Quench Dynamics in a Model With Tuneable Integrability Breaking - F.H.L. Essler

8/13/2019 Quench Dynamics in a Model With Tuneable Integrability Breaking - F.H.L. Essler

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Quench Dynamics in a Model with Tuneable Integrability Breaking

F.H.L. Essler, 1 S. Kehrein, 2 S.R. Manmana, 2 and N.J. Robinson 11 The Rudolf Peierls Centre for Theoretical Physics,

Oxford University, Oxford, OX1 3NP, United Kingdom 2 Institut f ur Theoretische Physik, Georg-August-Universit at G ottingen, D-37077 G ottingen, Germany

(Dated: November 20, 2013)

We consider quantum quenches in an integrable quantum chain with tuneable integrability break-ing interactions. In the case where these interactions are weak, we demonstrate that at intermediatetimes after the quench local observables relax to a prethermalized regime, which can be describedby a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. Wepresent explicit expressions for the approximately conserved charges characterizing this ensemble.We do not nd evidence for a crossover from the prethermalized to a thermalized regime on thetime-scales accessible to us. Increasing the integrability-breaking interactions leads to a behaviourthat is compatible with eventual thermalization.

PACS numbers: 02.30.Ik,03.75.Kk,05.70.Ln

I. INTRODUCTION

Important advances in manipulating cold atomic gaseshave allowed recent experiments [16] to realize essen-tially unitary time-evolution for extended periods of time.Stimulated by such experiments, there has been im-mense theoretical effort (see e.g. [ 7] for a recent re-view) to understand fundamental questions about thenon-equilibrium dynamics of quantum systems: Do ob-servables in a subsystem relax to stationary values? If so, can expectation values be reproduced with a thermaldensity matrix? What governs how and to which valuesobservables relax?

It is generally accepted that conservation laws and di-mensionality play important roles in the time-evolutionof isolated quantum systems. This is highlighted by the

ground-breaking experiments of Kinoshita, Wenger andWeiss [2]. There, it was found that a three-dimensionalcondensate of 87Rb atoms driven out of equilibriumrapidly relaxed to a thermal state (thermalized), whilsta condensate constrained to move in a single spatial di-mension relaxed slowly to a non-thermal ensemble. Itis thought that the presence of additional (approximate)conservation laws in the one-dimensional case lies at theheart of this difference.

Theoretical investigations on translationally invariantmodels have established two central paradigms for thelate time behaviour after a quantum quench: (1) subsys-tems thermalize and are then described by a Gibbs en-semble (GE) [8]; (2) subsystems do not thermalize, butat late times after the quench are described by Gener-alized Gibbs ensembles (GGE). There is substantial ev-idence [928] that the latter case applies quite generallyto quenches in quantum integrable models, as suggestedin a seminal paper by Rigol et al [29].

The dichotomy in the dependence of stationary be-haviour after a quench on integrability then poses an in-triguing question: what happens if integrability is weaklybroken? Does the system thermalize, and if so, how fastdoes it relax? Might there be an intermediate time scale

still governed by the physics of integrability?Early numerical studies [30] suggested that even with

an integrability breaking term the system does not ther-

malize on the accessible time scales and system sizes.Studies using dynamical mean eld theory (DMFT) [ 31]or a DMFT-like ( d ) limit [32] showed that on inter-mediate time scales the system approaches a non-thermalquasi-stationary state (a prethermalization plateau). Atlater times the system is expected to thermalize. Prether-malization plateaux have also been observed in a non-integrable quantum Ising chain with long-range interac-tions [33]. It has been suggested recently [34], that thetime scale for integrability breaking (leaving the prether-malization plateau) is not necessarily related to thestrength of the integrability breaking term. Experimen-tal evidence for the prethermalization plateau in systems

of bosonic cold atoms was reported in Refs. [ 6, 35, 36].In this work we study the effects of integrabilitybreaking interactions on the dynamics following a quan-tum quench. Our setup allows us to compare inte-grable quantum quenches to quenches where an addi-tional integrability-breaking interaction is added to thepost-quench Hamiltonian. By combining analytical cal-culations with time-dependent density matrix renormal-ization group (t-DMRG) results we demonstrate the ex-istence of a prethermalization plateau in the sense thatlocal observables relax to non-thermal values at inter-mediate times. We characterize this prethermalizationplateau in terms of a statistical description that we calldeformed GGE.

This paper is organized as follows. In Sec. II we intro-duce the model under study. In section III we considerintegrable quenches and compare the observed station-ary behaviour to thermal and generalized Gibbs ensem-bles. The continuous unitary transformation techniqueis introduced and used to study a weakly non-integrablequench of the model in Sec. IV. In Sec. V we establishthe existence of the prethermalized regime and describethe approximately stationary behaviour in this regimeby constructing a deformed GGE. The dynamics in

a r X i v : 1 3 1 1 . 4 5 5 7 v 1 [ c o n d - m a t . s t a t - m e c h ] 1 8 N o v 2 0 1 3


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the presence of strong integrability-breaking interactionsis studied numerically in Sec. VI. Sec. VII contains asummary and discussion of our main results. Technicaldetails underpinning our analysis are consigned to severalappendices.

II. THE MODEL

We consider the following Hamiltonian of spinlessfermions with dimerization and density-density interac-tions

H (, U ) = J L

l=1

1 + ( 1)l cl cl+1 + h .c.

+ U L

l=1

cl clcl+1 cl+1 , (1)

with periodic boundary conditions. Here {cl , cj } = l,jand we restrict our attention to the parameter regime

J > 0, U 0 and 0 < < 1. An important characteristicof H (, U ) is that fermion number is conserved by virtueof the U (1) symmetry

cj ei cj , [0, 2]. (2)We note that the Hamiltonian ( 1) is equivalent to adimerized spin-1/2 Heisenberg XXZ chain as can beshown by means of a Jordan-Wigner transformation.

A. Peierls Insulator

The special case H (, 0) describes a Peierls insulatorand can be solved by means of a Bogoliubov transforma-tion

cl = 1 L k> 0 =

(l, k| )a (k) . (3)

Here a (k) are fermion annihilation operators fullling

{a (k), a (q )}= 0 , {a (k), a (q )}= , k,q . (4)

The coefficients are chosen as

(l, k| ) = e ikl u (k, ) + v (k, )(1)l , (5)

where

v (k, ) = 1 +2J cos(k) (k)

2J sin(k)

2 1/ 2

,

u (k, ) = iv (k)2J cos(k) (k)

2J sin(k) , (6)

(k, ) = 2 J 2 + (1 2)cos2(k) . (7)Finally, k> 0 is a shorthand notation for the momentumsum

k> 0

f (k) =L/ 2

n =1f

2nL

. (8)

In terms of the Bogoliubov fermions the Peierls Hamil-tonian is diagonal

H (, 0) =k> 0

(k, )a (k)a (k). (9)

B. Integrability Breaking Interactions

Adding interactions to the Peierls Hamiltonian leadsto a theory that is not integrable. An exception is thelow-energy limit for | | 1, which is described by aquantum sine-Gordon model [37]. In the following wewill be interested in the regime 0 .4 0.8, which is faraway from this limit. It is useful to express the density-density interaction in H (, U ) in terms of the Bogoliubovfermions diagonalizing H (, 0)

H int = U L

l=1

cl cl cl+1 cl+1 = U

k j > 0

V 1 2 3 4 (k1 , k2 , k3 , k4)a 1 (k1)a 2 (k2)a

3 (k3)a 4 (k4) ,

V (k ) = 1L2

l

1 (l, k1| ) 2 (l, k2| ) 3 (l + 1 , k3| ) 4 (l + 1 , k4| ) ,

= 1L

ei (k3 k 4 ) k1 + k 3 ,k 2 + k4 [w 1 2 (k1 , k2)w 3 4 (k3 , k4) x 1 2 (k1 , k2)x 3 4 (k3 , k4)]+ k1 + k3 + ,k 2 + k4 [x 1 2 (k1 , k2)w 3 4 (k3 , k4) w 1 2 (k1 , k2)x 3 4 (k3 , k4)] . (10)

Here we have denedw (k, p) = u (k, )u ( p, ) + u v, (11)x (k, p) = u

(k, )v ( p, ) + u

v. (12)

III. INTEGRABLE QUANTUM QUENCHES

We rst consider a quantum quench of the dimerizationparameter in the limit of vanishing interactions U = 0.


