Dynamics of Interacting Quantum Systems - uni-goettingen.de · (ii) Systems without invariance under time reversal (atom in an external magnetic field) Hamiltonians are Hermitian)

Lea F. Santos, Yeshiva University Bad Honnef, Germany, 2015

Lea F. Santos

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

llabo

rato

rs

Dynamics of Interacting Quantum Systems

Marco Távora (Yeshiva University) E. Jonathan Torres-Herrera (Puebla, Mexico)

Ø  Isolated quantum systems with interacting particles.

Ø  Analysis of quench dynamics. Experiments: NMR, optical lattices, trapped ions


Coherent Evolution in Experiments

Optical Lattices Crystals formed by interfering laser beams Ultracold atoms play the role of electrons in solid crystal

Ø  highly controllable systems – interactions, level of disorder, 1,2,3D

(simple models) Ø  quasi-isolated -- study evolution for very long time

Greiner (Harvard) Weiss (Penn Sate)

Bloch (Max Planck) Esslinger (ETH)



NMR Optical Lattices Crystals formed by interfering laser beams Ultracold atoms play the role of electrons in solid crystal





Solid state NMR: nuclear positions are fixed; They are collectively addressed with magnetic pulses; Very slow relaxation

Cappellaro (MIT) Ramanathan (Dartmouth)



NMR Optical Lattices Crystals formed by interfering laser beams Ultracold atoms play the role of electrons in solid crystal





Solid state NMR: nuclear positions are fixed; They are collectively addressed with magnetic pulses; Very slow relaxation

Cappellaro (MIT) Ramanathan (Dartmouth)

Ions trapped via electric and magnetic fields. Laser used to induce couplings. Isolated from an external environment.

Monroe (Maryland)

Blatt (Innsbrück)

Ion Traps


1D spin-1/2 SYSTEMS


Flip-flop = hopping

H =J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

#

J4(!1

x! 2x +!1

y! 2y ) |!">= J

2|"!>

bn+bn+1 + h.c.( )

Holstein-Primakoff

Jordan-Wigner

Integrable systems:

XXZ and XX models (1D)


Ising interaction

H =J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

#

Two-body interaction

H = V bn+bn !

12

"

#$

%

&' bn+1

+ bn+1 !12

"

#$

%

&'! t bn

+bn+1 + h.c.( )(

)*

+

,-

n=1

L

.

Map into hardcore bosons or spinless fermions:

|!!>, |""># +J$ / 4

|!">, |"!>#%J$ / 4

(bn+bn )(bn+1

+ bn+1)

Holstein-Primakoff

Jordan-Wigner

Integrable systems:



H =J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

#Integrable systems:


Chaotic systems:

NNN model +++Δ=+ +++

−

=∑ )(4 111

1

1

yn

yn

xn

xn

zn

zn

L

nNNNNN

JHH σσσσσσ

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )

Chaotic Spin-1/2 Systems


H =J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

#Integrable systems:


Chaotic systems:

NNN model +++Δ=+ +++

−

=∑ )(4 111

1

1

yn

yn

xn

xn

zn

zn

L

nNNNNN

JHH σσσσσσ

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )

Chaotic Spin-1/2 Systems

H = !n" nz

n=1

L

! +J4("" n

z" n+1z +" n

x" n+1x +" n

y" n+1y )

n=1

L#1

!Disordered or Impurity models


Level Repulsion & Rigid Spectrum

Bohigas and

Giannoni (1984)

and also Mehta’s

Book

Average spacing

= 1


Quantum Chaos: Level Repulsion

121 EEs −=

232 EEs −=

343 EEs −=

454 EEs −=

1E

2E3E4E5E

Wigner-Dyson distribution (time reversal symmetry)

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

4exp

2)(

2sssPWDππ

Level repulsion

Level spacing distribution

(i)  Time-reversal invariant systems with rotational symmetry : Hamiltonians are real and symmetric Gaussian Orthogonal Ensemble (GOE)

(ii)  Systems without invariance under time reversal (atom in an external magnetic field) Hamiltonians are Hermitian) Gaussian Unitary Ensemble (GUE)

(iii) Time-reversal invariant systems, half-integer spin, broken rotational symmetry Gaussian Sympletic Ensemble (GSE)

Level repulsion = quantum chaos


Quantum Chaos: Level Repulsion

121 EEs −=

232 EEs −=

343 EEs −=

454 EEs −=

1E

2E3E4E5E

Level spacing distribution Hardcore bosons with NNN couplings

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

Spinless fermions with NNN couplings

LFS & Rigol PRE 80 036206 (2010)

H = t bn+bn !

