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Time-dependent correlation functions in an

arXiv:hep-lat/9303011v1 23 Mar 1993Time-dependent correlation functions in anone-dimensional asymmetric exclusion processGunter Sch¨utzDepartment of Physics, Weizmann Institute,Rehovot 76100, IsraelWe continue our studies [1] of an one-dimensional anisotropic exclusion process with parallel dy-namics describing particles moving to the right on a chain of L sites. Instead of considering periodicboundary conditions with a defect as in [1] we study open boundary conditions with injection ofparticles with rate α at the origin and absorption of particles with rate β at the boundary.

Weconstruct the steady state and compute the density profile as a function of α and β. In the largeL limit we find a high density phase (α > β) and a low density phase (α < β).

In both phases thedensity distribution along the chain approaches its respective constant bulk value exponentially ona length scale ξ. They are separated by a phase transition line where ξ diverges and where thedensity increases linearly with the distance from the origin.

Furthermore we present exact expres-sions for all equal-time n-point density correlation functions and for the time-dependent two-pointfunction in the steady state. We compare our results with predictions from local dynamical scalingand discuss some conjectures for other exclusion models.PACS numbers: 05.40.+j, 05.70.Ln, 64.60.Ht

I. INTRODUCTIONWe study an one-dimensional totally asymmetric exclusion model where particles areinjected stochastically at the origin of a chain of L sites, move to the right according torules defined below and are removed at its end, again according to stochastic rules. Eachsite of the chain can be occupied by at most one particle.

Among the interesting featuresof such exclusion models is the occurence of various types of phase transitions which arisefrom the interplay of the bulk dynamics with the boundary conditions [1]- [5] and their closerelationship to vertex models [6], growth models [7], and, in the continuum limit, to theKPZ equation [8] and the noisy Burger’s equation.Exclusion models can be divided into four classes according to the dynamics (parallel orsequential) and the boundary conditions (periodic with conservation of the number of parti-cles and (possibly) a defect or open with injection and absorption of particles). According tothis classification we call them p/p-models (parallel, periodic), p/o-models (parallel, open),s/p-models (sequential, periodic) and s/o-models (sequential, open).

A s/p-model with adefect has been studied numerically by Janowsky and Lebowitz [3], without a defect it wassolved by a Bethe ansatz by Gwa and Spohn [9]. The s/o-model was studied numerically byKrug [2], later the exact solution with the full phase diagram was found [4,5].

In additionto the density profile all equal-time n-point density correlation functions in the steady statewere determined [10]. Previously we solved a p/p-model with a defect [1] with Bethe ansatzmethods and obtained the density profile and the equal-time two-point correlation functionin the steady state.Here we discuss the p/o-model with the same parallel dynamics as in [1] but with openboundary conditions where particles are injected at the origin with a rate α and are removedat the (right) boundary with rate β.

The bulk dynamics of our model are deterministic anddefined as follows: Each site x on the ring (1 ≤x ≤L) is either occupied (τx(t) = 1) orempty (τx(t) = 0) at time t. The time evolution consists of two half time steps. In the firsthalf step we divide the chain with L sites (L even) into pairs of sites (2,3), (4,5), .

. ., (L, 1).1

If both sites in a pair are occupied or empty or if site 2x is empty and site 2x + 1 occupied,they remain so at the intermediate time t′ = t + 1/2. If site 2x is occupied and site 2x + 1empty, then the particle moves with probability 1 to site 2x + 1, i.e.,τ2x(t′)= τ2x(t)τ2x+1(t)τ2x+1(t′) = τ2x(t) + τ2x+1(t) −τ2x(t)τ2x+1(t) .

(1)These rules are applied in parallel to all pairs except the pair (L,1). In this pair representingthe boundary (site L) and the origin (site 1) resp.particles are absorbed and injectedaccording to the following stochastic rules.

If site 1 was empty at time t then it remainsso with probability 1 −α and becomes occupied with probability α at time t′. If site 1was occupied at time t then it remains occupied with probability 1.

These two rules areindependent of the occupation of site L. On the other hand, if site L was occupied at timet it remains so with probability 1 −β and becomes empty with probability β. If site Lwas empty, it remains empty with probability 1.