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The system is initially prepared in the ground state |0of H ( i , 0), and at time t = 0 the dimerization is suddenlyquenched from i to f . At times t > 0 the system evolvesunitarily with the new Hamiltonian H ( f , 0).

The diagonal form of our initial Hamiltonian is

H ( i , 0) = = k> 0

(k, i )b (k)b (k), (13)

and describes two bands of noninteracting fermions. Theground state is obtained by completely lling the band, i.e.

|0 =k

b (k)|0 , (14)

where |0 is the fermion vacuum dened by b (k)|0 = 0, = , k (0, ]. At times t > 0 the system is in thestate|0(t) = e iH (f ,0) t |0 . (15)

The new Hamiltonian is diagonalized by the Bogoliubovtransformation ( 3)

H ( f , 0) = = k> 0

(k, f )a (k)a (k), (16)

and by virtue of ( 3) the Bogoliubov fermions a (k), a (k)are linearly related to b (k), b (k). Using this relationit is a straightforward exercise to obtain an explicit ex-pression for the time evolution of the fermion Greensfunction

G0( j, , t) = 0(t)|cj c |0(t)

= 1L k> 0

( j, k | f ) ( , k| f ) ei ( (k ) (k )) t S (k)S

(k)

(17)

where

S (k) = u (k, f )u

(k, i ) + u v. (18)The late-time behaviour can be determined by a sta-

tionary phase approximation, which gives

limt

G0( j, , t)g1( j, ) + g2( j, )t 3/ 2 + . . . (19)

A. Generalized Gibbs Ensemble (GGE)

The stationary state of the dimerization quench isdescribed by a GGE [29]. We now briey review theconstruction of the GGE following Refs [911]. In thethermodynamic limit the system after the quench pos-sesses an innite number of local conservation laws I (n )a(a = 1, 2, 3, 4, n

N )

[I (n )a , I (m )b ] = 0 , I

(1)1 = H ( f , 0). (20)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 2 4 6 8 10Time t

G (50, 51)G (50, 49)

iG (50, 52) iG (50, 48)

FIG. 1. Greens function G0 ( j, l, t ) for a quench with i =0.75, f = 0 .25 and a lattice with L = 100 sites.

An explicit construction of these conservation laws is pre-sented in Appendix A. Given these conserved quantitieswe dened a density matrix

GGE = 1Z GGEexp

4

a =1 j 1

( j )a I ( j )a , (21)

where Z GGE ensures normalization [ 38]. The Lagrangemultipliers are xed by the requirements that the expec-tation values of the conserved quantities are the same inthe initial state and in the GGE

limL

1L

0|I ( j )a |0 = limL 1L

tr GGE I ( j )a . (22)

We then bipartition the system into a segment B of con-tiguous sites and its complement A and form the reduceddensity matrix

GGE ,B = tr A [ GGE ] . (23)

On the other hand the reduced density matrix of segmentB after our quantum quench is simply

B (t) = tr A |0(t) 0(t)| . (24)At late times after the quench it can be shown by usingfree fermion techniques (see e.g. [ 10]) that

limt

limL

B (t) = GGE ,B . (25)

An alternative [9, 13, 29] but equivalent [11] constructionof the GGE is based on the mode occupation numbers

n (k) = a (k)a (k). (26)

By construction these commute with H ( f , 0) and amongthemselves, and we can express the density matrix in theform

GGE = 1Z GGE

exp k> 0 =

( )k n (k) . (27)


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The Lagrange multipliers are xed by the conditions

0|n (k)|0 = tr [ GGE n (k)] , (28)which are solved by

e (+)k = |S

+ (k)|2

1

|S + (k)

|2 ,

e ( )k = |S

(k)|2

1 |S (k)|2 . (29)

Here the functions S (k) are dened in ( 18).

B. GGE vs. thermal expectation values

In the following it will be important to quantify the dif-ference between the GGE constructed above and a Gibbsensemble (GE)

G = 1Z G

exp( eff H ( f , 0)) , (30)

constructed by requiring that the average thermal energydensity is equal to the energy density in the initial state

limL

1L

0|H ( f , 0)|0 = limL 1L

tr [ G ( eff ) H ( f , 0)] .(31)

Using the fact that the fermions diagonalizing H ( f , 0)and H ( i , 0) are linearly related by

a (k) = S (k) b

(k), (32)

we can rewrite ( 31) in the form

k> 0+ (k, f ) |S (k)|2 |S + (k)|2

=k> 0

+ (k, f ) tanh eff 2

+ (k, f )|2 . (33)

1. Mode Occupation Numbers

In order to exhibit the difference between Gibbs andgeneralized Gibbs ensembles it is useful to consider themode occupation numbers, which are given by

n ( p) =

11+exp eff (k, f )

for GE,1

1+exp ( )k for GGE.

(34)

Clearly the mode occupation numbers shown inFigs. 2 & 3 are very different in the two ensembles.

0

0.02

0.040.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.5 1 1.5 2 2.5 3

n +

( k )

Momentum k

GE = 2 . 95782GGE

FIG. 2. Comparison between the mode occupation num-bers n + (k) for Gibbs and generalized Gibbs ensembles fora quench with i = 0 .75, f = 0 .25. The effective inversetemperature for this quench is eff = 2 .95782J .

0.8

0.85

0.9

0.95

1

1.05

0 0.5 1 1.5 2 2.5 3

n

( k )

Momentum k

GE = 2 . 95782GGE

FIG. 3. Comparison between the mode occupation num-bers n (k) for Gibbs and generalized Gibbs ensembles fora quench with i = 0 .75, f = 0 .25. The effective inversetemperature for this quench is eff = 2 .95782J .

2. Greens Function

As has been emphasized in [ 10], as we are dealing withthe non-equilibrium dynamics of an isolated quantum sys-tem, we should focus on the expectation values of local(in space) operators, as descriptions in terms of statis-tical ensembles most naturally apply to them (see also[11, 39]). We therefore consider the fermionic Greensfunction in position space, and furthermore focus on itsshort-distance properties. The Greens functions in theGGE and thermal ensembles are

cj cl = 1L p> 0

( j, p| f ) (l, p| f ) n ( p) , (35)

where the mode occupation numbers are given by ( 34).In Fig. 4 we show a comparison between the results forthe fermion Greens function calculated in the appropri-ate Gibbs and generalized Gibbs ensembles. We observe


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-0.1

0

0.1

0.2

0.3

0.4

0.5

-10 -5 0 5 10

G

( L / 2 , L / 2 + j , t

)

j

GE = 2 .95782GGE

FIG. 4. Greens function cL/ 2 cL/ 2+ j calculated in the Gibbsand generalized Gibbs ensembles for a quench with i = 0 .75, f = 0 .25 and a lattice with L = 100 sites. The effectiveinverse temperature for this quench is eff = 2 .95782J .

that in contrast to the mode occupation numbers, the dif-

ference between the short-distance behaviour of Greensfunction in the two ensembles is fairly small.

IV. QUENCHING TO A WEAKLYINTERACTING MODEL

We now modify our quantum quench as follows. Westill start out our system in the ground state |0 of thepure Peierls Hamiltonian H ( i , 0), but we now quench toH ( f , U ), where we consider U/t to be small compared tomin( i , f ). Our main interest is to quantify how a non-zero value of U modies the dynamics after the quench.

To tackle the quench problem in the non-integrableweakly interacting model we employ the continuous uni-tary transformation (CUT) technique [40, 41] which hasbeen applied extensively to non-equilibrium problems(see, for example, Refs. [ 32, 42]). We provide a brief overview of the CUT technique for out-of-equilibriummany-body systems and proceed to calculate the time-dependent Greens function and the four-point function.

A. Time-evolution of observables by CUT

For a non-integrable interacting model it is no longerpossible to calculate the time-evolution induced by theHamiltonian ( 1) exactly. We use the CUT technique toobtain a perturbative expansion in U of the time-evolvedobservables.

