Jan de Gier and Fabian H L Essler- Exact spectral gaps of the asymmetric exclusion process with open boundaries

8/3/2019 Jan de Gier and Fabian H L Essler- Exact spectral gaps of the asymmetric exclusion process with open boundaries

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J.Stat.Mech.(2006)P120

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ournal of Statistical Mechanics:An IOP and SISSA journal

Theory and Experiment

Exact spectral gaps of the asymmetricexclusion process with open boundaries

Jan de Gier1 and Fabian H L Essler2

1

ARC Centre of Excellence for Mathematics and Statistics of ComplexSystems, Department of Mathematics and Statistics, The University ofMelbourne, 3010 VIC, Australia2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 KebleRoad, Oxford OX1 3NP, UKE-mail: [email protected] and [email protected]

Received 25 September 2006Accepted 18 November 2006Published 14 December 2006

Online at stacks.iop.org/JSTAT/2006/P12011

doi:10.1088/1742-5468/2006/12/P12011

Abstract. We derive the Bethe ansatz equations describing the completespectrum of the transition matrix of the partially asymmetric exclusion processwith the most general open boundary conditions. By analysing these equations indetail for the cases of totally asymmetric and symmetric diffusion, we calculatethe finite-size scaling of the spectral gap, which characterizes the approach tostationarity at large times. In the totally asymmetric case we observe boundaryinduced crossovers between massive, diffusive and KPZ (KardarParisiZhang)scaling regimes. We further study higher excitations, and demonstrate theabsence of oscillatory behaviour at large times on the coexistence line, whichseparates the massive low and high density phases. In the maximum currentphase, oscillations are present on the KPZ scale t L3/2. While independentof the boundary parameters, the spectral gap as well as the oscillation frequencyin the maximum current phase have different values compared to the totallyasymmetric exclusion process with periodic boundary conditions. We discuss apossible interpretation of our results in terms of an effective domain wall theory.

Keywords: integrable spin chains (vertex models), quantum integrability (Betheansatz), driven diffusive systems (theory), exact results

c2006 IOP Publishing Ltd and SISSA 1742-5468/06/P12011+45$30.00
mailto:[email protected]:[email protected]://stacks.iop.org/JSTAT/2006/P12011http://dx.doi.org/10.1088/1742-5468/2006/12/P12011http://dx.doi.org/10.1088/1742-5468/2006/12/P12011http://dx.doi.org/10.1088/1742-5468/2006/12/P12011http://stacks.iop.org/JSTAT/2006/P12011mailto:[email protected]:[email protected]


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Exact spectral gaps of the asymmetric exclusion process with open boundaries

Contents

1. Relation to the spin-1/2 Heisenberg XXZ chain with open boundaries 51.1. Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Bethe ansatz for the XXZ Hamiltonian 8

3. Bethe ansatz equations for the generic PASEP 9

4. Symmetric exclusion process (SEP) 10

5. Totally asymmetric exclusion process (TASEP) 12

5.1. Stationary state phase diagram . . . . . . . . . . . . . . . . . . . . . . . . 125.2. Analysis of the Bethe ansatz equations . . . . . . . . . . . . . . . . . . . . 13

6. Low and high density phases 166.1. Small values of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.1.1. Coexistence line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2. Massive phase II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7. Maximum current phase 207.1. Leading behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2. Subleading corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3. Coefficient of the L3/2 term . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.3.1. Extrapolation procedure. . . . . . . . . . . . . . . . . . . . . . . . . 23

8. Other gaps and complex eigenvalues 248.1. Massive phase I: < c, < c, = . . . . . . . . . . . . . . . . . . . . 248.2. Coexistence line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3. Maximum current phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9. Summary and conclusions 319.1. Dynamical phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2. Domain wall theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.3. Higher excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Acknowledgments 35

Appendix A. Analysis of the AbelPlana formula 36

Appendix B. Expansion coefficients 37

Appendix C. Complex/real excited states and their Bethe root distributions 38

Appendix D. Excited states of the TASEP on a ring 39D.1. Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40D.2. Particlehole excitations . . . . . . . . . . . . . . . . . . . . . . . . . 40D.3. Multiple particlehole excitations . . . . . . . . . . . . . . . . . . . . 42

References 44

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Figure 1. Transition rates for the partially asymmetric exclusion process.

The partially asymmetric simple exclusion process (PASEP) [1, 2] is a model describingthe asymmetric diffusion of hard-core particles along a one-dimensional chain with L sites.Over the last decade it has become one of the most studied models of non-equilibrium

statistical mechanics, see [3, 4] for recent reviews. This is due to its close relationshipto growth phenomena and the KPZ equation [5], its use as a microscopic model fordriven diffusive systems [6] and shock formation [7], its applicability to molecular diffusionin zeolites [8], biopolymers [9][11], traffic flow [12] and other one-dimensional complexsystems [13].

At large times the PASEP exhibits a relaxation towards a non-equilibrium stationarystate. An interesting feature of the PASEP is the presence of boundary induced phasetransitions [14]. In particular, in an open system with two boundaries at which particlesare injected and extracted with given rates, the bulk behaviour in the stationary stateis strongly dependent on the injection and extraction rates. Over the last decade many

stationary state properties of the PASEP with open boundaries have been determinedexactly [3, 4, 15, 16], [19][22].

On the other hand, much less is known about its dynamics. This is in contrast tothe PASEP on a ring for which exact results using Bethes ansatz have been available fora long time [23][25]. For open boundaries there have been several studies of dynamicalproperties based mainly on numerical and phenomenological methods [26][30]. Veryrecently a real space renormalization group approach was introduced, which allows forthe determination of the dynamical exponents [31].

In this work, elaborating on [32], we employ Bethes ansatz to obtain exact results forthe approach to stationarity at large times in the PASEP with open boundaries. Uponvarying the boundary rates, we find crossovers in massive regions, with dynamic exponentsz = 0, and between massive and scaling regions with diffusive (z = 2) and KPZ (z = 3/2)behaviour.

The dynamical rules of the PASEP are as follows, see figure 1. At any given time teach site is either occupied by a particle or empty and the system evolves subject to thefollowing rules. In the bulk of the system (i = 2, . . . , L 1) a particle attempts to hopone site to the right with rate p and one site to the left with rate q. The hop is prohibitedif the neighbouring site is occupied. On the first and last sites these rules are modified.If site i = 1 is empty, a particle may enter the system with rate . If on the other handsite 1 is occupied by a particle, the latter will leave the system with rate . Similarly, ati = L particles are injected and extracted with rates and respectively.

With every site i we associate a Boolean variable i, indicating whether a particle ispresent (i = 1) or not (i = 0). Let |0 and |1 denote the standard basis vectors in C2.doi:10.1088/1742-5468/2006/12/P12011 3


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A state of the system at time t is then characterized by the probability distribution

|P(t) =

P(|t)|, (0.1)

where

| = |1, . . . , L =Li=1

|i. (0.2)

The time evolution of |P(t) is governed by the aforementioned rules and as a result issubject to the master equation

ddt

|P(t) = M|P(t), (0.3)

where the PASEP transition matrix M consists of two-body interactions only and is givenby

M =k

Ik1 M ILk1 + m1 IL1 + IL1 mL. (0.4)Here, I is the identity matrix on C2 and

M : C2 C2 C2 C2 is given by

M = 0 0 0 00 q p 00 q p 00 0 0 0

. (0.5)The boundary contributions m1 and mL describe injection (extraction) of particles withrates and ( and ) at sites 1 and L. In addition, as a tool to compute currentfluctuations [33, 34], we introduce a fugacity e conjugate to the current on the first site.The boundary contributions then are

m1 = e

e

, mL = . (0.6)Strictly speaking, with the inclusion of the matrix M is no longer a transition matrixof a stochastic process. In the following however, we will still refer to M as the transitionmatrix of the PASEP also for non-zero values of .

At = 0, the matrix M has a unique stationary state corresponding to eigenvaluezero. For positive rates, all other eigenvalues of M have non-positive real parts. Thelarge time behaviour of the PASEP is dominated by the eigenstates of M with the largestreal parts of the corresponding eigenvalues. In the next sections we will determine theeigenvalues of M using Bethes ansatz. The latter reduces the problem of determiningthe spectrum of M to solving a system of coupled polynomial equations of degree 3L

1.