These two rules are independent of theoccupation of site 1. This means that opposed to the models with sequential dynamicsstudied in refs.

[2,4,5,10] simultaneous injection and absorption is allowed with probabilityαβ. We haveτ1(t′) = 1 with probability τ1(t) + α(1 −τ1(t))τ1(t′) = 0 with probability (1 −α)(1 −τ1(t))τL(t′) = 1 with probability (1 −β)τL(t)τL(t′) = 0 with probability 1 −(1 −β)τL(t) .

(2)In the second half step t + 1/2 →t + 1 the pairing is shifted by one lattice unit such thatthe pairs are now (1,2), (3,4), ... (L −1, L). Here rules (1) are applied in all these pairs,there is no injection and absorption in the second half time step.∗∗Note that we reverse the order of the choice of pairs as compared to [1].

There the pairs were2

In the mapping of ref. [6] this model is equivalent to a two-dimensional four-vertex modelin thermal equilibrium with a defect line where other vertices, not belonging to the groupdefining the 6-vertex model or 8-vertex model, have non-vanishing Boltzmann weights.

Thetwo steps describing the motion of particles define the diagonal-to-diagonal transfer matrixT(α, β) in the vertex model (see appendix). The pairing is chosen as in [6] but the hoppingprobabilities are different.For the Bethe ansatz solution of the p/p-model with a defect [1] the conservation ofthe number of particles was crucial and we cannot repeat the calculation here, where theparticle number is not conserved.

However, since we are only interested in the steady state,we can construct the steady state explicitly for small lattices and then try to guess its generalform for arbitrary length L. This method was succesfully applied in refs. [4] and [5] andled to exact expressions for the particle current and the density profile for arbitrary valuesof the injection and absorption rates.

Only after guesswork produced the correct results,they were actually proven (see also [10]). It turns out that also here we can guess rulesfor the construction of the steady state.

Instead of proving them we verified our conjecturefor lattices of up to 14 sites. In the same way we guessed and verified expressions for thedensity profile (the one-point density correlator ⟨τx ⟩in the steady state) eq.

(18) and theequal-time n-point density correlation function (30). The simple form of these correlationfunctions then allowed for a conjecture of the time-dependent two-point function ⟨τxT tτy ⟩(37) - (41).

(In this expression T t denotes the t-th power of the transfer matrix T(α, β)).The mapping to the vertex model allows for an independent verification of this result.Among other things the time-dependent two-point function is of interest for the studyof local dynamical scaling in the absence of translational invariance. Dynamical scaling ina 1+1 dimensional system with translational invariance both in space and time directionimplies that the two-point function G(r, t) behaves under a global rescaling λ of the spacechosen as (1,2), (3,4), ... in the first half time step.3

and time coordinates as [11]G(λr, λzt) = λ−2xG(r, t) . (3)In this expression r denotes the distance in space direction, t is the distance in time direction,z is the dynamic critical exponent and x is the scaling dimension.

From (3) follows that thecorrelation function has the formG(r, t) = t−2x/z Φ( rzt )(4)with the scaling function Φ which is not determined by global dynamical scaling. By ex-tending the concept of global rescaling to local, space-time dependent rescaling, it has beenshown that for the special case z = 2 the correlation function G(r, t) is of the form [12,13]G(r, t) = at−x e−br22t(5)with some constants a and b, i.e., Φ(r2/t) = a exp (−br2/t).

The (connected) density cor-relation function in the probabilistic symmetric p/p-model without defect computed in [6]is indeed of this form with critical exponent x = 1/2. Since we study the steady state wehave translational invariance in time direction, but due to the open boundary conditionstranslational invariance is broken in space direction.

In sec. 5 we show that the form of⟨τxT tτy ⟩for large L on the critical line α = β and in the scaling regime close to it resembles(5) with x = 0 (sec.

5), i.e., one has z = 2, but there are additional pieces that arise fromthe breaking of translational invariance.The paper is organized as follows. In sec.

2 we present our conjectured rules for theconstruction of the steady state and exact expressions for the current and the density profile.In sec. 3 we study the limit of large L and derive the phase diagram.