The central idea of the CUT method is to constructa sequence of innitesimal unitary transformations, cho-sen such that the Hamiltonian becomes successively moreenergy-diagonal. A family of unitarily equivalent Hamil-tonians H (B ) characterized by the parameter B can be

constructed from the solutions of the differential equation

dH (B )dB

= (B ), H (B ) , (36)

where (B ) is the antihermitian generator of the unitarytransformation. Wegner [ 40] showed that the Hamilto-nian in the nal basis H (B = ) is energy diagonal if (B ) = [H 0(B ), H int (B )], where H 0 is the quadratic partof the Hamiltonian and H int is remainder. In practice(36) in used by expanding all operators in power seriesin an appropriate small parameter, which in our case willbe the interaction strength U .

Following the transformation with an appropriatechoice of generator, the Hamiltonian is energy diago-nal. This does not, however, remove all the interactionterms; energy-diagonal interactions remain. To performthe time-evolution we must introduce an additional ap-proximation: We normal order the interaction term withrespect to the initial state |0 and neglect the normal-ordered quartic (and higher order) terms

H (B =

) = H 0(B =

) + H int (B =

)

= H + : H int (B = ) : , U (t) exp(iH t) ,

where the time-evolution operator U (t) depends only onthe quadratic Hamiltonian H whose single particle en-ergies have O(U ) contributions. We expect this approx-imation to introduce a maximal timescale on which wecan trust our calculations. For this reason we extensivelycompare our CUT results to t-DMRG computations (seeSec. IV E). The procedure for calculating the approxi-mate time-evolution of observables is shown schemati-cally in Fig. 5.

FIG. 5. A schematic of the CUT method for nding the ap-proximate time-evolution of the operator O to order U .

B. The canonical generator and ow equations forthe Hamiltonian

We start by constructing the canonical generator of the unitary transformation [ 41] given by

(B ) = [H 0(B ), H int (B )]. (37)


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The ow-dependent operators are dened by

H 0(B ) = = k> 0

(k|B )a (k)a (k), (38)

H int (B ) =k j > 0

V (k |B )a 1 (k1)a 2 (k2)a 3 (k3)a 4 (k4)+ . . . . (39)

where the parameters in the Hamiltonian have been pro-moted to functions of the ow parameter B and werethe dots indicate terms sextic and higher in creation andannihilation operators. The canonical generator is givenby

= U k j > 0

W (k |B )a 1 (k1)a 2 (k2)a 3 (k3)a 4 (k4)+ O(U 2), (40)

where

W (k |B ) = V (k |B )E (k |B ),E (k |B ) = 1 (k1|B ) 2 (k2|B )

+ 3 (k3|B ) 4 (k4|B ).By inserting the canonical generator ( 40) and the ow

Hamiltonian

H (B ) = H 0(B ) + H int (B ) , (41)

into the ow equation ( 36) and integrating the resultingdifferential equations, we nd the ow-dependent singleparticle energies and interaction vertices

(k|B ) = (k|B = 0) , (42)V (k |B ) = V (k |B = 0) e BE

2 (k ) . (43)

Setting B = we obtain the Hamiltonian in the energy-diagonal basis

H (B = ) = = k> 0 (k)a (k)a (k) +

k j > 0V (k )a 1 (k1)a 2 (k2)a 3 (k3)a 4 (k4) + O(U 2) , (44)

where indeed the interaction vertices conserve energy

V (k ) V (k |B = ) = V (k ) E (k ) ,0 . (45)We note that to leading order in U the single particleenergies (k) remain unchanged by the unitary trans-formation. Having found the energy-diagonal form of theHamiltonian to leading order we now consider the unitarytransformation induced by the canonical generator ( 40)on the Greens function.

C. Greens function

Our main objective is to determine the fermion Greensfunction on the time-evolved initial state

G( j, l ; t) = 0(t)|cj cl |0(t) . (46)

Using the expression for the original fermions in terms of the Bogoliubov fermions a (k), we see that

cj cl = 1L

k,q> 0 , =

( j, k | f ) (l, q | f )

n (k, q |B = 0) , (47)where ( j, k

| ) are dened in Eq. ( 5) and n (k, q

|B =

0) = a (k)a (q ). Hence the basic objects we need tocalculate are expectation values of n ( p, q |B = 0). Thisis done by following the procedure set out in Fig. 5. Theow equations

dn ( p, q |B )dB

= (B ), n ( p, q |B ) (48)are easily constructed to order O(U ) and integratingthem gives

n (k, p|B ) = n (k, p|B = 0) + U qj > 0

N (q |k,p,B )a 1 (q 1)a 2 (q 2)a 3 (q 3)a 4 (q 4) + O(U 2), (49)where we have dened

N (q |k,p,B ) = q4 ,p 4 , V 1 2 3 (q 1 , q 2 , q 3 , k|B ) + q2 ,p 2 , V 1 3 4 (q 1 , k ,q 3 , q 4|B ) q3 ,k 3 , V 1 2 4 (q 1 , q 2 ,p ,q 4|B ) q1 ,k 1 , V 2 3 4 ( p, q 2 , q 3 , q 4|B ) ,

V (q |B ) = 1e B [E (q )]

2

E (q ) V (q ). (50)

1. Approximate Time-evolution

In the next step of the procedure sketched in Fig. 5we consider the time-evolution induced by the B =

Hamiltonian ( 44). We approximate the time-evolution


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operator U (t) by U (t) = e iH (B = ) t e iH t , (51)

where the Hamiltonian H (B = ) has been replaced bythe free fermion HamiltonianH =

= k> 0

(k)a (k)a (k),

with single particle energies

(k) = (k) + UP (k) . (52)

The additional term P (k) is given by

P (k) =, q> 0

V (k ,k ,q ,q ) + V (q ,q ,k ,k)

V (k ,q ,q ,k) V (q ,k ,k ,q ) n (q ) ,(53)

where V (k ) is dened in Eq. ( 45). The expectation val-ues n (q ) = 0|n (q, q )|0 taken in the initial stateare given by

n (k) = |S (k)|2

n++ (k) = |S

+ (k)|2

n+ (k) = S + (k)S (k)

n + (k) = S (k)S + (k)

,

(54)

where functions S (k) are dened by Eq. (18). The cor-rection to the single particle energies P (k) arises fromnormal ordering the interaction term with respect to theinitial state |0 . The normal ordering prescription forthe quartic term is given by

a 1 a 2 a 3 a 4 = : a

1 a 2 a

3 a 4 : +n 1 2 (k1) k1 ,k 2 : a

3 a 4 : +n 3 4 (k3) k 3 ,k 4 : a

1 a 2 :

n 1 4 (k1) k1 ,k 4 : a 3 a 2 : [n 3 2 (k3) 2 , 3 ] k2 ,k 3 : a 1 a 4 :+ n 1 2 (k1)n 3 4 (k3) k1 ,k 2 k3 ,k 4 n 1 4 (k1)[n 3 2 (k2) 2 , 3 ] k 1 ,k 4 k2 ,k 3 , (55)

The normal-ordered quartic interaction term on the righthand side of ( 55) has been neglected for the time evolu-tion in Eq. ( 51). Following this approximation, the time-evolution of fermion operators results only in additionalphase factors

U (t)a (k) U (t) = ei (k ) t a (k). (56)

Using (56) in (49) provides an explicit expression for thetime-evolved operators n (k, p|B = , t ). In the nalstep shown in Fig. 5 we reverse the CUT. Integratingback to the initial basis B = 0, and then taking theexpectation value with respect to the initial state |0we obtain

n ( p, q |B = 0, t ) = p,q ei ( ( p) (q)) t n ( p)+ U c ( p, q |t) + O(U 2), (57)

Here the order U piece is

c ( p, q |t) =q,r> 0

N (r,r,q,q | p, q |t)n 1 2 (r )n 3 4 (q ) N (r,q,q,r | p, q |t)n 1 4 (r ) n 3 2 (q ) 2 , 3 . (58)

where we have dened

N (k | p, q |t) = N (k | p, q, B = ) ei

E (k ) t ei ( ( p) (q)) t ,E (k ) = 1 (k1)

2 (k2) + 3 (k3)

4 (k4) . (59)

Substitution of the observables ( 57) into Eq. ( 47) and imposing the momentum conserving delta-functions in thevertices ( 10) gives the time-dependent Greens function

G( j, l ; t) = 0(t)|cj cl |0(t)

= 1L

k> 0 , =

( j, k | f ) (l, k| f ) ei ( (k ) (k )) t n (k) + Uc (k, k|t) + O(U 2). (60)The remaining momentum sum k> 0 has to be evaluated numerically.

D. CUT results for the Greens function

We rst comparing the U

= 0 CUT results to the ex-

actly solvable U = 0 case. Figures 6 and 7 show the

nearest-neighbour and next-nearest-neighbour Greens


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functions for the quench i = 0.8 f = 0.4 forseveral values of U . With increasing U the periodic-ity of the oscillations and the asymptotic value of thenearest neighbour Greens function are continuously de-formed away from the non-interacting result. The next-nearest-neighbour Greens function is an imaginary quan-tity that decays asymptotically to zero for both the non-interacting and CUT result.

-0.5

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

0 5 10 15 20

G (

L 2 , L 2

+ 1 )

Time t

U = 0 . 00U = 0 . 05U = 0 . 10U = 0 . 15

FIG. 6. Comparison of exact (solid) U = 0 nearest-neighbourGreens function G (L/ 2, L/ 2 + 1) = cL/ 2 cL/ 2+1 with theCUT results for the quench i = 0 .8 = 0 .4 and U i = 0 U on the L = 100 chain.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 5 10 15 20

i G (

L 2 ,

L 2

+ 2 )

Time t

U = 0 . 00U = 0 . 05U = 0 . 10U = 0 . 15

FIG. 7. Comparison of exact (solid) U = 0 next-nearest-neighbour Greens function G (L/ 2, L/ 2 + 2) = cL/ 2 cL/ 2+2with the CUT results for the quench i = 0 .8 = 0 .4 and

U i = 0 U on the L = 100 chain.

In Figs. 8 and 9 we show the fermion Greens function

G(L/ 2, L/ 2 + j ) = cL/ 2cL/ 2+ j for separations j = 1 , 2

for the quench i = 0 .75 = 0 .5 and U i = 0 U = 0.15 for the L = 200 chain. In both cases the long-time decay of the CUT result is compatible with the non-interacting t 3/ 2 power-law decay. This is a consequenceof the fact that the CUT result ( 60) has the same generalt-dependence as the non-interacting case ( 17).

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0.1 1 10

| G (

L 2 ,

L 2 +

1 , t

)

G (

L 2 ,

L 2

+ 1 , t

) |

Time t

CUT U=0.150 . 025 t 3 / 2

FIG. 8. A comparison of the CUT Greens function|G (100, 101, t ) G (100, 101, t )| and the free fermionasymptotic form, Eq. (19), on the L = 200 chain for thequench i = 0 .75 = 0 .5 and U i = 0 U = 0 .15. Theprefactor of the power law t 3 / 2 is used as a t parameter.The revival time of the L = 200 chain is t 50 and theasymptotic value G (100, 101, t ) = 0.482275.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0.1 1 10

| G (

L 2 ,

L 2

+ 2 , t

) |

Time t

CUT U=0.150 . 075 t 3 / 2

FIG. 9. A comparison between the free fermion asymptoticform of the Greens function, Eq. ( 19), and the CUT resultfor the quench i = 0 .75 = 0 .5 and U i = 0 U = 0 .15on the L = 200 chain. The prefactor of the power law t 3 / 2

is used as a t parameter.

E. Accuracy of the CUT approach: comparison totime-dependent density matrix renormalization

group at small U/t

In order to assess the accuracy of the CUT approachwe have carried out extensive comparisons to numericalresults obtained by the time-dependent density matrixrenormalization group (t-DMRG) algorithm. As is cus-tomary in density matrix renormalization group studies,we impose open boundary conditions. We have carriedout computations for systems up to L = 200 lattice sites,but for the puposes of comparing to our CUT results wechoose a system size of L = 50. Up to 1500 density ma-trix states were kept in the course of the time evolution,


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9

and a discarded weight of = 10 9 was targetted. In or-der to assess the accuracy of the results at later times, wecarried out comparisons to results obtained with a targetdiscarded weight of = 10 11 , and in addition comparedto simulations using different time steps of t = 0 .005or t = 0.01, respectively. Some details are presentedin Appendix B. As shown there, the difference betweenthe results at the end of the time evolution is 10 4 orsmaller for L = 100 sites, which means t-DMRG errorsare negligible in our comparison to the CUT results.

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

0 2 4 6 8 10

G (

L 2 , L 2

+ 1 )

Time t

CUTtDMRG

FIG. 10. Comparison of the CUT and t-DMRG results forG (L/ 2, L/ 2 + 1) = cL/ 2 cL/ 2+1 for the quench i = 0 .75 = 0 .5 and U i = 0 U = 0 .15 on a L = 50 chain. Therevival time for the L = 50 system is r 13.

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

0 2 4 6 8 10

G (

L 2 , L 2

+

1 )

Time t

CUTtDMRG

FIG. 11. Comparison of the CUT and t-DMRG results forG (L/ 2, L/ 2 + 1) = cL/ 2 cL/ 2+1 for the quench i = 0 .75 = 0 .5 and U i = 0 U = 0 .25 on a L = 50 chain.

The revival time r for measurements in the centre of anite chain of noninteracting particles is L/ 2vmax , whereL is the system size and vmax is the maximal velocity. Inthe small- U regime of interest here we can obtain a goodestimate of r by calculating it in the U = 0 limit. Theestimate can be improved by searching for features as-sociated with revivals at times close to the free fermionestimate. Finally, we carry out a comparison betweenCUT and t-DMRG results only for times t sufficientlysmaller than r . We note that as far as the t-DMRGcomputations are concerned, we have been able to reach

times 200 for system size L = 50. Whilst for shortenough times the error in the observable can be estimatedas , at longer times, even if the discarded weight iskept constant, the accumulation of errors in the course of the sweeps needs to be taken into account. Therefore, forthe situations in which times > 20 are discussed, a moredetailed error analysis is necessary, which is presented inAppendix B. In Figs. 10-12 we show a comparison of

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

0 2 4 6 8 10

G (

L 2 , L 2

+ 1 )

Time t

CUTtDMRG

FIG. 12. Comparison of the CUT and t-DMRG results forG (L/ 2, L/ 2 + 1) = cL/ 2 cL/ 2+1 for the quench i = 0 .75 = 0 .5 and U i = 0 U = 0 .5 on a L = 50 chain.

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 2 4 6 8 10

G

t D M R G

( L 2

, L 2 + 1 )

G

C U T

( L 2

, L 2 + 1 ) / U 2

Time t

U = 0 . 15U = 0 . 25

U = 0 . 5

FIG. 13. Rescaled difference between the t-DMRG and CUTdata for G (25, 26) and different values of U .

the CUT and t-DMRG results for the time-dependenceof the nearest-neighbour Greens function G(25, 26) forthe length L = 50 chain. We quench the dimerization pa-rameter i = 0.75 = 0.5 and the interaction strengthU = 0 U = 0.15, 0.25, 0.5, respectively. There is good,quantitative agreement between the CUT and t-DMRGresults provided U is small. The remaining discrepan-cies have their origin in the order O(U 2) corrections tothe CUT results as is shown in Fig. 13, where we plotthe rescaled difference between the t-DMRG data andthe CUT result for three values of U . The oscillatorynature of these differences can be explained as a beatfrequency arising from subtracting two oscillatory data


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10

sets where the frequencies dont match exactly.

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 2 4 6 8 10

i

G (

L 2 ,

L 2

+ 2 )

Time t

CUTtDMRG

FIG. 14. G (L/ 2, L/ 2 + 2) for the quench i = 0 .75 = 0 .5,U i = 0 U = 0 .15 on a L = 50 chain.

Figs 14-16 show that the good agreement betweenCUTs and t-DMRG is not limited to the nearest-neighbour Greens function by comparing results for

(cL/ 2cL/ 2+ j )( t) with j = 2, 3, 4 for the case of U = 0.15.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 2 4 6 8 10

G (

L 2 , L 2

+ 3 )

Time t

CUTtDMRG

FIG. 15. G (L/ 2, L/ 2 + 3) after the quench i = 0 .75 =0.5, U i = 0 U = 0 .15 on a L = 50 chain.