Using these equations, the spectrum of M can be studied numerically for very large L,and, as we will show, analytic results can be obtained in the limit L .doi:10.1088/1742-5468/2006/12/P12011 4


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1. Relation to the spin-1/2 Heisenberg XXZ chain with open boundaries

It is well known that the transition matrix M is related to the Hamiltonian H of the openspin-1/2 XXZ quantum spin chain through a similarity transformation

M = pq U1 HXXZU, (1.1)where U and HXXZ are given by (see e.g. [19, 20]),

U =Li=1

1 00 Qj1

, (1.2)

HXXZ =

1

2

L1

j=1 [xj

xj+1 +

yj

yj+1

zj

zj+1 + h(

zj+1

zj ) + ] + B1 + BL. (1.3)

The parameters and h, and the boundary terms B1 and BL are related to the PASEPtransition rates by

B1 =1

2

pq

+ + ( )z1 2e1 21e+1

,

BL =1

2

pq

+ ( )zL 2QL1L 21QL+1+L

,

= 12

(Q + Q1), h =1

2(Q Q1), Q =

q

p.

(1.4)

Here xj , yj and

zj are the usual Pauli matrices,

j = (

xj iyj )/2, and is a free gauge

parameter on which the spectrum does not depend.The expression in (1.3) is the Hamiltonian of the ferromagnetic UQ(SU(2)) invariant

quantum spin chain [35] with added boundary terms B1 and BL. We note thatthe boundary terms contain non-diagonal contributions (1 ,

L ) with L dependent

coefficients. In the absence of the boundary terms the spectrum of the Hamiltonianis massive, i.e. there is a finite gap between the absolute ground states and the lowestexcited states. As is shown below, the boundary terms lead to the emergence of severaldifferent phases. Some of these phases exhibit a spectral gap in the limit L in thesense that there is a finite gap in the real part of the eigenvalue of the transition matrix.

Other phases feature gapless excited states.In order to make contact with the literature, we start by relating the PASEP

parameters to those used in previous analyses of the spin-1/2 XXZ quantum spin chainwith open boundaries. For the latter we employ notation similar to that of [36], in whichthe XXZ Hamiltonian reads (up to a constant shift in energy)

H = 12

L1i=1

[xi xi+1 +

yi

yi+1 cos (zi zi+1 1)] + B1 + BL,

B1 = sin cos + cos

i

2(cos cos ) z1 + cos 1x1 + sin 1y1 sin

,

BL = sin cos + + cos +

i2

(cos + cos +) zL + cos 2 xL + sin 2 yL sin +.(1.5)

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Comparing this with the Hamiltonian (1.3) arising from the PASEP, we are led to thefollowing identifications

Q = ei,

= iei,

= iei+,

ei1 =

e, ei2 =

QL1.

(1.6)

Furthermore, are determined via the equations

pq=

sin

cos + cos ,

pq=

sin

cos + + cos +. (1.7)

As the spectrum ofH cannot depend on the gauge parameter , it follows from (1.6) thatthe spectrum depends on 1 and 2 only via 1 2.

Although it has been known for a long time that H is integrable [37, 38], the off-diagonal boundary terms have so far precluded a determination of the spectrum of Hby means of e.g. the algebraic Bethe ansatz [39]. However, recently a breakthrough wasachieved [40][42] in the case where the parameters satisfy a certain constraint. In theabove notation this constraint takes the form

cos(1 2) = cos((2k + 1) + + +). (1.8)Here k is an integer in the interval

|k

| L/2. In terms of the PASEP parameters the

constraint reads

(QL+2k e)(e QL2k2) = 0. (1.9)For given k and the constraint (1.9) can be satisfied in two distinct ways. Eitherone can choose Q as a root of the equation QL+2k = e, or one can impose a relationon the boundary and bulk parameters such that the second factor in (1.9) vanishes.Curiously precisely under the latter conditions, the DEHP algebra [15] for the PASEPfails to produce the stationary state [20, 43].

Either choice results in a constraint on the allowed rates in the PASEP. In particularthis implies that it is not possible to calculate current fluctuations for general values of

the PASEP transition rates from the Bethe ansatz equations presented below. However,in the case = 0 it is possible to satisfy the constraint (1.9) for arbitrary values of thePASEP parameters by choosing k = L/2. As the condition (1.8) at first sight appearsnot to have any special significance in terms of the XXZ chain, it is rather remarkablethat it can be identically satisfied for the PASEP.

1.1. Symmetries

The spectra of M and H are invariant under the particlehole and leftright symmetriesof the PASEP:

Particlehole symmetry

, , p q, . (1.10)

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Leftright symmetry

, , p q, . (1.11)Both particlehole and leftright symmetries leave the two factors in (1.9) invariantindividually. There is a third symmetry which leaves (1.9) invariant, but interchangesthe two factors

GallavottiCohen symmetry [44, 45] for the PASEP

e

Q2L2e. (1.12)

The combination of (1.12) with a redefinition of the gauge parameter,

eQL+1

, (1.13)

corresponds to the interchange 1 2. It is shown in the next section that if theconstraint (1.8) is satisfied, the spectrum ofH no longer depends on the difference 12.Hence (1.12) is not only a symmetry of the constraint equation (1.8), but also of thespectrum of H.

The GallavottiCohen symmetry (1.12) implies the fluctuation theorem [46, 47] forthe probability PL(J, t) to observe a current J on the first site at time t,

PL(J, t)PL(J, t)

Q2L+2

Jt(t ). (1.14)

We emphasize the L dependence in (1.12) and (1.14), which disappears for symmetrichopping, Q = 1 [48]. It is further clear from (1.12) that the current vanishes J = 0 whenthe detailed balance condition

Q2L+2 = 1 (1.15)

is satisfied [43]. Precisely the same GallavottiCohen symmetry as in (1.12) was observed

for the zero-range process (ZRP) with open boundaries [49]. This model is equivalent tothe PASEP on an infinite lattice but with a fixed number of L + 1 particles and particledependent hopping rates.

As a final remark, we note that it was realized in [36] that the constraint (1.8) hasin fact an algebraic meaning and corresponds to the non-semisimplicity of an underlyingTemperleyLieb algebra, and implies the existence of indecomposable representations.Condition (1.8) can be interpreted as a generalized root of unity condition for this algebra,and it implies certain additional symmetries for the XXZ chain. It would be of interestto understand the impact of these symmetries on the spectrum of the PASEP with openboundaries3.

3 However, we have checked that for the TASEP with open boundaries the spectrum consists of singlets of theTemperleyLieb algebra only and hence the transition matrix is diagonalizable (see also [50]).

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2. Bethe ansatz for the XXZ Hamiltonian

The first Bethe ansatz results pertaining to the spectrum of H (1.5) were reportedin [40, 41]. Subsequently it was noted on the basis of numerical computations in [42]and an analytical analysis for a special case in [51], that these initial results seemedincomplete. Instead of one set of Bethe ansatz equations, one generally needs two. Furtherdevelopments were reported in [52, 53].

In order to simplify the following discussion we introduce the notation

a =2sin sin

cos + cos . (2.1)

When the constraint (1.8) is satisfied for some integer k, the eigenvalues of H can be

divided into two groups, E1(k) and E2(k)

E1(k) = a a+ L/21kj=1

2sin2

cos2uj cos , (2.2)

E2(k) = L/2+kj=1

2sin2

cos2vj cos . (2.3)

Here the complex numbers {ui} and {vi} are solutions of the coupled algebraic equations(i = 1, . . . , L 1)

w(ui)2L = K(ui )K+(ui +)

K(ui )K+(ui +)L/21kj=1j=i

S(ui, uj)S(ui, uj) , (2.4)

w(vi)2L =

K(vi)K+(vi)

K(vi)K+(vi)L/2+kj=1j=i

S(vi, vj)

S(vi, vj) , (2.5)

where

w(u) =sin(/2 + ui)

sin(/2

ui)

, S(u, v) = cos2v cos(2 + 2u),

K(u) = cos + cos( + + 2u).

(2.6)

We will now rewrite these equations in terms of the PASEP parameters for which itis convenient to employ the notation

z = e2 iu, = e2 iv, (2.7)and

v, = p q + , (2.8)

, =1

2 v,v2, + 4 . (2.9)We then find that,

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equations (2.2) and (2.4) are rewritten as

pqE1(k) = + + + +L/21kj=1

p (Q2 1)2

zj(Q zj)(Qzj 1) , (2.10)

zjQ 1Q zj

2LK1(zj) =

L/21kl=j

zjQ2 zl

zj zlQ2zjzlQ

2 1zjzl Q2 , j = 1 . . . L 1. (2.11)

Here K1(z) = K1(z, , )K1(z , , ) and

K1(z, , ) =(z + Q+,)(z + Q

,)

(Q+,z + 1)(Q,z + 1)

. (2.12)

The second set of equations, (2.3) and (2.5), becomes

pqE2(k) =

L/2+kj=1

p (Q2 1)2 j(Q j)(Qj 1) , (2.13)

jQ 1Q j

2LK2(j) =

L/2+kl=j

jQ2 l

j lQ2jlQ

2 1jl Q2 , j = 1 . . . L 1, (2.14)

where K2() = K2( , , )K2( , , ) and

K2(z, , ) =(+,+ Q)(

,+ Q)

(Q + +,)(Q + ,)

. (2.15)

As we have mentioned before, in the case of the PASEP, and for = 0, theconstraint (1.9) can be satisfied by either considering symmetric hopping, Q = 1, orby choosing k = L/2.