In sec. 4 we presentexpressions for the equal-time n-point density correlation function.

They turn out to bereducible to a sum of one-point functions through associative fusion rules of the densityoperators. In particular we study the two-point function in the scaling regime.

In sec. 5we compute the time-dependent two-point correlator.

Again we put our emphasis on the4

vicinity to the phase transition line. In Sec.

6 we summarize our main results and discussour results in the context of other exclusion models. In the appendix we discuss the mappingto a two-dimensional vertex model.II.

CONSTRUCTION OF THE STEADY STATEBefore we discuss the construction of the steady state we introduce some useful notations.In anticipation of the correspondence of the model to a vertex model discussed in refs. [1,6]and in the appendix we denote a state of the system with N particles placed on sitesx1, .

. .

, xN and holes everywhere else by | x1, . .

. , xN ⟩.

The transfer matrix T(α, β) (A1)acting on the space of states spanned by these vectors acts as time evolution operator andencodes the dynamics and the boundary conditions of the system as defined by eqs. (1)and (2).

The steady state is the (right) eigenvector with eigenvalue 1 of the transfer matrixT(α, β) and we denote it by| 1 ⟩=LXN=0X{x}ΨN(x1, . .

. , xN)| x1, .

. .

, xN ⟩. (6)Here the N-particle “wave function” ΨN(x1, .

. .

, xN) is the unnormalized probability offinding the particular configuration | x1, . .

. , xN ⟩of N particles in the steady state.

Wedenote the state with no particles by | ⟩and the corresponding wave function by Ψ0. Thesummation runs over all states of N particles (0 ≤N ≤L) and all possible configurations{x} = {x1, .

. .

, xN} and one has T(α, β)| 1 ⟩= | 1 ⟩. The normalized probabilities are givenbypN(x1, .

. .

, xN) = ΨN(x1, . .

. , xN)/ZL(7)withZL =LXN=0X{x}ΨN(x1, .

. .

, xN) . (8)The transfer matrix T(α, β) has a left eigenvector ⟨1 | with eigenvalue 1 given by5

⟨1 | =LXN=0X{x}⟨x1, . .

. , xN | ,(9)where ⟨x1, .

. .

, xN | is the transposed vector to | x1, . .

. , xN ⟩.

Defining a scalar product inthe standard way (i.e., ⟨x1, . .

. , xN |y1, .

. .

, yM ⟩= 1 if the sets {x} and {y} are identical and0 else) one can write ZL as the scalar product ZL = ⟨1 | 1 ⟩.Furthermore we denote the projection operator on particles on site x by τx:τx| x1, . .

. , xN ⟩=| x1, .

. .

, xN ⟩if x ∈{x1, . .

. , xN}0else.

(10)The projector on holes is σx = 1 −τx. Expectation values ⟨τx1 .

. .

τxk ⟩of the operators τxand their products in the steady state can conveniently written in the form⟨τx1 . .

. τxk ⟩= ⟨1 |τx1 .

. .

τxk| 1 ⟩/ZL . (11)Taking the scalar product with the left eigenvector ⟨1 | and dividing by the normalizationsum ZL is equivalent to a summation over all probabilities pN(y1, .

. .

, yN) with {x1, . .

. , xk} ∈{y1, .

. .

, yN}. This is the definition of an expectation value in the steady state.The particle current j is a conserved quantity in the bulk since only the origin and theboundary act as a source or sink of particles.

It is given by the correlator [1] (see (A6))j = ⟨τ2xσ2x+1 ⟩. (12)Now we discuss the construction of the steady state.

In [1] we derived the importantresult that for the deterministic dynamics defined by (1) one hasτ2x−1σ2y| Λ ⟩= 0(13)for 1 ≤x ≤L/2 and x ≤y ≤L/2 and any right eigenvector | Λ ⟩of the transfer matrix.This simplifies the construction of the steady state considerably: If in a state | x1, . .

. , xN ⟩one of the xi is odd, then it has a non-vanishing weight ΨN(x1, .

. .

, xN) only if all even xjwith xi < xj ≤L are also contained in the set {x1, . .

