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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