3. Bethe ansatz equations for the generic PASEP

Inspection of the second set of equations (2.13) for the choice k = L/2 reveals that thereexists an isolated level with energy E0 = 0. This is readily identified with the stationarystate of the PASEP. Furthermore, all other eigenvalues E of the transition matrix Mfollow from the first set of equations (2.10) and (2.11), and are given by

E= L1j=1

p(Q

2

1)2

zj(Q zj)(Qzj 1) , (3.1)

where the complex numbers zj satisfy the Bethe ansatz equationszjQ 1Q zj

2LK(zj) =

L1l=j

zjQ2 zl

zj zlQ2zjzlQ

2 1zjzl Q2 , j = 1 . . . L 1. (3.2)

Here K(z) = K(z, , )K(z , , ) and

K1(z, , ) =(z + Q+,)(z + Q

,)

(Q+,z + 1)(Q,z + 1)

. (3.3)

In order to ease notation we set from now on, without loss of generality, p = 1 and henceQ =

q.

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4. Symmetric exclusion process (SEP)

The limit of symmetric diffusion is quite special and we turn to it next. We can obtainthis limit either by taking Q 1 and leaving k unspecified in the equations (2.10), (2.11)and (2.13), (2.14), or by setting k = L/2 and studying the limit of the equations (3.1),(3.2) for the generic PASEP. The two procedures lead to the same results and we willfollow the second for the time being.

Taking the limit Q 1 in the equations (3.1), (3.2) for the general PASEP we observethat

zQ 1Q z 1,

zjQ2 zl

zj Q2zl 1,zjzlQ

2 1zjzl Q2

1, K(z, , )

z +

z + .

(4.1)

We conclude that in this limit the spectral parameters z must fulfilz +

z +

z +

z + = 1. (4.2)

It is easy to see that the only solutions to this equation are z = 1. Having establishedthe leading behaviour of the roots of the Bethe ansatz equations in the limit Q 1 wenow parametrize

zj = 1 + ij(Q2 1), (4.3)then substitute (4.3) back into the Bethe ansatz equations (3.2), (3.1) and finally take thelimit Q

1. Choosing the plus sign in (4.3), we obtain with c(x) = (x + 2)/2x

j i/2j + i/2

2L j + i c( + )j i c( + )

j + i c(+ )

j i c(+ )

=L1l=j

j l ij l + i

j + l ij + l + i

, j = 1, . . . , L 1, (4.4)

E= L1j=1

1

2j + 1/4. (4.5)

On the other hand, choosing the minus sign in (4.3) we arrive at

1 =

L1l=j

j l ij l + i

j + l ij + l + i

, j = 1, . . . , L 1, (4.6)E= . (4.7)

We observe that for this choice of sign in (4.3) we obtain only a single energy level.For both choices there is a subtlety: we have implicitly assumed that the j remain

finite when we take the limit Q 1 and this need not be the case. A numerical analysis ofthe Bethe ansatz equations (3.2) for Q 1 shows that we have to allow one or several jto be strictly infinite, in which case they are taken to drop out of (4.4)4. This then leadsto the following set of Bethe ansatz equations, in which we only allow solutions where all

4 We have verified this prescription by solving the Bethe ansatz equations numerically for small systems in thevicinity of the limit Q 1.

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spectral parameters j are finite

j i/2j + i/2

2L j + i c( + )j i c( + )

j + i c(+ )j i c(+ )

=Nl=j

j l ij l + i

j + l ij + l + i

, j = 1, . . . , N , (4.8)

E= N

j=1

1

2j + 1/4. (4.9)

Here N is allowed to take the values 1, 2, . . . L 1.Curiously, the limit of symmetric exclusion can be described by a second set of Bethe

ansatz equations. Taking the limit Q 1 of (2.13), (2.14) and leaving k unspecified wearrive at E= 0 or

j i/2j + i/2

2L j + i d( + )j i d( + )

j + i d(+ )

j i d(+ )

=Nl=j

j l ij l + i

j + l ij + l + i

, j = 1, . . . , N , (4.10)

E= N

j=11

2j + 1/4. (4.11)

Here

d(x) = c(x) = x 22x

. (4.12)

Apart from a constant shift in energy the two sets of equations are related by thesimultaneous interchange and . Importantly, equations (4.10) coincidewith the Bethe ansatz equations derived in [54] by completely different means. Numericalstudies of small systems suggest that either set (4.4) or (4.8) gives the complete spectrumof the Hamiltonian.

Interestingly, the Bethe equations (4.8) and (minus) the expression for the energy (4.9)

are identical to the ones for the open Heisenberg chain with boundary magneticfields [55, 56]5

H = 2L1j=1

Sj Sj+1 1

4

1

Sz1 +

1

2

1

+

SzL +

1

2

, (4.13)

where = 1/( + ) and + = 1/(+ ), if the reference state in the Bethe ansatzis chosen to be the ferromagnetic state with all spins up. On the other hand, the Betheequations (4.10) and (minus) the expression for the energy (4.11) are obtained when thereference state is chosen as the ferromagnetic state with all spins down. This on theone hand shows the equivalence of the two sets of Bethe equations and on the other

5 We note that for symmetric diffusion the eigenvalues of the transition matrix are simply minus the correspondingeigenvalues of the Heisenberg Hamiltonian.

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hand establishes the fact, that the spectrum of the SEP with open, particle number non-conserving boundary conditions (which corresponds to the z-component of total spin Sz

not being a good quantum number in the spin-chain language) is identical to the spectrumof the open Heisenberg chain with boundary magnetic fields, for which Sz is a conservedquantity. This spectral equivalence is more easily established by means of a similaritytransformation, see section 6.5.1 [4] and [57] for the case at hand and [58] for a generaldiscussion on spectral equivalences for conserving and non-conserving spin chains.

The first excited state for the SEP occurs in the sector N = 1 of(4.10). The solutionto the Bethe ansatz equations for large L is

1 =L

1

[d( + ) + d(+ )]

12L

6L2 d( + ) 2d3( + ) + d(+ ) 2d3(+ )+ . (4.14)The corresponding eigenvalue of the transition matrix scales like L2 with a coefficientthat is independent of the boundary rates

E1(L) = 2

L2+ O(L3). (4.15)

We conclude that for the SEP the large time relaxational behaviour is diffusive anduniversal.

5. Totally asymmetric exclusion process (TASEP)

We now turn to the limit of totally asymmetric exclusion Q = 0 and set = = 0 inorder to simplify the analysis.

5.1. Stationary state phase diagram

The stationary state phase diagram for the TASEP was determined in [15, 16, 59] andfeatures four distinct phases, see figure 2:

(1) A low density phase for < 1/2, < .In the low density phase the current in the thermodynamic limit is equal to J =(1 ) and the density profile in the bulk is constant = .

(2) A high density phase for < 1/2, < .

Here the current in the thermodynamic limit is equal to J = (1 ) and the densityprofile in the bulk is constant = 1 .

(3) A coexistence line at = < 1/2.On the coexistence line the current is equal to J = (1 ), but the density profileincreases linearly in the bulk (x) = + (1 2)x.

(4) A maximal current phase at , > 1/2.This phase is characterized by the current taking the maximal possible value J = 1/4and the bulk density being constant and equal to = 1/2.

In [16] a further subdivision of the low and high density phases was proposed on the basisof differences in the behaviour of the density profile in the vicinity of the boundaries. In

the high density phase this distinction corresponds to the parameter regimes < 1/2 and > 1/2, respectively.

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Figure 2. Stationary phase diagram determined by the current of the TASEP.

5.2. Analysis of the Bethe ansatz equations

In order to determine the exact value of the spectral gap we will now analyse (3.1) and (3.2)in the limit L .

After rescaling z Qz and setting = = 0, the Q 0 limit of equations (3.1)and (3.2) reads

E= L1l=1

zlzl 1 , (5.1)

(zj 1)2zj

L= (zj + a)(zj + b)

L1l=j

(zj z1l ). (5.2)

In order to ease notation in what follows, we introduce

g(z) = ln

z

(z 1)2

, (5.3)

gb(z) = ln z1 z2+ ln(z + a) + ln(z + b), (5.4)where

a =1

1, b = 1

1. (5.5)

The central object of our analysis is the counting function [60][62],

iYL(z) = g(z) +1

Lgb(z) +

1

L

L1l=1

K(zl, z), (5.6)

where K(w, z) is given byK(w, z) = ln w + ln(1 wz). (5.7)

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Figure 3. Reciprocal root distributions for (a) = = 0.3 and (b) = = 0.7,both with L = 2n = 398.