. , xN}.6

Using this it is easy to construct the steady state explicitly for small L. We discoveredthat the unnormalized probabilities Ψ(L+2)N(x1, . .

. , xN) in the chain with L + 2 sites can beconstructed recursively out those of the chain with L sites according to the following rules:Rule 1: (0 ≤N ≤L, all {x})Ψ(L+2)N(x1 + 2, x2 + 2, .

. .

, xN + 2) = β2(1 −α)Ψ(L)N (x1, x2, . .

. , xN)Rule 2: (0 ≤N ≤L, all {x})Ψ(L+2)N+2 (2, x1, .

. .

, xN−1, L + 1, L + 2) = α2(1 −β)Ψ(L)N (2, x1, . .

. , xN−1)Rule 3: (0 ≤N ≤L/2, {x1, .

. .

, xL/2} ̸= {2, 4, 6, . .

. , L −2, L})Ψ(L+2)N+1 (2, x1 + 2, x2 + 2, .

. .

, xN + 2) = αβ2Ψ(L)N (x1, x2, . .

. , xN)Rule 4: (L/2 ≤N ≤L, {x1, .

. .

, xL/2} ̸= {2, 4, 6, . .

., L −2, L})Ψ(L+2)N+1 (x1, x2, . .

. , xN, L + 2) = α2βΨ(L)N (x1, x2, .

. .

, xN)Rule 5: ({x1, . .

. , xL/2} = {2, 4, 6, .

. ., L −2, L})Ψ(L+2)L/2+1(2, 4, .

. .

, L, L + 2) = αβ(α + β)Ψ(L)L/2(2, 4, . .

. , L) −(αβ)L/2+3(14)These rules together with (13) and the initial conditionsΨ(2)0= β2(1 −α),Ψ(2)1 (2) = αβ,Ψ(2)2 (1, 2) = α2(1 −β)(15)define recursively all quantities Ψ(L+2)N(x1, .

. .

, xN) in the chain with L + 2 sites.Based on these rules we constructed the steady state up to L = 14 and verified that it has7

indeed eigenvalue 1 of T(α, β). In a next step we computed the sum over all ΨN(x1, .

. .

, xN)and concluded that the normalization ZL (8) is given byZL = (1 −β)αLL/2−1Xk=0 βα!k+ (1 −α)βLL/2−1Xk=0 αβ!k+ (αβ)L/2=(1 −β)αL+1 −(1 −α)βL+1α −βα ̸= βαL (1 + L(1 −α))α = β. (16)This result was again checked explicitly up to L = 14.Going one step further we consider the average density ⟨τx ⟩at site x defined by (11).We found the following exact expressions for the even and odd sublattices resp.

:ZL⟨τ2x ⟩=(1 −β)αL2x−1Xk=0 βα!k+ (1 −β)αL+1LXk=2x βα!k+ (αβ)L+1αL+1 (1 + L(1 −α)) + 2xαL(1 −α)2ZL⟨τ2x−1 ⟩=(1 −β)2αL2x−3Xk=0 βα!k+ (1 −β)αL+2−2xβ2x−2αL+1(1 −α) + (2x −1)αL(1 −α)2. (17)On the r.h.s.

of (17) the upper expressions are valid for α ̸= β, while the lower expressionsare valid for α = β. Performing the summations we arrive at the main result of this section⟨τ2x ⟩=α + (1 −α)1 −( βα)2x1 −1−α1−β( βα)L+1 α ̸= βα + (1 −α)22x1 + L(1 −α)α = β⟨τ2x−1 ⟩=(1 −β)1 −1−α1−β ( βα)2x−11 −1−α1−β( βα)L+1α ̸= β(1 −α)22x1 + L(1 −α) +α(1 −α)1 + L(1 −α) α = β.

(18)The anisotropy between the even and odd sublattices is a consequence of the parallelupdating mechanism [1] and related to the net particle current (12) for which we found(again, by direct evaluation on small lattices and guessing the general form for arbitrary L)8

j =α −α −β1 −1−α1−β( βα)L+1 α ̸= βα −α(1 −α)1 + L(1 −α)α = β. (19)The fact that this quantitity is independent on x as it should be is an additional non-trivialcheck for our conjectures.Finally note that by the definition of the model there is the particle-hole symmetry (A5):Changing particles into holes, reflecting site x into site L + 1 −x and exchanging α and βleaves the system invariant.