12

"

#$

%

&' bn+1

+ bn+1 !12

"

#$

%

&'! t bn

+bn+1 + h.c.( )(

)*

+

,-

n=1

L

.

t ' bn+bn !

12

"

#$

%

&' bn+2

+ bn+2 !12

"

#$

%

&'! t ' bn

+bn+2 + h.c.( )(

)*

+

,-

n=1

L

.


Quantum Chaos: Rigidity

121 EEs −=

232 EEs −=

343 EEs −=

454 EEs −=

1E

2E3E4E5E

Level spacing distribution Hardcore bosons with NNN couplings

0 2 4 s

0 2 4 s

0 2 4 s

0 2 4 s

LFS & Rigol PRE 80 036206 (2010)

Level Number Variance

0.64

0.32 0.16 0.64

0.32

0.16


Two-Body Interaction vs Full Random Matrices

Density of States (Energy Distribution)

E!

!

French & Wong, PLB (1970)

Two-body interactions: Gaussian

E E!Wigner (1957)

Full random matrices: semicircular

!

Full random matrices: Matrices filled with random numbers and respecting the symmetries of the system. Wigner in the 1950’s used random matrices to study the spectrum of heavy nuclei (atoms, molecules, quantum dots)


Density of States (Energy Distribution)

E!

!


Two-body interactions: Gaussian

E!

Basis dependent

Participation Ratio

PR(! ) ! 1

| ci(! ) |4

i=1

D

"

! (" ) = ci(" )#i

i=1

D

!

E E!Wigner (1957)

Full random matrices: semicircular

!

PR

Two-Body Interaction vs Full Random Matrices


Thermalization

Time evolution of a generic observable:

Infinite time average: (generic system with nondegenerate and incommensurate spectrum)

!O(t)" = limt#$

1t

O(! )d!0

t

% = |C"ini

"

& |2 O"" =Odiag

(diagonal ensemble)

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

!

$ |2 O!! + C!ini*

!%"

$ C"iniei(E!&E" )tO!"

〉〈= βααβ ψψ || OO

|!(0)" = C!ini

!

# |"! "Initial state:


Thermalization

Will the system thermalize? Will the predictions from the infinite time average coincide with the predictions of the microcanonical ensemble?

Odiag ! |C!ini

!

" |2 O!!=?# $% Omicro !

1NE0 ,&E

O!!!

|E0'E! |<&E

"

depends on the initial conditions depends only on the energy

Infinite time average: !O(t)" = |C!ini

!

# |2 O!! =Odiag

O!! = !"! |O |"! "Eigenstate Expectation Value:


Thermalization

Will the system thermalize? Will the predictions from the infinite time average coincide with the predictions of the microcanonical ensemble?

Odiag ! |C!ini

!

" |2 O!!=?# $% Omicro !

1NE0 ,&E

O!!!

|E0'E! |<&E

"

depends on the initial conditions depends only on the energy Equation holds for all initial states that are narrow in energy when…

ETH: the expectation values of few-body observables do not fluctuate for eigenstates close in energy

ααO Deutsch, PRA 43, 2046 (1991) Srednicki, PRE 50, 888 (1994) Rigol et al, Nature 452, 854 (2008)

Full random matrices: chaotic eigenstates = (pseudo)random vectors

! (" ) = ci(" )#i

i=1

D

!

Infinite time average: !O(t)" = |C!ini

!

# |2 O!! =Odiag

O!! = !"! |O |"! "Eigenstate Expectation Value:

Zelevinsky, (1990’s) Flambaum, Izrailev Casati, Borgonovi Von Neumann (1929)


QUENCH DYNAMICS


Quench

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

!

# e$iE! t |! "Initial state

Hinitial

| n!H final

! , E!quench


Overlap between the initial state and the evolved state

Survival Probability (Fidelity)

This is not the Loschmidt Echo: FLE (t) = !(0) | eiH2te"iH1t |!(0)#2

F(t) = !(0) |!(t)"2= !(0) | e#iH finalt |!(0)"

22




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

!

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

!

# e$iE! t |! "

Eigenvalues and eigenstates of the final Hamiltonian

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2




Eigenvalues and eigenstates of the final Hamiltonian

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

-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 |2

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

!

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

!

# e$iE! t |! "

E!

C!ini 2

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Quench Dynamics

Chaotic

NNN model

H final =J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

# +Hini =

J4(!! n

z! n+1z +! n

x! n+1x +! n

y! n+1y )

n=1

L"1

#

XXZ model

Integrable

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

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

quench parameter


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: C!ini

"E

2


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

" = 0.4512!

!ini

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

Lorentzian

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


LDOS:

F(t) = exp(!"init)

C!ini

"E

2


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


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

= C!ini

!

#2e$iE!t

2

~1$" ini2 t2

C!ini

"E

2


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(!! 2init

2 )

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


LDOS:

C!ini

"E

2


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

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


E!

!

Density of States

Gaussian

LDOS: F(t) = exp(!! 2

init2 )

C!ini

"E

2


Decay slows down away from the middle of the spectrum

E!

!


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


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

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


Integrable to

chaotic

L=18, 6 up spins

Torres, Manan, LFS NJP 16 (2014)




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

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



Integrable to

chaotic

L=18, 6 up spins




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

HXX!" #" HXXZ

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

"

Integrable to

integrable

! = 0.4

! = 0.2

! =1.5


-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


HXXZ!! "! HXXZ +!HNNN HXX

!" #" HXXZ

Integrable to

integrable

Integrable to

chaotic

L=18, 6 up spins




-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

-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


Néel state: Strong perturbation and E~0

!(0) ="#"#"#"#

Néel state

! = 0.5,! =1

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )

Hinitial (!"#) $ H final = HNN +HNNN

Very strong perturbation

E ! (0) = Neel H final Neel =J"4[#(L #1)+ !(L # 2)] ~ 0


Néel state: Gaussian LDOS

!(0) ="#"#"#"#

Néel state

! = 0.5,! =1

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )


Strong perturbation

E ! (0) ~ 0

-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!L =16,Sz = 0

PR! (0) "1

|C!(! (0)) |4

!=1

Dim

#~ Dim

6

This is a chaotic initial state


Néel state: short-time Gaussian decay

!(0) ="#"#"#"#

Néel state

! = 0.5,! =1

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )


-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!L =16,Sz = 0

IPR! (0) = C!ini 4 =

1PR! (0)

~ 6Dim

0 5 10 15 20Jt

10-6

10-4

10-2

100

F

! ini =J2

L !1

The decay is Gaussian There is no transition to exponential

Open boundaries


Néel state: short-time Gaussian decay

The decay is Gaussian There is no transition to exponential

! = 0.5,! =1Sz = 0

L=10 L=20 L=30 L=40


! ini =J2

L !1

Jt

F

Jt=2 Hopping: J/2


Néel state: long-time powerlaw decay

!(0) ="#"#"#"#

Néel state

! = 0.5,! =1

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )

L = 24,Sz = 0

Long-time decay is powerlaw

0.1 1 10 100 1000

10�9

10�7

10�5

0.001

0.1

t

F�t�

Jt

F t!2

exp(!! 2init

2 )

-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!

L =16,Sz = 0


Néel state: long-time powerlaw decay

!(0) ="#"#"#"#

Néel state

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )


0.1 1 10 100 1000

10�9

10�7

10�5

0.001

0.1

t

F�t�

Jt

F t!2

exp(!! 2init

2 )

0 5 10 15 20Jt

0

0.4

0.8

1.2

f(t)

! = 0.5,! =1

f (t) = ! 1LlnF(t)

! ln(t!2 ) / L L=24 L=22

-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 8

0

0.1

0.2

0.3

0 2 4 6

J-1

E!

0

0.5

1

1.5

2

2.5P

DW

(a) (c)

(d) (e) (f)

(b)

C!ini 2

E!

L =16

L = 24,Sz = 0Kehrein, Fagotti


Brody et al Rev. Mod. Phys (1981)

As the number of particles interacting simultaneously increases, the DENSITY OF STATES changes from Gaussian to semicircle.

DENSITY OF STATES of full random matrices is semicircle and so is the LDOS.


Dynamics under full random matrices

Distribution of for initial state projected into random matrices: semicircular

C!ini 2

0 1 2 3 4 5 610-6

10-4

10-2

100

FSC

-2 -1 0 1 20

0.1

0.2

0.3

0.4

PiniSC

0 5 10 15 20 2510-3

10-2

10-1

100

FBW

-3 -2 -1 0 10

1

2

3

PiniBW

0 1 2 3 4 5 6Jt

10-4

10-2

100

FG

-6 -4 -2 0 2 4

J-1E

0

0.1

0.2

0.3

0.4

0.5

PiniG

(a)

(c)

(e)

(b)

(d)

(f)

E E time E!

C!ini 2




LDOS

J1(2! init)2

! 2init

2

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Dynamics under full random matrices

Distribution of for initial state projected into random matrices: semicircular

C!ini 2

0 1 2 3 4 5 610-6

10-4

10-2

100

FSC

-2 -1 0 1 20

0.1

0.2

0.3

0.4

PiniSC

0 5 10 15 20 2510-3

10-2

10-1

100

FBW

-3 -2 -1 0 10

1

2

3

PiniBW

0 1 2 3 4 5 6Jt

10-4

10-2

100

FG

-6 -4 -2 0 2 4

J-1E

0

0.1

0.2

0.3

0.4

0.5

PiniG

(a)

(c)

(e)

(b)

(d)

(f)

E E time E!

C!ini 2

LDOS

F(t) =J1(2! init)

2

! 2init

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2

t!3




LONG-TIME BEHAVIOR


Long-time: Powerlaw Decay

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2

F(t) = Pini (E)e!iEt dE

Ecut

"

#2

$ Aexp(!btq )

Khalfin (JETP, 1957)

0 < q <1

Ersak (1969) Fleming (1973) Fonda et (1978) Knight (1977) Sluis & Gislason PRA (1991) Muga, del Campo (2009) Del Campo, arXiv 1504.0120 (CS model)


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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Ecut

"

#2

$ Aexp(!btq )


Asymptotic Expansions:

*)limE!EcutPini (E)> 0" F(t)# t$2

0 < q <1

Powerlaw Decay: Gaussian LDOS

Pini (E)dE =1!"

"

# , F(t)$t$" 0


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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Ecut

"

#2

$ Aexp(!btq )


Asymptotic Expansions:

F(t) = 12!" ini

2e!(E!Eini )

2 /2" ini2

e!iEt dE0

"

#2

$t$"

12!" ini

2e!Eini

2 /2" ini2

e!iEt dE0

"

#2

%1t2

-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 < q <1 LDOS


Pini (E)dE =1!"

"

# , F(t)$t$" 0




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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Ecut

"

#2

$ Aexp(!btq )


Asymptotic Expansions

0.1 1 10 100 1000

10�9

10�7

10�5

0.001

0.1

t

F�t�

Jt

F t!2

exp(!! 2init

2 )

Chaotic model with NNN couplings

0 < q <1



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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Ecut

"

#2

$ Aexp(!btq )



! t"2

*)Pini (E) = (E !Emin )! g(E)" F(t)# t!2(1+! )

0 1 2 3 4 5 610-6

10-4

10-2

100

FSC

-2 -1 0 1 20

0.1

0.2

0.3

0.4

PiniSC

0 5 10 15 20 2510-3

10-2

10-1

100

FBW

-3 -2 -1 0 10

1

2

3

PiniBW

0 1 2 3 4 5 6Jt

10-4

10-2

100

FG

-6 -4 -2 0 2 4

J-1E

0

0.1

0.2

0.3

0.4

0.5

PiniG

(a)

(c)

(e)

(b)

(d)

(f)

E E!

C!ini 2

LDOS

0 < q <1

Powerlaw Decay: Semicircle LDOS



Powerlaw Decay: Semicircle LDOS

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

ini

!

#2e$iE! t

2

% Pini (E)e$iEt dE

$&

&

'2


Ecut

"

#2

$ Aexp(!btq )

Khalfin (JETP, 1957) 0 < q <1


Pini (E) =1

! ini1! E

2! ini

"

#

$$

%

&

''

2

( (E ! 2! ini )1/2 (E + 2! ini )

1/2

0 1 2 3 4 5 610-6

10-4

10-2

100

FSC

-2 -1 0 1 20

0.1

0.2

0.3

0.4

PiniSC

0 5 10 15 20 2510-3

10-2

10-1

100

FBW

-3 -2 -1 0 10

1

2

3

PiniBW

0 1 2 3 4 5 6Jt

10-4

10-2

100

FG

-6 -4 -2 0 2 4

J-1E

0

0.1

0.2

0.3

0.4

0.5

PiniG

(a)

(c)

(e)

(b)

(d)

(f)

E time

t!3

*)Pini (E) = (E !Emin )! g(E)" F(t)# t!2(1+! )



!(0) ="#"#"#"#Néel state J

4n=1

L

! (! nx! n+1

x +! ny! n+1

y ) [XX model]

t!1/2It suggest that the LDOS Pini(E) is not well behaved. (not Gaussian)

Jt

F

Powerlaw decay for the XX model: sparse and fractal LDOS

Zanardi, Sirker, Haque

F(t) = limL!"

eL2! dk lncos2(Jt cosk)

#! /2

! /2

$


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 92, 01420 (2015)

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

!

! = IPRini

H final = hnSnz

n=1

L

! + J(SnxSn+1

x + SnySn+1

y + SnzSn+1

z )n=1

L

! multifractal fluctuations

Powerlaw decay for a disordered system: sparse LDOS


Powerlaw Exponent

Generalized dimension Multifractal dimension

!IPRini = C!ini 4

!

! "1

(Dim)"# " <1

Torres & LFS PRB 92, 01420 (2015)

ln(Dim)


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"!

Torres & LFS PRB 92, 01420 (2015)


Powerlaw decay: tail and filling of the LDOS

The long-time powerlaw decay of F(t) in many-body quantum systems depends on the tails and filling of the LDOS. •  Well filled LDOS (chaos): between t-2 and t-3 depending on the number of

particles that interact simultaneously.

•  Sparse LDOS: energy of the initial state, level of disorder, symmetries, regime.

Rigol arXiv:1511.04447


Real Systems: Banded Matrices

H =J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+ " J4n=1

L!2

" (#! nz! n+2

z +! nx! n+2

x +! ny! n+2

y )

H = H0 + !V

Torres & LFS PRE 89 (2014)


Band Random Matrix

Wigner Band Random Matrix


1 +v +v 0+v 2 !v +v+v !v 3 +v0 +v +v 4

0 0 0 00 0 0 0!v 0 0 0+v +v 0 0

0 0 !v +v0 0 0 +v0 0 0 00 0 0 0

5 +v !v 0+v 6 +v +v!v +v 7 +v0 +v +v 8

"

#

$$$$$$$$$$

%

&

''''''''''

Wigner, Ann. Math. (1955)

q = v2

bs

q>>1: semicircle q<<1: Lorentzian


Band Random Matrix

Wigner Band Random Matrix


1 +v +v 0+v 2 !v +v+v !v 3 +v0 +v +v 4

0 0 0 00 0 0 0!v 0 0 0+v +v 0 0

0 0 !v +v0 0 0 +v0 0 0 00 0 0 0

5 +v !v 0+v 6 +v +v!v +v 7 +v0 +v +v 8

"

#

$$$$$$$$$$

%

&

''''''''''

q = v2

bs

q>>1: semicircle q<<1: Lorentzian

Powerlaw Band Random Matrix

Hnm2 =

1

1+ n!mb

2


Powerlaw Band Random Matrix

Hnm2 =

1

1+ n!mb

2

t!3

t!2

t!(<1)

b=0

b=Dim


Few-body Observables


Dynamics of few-body observables

When:

[Hinitial,O]= 0

!O(t)" = F(t)O(0)+ !n |O | n" !n | e#iH finalt | ini"n$ini%

2

When initial state is a site-basis vector:

magnetization, spin-spin correlation in z, structure factor in z, interaction energy

!O(t)" = !#(t) |O |#(t)" Hinitial n = !n n

H final ! = E! !


Dynamics of few-body observables

When:

[Hinitial,O]= 0

!O(t)" = F(t)!O(0)+ !n |O | n" !n | e#iH finalt | ini"! "## $##n$ini

%2

When initial state is a site-basis vector:

magnetization, spin-spin correlation in z, structure factor in z, interaction energy

!O(t)" = !#(t) |O |#(t)" Hinitial n = !n n

H final ! = E! !

C!nC!

inie!iE!t!

"2

C!ini 2 e!iE!t

!

"2


Observables: Full Random Matrix

!O(t)" = F(t)O(0)+ !n |O | n" !n | e#iH finalt | ini"! "## $##n$ini

%2

C!nC!

inie!iE!t!

"2

ML/2z (0)

J1(2! init)2

! 2init

2

F(t) =J1(2! init)

2

! 2init

2

ML/2z (0)F(t)

ML/2z (t)

ML/2z (t)

Jt

infinite time average


Observables: Band Random Matrix

!O(t)" = F(t)O(0)+ !n |O | n" !n | e#iH finalt | ini"! "## $##n$ini

%2

C!nC!

inie!iE!t!

"2

0.01 0.1 1 10Jt

10-410-310-210-1100

Mz L/2(t)

ML/2z (0)F(t)

ML/2z (t)


Observables: Integrable models

!O(t)" = F(t)O(0)+ !n |O | n" !n | e#iH finalt | ini"2

n$ini%

0.1 1 10Jt

10-510-410-310-210-1

Mz L/2(t)

!(0) ="#"#"#"#

Néel state XX model

ML/2z (t) = J 0 (2t) / 2

L = 24,Sz = 0

t!1/2

Schütz, Mitra


Observables: Integrable models

!O(t)" = F(t)O(0)+ !n |O | n" !n | e#iH finalt | ini"2

n$ini%

0.1 1 10Jt

10-510-410-310-210-1

Mz L/2(t)

!(0) ="#"#"#"#

Néel state XX model

ML/2z (t) = J 0 (2t) / 2

!(0) ="#"#"#"#

Néel state XXZ model

Barmettler et al, NJP (2010) L = 24,Sz = 0

ms (t)! e"t/! 2 cos("t +#)

t!1/2

Mossel & Caux, NJP (2010)

DW, ! =1.5 F(t)" e#t/!Schütz, Mitra


2 4 6 8 10Jt

10-6

10-4

10-2

100Mz L/2(t)

Observables: Chaotic spin-1/2 system

L = 24,Sz = 0

!(0) ="#"#"#"#

Néel state

J4n=1

L!1

" (#! nz! n+1

z +! nx! n+1

x +! ny! n+1

y )+

+!J4n=1

L!2

" (#" nz" n+2

z +" nx" n+2

x +" ny" n+2

y )

ML/2z (t) = A

te!t/!sin("t +#)

0.1 1 10 100 1000

10�9

10�7

10�5

0.001

0.1

t

F�t�

Jt

F t!2

exp(!! 2init

2 )


Conclusions

Ø  The dynamics of the system depends on the interplay between the initial state and the final Hamiltonian.

F(t) for short- and long-times Ø  The shape and width of the LDOS: short-time decay of F(t). (No dependence on the regime)

Tavora, Torres, LFS, arXiv soon

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

•  Lorentzian LDOS: exponential decay. •  Gaussian LDOS: Gaussian decay. •  Semicircle LDOS: Bessel2/t2 •  Bimodal LDOS: uncertainty relation bound •  Filled LDOS: t-2 , t-3 due to bounds in the spectrum. •  Fragmented LDOS: small powerlaw exponent due to correlations.

Ø  The filling of the LDOS: long-time decay of F(t) (Dependence on the regime) PRB 92 (2015)

Ø  Observables: no general picture yet. C!nC!

inie!iE!t!

"2


Excited State Quantum Phase Transition with trapped ions: long-range interaction

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)

N=2000, L=0

! = 0.6

Ek/N 0 0.2 0.4 0.6 0.8

Ckn

!E

2

LFS & Pérez-Bernal PRA 92, 050101R

(2015)

|!(0)" = n = 0 = Ckn=0

k# |!k "

! = 0Lipkin H = (1!! )nt +

!N(t+s+ s+t)2

n=1332

!(0) |!(t)"22

0 0.2 0.4 0.6 0.8 1Control Parameter

0

0.2

0.4

0.6

0.8

E k/N (a

.u.)

0 0.2 0.4 0.6 0.8Normalized Energy (a.u.)

0

5

10

Den

sity

of S

tate

s

0 0.4 0.8

0.41 0.42

0 0.4 0.8 0 0.4 0.80 0.4 0.80

0.04

0.08

0.12

PR(k

)U

(3)/N

0 0.4 0.80

0.1

0.2

PR(k

)SO

(4)/N

0.41 0.42

0 0.4 0.8

0 0.4 0.8Normalized Excitation Energy (Ek/N, a.u.)

0

0.1

0.2

0 0.4 0.8Normalized Excitation Energy (Ek/N, a.u.)

c = 0.2 = 0.3 = 0.4

= 0.0

= 0.6 = 0.9

= 0.6 = 0.9

PR vs E/N

Documents

Dynamics of Interacting Quantum Systems - uni-goettingen.de · (ii) Systems without invariance under time reversal (atom in an external magnetic field) Hamiltonians are Hermitian)