Using the counting function, the Bethe ansatz equations (5.2) can be cast in logarithmicform as

YL(zj) =2

LIj, j = 1, . . . , L 1. (5.8)

Here Ij are integer numbers. The eigenvalues (5.1) of the transition matrix can beexpressed in terms of the counting function as

E= L limz1

i YL(z) g(z)

1

Lgb(z)

. (5.9)

Each set of integers {Ij |j = 1, . . . , L 1} in (5.8) specifies a particular excited state.In order to determine which set corresponds to the first excited state, we have calculatedthe eigenvalues of the transition matrix numerically for small systems of up to L = 14 sitesfor many different values of and . By comparing these with the results of a numericalsolution of the Bethe ansatz equations, we arrive at the conclusion that the first excitedstate always corresponds to the same set of integers

Ij = L/2 + j for j = 1, . . . , L 1. (5.10)The corresponding roots lie on a simple curve in the complex plane, which approachesa closed contour as L . The latter fact is more easily appreciated by consideringthe locus of reciprocal roots z

1

j rather than the locus of roots zj. In figure 3 we presentresults for = = 0.3 and = = 0.7 respectively. The limiting shape of the curve isthat of the cardioid, which can be seen as follows. Assuming that the last term in (5.6)is approximately constant as L and using (5.8), (5.10) we find that

exp(g(zj)) = (z1/2j z1/2j )2 = e2ij/L (L ). (5.11)Here = (, ) is some constant. Parametrizing z = ei and multiplying (5.11) by itscomplex conjugate, we conclude that the roots lie on the curve defined by

2 ( + 2 cos ) + 1 = 0, (5.12)the defining equation of the cardioid.

The shape of the locus of inverse roots depends on the rates and . For example,in the case = = 0.7 shown in figure 3(b), a cusp is seen to develop at the intercept

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Figure 4. Sketch of the contour of integration C in (5.13). The open dotscorrespond to the roots z

jand is chosen close to z

L1

and avoiding poles ofcot(LYL(z)/2).

of the curve with the negative real axis (which occurs at z = 1 in the limit L ).In terms of the parameter characterizing the cardioid the cusp develops at = 4. Incontrast, no cusp occurs for = = 0.3 as shown in figure 3(a).6 We will see that thisdifference in the shape of the loci of inverse roots is reflected in a profoundly differentfinite-size scaling behaviour of the corresponding spectral gaps.

In order to compute the exact large L asymptotics of the spectral gap, we derive anintegro-differential equation for the counting function YL(z) in the limit L . As asimple consequence of the residue theorem we can write

1

L

L1j=1

f(zj) =

C1+C2

dz

4if(z)YL(z)cot

1

2LYL(z)

, (5.13)

where C = C1 + C2 is a contour enclosing all the roots zj, C1 being the interior and C2the exterior part, see figure 4. The contours C1 and C2 intersect in appropriately chosenpoints and . It is convenient to fix the end points and by the requirement

YL() = +

L, YL() =

L. (5.14)

Using the fact that integration from to over the contour formed by the roots is equalto half that over C2 C1 we find,

i YL(z) = g(z) +1

Lgb(z) +

1

2

K(w, z)YL(w) dw

+1

2

C1

K(w, z)YL(w)

1 eiLYL(w) dw +1

2

C2

K(w, z)YL(w)

eiLYL(w) 1 dw, (5.15)

where we have chosen the branch cut of K(w, z) to lie along the negative real axis.Our strategy is to solve the integro-differential equation (5.15) by iteration and then

use the result to obtain the eigenvalue of the transition matrix from equation (5.9).

6 This corresponds to having > 4.

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6. Low and high density phases

In the low and high density phases the locations of the end points and are such thata straightforward expansion of (5.15) in inverse powers of L is possible (see e.g. [63, 64]and appendix A). The result is

i YL(z) = g(z) +1

Lgb(z) +

1

2

K(w, z)YL(w) dw

+

12L2

K(, z)

YL()

K(, z)

YL()

+ O(L4)

= g(z) +1

L

gb(z) +1

2 z+c

z

c

K(w, z)YL(w) dw

+1

2

zc

K(w, z)YL(w) dw +1

2

zc

K(w, z)YL(w) dw

+

12L2

K(, z)

YL()

K(, z)

YL()

+ O(L4), (6.1)

where the derivatives of K are with respect to the first argument. We note that here wehave implicitly assumed that YL() is non-zero and of order O(L0). In order to find theeigenvalue of the transition matrix (5.9) up to second order in inverse powers of L, wewill need to solve (6.1) perturbatively to third order. Substituting the expansions

YL(z) =n=0

Lnyn(z), = zc +n=1

Ln(n + in), (6.2)

back into (6.1) yields a hierarchy of equations for the functions yn(z) of the type

yn(z) = gn(z) +1

2i

z+czc

K(w, z)yn(w) dw, (6.3)

where zc = zc i0. The integral is along the closed contour following the locus of theroots. The first few driving terms gn(z) are given by

g0(z) = ig(z),g1(z) = igb(z) + 1 + 1K(zc, z),g2(z) = 2 + 2K(zc, z) + 2K

(zc, z),

g3(z) = 3 + 3K(zc, z) + 3K(zc, z) + 3K

(zc, z).

(6.4)

We recall that the functions g and gb are defined in (5.3) and (5.4) and K(zc, z) =ln(z z1c ). The coefficients n, n, n and n are given in terms of n, n definedby (6.2) as well as derivatives of yn evaluated at zc. Explicit expressions are presented inappendix B.

These coefficients as well as zc are determined self-consistently by solving (6.3) andthen imposing the boundary conditions (5.14).

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6.1. Small values of and

When and are small we assume that the singularities in gb(z), i.e. the pointsa = 1 1/ and b = 1 1/, lie outside the contour of integration. From thedistribution of the reciprocal roots, figure 3(a), we further infer that for small values of and the roots lie inside the unit circle. In particular we assume that zc = 1 and thatthe points 1 lie outside the contour of integration.

We now proceed to solve (6.3) and then verify a posteriori that the above assumptionshold.

The equation (6.3) for n = 0 is solved by the simple ansatz

y0(z) = 0 + g0(z). (6.5)

Substituting the ansatz into the integro-differential equation (6.3) for n = 0 we find that

0 = 12

z+czc

( ln w + ln(1 wz))

1

w 2

w 1

dw

= i( ln(zc) + 2 ln(1 zc)). (6.6)This in turn implies that the zeroth-order term in the expansion of the counting functionis given by

y0(z) = i ln z

zc

1 zc1 z

2. (6.7)

In order to derive this result we have made use of the following simple but useful identity(C denotes the contour of integration from zc to z

+c )

1

2i

z+czc

ln w

w + xdw =

ln(1 + zc/x) ifx outside C,ln(x zc) ifx inside C. (6.8)

The integro-differential equations for n = 1, 2, 3 are solved in an analogous manner, withthe results

y1(z) = i ln z

zc

1 z2c1 z2

zc z1cz z1c

i1 z + azc + a

z + b

zc + b

+

1 iln(ab(

zc)i1), (6.9)

y2(z) = 2 ln

zc z1cz z1c

+

2z2c

1

z z1c 1

zc z1c

+ 2 2 ln(zc) 2

zc, (6.10)

y3(z) = 3 ln

zc z1cz z1c

+

3z2c

3z3c

1

z z1c 1

zc z1c

3

z3c z

(z

z1c )

2 zc

(zc

z1c )

2+ 3 3 ln(zc)

3zc

3z2c

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The coefficients n, n, n and n are given in terms ofn, n and derivatives ofyn at zc, seeappendix B and we now proceed to determine them. Substituting the expansions (6.2) intoequation (5.14), which fixes the end points and , we obtain a hierarchy of conditionsfor yn(zc), e.g.

YL() = y0() +1

Ly1() +

1

L2y2() +

= y0(zc) +1

L[y1(zc) + y

0(zc)(1 + i1)] +

= L

. (6.12)

Solving these order by order we obtain

1 = 2i,3 = 2 = 2 = 1 = 0,

3 = zc3 = zc(zc 1)22 = i2z2c

zc 1zc + 1

2,

(6.13)

with 3 undetermined. Furthermore, the intersect of the solution curve with the negativereal axis in the limit L is found to be

zc = 1ab

. (6.14)

Having determined the counting function, we are now in a position to evaluate thecorresponding eigenvalue of the transition matrix from equation (5.9)

E1(L) = 2zc1 zc

2

zc z1c1

L2+ O(L3)

= + 21 +

ab

2

ab 1/ab

1

L2+ O(L3). (6.15)

We conclude that to leading order in L, the eigenvalue of the transition matrix withthe second largest real part is a non-zero constant. This implies an exponentially fastrelaxation to the stationary state at large times. We note that due to the symmetry ofthe root distribution corresponding to (5.10) under complex conjugation

E1 is in fact real.