All our results are indeed invariant under this operation.III. THE PHASE DIAGRAMFrom eqs.

(18) one realizes that the system changes its behaviour if α = β. The averagedensities on the even and odd sublatticesρ(even) = 2LL/2Xx=1⟨τ2x ⟩,ρ(odd) = 2LL/2Xx=1⟨τ2x−1 ⟩(20)have a discontinuity in the thermodynamic limit L →∞at α = β ̸= 0, 1.† One finds from(18)ρ(even) =αα < β1α > βρ(odd) =0α < β1 −βα > β(21)†If α or β is either 0 or 1 the system is trivial in the sense that the density profile is exactly constant(even in finite systems) and all correlation functions can be obtained without any calculation (seeappendix).

Therefore we exclude these cases from our discussion.9

On the phase transition line α = β one obtains ρ(even) = (1 + α)/2 and ρ(odd) = (1 −α)/2resp. For α < β (more particles are absorped than injected) the system is in a low densityphase with total average density ρ = 12, (ρ(even) + ρ(odd)) = α/2 < 12, while for α > β it is ina high density phase with ρ = 1 −β/2 > 12 (Fig.

1).01/211/21αβABFIG. 1.

Phase diagram of the model in the α −β plane. Region A is the low density phase andregion B the high density phase.

The phases are separated by the curve α = β.In the thermodynamic limit L →∞the current j (19) is given byj = min (α, β) . (22)There is no discontinuity at α = β in the current, but its first derivatives w.r.t.

α and β arediscontinous. The discontinuity of ρ at the phase transition line α = β and eq.

(22) remindus of the s/o-model [4,5,10]. In this model in the region α, β < 1/2 the phase diagram showsa low density phase AI and a high density phase BI separated by a phase transition line atα = β [5].

Also in this model the density and the first derivatives of the current w.r.t. theinjection and absorption rates α and β have a discontinuity at the phase transition line.In terms of the sublattice densities the current j is given by j = ρ(even)−ρ(odd) for all α, β.In terms of the total average density ρ the current satisfies j = 2ρ if ρ < 1/2 (low densityphase) and j = 2(1 −ρ) if ρ > 1/2 (high density phase).

These are the same relations as inthe p/p-model in the respective phases [1].10

Now we turn to a discussion of the density profile. We first study the case α < β andL →∞.

Defining the decay length ξ byξ−1 = ln βα(23)one obtains from (18) the density profile up to corrections of order exp (−L/ξ)⟨τ2x ⟩= α + (1 −β)e−(L+1−2x)/ξ⟨τ2x−1 ⟩= (1 −β)e−(L+2−2x)/ξ . (24)The profile decays exponentially with increasing distance from the boundary to its respectivebulk values ρ(even)bulk= α and ρ(odd)bulk = 0.

This the low density phase of the system.In the high density phase α > β which is related to the low density phase by the particle-hole symmetry the profile is given by⟨τ2x ⟩= 1 −(1 −α)e−2x/ξ⟨τ2x−1 ⟩= 1 −β −(1 −α)e−(2x−1)/ξ . (25)The bulk densities are ρ(even)bulk= 1 and ρ(odd)bulk = 1 −β.On approaching the phase transition line α = β the decay length ξ diverges.

On the linethe profile is linear and up to corrections of order L−1 given by⟨τ2x ⟩= α + (1 −α)2xL⟨τ2x−1 ⟩= (1 −α)2x −1L. (26)An explanation for the shape of the profile in the two phases and on the phase transitionline will be given in the next section.IV.

EQUAL-TIME CORRELATION FUNCTIONSHaving found exact expressions for the current and the density profile we proceed calcu-lating the n-point equal-time density correlation function ⟨τx1 . .

