) Huijie Guan (Rutgers University, USA) Natan Andrei (Rutgers University, USA) Marcos Rigol (Penn State University, USA) co

llabo

rato

rs F. Pérez-Bernal (U. Huelva, Spain)

Luca Celardo (U. Brescia, Italy) Fausto Borgonovi (U. Brescia, Italy)


Lea F. Santos

Department of Physics, Yeshiva University, New York, NY, USA

Initial state +

Hamiltonian

Ø  Isolated quantum systems with interacting particles.

Ø  Analysis of quench dynamics. Experiments:

NMR, optical lattices, trapped ions

E. Jonathan Torres-Herrera (U. Puebla, Mexico) Huijie Guan (Rutgers University, USA) Natan Andrei (Rutgers University, USA) Marcos Rigol (Penn State University, USA) co

llabo

rato

rs F. Pérez-Bernal (U. Huelva, Spain)

Luca Celardo (U. Brescia, Italy) Fausto Borgonovi (U. Brescia, Italy)

Dynamics of Interacting Quantum Systems: Effects of Symmetries, Perturbation Strength,

and Initial States


Same Hamiltonian Different Initial States

Hamiltonian with strong long-range interaction

(trapped ions)


Trapped ions: long-range interactions

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

H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!



H = B ! nz

n! +

J| n"m |"

! nx! 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)

L=100, excitation on 50

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

! = 3Magnetization

in z of each site

Violation of the Lieb-Robinson

bound

! = 0.7



H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!

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

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


!!..!!"!!..!!X

! = 3Magnetization in

x of each site

LFS, Celardo, Borgonovi arXiv:1507.xxxx



H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!




! = 3 ! = 0Magnetization in

x of each site

Localization

!!..!!"!!..!!XLFS, Celardo, Borgonovi

arXiv:1507.xxxx



H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!



! = 0

Magnetization in x of each site

Localization

!!..!!"!!..!!XLFS, Celardo, Borgonovi

arXiv:1507.xxxx


Energies 10 20 30 40 50 60

Mx = ! nx / 2

n=1

L

!2JMx2


Different Hamiltonians (integrable vs chaotic) Similar Initial States

Hamiltonian with short-range interaction

(optical lattices)


Integrable vs chaos (1D)

XX model

Chaotic NNN model

H = Jn=1

L!1

" (#SnzSn+1

z + SnxSn+1

x + SnySn+1

y )+

H = J(SnxSn+1

x + SnySn+1

y )n=1

L!1

"

Integrable

H = J(!SnzSn+1

z + SnxSn+1

x + SnySn+1

y )n=1

L"1

#

XXZ model

H = J(!SnzSn+1

z + SnxSn+1

x + SnySn+1

y )n=1

L"1

#

XXZ model

Integrable

Integrable

+! Jn=1

L!2

" (#SnzSn+2

z + SnxSn+2

x + SnySn+2

y )


Level Spacing Distribution

Integrable XXZ model

Chaotic NNN model

H = J(!SnzSn+1

z + SnxSn+1

x + SnySn+1

y )n=1

L"1

# H = Jn=1

L!1

" (#SnzSn+1

z + SnxSn+1

x + SnySn+1

y )+

+! Jn=1

L!2

" (#SnzSn+2

z + SnxSn+2

x + SnySn+2

y )


Level Spacing Distribution: Integrable Models

! = 0.1 ! = 0cos(! / 2)

! = 0.5cos(! / 3)

! = cos(!" / N )Roots of units

Kudo and Deguchi J. Phys. Soc. Jpn (2005)


Density of States

Chaotic NNN model

H = Jn=1

L!1

" (#SnzSn+1

z + SnxSn+1

x + SnySn+1

y )+

Integrable

H = J(!SnzSn+1

z + SnxSn+1

x + SnySn+1

y )n=1

L"1

#XXZ model

+! Jn=1

L!2

" (#SnzSn+2

z + SnxSn+2

x + SnySn+2

y )

-6 -4 -2 0 2 4 6E

0

0.1

0.2

(E)

-6 -4 -2 0 2 4 6E

-6 -4 -2 0 2 4 6E

(a) (b) (c)

-6 -4 -2 0 2 4 6E

0

0.1

0.2

(E)

-6 -4 -2 0 2 4 6E

-6 -4 -2 0 2 4 6E

(a) (b) (c)

! = 0.5

! = 0.5! =1

0.2

0.1

0


Overlap between the initial state and the evolved state

F(t) = !(0) |!(t)"2= C!

ini

!

#2e$iE! t

2

Survival Probability (Fidelity)

|!(0)" = ini = C!ini

!

# |"! " |!(t)" = C!ini

!

# e$iE!t |"! "

Eigenvalues and eigenstates of the final Hamiltonian




|!(0)" = ini = C!ini

!

# |"! " |!(t)" = C!ini

!

# e$iE!t |"! "


F(t) = !(0) |!(t)"2= C!

ini

!

#2e$iE!t

2

% Pini (E)2 e$iEt dE

$&

&

'2

Fourier transform of the weighted energy distribution of the initial state of the LDOS (local density of states), strength function



F(t) = !(0) |!(t)"2= C!

ini

!

#2e$iE!t

2

% Pini (E)2 e$iEt dE

$&

&

'2

Fourier transform of the weighted energy distribution of the initial state of the LDOS (local density of states), strength function


|!(0)" = ini = C!ini

!

# |"! " |!(t)" = C!ini

!

# e$iE!t |"! "


-6 -4 -2 0 2 4E

0

0.1

0.2

0.3

0.4

|C |2

0

0.1

0.2

0.3

0.4

|C |2P


Perturbation increases Fidelity decays faster

Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN

-6 -4 -2 0 2 4

J-1

E!

0

1

2

3

4

5

P!

0 1 2 3 4

Jt

10-4

10-3

10-2

10-1

100

F

" = 0.2

delta function

Slow decay

L=16, 8 up spins, T=7.1 ! = 0.5

F(t) = Cini (E)2e!iEt dE

!"

"

#2

LDOS:

Integrable Chaotic


Exponential decay

-6 -4 -2 0 2 4

J-1

E!

0

0.2

0.4

0.6

0.8

1

P!

0 1 2 3 4

Jt

10-4

10-3

10-2

10-1

100

F

" = 0.45 F(t) = exp(!"init)12!

!ini

(Eini "E)2 +!2ini / 4

Lorentzian

F(t) = 12!

!ini

(Eini "E)2 +!2ini / 4

e"iEt dE"#

#

$2

L=16, 8 up spins, T=7.1 ! = 0.5

C!ini 2


LDOS:


Exponential decay

-6 -4 -2 0 2 4

J-1

E!

0

0.2

0.4

0.6

0.8

1

P!

0 1 2 3 4

Jt

10-4

10-3

10-2

10-1

100

F

" = 0.45

F(t) = exp(!"init)12!

!ini

(Eini "E)2 +!2ini / 4

Lorentzian

L=16, 8 up spins, T=7.1 ! = 0.5

C!ini 2


LDOS: F(t) = !(0) |!(t)"2

= C!ini

!

#2e$iE!t

2

~1$" ini2 t2


Faster than exponential: Gaussian

-6 -4 -2 0 2 4

J-1

E!

0

0.1

0.2

0.3

0.4

0.5

P!

0 1 2 3 4

Jt

10-4

10-3

10-2

10-1

100

F

" = 1

Strong perturbation regime (global quench)

Gaussian

12!" ini

2e!(E!Eini )

2

2" ini2 F(t) = exp(!! 2

init2 )

L=16, 8 up spins, T=7.1 ! = 0.5

C!ini 2


LDOS:


Gaussian decay Gaussian DOS & LDOS

-6 -4 -2 0 2 4

J-1

E!

0

0.1

0.2

0.3

0.4

0.5

P!

0 1 2 3 4

Jt

10-4

10-3

10-2

10-1

100

F

" = 1

Strong perturbation regime (global quench)

12!" ini

2e!(E!Eini )

2

2" ini2 F(t) = exp(!! 2

init2 )

L=16, 8 up spins, T=7.1 ! = 0.5

C!ini 2


E!

!

Density of States

Gaussian

LDOS:


Decay slows down away from the middle of the spectrum

E!

!

French & Wong, PLB (1970)

DENSITY OF STATES of systems with 2-body interactions

-6 -4 -2 0 2E

0

0.2

0.4

0.6

Pini

0 1 2 3 4 5t

10-3

10-2

10-1

100

F

-3.5 -3 -2.5 -2 -1.5Eini

0

1

2

3

1

-3.5 -3 -2.5 -2 -1.5Eini

0

1

2

2

(a) (b)

(c) (d)

LDOS: skewed Gaussian

L=18, 6 up spins

HXXZ!! "! HXXZ +!HNNN

! =1

Torres & LFS PRA 90 (2014)

C!ini 2

Integrable Chaotic


0 2 4 6 8 10

0.2

0.40.60.81

F

0 2 4 6 8 10

0 3 6 9 1210-3

10-2

10-1

100

F

0 3 6 9 12

0 1 2 3Jt

10-4

10-2

100

F

0 1 2 3Jt

µ = 0.2 = 0.2

µ = 0.4 = 0.4

= 1µ = 1.5

Exponential and Gaussian F(t) Hfinal: Chaotic or Integrable

! = 0.4

! = 0.2

! =1.5

J(SnxSn+1

x + SnySn+1

y )n=1

L!1

"#$

J(SnxSn+1

x + SnySn+1

y +$SnzSn+1

z )n=1

L!1

"

HXXZ!! "! HXXZ +!HNNNHXX

!" #" HXXZ

Integrable to

integrable

Integrable to

chaotic

L=18, 6 up spins

Torres, Manan, LFS NJP 16 (2014)




Exponential and Gaussian F(t) Hfinal: Chaotic or Integrable

J(SnxSn+1

x + SnySn+1

y )n=1

L!1

"#$

J(SnxSn+1

x + SnySn+1

y +$SnzSn+1

z )n=1

L!1

"

HXXZ!! "! HXXZ +!HNNNHXX

!" #" HXXZ

Integrable to

integrable

Integrable to

chaotic

L=18, 6 up spins

Torres, Manan, LFS NJP 16 (2014)



-1 -0.5 00246

8

P

-1 -0.5 0

-2 -1 0 10

1

2

P

-2 -1 0 1

-4 -2 0 2 4

J-1E

0

0.1

0.2

0.3

0.4

P

-4 -2 0 2 4

J-1E

µ = 0.2 = 0.2

µ = 0.4 = 0.4

= 1µ = 1.5

! = 0.4

! = 0.2

! =1.5


Lower bound achieved

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

Pini

0 5 10 15 20 2510

-4

10-2

100

F

-6 -4 -2 0 2 4 6

E

0

0.1

0.2

0.3

0.4

0.5

Pini

0 1 2 3 4 5 6

t

10-4

10-2

100

F

(a) (b)

(c) (d)

L=16, 8 up spins ! = 0.48

F(t) = cos2 (dt / 2)exp(!! 2t2 )d=8


Hinitial = HXXZ = J(SnxSn+1

x + SnySn+1

y +!SnzSn+1

z )n=1

L"1

# $%$ H final = HXXZ + dJSL/2z

C!ini 2

impurity model

F(t) ! cos2 (! init)

Mandelstam-Tamm relation

! H! A ![H,A]2i

=12d Adt

LDOS

Lower bound


Long-time decay

0 5 10 15 20Jt

10-6

10-4

10-2

100

FF(t) = C!ini

!

!4+ C!

ini

!""

!2C"

ini 2 ei(E!#E" )t

F(t) = !(0) |!(t)"2= C!

ini

!

#2e$iE!t

2

! F = C!ini

!

"4= IPRini

!"!"!"!"Néel state !"! HXXZ +!HNNN


Long-time decay

!"!"!"!"Néel state !"! HXXZ +!HNNN

C(t) = 1t

F(! )0

t! d! " t#"

L=16, 8 up spins


Many-body localization

DISORDER &

POWERLAW


Disorder: Random Magnetic Fields

H final = hnSnz

n=1

L

! + J(SnxSn+1

x + SnySn+1

y + SnzSn+1

z )n=1

L

!

n-2 n-1 n n+1 n+2

Anderson localization Many-body localization

Ising model

Disordered XXZ model Hinitial = JSnzSn+1

z

n=1

L

!

Schreiber et al arXiv:1501.05661


Survival Probability vs LDOS

L=16, 8 up spins

100 102 104t

10-4

10-3

10-2

10-1

100

<F(t)>

0

0.1

0.2

0.3

-4 -2 0 2 4E

0

1

2

0

0.4

0.8

(a)

(b)

(c)

(d)

J

h=0.5

Torres & LFS PRB(2015) - arXiv:1501.05662

h=1.5

h=2.7

100 102 104t

10-4

10-3

10-2

10-1

100<F(t)>

0

0.1

0.2

0.3

-4 -2 0 2 4E

0

1

2

0

0.4

0.8

(a)

(b)

(c)

(d)

h=1.5

h=2.7

C!ini 2

100 102 104t

10-4

10-3

10-2

10-1

100

<F(t)>

0

0.1

0.2

0.3

-4 -2 0 2 4E

0

1

2

0

0.4

0.8

(a)

(b)

(c)

(d)

C!ini 2

100 102 104t

10-4

10-3

10-2

10-1

100

<F(t)>

0

0.1

0.2

0.3

-4 -2 0 2 4E

0

1

2

0

0.4

0.8

(a)

(b)

(c)

(d)C!ini 2

h=0.5

h=2.7 C!

ini 4

!

! = IPRiniH final = hnSn

z

n=1

L

! + J(SnxSn+1

x + SnySn+1

y + SnzSn+1

z )n=1

L

! multifractal fluctuations


Powerlaw Exponent

10-1 100 101 102 103t

10-4

10-3

10-2

10-1

100

<F(t)>

10-1 100 101 102 103t

10-1

100(a) (b)h=1.0 h=2.7

L=16 L=14 L=12 L=10

J J

Generalized dimension Multifractal dimension

!IPRini = C!ini 4

!

! "1

(Dim)"# " <1

F(t) = dE! dE" C!ini 2 C"

ini 2!! ei(E""E! )t # d#ei#t |# |$"1! #1t$

F(t) ! t"!



Time-Averaged Survival Probability

C(t) = 1t

F(! )0

t! d! " t#"

L=16 L=14 L=12 L=10

10-1 100 101 102 103t

10-3

10-2

10-1

100

C(t)

10-1 100 101 102 103t

10-1

100(a) (b)h=1.0 h=2.7

J J

PRtypini ! exp lnPRini( )" (Dim)! typ


IPRini = C!ini 4

!

! "1

(Dim)"# " <1


Excited State Quantum Phase Transition


Summary

H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!

!!..!!"!!..!!z !!..!!"!!..!!x! <1

0

100

200 Same Hamiltonian,

different initial states


Summary

H = B ! nz

n! +

J| n"m |"

! nx! m

x

n<m!

!!..!!"!!..!!z !!..!!"!!..!!x! <1

0

100

200 Same Hamiltonian,

different initial states

Similar initial states, different Hamiltonians

0 2 4 6 8 10

0.2

0.40.60.81

F

0 2 4 6 8 10

0 3 6 9 1210-3

10-2

10-1

100

F

0 3 6 9 12

0 1 2 3Jt

10-4

10-2

100

F

0 1 2 3Jt

µ = 0.2 = 0.2

µ = 0.4 = 0.4

= 1µ = 1.5

HXXZ + !HNNNHXXZintegrable chaotic

-8 -4 0 4 80

0.1

0.2

0.3

0.4

0 2 40

0.6

1.2

2 4 6

J-1

E!

0

0.3

0.6

0.9

1.2

0 2 4 6

J-1

E!

0

0.2

0.4

0.6

0.8-8 -4 0 4 8

0

0.2

0.4

0.6

PN

S

-8 -4 0 4 80

0.1

0.2

0.3

0 2 4 6

J-1

E!

0

0.5

1

1.5

2

2.5

PD

W

(a) (c)

(d) (e) (f)

(b)

C!ini 2

E!

! = 0.5

Gaussian LDOS


Dynamics at an ESQPT

Same Hamiltonian, initial states with similar energy,

but different structures

LFS & Pérez-Bernal arXiv: 1506.06765

Lipkin

H = (1!! ) N2+ " n

z

n=1

N

"#

$%

&

'(!

4!N" n

x" mx

n,m=1

N

"

L=2000






Lipkin

H = (1!! ) N2+ " n

z

n=1

N

"#

$%

&

'(!

4!N" n

x" mx

n,m=1

N

"! = 0.6

L=2000





Lipkin

H = (1!! ) N2+ " n

z

n=1

N

"#

$%

&

'(!

4!N" n

x" mx

n,m=1

N

"LDOS


! = 0.6

L=2000





Lipkin

H = (1!! ) N2+ " n

z

n=1

N

"#

$%

&

'(!

4!N" n

x" mx

n,m=1

N

"LDOS


U ! = 0.6

L=2000

U(n+1) QPT

U(n) SO(n+1)


Conclusions

Ø  Equilibration process depends on the structure of the initial state with respect to the final Hamiltonian that evolves it.

Ø The shape and width of the LDOS: short-time decay of the survival probability

PRA 89 (2014) PRA 90 (2014) NJP 16 (2014) PRE 89 (2014)

•  Lorentzian LDOS >>> exponential decay. •  Gaussian LDOS >>> Gaussian decay. •  Bimodal LDOS >>> uncertainty relation bound •  Fragmented LDOS >>> powerlaw behavior at long times. PRB (2015)

arXiv:1501.05662 arXiv:1506.08904

THANK YOU

Ø  The filling of the LDOS: long-time decay of F(t)

Ø  Presence of a critical point affects the LDOS arXiv:1506.06765

lea f. santos - crm. · pdf filelea f. santos, yeshiva university beyond integrability 2015...

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