The domain of validity of (6.15) is determined by the initial assumption that a and blie outside the contour of integration, i.e.

a < zc and b < zc. (6.16)The parameter regime in and in which (6.15) is valid is therefore bounded by thecurves

c =

1 +

1 1/31

, 0 < 12

,

c = 1 + 1 1/31 , 0 < 12 .

(6.17)

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6.1.1. Coexistence line. On the line = , the leading term in (6.15) vanishes and theeigenvalue with the largest real part different from zero is therefore

E1(L) = 2(1 )1 2

1

L2+ O(L3). (6.18)

This equation holds for fixed < 1/2 and L . We note that there is a divergencefor 1/2, signalling the presence of a phase transition. The inverse proportionalityof the eigenvalue (6.18) to the square of the system size implies a dynamic exponentz = 2, which suggests that the dominant relaxation at large times is governed by diffusivebehaviour. As shown in [29, 30] the diffusive nature of the relaxational mechanism can infact be understood in terms of the unbiased random walk behaviour of a shock (domainwall between a low and high density region). Our results (6.15) and (6.18) for the phase

domain given by (6.17) and the coexistence line agree with the relaxation time calculatedin the framework of a domain wall theory (DWT) model in [30]. As we will show next,this is in contrast to the massive phases beyond the phase boundaries (6.17), where theexact result will differ from the DWT prediction. In section 9 we present a modified DWTframework that allows us to understand the phase boundaries (6.17).

6.2. Massive phase II

In this section we will treat the case b > zc with a < zc as before. The case a > zcwith b < zc is obtained by the interchange in the relevant formulae.

We need to solve the same integral equation (6.3) as before; in particular the driving

terms gn defined in (6.4) remain unchanged. However, when iterating the driving termg1(z) there is an extra contribution because the pole at b now lies inside the integrationcontour, see (6.8). As a result the solution for y1(z) is now of the form

y1(z) = i ln z

zc

1 z2c1 z2

zc z1cz z1c

i1 z + azc + a

z + b

zc + b

z + 1/b

zc + 1/b

+ 1 i ln

a(zc)i1

. (6.19)

The forms of the solutions for yn for n 2 remain unchanged. The coefficients of thevarious terms are again fixed by imposing the boundary condition (5.14), with the result

1 = 3i,3 = 2 = 2 = 1 = 0,

3 = zc3 =3

2zc(zc 1)22 = 4i2z2c

zc 1zc + 1

2,

(6.20)

and 3 again remains undetermined. We note that the difference in the functional form ofy1(z) compared to (6.9) affects the coefficients in all subleading contributions y2(z), y3(z)etc. The intersect of the locus of roots with the negative real axis in the limit L isnow given by

zc = 1

abc = a1/3

= 1 1/3

, (6.21)

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where we have defined

bc = 1 cc . (6.22)

The eigenvalue of the transition matrix with the largest non-zero real part is againdetermined from (5.9)

E1(L) = 1 + 2zc1 zc

42

zc z1c1

L2+ O L3

= c + 21 + a1/3

42

a1/3 a1/31

L2+ O(L3). (6.23)

The result (6.23) is valid in the regime

0 < 1/2, c 1, (6.24)and is seen to be independent of . Hence in this phase the relaxation to the stationarystate at large times is independent of the extraction rate at the right-hand boundary ofthe system.

7. Maximum current phase

In the maximum current phase , > 1/2 the above analysis of the integro-differentialequation (5.15) breaks down. The primary reason for this is that the locus of roots now

closes at zc = 1 for L and this precludes a Taylor expansion of the kernel K(w, z)around w = . A more detailed discussion of the complications arising from zc = 1 ispresented in appendix A.

In order to determine the large L behaviour of the eigenvalues of the transition matrixwith the largest real parts in the maximum current phase, we have resorted to a directnumerical solution of the Bethe ansatz equations (3.2) for lattices of up to L = 2600 sites.In order to facilitate a finite-size scaling analysis it is necessary to work with quadrupleprecision (32 digits in C) at large L. We find that the leading behaviour of the spectralgap in the limit L in the maximum current phase is independent of the boundaryrates and .

7.1. Leading behaviour

In figure 5 we plot the numerical results for eigenvalue E1 of the first excited state for = = 0.7 as a function of the inverse system size L1 on a double-logarithmic scale.The almost straight line suggests an algebraic behaviour at large L

E1(L) eLs (L ). (7.1)A simple least-square fit of the graph in figure 5 in the region 2200 L 2600 to astraight line gives a slope of s 1.493, which is close to the expected value 3/2.

A better result is obtained by extrapolating our finite-size data as follows. We first

divide the data set for E1(L) into bins containing 11 data points each. The kth bin Bk isdefined by taking 20k L 20(k + 1). Within each bin we fit the data for E1(L) to adoi:10.1088/1742-5468/2006/12/P12011 20


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Figure 5. Double-logarithmic plot ofE1(L) as a function of 1/L for = = 0.7.

Figure 6. Polynomial fit to the exponents sk of (7.2) plotted against 1/k for = = 0.7. Extrapolation gives an exponent s = 1.5.

functional form

E1(L) ekLsk . (7.2)We have implemented this procedure for 100 k 128 and obtained a sequence ofexponents sk. We observe that the following least-square fit of the sequence sk to apolynomial gives excellent agreement

sk

1.499 9949

0.782 2533k1 + 3.801 4387k2. (7.3)

Finally, we extrapolate to k = and obtains 1.499 9949. (7.4)

This is very close to the expected result s = 3/2. The polynomial fit as well as theextrapolation is shown in figure 6.

7.2. Subleading corrections

Having established that the leading behaviour of E1(L) at large L is as L3/2, we nowturn to subleading corrections. Assuming that

E1(L) eL3/2 f Ld, (L ), (7.5)

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Figure 7. Double-logarithmic plot of L as a function of 1/L for = = 0.7and 800

L

2600.

we wish to determine the value of the exponent d. To this end, we define the sequence

L =L

2[(L + 2)3/2E1(L + 2) L3/2E1(L)]. (7.6)

If (7.5) is correct, we expect L to be proportional to L3/2d at large L. In figure 7 we plot

L as a function of L1 in a double-logarithmic plot. The result is well approximated

by a straight line with slope 0.95, which suggests that the exponent of the subleadingcorrections is d = 5/2.

A more accurate estimate of the exponent can be obtained by extrapolating the finite-size data along the same lines as before. We group the data for sL into bins of 11 pointseach and carry out a least-square fit for each bin to a functional form

L ekLsk . (7.7)The resulting sequence of exponents sk is described very well by the polynomial least-square fit

sk 0.998 755 255 4.329298118k1 + 8.363 104 997k2. (7.8)Extrapolation gives s 0.998 755 255, which is very close to 1. This suggests that thesubleading corrections to

E1 indeed scale like L

5/2. The polynomial fit to the sequence

of exponents sk and the extrapolation to k = are shown in figure 8.

7.3. Coefficient of the L3/2 term

Having established that asymptotically the dependence on L of the eigenvalue is given by

E1(L) eL3/2 f L5/2 (L ), (7.9)we now determine the coefficient e. We again arrange the data for E1(L) into bins20k L 20(k + 1) and within each bin perform least-square fits of E1(L) to thefunctional form

E1(L) ekL3/2 fkL5/2, 20k L 20(k + 1). (7.10)doi:10.1088/1742-5468/2006/12/P12011 22


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Figure 8. Polynomial fit to the exponents sk in (7.7) plotted against 1/k for = = 0.7.

Figure 9. Polynomial fit to the coefficients ek defined in (7.10) plotted against1/k for = = 0.7.

As is shown in figure 9, the resulting sequence of coefficients ek for the largest availablevalues of k (100 k 128) can be fitted very well to the polynomial

ek 3.578 0576 7.270 4902k2. (7.11)Extrapolation to k then gives the following result for the eigenvalue of the firstexcited state of the TASEP in the MC phase

E1(L)

3.578L3/2 +

O(L5/2). (7.12)

As we have already mentioned, the behaviour (7.12) is in fact independent of and throughout the maximum current phase and the coefficient is a universal number.

The numerical value for the gap is smaller than that of the half-filled TASEP on aring, where it is found that E1,ring(L) 6.509 . . . L3/2 [24, 25, 65].

7.3.1. Extrapolation procedure. In order to assess the accuracy of the numericalextrapolation procedure we have employed above, it is instructive to consider theanalogous analysis on the coexistence line. Here the exact analytical value for the spectralgap is known. For = = 0.3 equation (6.18) gives

E1(L) = 5.181 5423 . . . L2 + O(L3). (7.13)

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We have computed E1(L) on the coexistence line by a direct numerical solution of theBethe ansatz equations for lattices of up to L = 1800 sites. Fitting

E1(L) to the form

E1(L) ekLsk , 20k L 20(k + 1), (7.14)and using the same analysis as above, but for the somewhat smaller values 61 k 88,we find that the sequence of exponents sk is well approximated by the polynomial

sk 2.000 0967 0.377 2271k1 + 0.795 3103k2. (7.15)Extrapolation to k = gives excellent agreement with the exact result s = 2. Similarly,a least-square fit of the sequence of coefficients ek to a polynomial in 1/k gives

ek 5.181 4722 0.000 2060k1 + 0.000 0782k2. (7.16)

The extrapolated value e agrees with the exact result (7.13) to five significant digits.This is quite satisfactory.

8. Other gaps and complex eigenvalues

Having established the large L behaviour of the eigenvalue of M corresponding to thelowest excited state, we now turn to higher excited states. As M is not Hermitian,its eigenvalues are in general complex. A complex eigenvalue in turn leads to interestingoscillatory behaviour in the time evolution. Aspects of such behaviour have been discussedfor the KPZ equation [68, 69].

8.1. Massive phase I: < c, < c, =

Here the next lowest excitation is obtained by choosing the integers Ij in the Bethe ansatzequations (5.6) as

Ij = L/2 + j for j = 1, . . . , L 2 IL1 = L/2. (8.1)This choice of integers Ij corresponds to a hole between the last two roots. Hence

we will refer to this state as a hole state. As this choice is asymmetric with respect tothe interchange j L j, we expect the corresponding eigenvalue to be complex. Thisis indeed the case for small system sizes L. However, as the system size increases, the

eigenvalue becomes real at a certain finite value of L. This comes about in the followingway. For small L, the last root has a finite imaginary part and the distribution of theother roots is asymmetric with respect to the real axis, see figure 10(a). As L increases thelast root approaches the negative real axis until above a critical value of L its imaginarypart vanishes. The other roots are then arranged in complex conjugate pairs, so thatthe corresponding eigenvalue becomes real. An example of this is shown in figure 10(b).Further details regarding this phenomenon are given in appendix C.

Assuming that the root distribution remains as in figure 10(b) in the limit L ,we can compute the corresponding eigenvalue in the following way. As zL1 is an isolatedroot, we first write the counting function in (5.6) as

iYL(z) = g(z) + 1L

(gb(z) + K(zL1, z)) + 1L

L2l=1

K(zl, z). (8.2)

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Figure 10. Reciprocal root distributions corresponding to the hole state for = 0.3 and = 0.4; (a) L = 2n = 20 (b) L = 2n = 100.

We can now follow the same procedure as in section 5.2 and turn this into an integro-differential equation of the form (5.15). Importantly, the end points of the integrationcontour are again complex conjugates of one another. Compared to the equation for thelowest excitation the driving term has an extra contribution of order O(L1) and theboundary conditions that determine the end points and of the contour now read

YL() = +

L, YL() = 3

L. (8.3)

The root = zL1 is determined by the requirement that

YL() = . (8.4)

If we expand ,

and YL, but not , in powers of L1

, we can use the results ofsection 6. With

YL(z) =n=0

Lnyn(z), (8.5)

we find the same solutions for y0, y2 and y3 as in (6.7), (6.10) and (6.11). However, dueto the additional term proportional to L1 in (8.2), the solution for y1 differs from thatin (6.9) and is given by

y1(z) = i lnz

zc

1 z2c1

z2

zc z1cz

z1c

i1 z + a

zc + a

z + b

zc + b i ln

z 1/

zc 1/

+ 1 i ln1ab(zc)i1 . (8.6)

Only now will we expand ,

= 0 +n=1

Lnn, (8.7)

and use the definitions of the points , and above. Employing again theexpansion (6.2) and appendix B, we find that the intersect of the solution curve withthe negative real axis in the limit L

is given, as before, by

zc = 1ab

. (8.8)

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Furthermore, in leading order, the isolated root tends to zc in the limit, i.e. 0 = zc.Putting everything together we finally find that the eigenvalue for this solution is givenby

E2(L) = 2zc1 zc

42

zc z1c1

L2+ O(L3), (8.9)

with zc given in (8.8). It is instructive to compare E2(L) to the gap of the first excitedstate E1(L). The latter exhibits a non-analytic change at = c. Interestingly, we findthat to order O(L2)

limc

E2(L) = limc

E1(L) = limc

E1(L). (8.10)

In conclusion we find that in the massive phase MI also the second gap is real and givenby (8.9). The second gap may be a complex conjugate pair for small values of the systemsize. However, this pair merges at a finite value of L producing two real eigenvalues,the lowest of which is given by (8.9). This observation is consistent with the results ofDudzinski and Schutz [30], who computed the spectrum on the coexistence line = forsmall system sizes.

8.2. Coexistence line

As we have seen above, the second gap of the TASEP is real for large L, even though itmay be complex for small system sizes. On the coexistence line, the leading term of ( 8.9)

vanishes and we are left with a second diffusive mode with an eigenvalue that vanishesas O(L2). A priori we expect a whole band of diffusive modes on the coexistence line,some of which should be given by the domain wall theory of [30],

En(L) = n22

zc z1c1

L2+ O(L3), (8.11)

with zc given by (8.8). It is clearly of interest to know whether or not there are complexmodes as well, which would result in oscillatory behaviour in the relaxational dynamicsat large times.

We observed numerically, that the first complex excitation that does not become real

for a finite value of L in the massive phase MI corresponds to the choice of integers

Ij = L/2 + j for j = 1, . . . , L 3,IL2 = L/2 1, IL1 = L/2, (8.12)

i.e. there is one hole between the second- and third-last roots. The corresponding(reciprocal) root distribution again has an isolated root on the negative real axis, seefigure 11, which now lies inside the contour of integration.

Proceeding as in the preceding section, we write the counting function as

iYL(z) = g(z) +1

L(gb(z) + K(zL1, z)) +

1

L

L2

l=1 K(zl, z). (8.13)doi:10.1088/1742-5468/2006/12/P12011 26


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Figure 11. Reciprocal root distributions corresponding to the first complexeigenvalue Ec(L) for = = 0.3 and L = 1860.

Turning the sum into an integral we arrive at the following integro-differential equation

i YL(z) = g(z) +1

L(gb(z) + K(1, z) K(2, z))

+1

2

12

K(w, z)YL(w) dw

+1

2

C1

K(w, z)YL(w)

1 eiLYL(w) dw +1

2

C2

K(w, z)YL(w)

eiLYL(w) 1 dw. (8.14)

Here, 1 = zL1 is the isolated root and 2 corresponds to the position of the hole. Thecontribution due to the latter needs to be subtracted in order to cancel the contributionarising from the integral. The values of 1, 2 as well as the end points 1 and 2 of thecurve are determined self-consistently by the boundary conditions

YL(2) = + L

, YL(1) = L

,

YL(1) = , YL(2) = 4L

.(8.15)

Assuming an expansion of the form

YL(z) =

n=0

Lnyn(z), (8.16)

we can utilize the results of section 6. If we furthermore assume that 11 lies inside thecontour of integration, we find that y0(z) is again given by (6.7) and obtain the followingresult for y1(z)

y1(z) = i ln z

zc

1 z2c1 z2

zc z1cz z1c

i1 z + azc + a

z + b

zc + b

i ln

z 1/1zc 1/1

z 1zc 1+ iln

z 1/2zc 1/2

+ 1 i ln2ab(zc)i1 . (8.17)

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It is important to note, that expression (8.17) has been derived under the assumption thatz1 lies outside the contour of integration7. Hence, we cannot use (8.17) to determine 1via (8.15). However, it turns out that (8.17) already contains sufficient information fordetermining the leading order of Ec.

Using the definitions of the points 1, 2 and 2 above, the expansion (6.2) andappendix B, we find that the intersect of the solution curve with the negative real axis inthe limit L is now given by

zc = (ab)1/4. (8.18)The O(1) term of the eigenvalue corresponding to the choice (8.12) is real

Ec(L) = 1 + 3zc1

zc

+ o(1), (8.19)

where zc given in (8.18). The subleading corrections are complex. However, the O(1)contribution (8.19) to the eigenvalue does not vanish on the coexistence line. Hence thisexcitation does not play an important role at large times. We have checked that for = = 0.3 both (8.18) and (8.19) agree well with a direct numerical solution of theBethe ansatz equations. On the basis of the above analysis, we conjecture that for allgapless excitations on the coexistence line with eigenvalues that scale with system size as

E= eL2 + O(L3), (8.20)the coefficients e are real. In other words, the dominant large time relaxation on the

coexistence line does not have oscillatory modes.

8.3. Maximum current phase

Comparing again the numerical solutions of the Bethe ansatz equations to a directnumerical determination of the eigenvalues of the transition matrix for small system sizesL we find that the lowest excitations with non-vanishing imaginary part are characterizedby the set of integers

Ij = L/2 + j for j = 1, . . . , L 2 IL1 = L/2, (8.21)or

Ij = L/2 + j for j = 2, . . . , L 1 I1 = L/2. (8.22)For the first choice the imaginary part of the eigenvalue is positive, whereas the secondyields the complex conjugate eigenvalue. In contrast to the corresponding excitations inthe massive phase MI, we find that the gaps in the MC phase remain complex for allvalues of L.

We denote the complex eigenvalue corresponding to (8.21) by Ec(L). We havedetermined Ec(L) from a numerical solution of the Bethe ansatz equations for latticesof up to L = 1200 sites and find its leading large L behaviour to be independent of therates and . In what follows we therefore fix = = 0.7, in the understanding that

7 Otherwise the term ln(1 wz) in the kernel K(w, z) would produce an additional contribution in the integro-differential equation for y1(z).

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Figure 12. Reciprocal root distributions corresponding to the lowest complexeigenvalue Ec(L) for = = 0.7 and L = 2n = 400.

Figure 13. Double-logarithmic plots of the real and imaginary parts Ec asfunctions of 1/L.

the results for the large L asymptotics of the eigenvalue of M are universal throughoutthe maximum current phase.

The distribution of reciprocal Bethe roots for L = 400 and = = 0.7 is shown infigure 12. We observe that the root distribution is quite similar to that of the second-lowest eigenvalue in figure 3. The main difference is that now there is a slight gap betweenthe last two roots.

On the basis of the similarity of the root distribution to the one of the lowest excited

state, we expect that both real and imaginary parts of Ec(L) will be proportional to L3/2for large systems. This is indeed the case, as is shown in figure 13, where Re(Ec(L)) andIm(Ec(L)) are plotted as functions of L1 on a double-logarithmic scale for = = 0.7.Both curves are very close to straight lines with slope 3/2.

In order to determine the asymptotic form of Ec(L) more accurately, we repeat theanalysis we employed for the lowest excited state in the maximum current phase insection 7. We group the data points for Ec(L) in bins containing 11 points each andwithin each bin perform least-square fits to

Re(Ec(L)) Re(ek)Lsk ,

Im(Ec(L)) Im(ek)Ltk

, 20k L 20(k + 1).(8.23)

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Figure 14. Fits to the exponents sk and tk in (8.23) of the real and imaginaryparts ofEc(L) plotted against 1/k for = = 0.7.

The resulting sequences of exponents sk and tk in the range 30 k 59 are wellapproximated by the polynomials

sk 1.499 7697 0.737888549k1 + 3.369432984k2,tk 1.499 1041 0.423 377 041k1 + 6.133640280k2.

(8.24)

The extrapolated values at k = are very close to 3/2. The fits to the data as well asthe extrapolation to k = are shown in figure 14.

In order to determine the coefficients of the O(L3/2) terms, we carry out least-squarefits of the binned data for Ec(L) to

Ec(L) ekL3/2 fkL5/2, 20k L 20(k + 1). (8.25)

The resulting sequence of coefficients ek is well described by the polynomial fits (seefigure 15)

Re(ek) 8.468 7424 0.188 6671k1 12.277 915k2,Im(ek) 1.650 8588 + 0.126 3865k1 + 3.633 3854k2.

(8.26)

Extrapolation to the limit k then gives the following result for the energy of thefirst complex excited state of the TASEP in the MC phase

Ec(L) (8.47 1.65i)L3/2 + O(L5/2). (8.27)

The result (8.27) can be compared to excited states with complex eigenvalues in thehalf-filled8 TASEP on a ring. We are not aware of any explicit results in the literatureon complex eigenvalues, but they can be easily determined by employing the methodof[65]. We summarize some results in appendix D. The lowest excited state with complexeigenvalue found to be

Ec,ring(L) (17.1884 . . . 5.43662 . . . i)L3/2. (8.28)

8 It is natural to compare the half-filled case, as the average bulk density of the TASEP with open boundaries inthe maximum current phase is 1/2.

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Figure 15. Polynomial fits to the real and imaginary parts of the coefficient ekin (8.25) plotted against 1/k for = = 0.7.

9. Summary and conclusions

In this work we have used Bethes ansatz to diagonalize the transition matrix M forarbitrary values of the rates p, q, , , and that characterize the most general PASEPwith open boundaries. The resulting Bethe ansatz equations (3.1), (3.2) describe thecomplete excitation spectrum of M.

We have carried out detailed analyses of the Bethe ansatz equations for the limitingcases of symmetric and totally asymmetric exclusion and determined the exact asymptoticbehaviour of the spectral gap for large lattice lengths L. This gap determines the longtime (t L) dynamical behaviour of the TASEP. We emphasize that care has to be takenregarding time scales, and that our results below are not valid at intermediate times wherethe system behaves as for periodic boundary conditions [4].

9.1. Dynamical phase diagram

In the case of totally asymmetric exclusion and = = 0, we found that there are threeregions in parameter space where the spectral gap is finite and the stationary state isapproached exponentially fast. In addition there is one region (maximum current phase)and a line (coexistence line) where the gap vanishes as L . The resulting dynamicalphase diagram9 is shown in figure 16.

Recalling that

c =

1 +

1 1/31

, c =

1 +

1 1/31

, (9.1)

the leading asymptotic values of the spectral gap in the various regions of the phasediagram of the TASEP are as follows:

Massive phase MI: < c, < c, =

E1(L) = + 21 +

(1 )(1 )/ + O(L

2). (9.2)

9 We use the term phase to indicate a region in parameter space characterized by a particular type of relaxationalbehaviour.

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Figure 16. Dynamic phase diagram determined by the first eigenvalue gap ofthe TASEP. MI, MIIa and MIIb are massive phases, CL denotes the criticalcoexistence line and MC the critical maximal current phase.

The spectral gap does not vanish as L , which implies a finite correlation length andan exponentially fast approach to stationarity.

High density phase MIIb: < 1/2, c <

E1(L) = c + 21 + [(1 )/]1/3

+ O(L2). (9.3)

The spectral gap is finite and independent of . The relaxational behaviour is againexponentially fast.

The gap in the low density phase MIIa: < 1/2, c < is obtained by the exchange . We note that the subdivision of the massive high and low density phases into MIand MIIa,b is different from the one suggested on the basis of stationary state propertiesin [16].

Coexistence line (CL): = < 1/2

E1(L) = 2(1 )1 2 L

2 + O(L3). (9.4)

The gap vanishes like L2 for large systems, which corresponds to a dynamic exponentz = 2 and indicates relaxational behaviour of a diffusive type.

Maximal current phase (MC): , > 1/2

E1(L) 3.578L3/2 + O(L5/2). (9.5)

Here the gap is independent of the rates and . The dynamic exponent z = 3/2 isindicative of KPZ like behaviour [5]. We find that the magnitude of E1(L) is smaller thandoi:10.1088/1742-5468/2006/12/P12011 32


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half that of the lowest excited state for the TASEP with periodic boundary conditions

E1,ring(L) 6.50919 . . . L3/2

. (9.6)

The two gaps do not seem to be related in any obvious way.It is known [17] that by varying the bulk hopping rates one can induce a crossover

between a diffusive EdwardsWilkinson (EW) scaling regime [18] with dynamic exponentz = 2 and a KPZ regime [5] with z = 3/2. Here we have shown using exact methods thata crossover between phases with z = 2 and z = 3/2 occurs in the case where the bulktransition rates are kept constant, but the boundary injection/extraction rates are varied.

9.2. Domain wall theory

As shown in [29, 30] the diffusive relaxation (z = 2) is of a different nature than in theEW regime and is in fact due to the unbiased random walk behaviour of a shock (domainwall between a low and high density region) with right and left hopping rates given by

D =J

+ . (9.7)

Here, = and + = 1 are the stationary bulk densities in the low and highdensity phases respectively, and the corresponding currents are given by J = (1).Our results (9.2) and (9.4) for the massive phase MI and the coexistence line agree withthe relaxation time calculated in the framework of a domain wall theory (DWT) modelin [30]. This is in contrast to the massive phases M

II, where the exact result (9.3) differs

from the DWT prediction of [30]. More precisely, the DWT predicts that the relaxationalmechanism in the stationary high density phase ( < 1/2, > ) for < 1/2 is due tothe random walk of a (0|1) domain wall between a low and a high density segment. Onthe other hand, in the high density phase for > 1/2, the DWT predicts a relaxationalmechanism based on so-called (m|1) domain walls between a maximum current and alow density region. Our results for the gap exhibit a change of behaviour at = c ratherthan at = 1/2. We therefore propose the following modified DWT.

We assume that in the high density phase for large we can still use the DWT ratesas given in (9.7), but with an effective density eff. In contrast to [30] we do not takethe effective density equal to that of the maximum current phase, eff = 1/2, but instead

determine it below from using a monotonicity argument. Consider first figure 17, in whichwe plot the first gap as a function of = for constant = 0.3.

< We are in the low density phase. The gap in the infinite volume is finite in thisregion, but upon increasing we are driven towards the coexistence line where thegap vanishes and the relaxation is diffusive. Hence, E/ > 0 in this region.

> We are now in the high density region and we expect the gap to be finite again,decreasing with increasing . Hence, in this region we should have E/ 0.Initially, this is indeed the observed behaviour of the graph of (9.2), but we see that

at = c the graph of (9.2) has local minimum, where the expected behaviour breaksdown.

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Figure 17. The gap as a function of for = 0.3. The curve corresponds to the

function in (9.2) and the horizontal line to (9.3). The gap of the lowest excitation(red) is a combination of the curve (0 < < c) and the line (c < < 1).

Hence, for small values of and large values of + the domain wall theory of [30]correctly predicts the gap to be

E1(, +) = D+ D + 2

D+D, (9.8)

which is equal to (9.2). In the high density region, there is a critical value eff = c beyondwhich the gap is given by E1(eff, +), where eff is determined by

E1(, +) =eff = 0. (9.9)

The considerations above are of a sufficiently general nature to remain valid for thewider class of driven diffusive systems that can be described by effective domain walltheories. We emphasize that the DWT as discussed above pertains to the limit L only, and that a more careful analysis is necessary to correctly produce the differencein finite-size behaviour for the phases MI and MII, see (6.15) and (6.23). In particular,the boundary rates generally need to be chosen differently from the bulk rates. A moredetailed comparison of the Bethe ansatz solution and DWT is possible for the case of thePASEP with Q = 0, where DWT becomes exact for certain values of the rates [66]. This isbeyond the scope of the present work and will be discussed in a separate publication [ 67].

9.3. Higher excitations

Massive phase MI and coexistence line (CL). The second gap in this phase is given by

E2(L) = 2zc1 zc

42

zc z1c1

L2+ O(L3), (9.10)

where zc = 1/

ab. This result is in agreement with the second gap predicted by theDWT of [30]. In particular, on the coexistence line it corresponds to n = 2 of the DWTband of diffusive modes on the coexistence line,

En(L) =

n22

zc z1c1

L2+

O(L3). (9.11)

The first gap given in (9.4) corresponds to n = 1.

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We studied the possibility of an oscillating diffusive mode on the coexistence line andfound that the first excitation with non-zero imaginary part has a finite gap for L

Ec(L) = 1 + 3zc

1 zc + o(1), (9.12)

where zc = (ab)1/4. Although the subleading terms have non-zero imaginary parts, theleading term is real and does not vanish on the coexistence line. Hence this excitationdoes not play an important role at large times. We conjecture that the leading O(L2)terms of the eigenvalues of all diffusive modes on the coexistence line are real.

Maximum current phase MC. We determined the energy of the first complex excited state

in the MC phase by means of a direct numerical solution of the Bethe ansatz equations.We find it to be independent of and and given by

Ec(L) (8.47 1.65i) L3/2 + O(L5/2). (9.13)

Complex eigenvalues give rise to interesting oscillations in correlation functions. Suchbehaviour has been observed for the KPZ equation [68, 69], which is related to the TASEPwith periodic boundary conditions. In appendix D we have computed the low lyingcomplex excitations of the TASEP on the ring using the method of [ 65]. The lowest suchexcitation is given by

Ec,ring(L) (17.1884 . . . 5.43662 . . . i)L3/2. (9.14)

The result (9.14) confirms the prediction from the KPZ equation that both real andimaginary part scale with L3/2 [69]. According to (9.13) this scaling still holds whenboundaries are present. However, as was the case with the first gap, although (9.13) doesnot depend on the boundary rates and , its value does not seem to be related to (9.14)in a simple way.

A number of interesting open problems remain. In a forthcoming publication [67] wedetermine the spectral gaps for the case of partially asymmetric diffusion and , , , = 0from the Bethe ansatz equations (3.2). As we have discussed, the constraint (1.8) precludes

the determination of current fluctuations from the Bethe ansatz. It therefore would behighly desirable to obtain Bethe ansatz equations for the Heisenberg XXZ chain with themost general open boundary conditions. Furthermore, the existing Bethe ansatz solutionprovides information only about the spectrum, but not the eigenstates of the transitionmatrix. In order to study correlations functions [39] the knowledge of matrix elements ofthe spin operators between left and right eigenstates is required. In light of this it wouldbe very interesting to construct the eigenstates from the Bethe ansatz.

Acknowledgments

We are grateful to B Derrida, G M Schutz and R Stinchcombe for very helpful discussions.

This work was supported by the ARC (JdG) and the EPSRC under grant GR/R83712/01(FE).

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Appendix A. Analysis of the AbelPlana formula

In this appendix we sketch how to extract the finite-size correction terms from the integralexpression

i YL(z) = g(z) +1

Lgb(z) +

1

2

K(w, z)YL(w) dw

+1

2

C1

K(w, z)YL(w)

1 eiLYL(w) dw +1

2

C2

K(w, z)YL(w)

eiLYL(w) 1 dw. (A.1)

The main contributions to the correction terms in (A.1) comes from the vicinities of theend points , . Along the contour C1 the imaginary part of the counting function ispositive, Im(YL(w)) > 0, whereas along the contour C2 it is negative, Im(YL(w)) < 0.

As a result the integrands decay exponentially with respect to the distance from the endpoints. In the vicinity of , we therefore expand

YL(w) = YL() + YL()(w ) + . (A.2)

Assuming that YL() is O(1) (an assumption that will be checked self-consistently), wefind that the leading contribution for large L is given by

1

2

C1

K(w, z)YL(w)

1 eiLYL(w) dw YL()

2

0

K(w, z)

1 + eiLY

L()(w)

dw, ( ). (A.3)

Here we have used the boundary conditions (5.14). Carrying out the analogous analysis

for the integral along the contour C2, we arrive at the following expression for the leadingcontribution of the last two terms in (A.1)

A = 12

C1

K(w, z)YL(w)

1 eiLYL(w) dw +1

2

C2

K(w, z)YL(w)

eiLYL(w) 1 dw

=i

2L

0

1

1 + ex

K

+

i x

LYL(), z

K

i x

LYL(), z

dx

( ). (A.4)If the end points , are such that we can Taylor expand the kernels appearing in (A.4),we can simplify the expression further with the result

A K(, z)L2YL()

0

x1 + ex

dx ( )

= 12L2

K(, z)

YL() ( ). (A.5)

This is the leading EulerMacLaurin correction term that occurs in the low and highdensity phases, see (6.1). The key in the above derivation was the ability to expand

K

+

i x

LYL(), z

K

i x

LYL(), z

ln

LYL()(z

1 ) + i xLYL()(z

1 ) i x

(A.6)

in a power series in x. This is unproblematic as long as LYL()(z1

) is large, which

turns out to be the case in the low and high density phases as well as on the coexistenceline.

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In the maximum current phase the above analysis no longer holds. As mentioned inthe main text,

1 in the MC phase, and this has two consequences. Firstly, by virtue

of the end points , being close to stationary point, the derivative YL() is now foundto be of order L1/2 whereas YL () is of order one. As a result (cf [25]) we have to retainsubleading terms in the Taylor expansions of YL(w) and Y

L(w) around w = , e.g.

YL(w) YL() + YL()(w ) + 12YL ()(w )2. (A.7)Secondly, and more seriously, as 1 = O(L1/2) the Taylor expansion of the

integration kernel K(, z) breaks down near z = . Taking (A.7) into account andfollowing through the same steps as before, we find that the leading contributions ofthe last two terms in (A.1) in the maximum current phase are

A=

i

L

0

x + p

1 + exp(x2 + 2xp)K( + x,z) dx

iL

0

x ip1 + exp(x2 2ipx) K( + i x,z) dx (

). (A.8)

Here we have introduced

=

2i

LYL ()= O(L1/2), p = i

2YL()L = O(1). (A.9)

In order to make progress we would like to expand around x = 0. However, as z1 becomes of order O(L1/2) for z near , all the terms xnK(n)(, z) in the expansion areof the same order L0 for (n

1), and we would need to self-consis

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Jan de Gier and Fabian H L Essler- Exact spectral gaps of the asymmetric exclusion process with open boundaries