. τxn ⟩in the steady state.11

Examining the two-point function for small L we found the following exact relations for1 ≤x < y ≤L/2⟨τ2xτ2y ⟩= ⟨τ2x ⟩−α + α⟨τ2y ⟩⟨τ2xτ2y−1 ⟩= (1 −β)(⟨τ2x ⟩−α) + α⟨τ2y−1 ⟩⟨τ2x+1τ2y ⟩= ⟨τ2x+1 ⟩⟨τ2x−1τ2y−1 ⟩= (1 −β)⟨τ2x−1 ⟩. (27)The third of these equations is a simple consequence of (13) which says that whenever thereis a particle on an odd lattice site then all even lattice sites to its right must be occupiedas well.

The important result is that the two-point is completely determined by the one-point function and some constants! Going further we made the surprising observation thatthe n-point function can also be expressed in terms of one-point functions by repeatedlyfusing products of operators τxτy according to the fusing rules that are defined by (27) byomitting the averaging.

This fusion can be performed in arbitrary order until one reachesthe one-point level.The fusion rules implied by eqs. (27) can be simplified by using operators ηx defined byη2x = τ2x −α1 −α ,η2x−1 = τ2x−11 −β(28)instead of using the density operators τx.

In the bulk of the high density region both ⟨η2x ⟩and ⟨η2x−1 ⟩take the value 1, while in the bulk of the low density region both average valuesare 0. Expressing all τx in terms of the ηx the correlation functions (27) become⟨ηx1ηx2 ⟩= ⟨ηx1 ⟩(x2 > x1) .

(29)Fusion of n operators ηx1 . .

. ηxn gives ηxi with xi = min {x1, .

. .

, xn}. So the n-point corre-lation function is⟨ηx1 .

. .

ηxn ⟩= ⟨ηxi ⟩(xi = min {x1, . .

. , xn}) .

(30)This is the main result of this section.12

The form of the two-point function (29) can be understood by considering the steadystate as composed of “constituent profiles” with a region of constant low density up to somepoint x0 in the chain followed by a high density region beyond this “domain wall”. Suchan assumption explains why the correlator (29) does not depend on x2: In the low densityregion of density α on the even sublattice and 0 on the odd sublattice the operator ηx1 hasvanishing expectation value and therefore the whole expression ⟨ηx1ηx2 ⟩is zero if x1 is inthis region, independent of ηx2.

If, however, x1 is in a region of high density, then, accordingto our assumption, also x2 > x1 must be in region of high density. Thus, ηx1ηx2 again doesnot depend on x2 and takes the value 1.

We conclude that the product ηx1ηx2 is either 0 or1, depending on whether x1 is in a region of low or high density. This leads to the expression(29) for the expectation value of this product.The average value ⟨ηx ⟩itself contains the information about the position x0 of thedomain wall.

In the low density phase the density profile decays exponentially from aboveto its bulk value with increasing distance from the boundary. This means that the probabilityof finding the domain wall also decreases exponentially with the same decay length ξ with thedistance from the boundary.

The domain wall is caused by particles hitting the boundarywhere they get stuck with probability 1 −β and then cause other incoming particles topile up and create a region of high density (fig. 4 in the appendix).

On the other hand, inthe high density phase where the rate of injection is higher than the absorption rate thesituation is reversed. Here the probability of finding the domain wall decreases exponentiallywith the distance from the origin.

This can alternatively be explained either in terms ofholes through the particle-hole symmetry or in terms of particles being piled up from theboundary over the whole system up to a point close to the origin. On the phase transitionline injection and absorption are in balance and the probability of finding the domain wallis space-independent and the density profile is a linear superposition of the assumed stepfunction constituent profiles.

This leads to the observed linearly increasing average density(15).We conclude this section by studying the two-point function in more detail in the limit13

L →∞. We define the equal-time connected two-point function byGc(x1, x2; t = 0) = ⟨ηx1ηx2 ⟩−⟨ηx1 ⟩⟨ηx2 ⟩= ⟨τx1τx2 ⟩−⟨τx1 ⟩⟨τx2 ⟩(1 −α)m(1 −β)n.(31)where m = 2, n = 0 if both x1 and x2 are even, n = 2, m = 0 if both x1 and x2 are odd andm = n = 1 else.In what follows we restrict ourselves to the case where x1 and x2 are both odd, the mixedcorrelators can be computed analogously.

From (29) one obtains G(x1, x2; 0) = ⟨ηx1 ⟩(1 −⟨ηx2 ⟩) and inserting the expressions for the density profile (13) - (15) one obtains withx2 = x1 + 2r (r > 0)Gc(x1, x2; 0) =A(x2)e−2r/ξα < β˜A(x1)e−2r/ξα > βx1L (1 −x1L ) −x1LrL α = β. (32)The amplitudes of the exponential decay are given byA(x) = e−R/ξ 1 −e−R/ξ˜A(x) = 1 −α1 −β e−x/ξ 1 −1 −α1 −β e−x/ξ!

(33)where R = L + 1 −x measures the distance of site x from the boundary.The decay length ξ is identical with correlation length of the connected two-point func-tion. On the phase transition line the correlation function is constant for relative distances2r ≪L.Its amplitude depends on the position x1 in the bulk.A similar form of theconnected equal-time correlator was found in the p/p-model with a defect [1].V.

TIME-DEPENDENT CORRELATION FUNCTIONSIn this section we study the time-dependent two-point correlation function in the steadystateG(x1, x2; t) = ⟨ηx1T tηx2 ⟩(34)14

where T t denotes the t-th power of the transfer matrix T.‡ We define the direction of thetime evolution formally by T −tτx(t0)T t = τx(t0 + t), thus G(x1, x2; t) = ⟨ηx1(t0 + t)ηx2(t0) ⟩.The connected two-point function is defined by Gc(x1, x2; t) = G(x1, x2; t) −⟨ηx1 ⟩⟨ηx2 ⟩.The standard way of computing the correlation function (34) would be the insertion ofa complete set of eigenstates of T, evaluating the matrix elements ak(x1) = ⟨1 |ηx1| Λk ⟩and˜ak(x2) = ⟨Λk |ηx2| 1 ⟩and summing over ak˜akΛtk. Since we do not know the eigenstates andeigenvalues we take the alternative route using the commutator of [ ηx, T t ].

Since ⟨1 | is aleft eigenvector of T with eigenvalue 1 one has ⟨[ ηx, T t ]ηx2 ⟩= ⟨1 |(ηx1T t −T tηx1)ηx2| 1 ⟩=Gc(x1, x2; t). From this one obtains G(x1, x2; t).First we note that from the commutation relations (A7) of τx with T one obtainsτ2x−1 T = T (1 −σ2x−2σ2x−1)τ2xτ2x+1τ2x T = T (1 −σ2x−2σ2x−1(1 −τ2xτ2x+1)) .

(35)It is obvious that evaluating τx T t is not an easy task. By iterating relations (35) t timesnot only the number of terms in the products on the r.h.s.

but also the total number ofsuch multi-point correlators increases extremely fast with t. It is only the simplicity of themulti-point correlators (see eqs. (29) and (30)) that makes this approach promising.

Werestrict our discussion again to both x1 = 2y1 −1 and x2 = 2y2 −1 odd.By iterating (35) t times one finds that τ2y1−1T tτ2y2−1 is of the formτ2y1−1T tτ2y2−1 = T t {1 −(σ2y1−2tσ2y1−2t+1 . .

.) −(σ2y1−2t+2σ2y1−2t+3 .

. .

)−(. .

.) −(σ2y1−2t+2kσ2y1−2t+2k+1 .

. .) −(.

. .

)−σ2y1−4+2tσ2y1−3+2t} τ2y1−2+2tτ2y1−1+2tτ2y2−1(36)where the dots denote some complicated sums of products of operators τy acting on sites ybetween 2y1 −2t and 2y1 −1+2t. σy = 1−τy denotes the projector on holes and in order to‡Note that t = 1 corresponds to a distance of two lattice units in time direction in the underlyingvertex model (see appendix).15

avoid boundary effects one has to choose t < y1 −1. We first discuss the correlation functionoutside the light cone, then on the edges of the light cone and finally in its interior.A.

Correlation function outside the light coneWe want to evaluate ⟨τ2y1−1T tτ2y2−1 ⟩with 2y1 −1 ≥2y2 −1 + 2t. Recalling the fusionrule ⟨τ2x+1τ2y ⟩= ⟨τ2x+1 ⟩for y > x (27) one obtains ⟨τ2x+1σ2y ⟩= 0 for y > x. Sincethe fusion procedure is associative all terms on the r.h.s.

of (36) vanish when contractedwith τ2y2−1 except ⟨τ2y2−1τ2y1−2+2tτ2y1−1+2t ⟩= ⟨τ2y1−1−2tτ2y1−2+2tτ2y1−1+2t rangle .Usingalso ⟨τ2x−1τ2y−1 ⟩= (1 −β)⟨τ2x−1 ⟩for y > x one gets ⟨τ2y1−1−2tτ2y1−2+2tτ2y1−1+2t ⟩= (1 −β)⟨τ2y1−1−2t ⟩. Therefore we obtainG(x1, x2; t) = ⟨ηx2 ⟩(x1, x2 odd, x1 ≥x2 + 2t) .

(37)Now we study the correlator ⟨τ2y1−1T tτ2y2−1 ⟩with 2y1 −1 ≤2y2 −3 −2t. Here thefusion of τ2y1−1+2t in the r.h.s.

of (36) with τ2y2−1 yields (1 −β)τ2y1−1+2t and by taking theaverage value one obtains ⟨τ2y1−1T tτ2y2−1 ⟩= (1 −β)⟨τ2y1−1T t ⟩= (1 −β)⟨τ2y1−1 ⟩. We findG(x1, x2; t) = ⟨ηx1 ⟩(x1, x2 odd, x1 ≤x2 −2 −2t) .

(38)Eqs. (37) and (38) are no surprise.

The area defined by (37) and (38) is the exterior ofthe forward light cone of the particle at site x2. If x1 and x2 are chosen in this way andboth are in a region of uniform density (either in the bulk of the high density phase or inthe bulk of the low density phase) one has ηx1 = ηx2 = 1 or 0 and therefore the connectedcorrelation function Gc(x1, x2; t) is time-independent and 0 as one would expect.

In theboundary region of the low density phase where particles pile up and lead to a non-uniformdensity profile (or near the origin in the high density phase) it is still time-independent asit must be outside the light cone, but non-zero (see (32)). This is due to the hard-corerepulsion of the particles which behave as an incompressible liquid.16

B. Correlation function on the edges of the light coneOn the right edge of the light cone of the particle at site x2 defined by x1 = x2 −2 + 2t we can repeat the considerations that led to (37): All the pieces on the r.h.s.

of(36) containing τ2y2−1σ2y1−2t+2k = τ2y1+1−2tσ2y1−2t+2k with k ≥1 vanish as a result of thefusion rules and only the first two pieces in the sum remain. Although the term containingσ2y1−2tσ2y1+1−2t .

. .

τ2y1+1−2t does not vanish due to fusion with τ2y1+1−2t it is nevertheless 0since by definition σ2y1+1−2tτ2y1+1−2t = 0. ThereforeG(x2 −2 + 2t, x2; t) = ⟨ηx2 ⟩(x2 odd) .

(39)Consequently the connected correlation function on the odd sublattice vanishes also on theforward edge of the light cone if the two points are in a region of uniform density. This isa result of the asymmetry of the model: if the system is in a region of uniform low densityρ < 1/2 the odd sublattice is empty and the vanishing of the correlator is trivial.

In aregion of uniform high density the even sublattice is completely occupied and particles onthe odd sublattice effectively move only to the left (fig. 4 in the appendix) and are thereforeuncorrelated to particles on the right edge of their (forward) light cone.Due to the deterministic nature of the dynamics the particles on the odd sublatticemove with the velocity of light, i.e.

two lattice units per full time step as long as theyare in region of uniform high density.Thus we expect a singularity of the correlationfunction on the left edge of the light cone defined by x1 = x2 −2t: Indeed, choosing2y2 −1 = 2y1 −1 + 2t does not change the r.h.s. of (36) since τ 22y1−1+2t = τ2y1−1+2t andtherefore ⟨τ2y1−1T tτ2y1−1+2t ⟩= ⟨τ2y1−1T t ