GGE  and  applica+ons  for  integrable  models  

Garry  Goldstein  

With  Natan  Andrei  

Quenching and Time Evolution - Prepare an isolated quantum many-body system in state , typically eigenstate of - At turn on interaction , and evolve system with :

- Many experiments: cold atom systems, nano-devices, molecular electronics, photonics : new systems, old questions

|�0, ti = e�iHt|�0i

Mott insulator superfluid

Time evolution of observables:

- Manifestation of interactions

Newton’s cradle

Closed systems: quenching – long time limit, thermalization

•   Long  &me  limit  and  thermaliza&on:                                                                                                                                                        -­‐  is  there  a  limit                                                                                                                                                                                                                              -­‐  is  there  a  density  operator        such  that                                      ?                                                                                                        

Does  it  depend  on                                                          not  on        

t

- Diagonal matrix elements of physical operators do not fluctuate much around constant energy surface (ETH-eigenstate thermalization hypothesis, Deutsch 92, Srednicki 94) - Overlaps do not fluctuate on the energy surface for reasonable IC - Both fluctuate but are uncorrelated

•  Scenarios of thermalization (Rigol et al)

•   Thermaliza&on,  Integrability,  Non-­‐Boltzmannian  ensembles,  Rigol,  Cardy,  Cazalilla..      If  conserva+on  laws  are  present  –  how  do  they  affect    dynamics    of    thermaliza+on?  

•   Manifesta&on  of  interac&ons  in  the  &me  evolu&on  dynamics  

Time  evolu&on  and  sta&s&cal  mechanics:  

A↵↵ = h ↵|A| ↵i

|C↵|2 = |h ↵|�0i|2

GGE  Hypothesis  -­‐  Integrable  Models  •  Integrable  models  have  infinite  number  of  local  conserved  quan++es  Im  

•  Lieb  Liniger  model  •  GGE  –  Generalized  Gibbs  Ensemble  

•  GGE  Implies  GETH  (Generalized  Eigenstate  Thermaliza+on  Hypothesis)  

Imk = kmi∑

k

Θ T→∞( ) = Tr ρ̂Θ[ ]

ρ̂ = Z −1Exp − αmIm∑( ) Tr Imρ̂[ ] = Im t = 0( )Ini+al  condi+ons  

λ Θ λ = F I1,λ, I2,λ.....( ), F smooth

What  do  we  want  to  do?  

Φ Exp αQxy T( )( ) ΨQxy ≡ b+ z( )b z( )dz

x

y

∫

b+ x( )b x( ) = −∂∂α

∂∂x

Exp αQxy( )α=0

b+ x( )b x( )b+ y( )b y( ) = −12∂2

∂α 2∂2

∂x∂yExp αQxy( )

α=0

Exp αQxy T( )( )Generating function

Want to calculate

Here

We can compute:

How  are  we  going  to  do  it?  

 1)  Compute  the  eigenstates  –  Bethe  Ansatz  2)  Compute  the  ini+al  overlaps  –  Yudson  representa+on  3)  Compute  matrix  elements                                                      -­‐  Algebraic  Bethe  ansatz  4)  Sum  over  states  –  Turns  out  to  be  a  determinant      

Exp αQxy t( )( ) ≡ Φ eitHExp αQxy( )e−itH Ψ =

= Φq

k∑

q∑ q Exp αQxy( )

kk Ψ e

i q2−k 2( )t

q Exp αQxy( )k

Yudson  Representa+on  for  finite  sized  systems  

IN =1

N kn1,....knN( )n1<....<nN

∑ kn1.....knN kn1......knN

Exp ikiL( ) =ki − kj + icki − kj − icj≠i

∏

IN =1

N kn1,....knN( )n1....nN

∑ kn1.....knN kn1.....knN(

N k1....kN( ) = det M jk( )

M jk = δ jk L + 2cc2 + kj − k l( )

2∑#

$

%%

&

'

((−

2cc2 + kj − k k( )

2

Here  

Usual  

Yudson  

N kn1.....knN( ) = kn1.....knN kn1.....knNkn1.....knN( = eikni xi

i∏

( = dx1....dxNθ x1 < x2 < ...... < xN( )∫

Exp αQxy t( )( ) ≡ Φ eitHExp αQxy( )e−itH Ψ =

= Φq)

k∑

q∑ q Exp αQxy( )

kk( Ψ e

i q2−k 2( )t

Yudson  -­‐  Proof  

k1,....kN k1,....kN = kP1,....kPN kP1....kPN(P∈SN

∑

kP1.....kPN = S−1 ki − kj( ) k1....kNi, j( )∈P∏

k1,....kN = S−1 ki,kj( )i, j( )∈P∏ k1....kN(

P∈SN

∑

S k( ) = k − ick + ic

kn1.....knN( = eikni xii∏

Want  to  show  

But  

Lieb-­‐Liniger  final  answer  Exp αQxy T( )( ) =1+

i eα −1( )2π

dx1 dy1Exp −

i4T

y12 − x1

2( )"

#$

%

&'

y1 − x1⋅ Exp

i y1 − x1( ) x2T

"

#$

%

&'−Exp

i y1 − x1( ) y2T

"

#$

%

&'

"

#$$

%

&'' b+ y1( )Exp iπ ρ z( )dz

x1

y1

∫"

#$$

%

&''b x1( )∫

"

#

$$$$

%

&

''''

−

−eα −1( )

2

2π( )2dx1 dy1

Exp −i4T

y12 − x1

2( )"

#$

%

&'

y1 − x1⋅ Exp

i y1 − x1( ) x2T

"

#$

%

&'−Exp

i y1 − x1( ) y2T

"

#$

%

&'

"

#$$

%

&''∫

"

#

$$$$

%

&

''''

×

× dx2 dy2Exp −

i4T

y22 − x2

2( )"

#$

%

&'

y2 − x2⋅ Exp

i y2 − x2( ) x2T

"

#$

%

&'−Exp

i y2 − x2( ) y2T

"

#$

%

&'

"

#$$

%

&''∫

"

#

$$$$

%

&

''''

×

× sgn y2 − y1( )sgn x2 − x1( )b+ y1( )b+ y2( )Exp iπ ρ z( )dzx1

y1

∫"

#$$

%

&''Exp iπ ρ z( )dz

x2

y2

∫"

#$$

%

&''b x1( )b x2( ) −

−i2 e2α −1( )2π( )2 c

dx1 dy1Exp −

i4T

y12 − x1

2( )"

#$

%

&'

y1 − x1⋅Exp

i y1 − x1( ) x2T

"

#$

%

&'∫ dx2 dy2

Exp −i4T

y22 − x2

2( )"

#$

%

&'

T⋅Exp

i y2 − x2( ) x2T

"

#$

%

&'∫

"

#

$$$$

−

dx1 dy1Exp −

i4T

y12 − x1

2( )"

#$

%

&'

y1 − x1⋅Exp

i y1 − x1( ) y2T

"

#$

%

&'∫ dx2 dy2

Exp −i4T

y22 − x2

2( )"

#$

%

&'

T⋅Exp

i y2 − x2( ) y2T

"

#$

%

&'∫

%

&

''''

×

× sgn y2 − y1( )sgn x2 − x1( )b+ y1( )b+ y2( )Exp iπ ρ z( )dzx1

y1

∫"

#$$

%

&''Exp iπ ρ z( )dz

x2

y2

∫"

#$$

%

&''b x1( )b x2( ) +.....

In t = 0( ) = In t→∞( )

•  We obtained an expression valid for any initial state. Can examine for what initial states the system equilibrates, whether GGE emerges or not •  If GGE emerges it provides a shortcut: one can determine final state by setting •  Valid also when system does not equilibrate but tends to a nonequilibrium steady state (NESS)

Time evolution- flow chart

Expression valid for any time and any initial state

Initial state translational invariant example: Mott insulator

Equilibrates to GGE (c >0)

Initial state not translational invariant example: Domain wall

Equilibrates but not to GGE (c < 0)

Universal correlations (if low YY entropy)

System does not equilibrate: currents, local entropy production

- Conclusions:

- GGE fails when bound states (strings) are present: LL, XXZ, Hubbard, Anderson…

Does  the  GGE  work  for  other  models?  …No  The  long  +me  dynamics  of  models  with  bound    states  are  not  controlled  by  the  GGE  

( ) ( ) ( )∏ ∏∫<

+−×=Ψji

jj

xikji

xijNN xbekkZdxdxkk jj 0....,... 11

Here  k’s  are  complex  numbers    arranged  in  strings  in  the  complex    plane.  Each  string  is  a  bound  state  

The  strings  are  chosen  such  that    

( ) ,021 =−kkS

( )ickickkS

+

−=

As  such  the  wavefunc+on  is  normalisable  

Mul+  to  one  mapping  We  can  introduce    densi+es  of  string    excita+ons  

( )dkkL npρ

Number  of  strings  in    the  interval  [k,k+dk]  

Expecta+on  values  of    all  local  observables    are  determined  by    the  string  densi+es.  Aarac+ve  Lieb-­‐Liniger  conserva+on  laws   ( ) 0

00 02

2 i

n

j

lili

n

i

l

nl IjnicJ =−⎟

⎠

⎞⎜⎝

⎛ ∑∑∑=

−−∞

= =np

nl ofmomentsJ ρ

GGE  corresponds  to  a  single  eigenstate  

J  Mossel,  J.  S.  Caux,  J  Phys  A:  Math  There+cal,  45  255001  (2012)  

O:  local  operator  ( )( )

( ) ( ) ( )( )kkee

kkDTr

kkSLkkL

p

tGGE

ptYYpi

i!!

Θ∫∫⎥⎥⎦

⎤

⎢⎢⎣

⎡=Θ ∫

−− ρρρα

ρρ

ρ,,}{

( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )( )

( ) ( )( ) ( )( ) ( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛≅

≅⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=

kkkkk

kkkdk

dkkdkkdkkdkkdkkkS

pt

tpt

p

tp

ptp

tptYY

ρρρ

ρρρρ

ρ

ρρρρ

ρρ

lnln

!!!ln,

( )( )

( )∫ −++= q

qkccdqk pt ρ

πρ 22

221

Take  saddle  point:   ( )∑ =+−i

YYp

ii S

kk 0

δρδ

α

Yang-­‐Yang    Entropy  

Time evolution- interaction quench from a Mott state Quenching from a Mott insulator to a Lieb-Liniger Liquid: GGE

|�0i =1Y

j=�1

Z 1

�1' (x+ jl) b† (x) |0i

' (x) =e

�x

2/�

(⇡�/2)1/4

- For GGE (Caux): tr [⇥⇢GGE ] =D~k0

���⇥���~k0

E

L

Zdk⇢p (k) k

n = In (t = 0) .⇢p (k) =

�12

⇡12 l

exp

✓�k2�

2

◆

⇢t (k) =1

2⇡+

1

2⇡

ZdqK (k, q) ⇢p (q)

|~k0i

and:

=L

l

✓2

�

◆n2 n!

2n2

�n2 !�

K (k, q) =2c

c2 + (k � q)2

l �p�⇢t (k) ⇠=

1

2⇡

f (k) ⌘ ⇢p (k)

⇢t (k)⇠=

2

p⇡�

lexp

✓�k2�

2

◆

for

- The occupation probability

,

Example:

with the eigenstate saisfying:

with

t ! 1

Time evolution – interaction quench from a Mott state

Can compute various correlation functions: ⌦b† (0) b† (0) b (0) b (0)

↵ ⇠= 2

Zdk12⇡

Zdk22⇡

f (k1) f (k2)(k2 � k1)

2

(k2 � k1)2 + c2

+ .....

=

2

l2� 2

p⇡c2�

l2

"exp

✓�c2

4

◆Erfc

r�c2

4

!#

⌦b† (0) b† (0) b† (0) b (0) b (0) b (0)

↵ ⇠= 6Rdk1dk2dk3f (k1) f (k2) f (k3)

(k2�k1)2

(k2�k1)2+c2

(k3�k1)2

(k3�k1)2+c2

(k3�k2)2

(k3�k2)2+c2

⇠=1

l2c4�2

⇠=2

l2c2� ⌧ 1

c2� � 1

⇠=6

l3

⇠=9⇥ 2

92

l3c6�3

c2� ⌧ 1

c2� � 1

Strong suppression of three body decay rates, measurable through trap loss or third moment of particle number (Bouchoule‘10)

Suppression of density correlations, measurable by Time of Flight experiments

h⇢ (x) ⇢ (0)i ⇠=

⇢

2+

1

4⇡

2e

2l

2exp

✓�x

2

�

◆l �

p�

Gaussian decay of density-density function

1.

2.

3. for

Real  Experiments  Have  Parabolic  confining  poten+als  

Box  Bethe  Ansatz  

Quench  ac+on  for  +me  averaged  quan++es  “Time average diagonal ensemble”

Quench Action

Caux et al.

GGE  for  +me  averaged  quan++es  

For an initial state the GGE corresponds to a quasiparticle density Φ

The even conserved local charges form the GGE

Newton’s  cradle  ini+al  condi+ons  

V1 VnL

L1 Ln

Initialize several moving boxes within a larger container

Conservation laws read

Final quasiparticle density

Newton’s  cradle  ini+al  condi+ons  (BEC)  

V1 VnL

BEC BEC

Correla+on  Func+ons  

BEC

Ground state

For two boxes in their ground state:

For two boxes containing BEC’s

Conclusions  

Future  Trends  

1) Showed  finite  sized  Yudson  formula  2) Presented  formulas  for  correlators  for    Lieb  Liniger  and  Tonks  Girardeau  Gas  3)  Used  them  to  prove  GGE  4)  Showed  failure  of  GGE  for  systems  with  bound  states  5)  GGE  for  hard  walls  

1) Field  field  correlators    2) Two  +me  correla+on  func+ons    3) Imaginary  +me  formalism    4) Kubo  Formula  5) Other  in+al  states  6) What  operators  work  for  systems  with  bound  states?  

b+ x( )b y( ) T( )ρ x,T( )ρ y,T '( )

T→ iτ