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 2 4 6 8 10

i

G (

L 2 ,

L 2

+ 4 )

Time t

CUTtDMRG

FIG. 16. Comparison of the CUT and t-DMRG results forG (L/ 2, L/ 2 + 4) for the quench i = 0 .75 = 0 .5, U i =0 U = 0 .15 on a L = 50 chain.

F. CUT results for the four-point function

The procedure which we have outlined above for the

single-particle Greens function can be generalized to N -point functions. The next non-vanishing correlation func-tion is the four point function

(t)|cj cj c

l cl |(t) =

1L2 qj > 0 j =

0|A (q , t )|0

1 ( j, q 1) 2 ( j , q 2) 3 (l, q 3) 4 (l , q 4),(61)

where ( j, k ) are dened in Eq. ( 5) and

A (q , t ) = a 1 (q 1 , t )a 2 (q 2 , t )a 3 (q 3 , t )a 4 (q 4 , t ). (62)

Going to the B = basis by applying the CUT andthen time evolving with (51), we obtain

A (q , t |B = ) = ei E (q ) t a 1 (q 1)a 2 (q 2)a

3 (q 3)a 4 (q 4)

+ U k j > 0 j =

ei ( 1 (q1 ) 2 (q2 )) t N 3 4 (k |q 3 , q 4|t)a 1 (q 1)a 2 (q 2)a 1 (k1)a 2 (k2)a 3 (k3)a 4 (k4)

+ U

k j > 0 j =

ei ( 3 (q3 ) 4 (q4 )) t N 1 2 (k |q 1 , q 2|t)a 1 (k1)a 2 (k2)a 3 (k3)a 4 (k4)a 3 (q 3)a 4 (q 4)+ O(U 2), (63)

where E (q ) and N (k | p, q |t) are dened in Eq. ( 59). Taking the expectation value of Eq. ( 63) on the initial stateusing Wicks theorem and substituting in to Eq. ( 61) yields the real-space four-point function.

V. PRETHERMALIZED REGIME

The combination of CUT and t-DMRG results estab-lish that at intermediate times the fermion Greens func-

tion G( j, l, t ) after a quench ( i , 0) ( f , U ) decays in apower-law fashion with approximate exponent 3/ 2 to a


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11

stationary value, i.e.

G( j, , t) g( j, ) + O(t 3/ 2) , J t 0

(k)a (k)a (k). (73)

Clearly the mode occupation number operators n (k)commute with H , and hence constitute conservationlaws (to rst order in U ) within our CUT approach.Their pre-images under the CUT, accurate to order

O(U ), are simplyQ (k) = a (k)a (k) U

qj > 0N (q |k,k ,B = )

a 1 (q 1)a 2 (q 2)a 3 (q 3)a 4 (q 4). (74)By construction these operators approximately commutewith one another

[Q (k), Q ( p)] = O(U 2). (75)However, the commutator with the Hamiltonian is in fact

[Q (k), H ( f , U )] = O(U ), (76)i.e. the charges ( 74) are not (approximately) conservedon an operator level, but only their expectation valueswith respect to |0(t) are (approximately) time inde-pendent. This is a fundamental difference to the proposalput forward in Ref. [31] for describing prethermalizationplateaux. The charges Q (k) have a very transparentphysical meaning: they are the number operators for ap-proximately conserved quasiparticles, and the quarticterms describe the leading contribution to the dressingof the non-interacting fermions.

2. Approximate description by a Deformed GGE

It is natural to attempt a description of the prether-malized regime in terms of a statistical ensemble of theform

PT = 1Z PT

expk,

( )k Q (k) . (77)


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12

Here the Lagrange multipliers ( )k are xed by the re-quirements

tr [ PT Q (k)] = 0|Q (k)|0 . (78)

The left-hand-side of ( 78) is most easily evaluated in theB = basis, where it becomes

1Z PT

tr e k, ( )k a

(k )a (k ) a (k)a (k) =

1

1 + e ( )k

.

(79)

The right-hand-side of ( 78) is equal to

n (k) U qj > 0

N (q |k, B = ) [n 1 2 (q 1)n 3 4 (q 3) q1 ,q 2 q3 ,q 4 + n 1 4 (q 1) [ 2 , 3 n 3 2 (q 2)] q1 ,q4 q2 ,q 3 ] . (80)

Equating ( 80) with ( 79) and using ( 54) we obtain anexplicit expression for the Lagrange multipliers ( )k . Thefermion Greens function evaluated with respect to thedensity matrix ( 77) is

GPT ( j, ) = tr PT cj c

= 1L q> 0 =

( j, q | f ) ( , q | f )

tr PT a (q )a (q ) . (81)

We wish to show that this is equal to the innite timelimit of the CUT result up to order O(U 2) corrections,i.e.

GPT ( j, ) = limt G( j, ; t) + O(U 2). (82)

The trace in ( 81) is most easily evaluated in the B = basis

tr PT a (q )a (q ) = 1Z PT

tr e k, ( )k a

(k )a (k ) n, (q, q |B = )

= n (q ) U k1 , 2 > 0

N (k1 , k1 , k2 , k2|q,q ,B = )n 1 2 (k1)n 3 4 (k2)[1 1 , 2 3 , 4 ]

U k1 , 2 > 0

N (k1 , k2 , k2 , k1|q,q ,B = )n 1 4 (k1) 2 , 3 n 3 2 (k2) [1 1 , 4 2 , 3 ] . (83)

Substituting ( 83) into ( 81) we obtain an expression thatindeed agrees with the innite time limit of ( 60) in thethermodynamic limit L . This establishes ( 82).Hence the Greens function G( j, ) (for xed j, in thethermodynamic limit) on the prethermalization plateauis described by the GGE ( 77) with deformed charges ( 74).This observation is consistent with a description of local

observables on the prethermalization plateau in terms of a deformed GGE. On the other hand there are non-localoperators, n+ (k) being a simple example, which in factdo not relax at intermediate times and are therefore notdescribed by the ensemble PT (without time-averaging).

3. Deformed GGE description of the four-point function

The preceding section shows that the value of the

Greens function on the prethermalization plateau isgiven by the deformed GGE P T . We now show that thedeformed GGE also reproduces the t expectationvalue of the CUT result for the four-point function ( 61).We wish to calculate

tr P T cj cj cl cl =

1L2 qj > 0 j =

1 ( j, q 1) 2 ( j , q 2)

3 (l, q 3) 4 (l , q 4)tr P T a 1 (q 1)a 2 (q 2)a

3 (q 3)a 4 (q 4) , (84)


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13

with P T given in (77). As in the previous section, this trace is most easily performed in the B = basistr P T A (q ) =

1Z P T

tr e k, ( )k a

(k )a (k ) A (q , B = )

= 1Z P T

tr e k, ( )k a

(k )a (k ) A (q )

+ U Z P T k j > 0

N 3 4 (k |q 3 , q 4 , )tr e k, ( )k a

(k )a (k ) a 1 (q 1)a 2 (q 2) A (k )

+ U Z P T k j > 0

N 1 2 (k |q 1 , q 2 , )tr e k, ( )k a

(k )a (k ) A (k )a 3 (q 3)a 4 (q 4) + O(U 2), (85)

0.0001

0.001

0.01

10

| c

j 1

c j 2

c l

1

c l

2 C

U T

c j

1

c j 2

c l

1

c l

2

d G G E

|

Length L

j 1 = 2, j 2 = 2, l 1 = 4, l 2 = 4 j 1 = 2, j 2 = 2, l 1 = 5, l 2 = 5 j 1 = 1, j 2 = 2, l 1 = 3, l 2 = 4 j 1 = 1, j 2 = 5, l 1 = 8, l 2 = 10

FIG. 17. The L dependence of the difference between thedeformed GGE and the t CUT result for the four pointfunction for a number of separations. The solid lines are linearts cL 1 to the data.

where A (k ) = a 1 (q 1)a 2 (q 2)a 3 (q 3)a 4 (q 4). The GGEexpectation values are easily calculated using Wicks the-orem and ( 78). Retaining only terms up to O(U ) andsubstituting the result back into ( 84), we obtain the de-formed GGE value for the four-point function on theprethermalization plateau.

In Fig. 17 we plot the difference between the deformedGGE result obtained in this way and the stationary valueof the CUT result (found by projecting on to the station-ary terms of Eq. ( 61)) for a number of system sizes andseparations. In all cases the difference between the CUTand deformed GGE results scales as 1L and vanishes inthe thermodynamic limit L

. This conrms that the

t stationary value of the CUT four-point function isreproduced by the deformed GGE ( 77). This is a rathernon-trivial check of our proposal that pre-thermalizationplateaux can be described in terms of a deformed GGE.

In Figs. 18 and 19 we present comparisons betweent-DMRG results and predictions of the deformed GGEfor nearest-neighbor and next-nearest-neighbor density-density correlation functions ( 84) for the quench i =0.8 f = 0 .4 and U = 0 0.4. Taking into accountthat U f is not particularly small, the observed agree-

0

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20

n

L 2

n L 2

+ 1

Time t

tDMRGdGGE

CUT

FIG. 18. Nearest neighbour density-density correlation func-tion n( L2 )n(

L2 + 1) for a quench from i = 0 .8 f = 0 .4

and U = 0 0.4 computed by t-DMRG for system sizeL = 100. For comparison we show CUT results for L = 40and the asymptotic value predicted by the L = 50 deformedGGE.

ment between the two results is quite satisfactory. Thissupports our assertion that the deformed GGE providesa good description of higher-order correlation functionson the prethermalization plateau. We see similarly goodagreement for all separations (up to 4 sites) that we ex-plicitly checked. The deformed GGE predictions and theCUT result of Fig. 18 are calculated for system sizesL = 40, 50 rather than L = 100, because the compu-tational cost of carrying out the momentum sums in theexpression for the four-point function ( 61) increases veryrapidly with system size.

VI. T-DMRG RESULTS FOR LARGER VALUESOF U AND ABSENCE OF THERMALIZATION

ON ACCESSIBLE TIMES SCALE

In this section we turn to numerical results obtainedfor quenches to nal Hamiltonians with both weak andstrong interactions, i.e., when U >

| i f |. As can beseen, in all cases the time evolution seems to reach aplateau and remains - on the accessible time scales - on


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14

0.256

0.257

0.258

0.259

0.26

0.261

0.262

0 5 10 15 20

n L 2

n

L 2

+ 2

Time t

tDMRGdGGE

FIG. 19. Next-nearest-neighbor density-density correlationfunction n( L2 )n(

L2 + 2) for a quench from i = 0 .8 f =

0.4 and U = 0 0.4 computed by t-DMRG for system sizeL = 100. The correlator relaxes to a stationary value inconsistent with the deformed GGE prediction (evaluated forL = 50).

-0.5

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

0 5 10 15 20 25 30

time t

L = 16 (ED, PBC)L = 100L = 200

FIG. 20. Time evolution of G (L/ 2, L/ 2+1) for quenches with i = 0 .8 f = 0 .4 and U i = 0 U = 0 .4 and system sizesL = 16, L = 100 and L = 200 sites. The data for L = 16are ED results for systems with periodic boundary conditions(PBC) and are seen to exhibit many revivals.

this plateau. This is observed for quenches starting froma non-interacting initial state as well as when U ini = 5.

A. Extent of prethermalization plateaux

The rst issue we want to address is the time scaleover which we observe prethermalization plateaux. InFigs. 10-15 results are shown only up to t 10 in orderto avoid revivals. The prethermalization plateau for U =0.4 persists to much later times of at least t 30, as canbe seen in Fig. 20, where we present data for L = 16, L =100, L = 200. On the accessible time scales there is nosign that the L = 200 system starts to deviate from the

plateau at late times.

B. Time averages

A standard method for extracting stationary valuesof observables from nite systems is to consider time-averaged quantities, e.g.

1T

T

0dt G(L/ 2, L/ 2 + 1) . (86)

For the L = 16 system shown in Fig. 20 the average overlong times is in good agreement with the plateau valuefor the L = 100 and L = 200 data. One question thatcan be asked is whether time averages may reveal signs of the system deviating from the prethermalization plateau.In order to investigate this issue, we have carried out t-DMRG simulations for a L = 50 system up to very latetimes t = 200. The results are shown in Fig. 21. Time

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 50 100 150 200

time t

FIG. 21. Time evolution of G (L/ 2, L/ 2 + 1) for quencheswith i = 0 .8 f = 0 .4 and U i = 0 U = 0 .4. L =50 site system up to t 200, with error bars estimated inAppendix B.

averages of the t-DMRG data do not reveal any signs of deviations from the plateau value at late times.

C. The role of interactions in the pre-quench andpost-quench Hamiltonians

In this section we present results for a variety of in-teraction strengths 0 .4 U 10 in the post-quenchHamiltonian, as well as for quenches from the groundstate at a nite value of the interactions. We providetwo benchmarks for comparison:

1. Gibbs Ensemble

One useful comparison is with the appropriate Gibbsensemble describing a putative thermal ensemble at late


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15

times. We have computed these by quantum Monte Carlo(QMC) using the ALPS collaboration [43] directed loopstochastic series expansion [44] code. Using the Jordan-Wigner transformation to map onto a spin model, theQMC calculations are performed in the grand canonicalensemble; the chemical potential and the effective tem-perature are xed to ensure the correct energy and num-ber densities (within the QMC error): these are given in

Table I. In the QMC simulations of the L = 100 chain we

U E/L G ( L2 , L

2 + 1) QMCError

0.4 -0.664373 3.0741 0.4 -0.46358 1.62 10 3

1 -0.589142 2.6494 1 -0.46247 2.98 10 4

2 -0.463757 2.0437 2 -0.44347 6.94 10 5

3 -0.338371 1.5882 3 -0.40153 6.49 10 5

4 -0.212986 1.2175 4 -0.34284 3.06 10 4

6 0.037784 0.7250 6 -0.23885 1.34 10 4

8 0.288550 0.4868 8 -0.17441 3.15 10 4

10 0.539324 0.3591 10 -0.13514 1.23 10 4

TABLE I. Summary of the effective temperature and chem-ical potential used in the QMC to calculate the Greensfunction G ( L2 ,

L2 + 1) on the L = 100 chain as presented in

Figs. 23-30. The energy density E/L is found by taking theexpectation value of the interacting Hamiltonian H (t > 0) att = 0 + .

perform 5 107 thermalization steps and perform mea-surements of the nearest-neighbour Greens function after1.5 108 sweeps.

2. Diagonal Ensemble

A second useful benchmark is provided by the diagonalensemble. Given an initial state |0 and a basis {|n }of energy eigenstates, the diagonal ensemble average of an observable O is dened as

ODE =n

n|O|n |n |0 |2 . (87)

For nite systems this equals the long-time average (overmany recurrences). We compute the diagonal ensemblefor a system of L = 16 sites by exact diagonalization

(ED).

3. Difference between diagonal and Gibbs averages

In Fig. 22 we show the difference between the expec-tations values of the nearest-neighbour Greens function

G(L/ 2, L/ 2 + 1) in the diagonal and Gibbs ensembles re-spectively for different values of U f . As the diagonalensemble is available only for system size L = 16, we

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5 6 7 8 9 10

t h e r m a l , t

i m e - a v e r a g e

U f

Difference ED dataDifference ED time av. to

QMC finite-T results

FIG. 22. Difference in the value of G (L/ 2, L/ 2+1) between -nite temperature results obtained with QMC ( L = 100) or ED(L = 16), respectively, to the time-average values obtained viaED for L = 16 for a quench with i = 0 .8 f = 0 .4 andU i = 0 U f as a function of U f . Finite size effects are lesspronounced for small values of U f , but prominent for U f > 1.

The intermediate region 1 U f < 8 is the best candidate toobtain thermalization on long time scales in this system.

display the quantities

cL/ 2cL/ 2+1 DE ,L=16 cL/ 2cL/ 2+1 Gibbs ,L=16 ,

cL/ 2cL/ 2+1 DE ,L=16 cL/ 2cL/ 2+1 Gibbs ,L=100 . (88)

We see that for small values U f the two averages areclose to one another, but for large U f they become verydifferent.

4. Results

As can be seen from Figs. 23, 29 and 30, the nearest-neighbour Greens function approaches plateaux valuesat late times, which are compatible with the diagonal en-semble (given that the latter was calculated for L=16 weexpect there to be nite-size effects), but not the Gibbsensemble.

On the other hand, the plateau for intermediate val-ues U 2 is compatible with a thermal ensemble on theaccessible time scales. We propose the following expla-nation for these observations:

1. The small- U regime is described by a pre-thermalization plateau as discussed in section V.It can be understood in terms of a deformation of the generalized Gibbs ensemble characterizing thestationary state of the U = 0 quench.

2. The large- U regime is also described by a pre-thermalization plateau, which now can be under-stood in terms of a deformation of the generalizedGibbs ensemble characterizing the stationary stateof the f = 0 quench. This corresponds to a quench


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-0.48

-0.475

-0.47

-0.465

-0.46

-0.455

-0.45

-0.445

-0.44

0 5 10 15 20

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 0 .4ED t-av U = 0 .4

QMC U = 0 .4

FIG. 23. Comparison of the t-DMRG, time-averaged (t-av) ED and QMC results for the nearest-neighbour Greensfunction at time t after the quench i = 0 .8 f = 0 .4U = 0 0.4. t-DMRG and QMC simulations are performedon the L = 100 chain, whilst ED studies the L = 16 chain.

-0.48

-0.475

-0.47

-0.465

-0.46

-0.455

-0.45

-0.445

-0.44

0 2 4 6 8 10 12 14

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 1ED t-av U = 1

QMC U = 1

FIG. 24. Comparison of the t-DMRG, time-averaged (t-av) ED and QMC results for the nearest-neighbour Greensfunction at time t after the quench i = 0 .8 f = 0 .4U = 0 1. t-DMRG and QMC simulations are performedon the L = 100 chain, whilst ED studies the L = 16 chain.

to the Heisenberg XXZ chain in the massive regime.Given that our initial state has a short correlationlength, GGE expectation values of local observablescould be calculated by the method of Ref. [23]. Inorder to test our interpretation, we have investi-

gated the dependence of the plateau value on f ( f = 0 corresponding to an integrable quench inthe XXZ chain). In Fig. 31 we show a compari-son between quenches to U f 1 and f = 0 or f > 0, respectively. The correlator clearly ap-proaches a plateau, the value of which is only veryweakly dependent on the integrability-breaking pa-rameter f , which supports our interpretation.

3. In the intermediate- U regime there is no prether-malization plateau, but the system relaxes slowly

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

-0.41

-0.4

0 2 4 6 8 10

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 2 . 0ED t-av U = 2 . 0

QMC U = 2 . 0


-0.43

-0.42

-0.41

-0.4

-0.39

-0.38

-0.37

-0.36

0 1 2 3 4 5 6 7

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 3 . 0ED t-av U = 3 . 0

QMC U = 3 . 0


towards a Gibbs ensemble.

5. Initial state dependence

A nal issue we would like to address is whether ourndings are sensitive to our particular choices of initialstate. In order to assess this question we have carriedout t-DMRG computations for quenches starting in theground state of strongly interacting Peierls insulators, i.e.Hamiltonians H ( i , U i > 0). Results for quenches of theform

( i = 0.8, U i = 5) ( f = 0 .4, U f ) (89)


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17

-0.38

-0.37

-0.36

-0.35

-0.34

-0.33

-0.32

-0.31

-0.3

-0.29

0 1 2 3 4 5 6 7

G ( L

2

, L 2 + 1 )

Time t


QMC U = 4 .0


-0.27

-0.26

-0.25

-0.24

-0.23

-0.22

-0.21

-0.2

-0.19

-0.18

0 1 2 3 4 5 6

G (

L 2 , L 2

+ 1 )

Time t


QMC U = 6 .0


with several values of U f are shown in Figs. 32 & 33.Here the expectation values of both the diagonal andGibbs ensembles have been computed for L = 16 sitesystems. Hence nite-size effects should be taken intoaccount when making comparisons to the t-DMRG data.

The observed behaviour is qualitatively very similar tothat seen for quenches starting in non-interacting groundstates. Observables relax to plateaux values that are in-compatible with thermalization when U f is either smallor large.

-0.28

-0.26

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

0 1 2 3 4 5 6

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 8 . 0ED t-av U = 8 . 0

QMC U = 8 . 0


-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0 1 2 3 4 5

G ( L

2

, L 2 + 1 )

Time t

tDMRG U = 10 . 0ED t-av U = 10 . 0

QMC U = 10 . 0


VII. CONCLUSIONS

Using a combination of anaytical calculations basedon the continuous unitary transform technique and time-dependent density matrix renormalization group com-putations we have established the existence of a robustprethermalization regime at intermediate times after aquantum quench to the weakly non-integrable interact-ing Peierls insulator Hamiltonian ( 1).

The CUT results allowed us to explicitly construct adeformed generalized Gibbs ensemble, which providesan approximate statistical description of the prether-malization plateau. The deformed GGE is constructedfrom charges Q (k) cf Eq. (74), that form a mutuallycommuting set but do not commute with the Hamilto-


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18

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 1 2 3 4 5 6

time t

L = 100, U i = 0, i = 0.8, U f = 10 f = 0.4

f = 0

FIG. 31. Comparison of t-DMRG results for the time evo-lution of G (L/ 2, L/ 2 + 1) for systems with L = 100 sites forquenches with initial U i = 0 , i = 0 .8 to values of U f = 10and f = 0 .4 or f = 0, respectively. As can be seen, theexpectation value for both cases is very similar.

-0.44

-0.43

-0.42

-0.41

-0.4

-0.39

-0.38

-0.37

0 2 4 6 8 10 12

G (

L 2 , L 2

+ 1 )

Time t

tDMRG U = 0 . 0ED Thermal U = 0 . 0

ED t-av U = 0 . 0tDMRG U = 0 . 2

ED Thermal U = 0 . 2ED t-av U = 0 . 2

FIG. 32. Greens function results from t-DMRG and ED forthe quench i = 0 .8 f = 0 .4 with U i = 5 to U = 0 , 0.2. Aswith Figs. 23-30 we see that the time-averaged (t-av) ED iscompatible (up to nite size effects) with the t-DMRG plateauvalue, whilst the thermal expectation is not.

nian (44). As such, the deformed charges are not con-served at the operator level; only the expectation values

Q (k) with respect to the time-evolved state |(t) areapproximately conserved. Our construction is thereforequite different from that of Ref. [ 31]. We expect that atvery late times the system will actually thermalize, butwe are not able to access sufficiently long times scaleswith either the perturbative CUT approach or t-DMRG.A possible approach to describe the dynamics at very latetimes might be through a quantum Boltzmann equation(see e.g. Refs. [45]).

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

-0.41

-0.4

-0.39

0 2 4 6 8 10 12

G (

L 2 , L 2

+ 1 )

Time t

tDMRG U = 0 . 5ED Thermal U = 0 . 5

ED t-av U = 0 . 5tDMRG U = 1 . 0

ED Thermal U = 1 . 0ED t-av U = 1 . 0

FIG. 33. Greens function results from t-DMRG and ED forthe quench i = 0 .8 f = 0 .4 with U i = 5 to U = 0 .5, 1.0.As with Figs. 23-30 we see that the time-averaged (t-av) ED iscompatible (up to nite size effects) with the t-DMRG plateauvalue.

ACKNOWLEDGEMENTS

We thank A. Chandran, M. Fagotti, M. Kolodrubetzand S. Sondhi for stimulating discussions. This work wassupported by the EPSRC under grants EP/I032487/1(FHLE and NJR) and EP/J014885/1 (FHLE).

Appendix A: Local Conservation Laws for H 0

To derive the local conservation laws for the non-interacting Hamiltonian H (, 0) we follow Appendix C

of Ref. [11]. Below we give the local conservation lawsand summarize the salient points of the derivation.The Hamiltonian can be written in the form

H 0 =2L 1

i,j =0

a iHij a j ,

where ai are Majorana fermions {a i , a j } = 2 i,j denedbya2n = cn + cn ,

a2n +1 = i(cn cn ),and

H is a skew-symmetric block-circulant matrix of the

form

H=Y 0 Y 1 . . . Y L 1

Y L 1 Y 0...

... . . .

...

Y 1 . . . . . . Y 0,

where Y n are 4 4 matrices with Y n = Y T L n andL = L/ 2. We dene the Fourier transform of the block


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19

matrices as

Y n jj = 1L

L

k=1

e2 ik

L n Y k

jj

with ( Y k ) jn = (Y k )nj .For free fermions a complete set of local conservationlaws can be given by fermion bilinears

I (r ) = 12

l,n

a l I ( r )ln an ,

where the matrices I ( r ) must satisfy

H, I ( r ) = 0 and I ( r ) , I ( r ) = 0. (A1)The problem of deriving local conservation laws has

now become the problem of nding a set of mutuallycommuting matrices that also commutes with the Hamil-tonian matrix H. At rst sight the complexity of theproblem does not seem to have been reduced, but we cannow utilise a useful property of the Hamiltonian matrixH: the projectors onto eigenvectors of block circulantmatrices are themselves block circulant matrices. Thismeans one can consider I ( r ) that are block circulant:

I ( r ) =

Y ( r )0

Y ( r )1 . . . Y

(r )L 1

Y ( r )L 1

Y ( r )0

......

. . . ...

Y ( r )1 . . . . . . Y

( r )0

.

Imposing Eqs. ( A1), we obtain the conditions (for all k)

Y k , Y

( r )

k = 0 ,Y

( r )

k , Y

( r )

k = 0 ,

where Y ( r )k is the Fourier transform of Y ( r ) .The construction of Y ( r )k is straightforward as

Y k = Aky ,

where

Ak = J (1 + ) + J (1 )cos2kL

x

J (1 )sin2kL

y .

So Y ( r )k takes the form

Y ( r )k = q ( r )k Ak

y + q ( r )k Ak1 2

+ ( r )k 1 2

y + ( r )k 1 2

1 2 ,

where the functions ( r )k , ( r )k , q

( r )k and q

( r )k are cho-

sen such that the Fourier transform satises ( Y k ) jn =

(Y k )jn .The ambiguity in choice of functions leads to differ-

ent representations of the conservation laws; followingRef. [11] we make a particular choice that ensures thereis a nite real-space range r0 of the conservation laws:

I ( r )ln = 0 for |l n| > r 0 . We consider the conservation

laws associated with each of the terms in Y ( r )k separately

and Fourier transforming back to real space we nd

I ( r )1 = L 1

n =0

J 2

(1 + ) c2n c2n 2r +3 + c2n c2n +2 r 1 + c

2n +1 c2n 2r +2 + c

2n +1 c2n +2 r 2 + H .c

L 1

n =0

J 2

(1 ) c2n c2n 2r +1 + c

2n c2n +2 r 3 + c

2n +1 c2n 2r +4 + c

2n +1 c2n +2 r + H .c. , (A2)

I ( r )2 =

L 1

n =0

J

2(1 + ) i c2n c2n 2r +1

c2n c2n +2 r +1 + c

2n +1 c2n 2r

c2n +1 c2n +2 r + H .c.

L 1

n =0

J 2

(1 ) i c2n c2n 2r 1 c

2n c2n +2 r +1 + c

2n +1 c2n 2r +2 c

2n +1 c2n +2 r +2 + H .c. , (A3)

I ( r )3 =L 1

n =0i c2n +2 r +2 c2n + c

2n +1 c2n +2 r +3 + H .c. , (A4)

I ( r )4 =L 1

n =0i c2n +2 r +2 c2n c

2n +1 c2n +2 r +3 + H .c. , (A5)


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20

where r is a measure of the locality of the conservationlaws and takes values 1 to L.

The local conservation laws I ( r )3 , I ( r )4 are independent

of the microscopic parameters of the theory; they arisefrom the 1 2

1 2 and 1 2y terms in Y ( r )k . The remain-

ing local conservation laws are dependent on the dimer-ization parameter . Energy conservation is also manifestin the set of local conservation laws with I (1)1

H 0 .

Appendix B: Error estimate for the t-DMRG

In this appendix, we estimate the error for the longtime simulations. In principle, the error in a given observ-able can be estimated by the discarded weight , and dueto the variational nature of the DMRG for ground state calculations, it is

[46]. At short times this providesa reasonable estimate for time-evolved quantities as well.On longer time scales a number of complications emerge.1) Due to the entanglement growth, the discarded weightgrows quickly in time [47]. This can be addressed by ad- justing the number of density matrix eigenstates, so that is smaller than a chosen threshold (in our case 10 9 or10 11 for some simulations, respectively). 2) The errordue to the Trotter decomposition becomes sizeable. 3)Errors incurred in the sweeping procedure accumulate.

In each DMRG step, the change of basis needed duringthe sweeps introduces an error as a result of the basistruncation. Hence, each sweep introduces an error Lfor a system of size L. This error is present at each timestep. After a certain time T , a simulation with a stepsize dt leads to an error T/dt L . This error is inaddition to the error in the observable due to the den-sity matrix truncation discussed above. At short times

the error due to the basis truncation dominates,but at later times other error sources can no longer beneglected. This can be seen by varying both the tar-get discarded weight and the time step. In Fig. 34 weshow the difference of runs with different parameters toa reference run with = 10 11 and dt = 0 .01. The er-ror between the results with a target discarded weightof 10 11 and 10 9 is seen to be roughly two order of magnitudes, as expected from the above estimate. Theerror bars shown in Figs. 21 and 35 are estimated on thebasis of the above considerations. The error bars growsignicantly towards the end of the time evolution, butstill permit us to make qualitative statements. For theruns considered, this indicates that on the time scalestreated the quasi-stationary state does not change, i.e.,the prethermalization plateau is still present. Togetherwith ED results obtained for small systems for times upto t = 1000, this indicates that thermalization happensat much larger time scales ( 100), if at all.

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-0.00025

-0.0002

-0.00015

-0.0001

-5e-05

0

5e-05

0.0001

0.00015

0.0002

0 2 4 6 8 10 12 14 16 18

a b

s o

l u t e d i f f

e r e n c e

t o r

e f

e r e n c e

time t

L = 100, ini = 0.8, = 0.4, U ini = 0, U = 0.4 = 1e-9, dt = 0.01 = 1e-11, dt = 0.005 = 1e-9, dt = 0.005

-0.0001

-8e-05

-6e-05

-4e-05

-2e-05

0

2e-05

4e-05

6e-05

8e-05

0 2 4 6 8 10 12 14

a b

s o

l u t e d i f f

e r e n c e

t o r

e f

e r e n c e

time t

L = 100, ini = 0.8, = 0.4, U ini = 0, U = 1 = 1e-9, dt = 0.01 = 1e-11, dt = 0.005 = 1e-9, dt = 0.005

-0.00015

-0.0001

-5e-05

0

5e-05

0.0001

0.00015

0 1 2 3 4 5 6 7 8 9 10

a b

s o

l u t e d i f f

e r e n c e

t o r

e f

e r e n c e

time t


-8e-05

-6e-05

-4e-05

-2e-05

0

2e-05

4e-05

6e-05

0 1 2 3 4 5 6

a b

s o

l u t e d i f f

e r e n c e

t o r

e f

e r e n c e

time t


-0.0003

-0.00025

-0.0002

-0.00015

-0.0001-5e-05

0

5e-05

0.0001

0.00015

0.0002

0.00025

0 1 2 3 4 5 6 7

a b

s o

l u t e d i f f e

r e n c e

t o r

e f

e r e n c e

time t


FIG. 34. Differences between runs with different parameters and different quenches ( L = 100 in all cases).

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-0.5

-0.495

-0.49

-0.485

-0.48

-0.475

-0.47

-0.465

0 10 20 30 40 50 60 70 80

time t

L = 200, i = 0.75, U i = 0, f = 0.5, U f = 0.15dataerror bar from estimateerror bars from comparing different runs

FIG. 35. (Color online) Error estimates for t-DMRG resultson the time evolution of G (L/ 2, L/ 2 + 1) for a system withL = 200 sites and a quench i = 0 .75 f = 0 .5 and U i =0 U f = 0 .5. The data is obtained using a time step of t = 0 .005 and a target discarded weight of = 10 9 . Thered error bars (lines) are obtained from the estimate discussedin the appendix, the blue ones (asterisks) are obtained by

comparing to the results of a run with time step t = 0 .01.The error estimate appears to be larger, but of similar orderof magnitude to the actual deviation between the results attimes 50. From this estimate we obtain at the end of thetime evolution a relative error of the order of 1.5%.

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Documents

Quench Dynamics in a Model With Tuneable Integrability Breaking - F.H.L. Essler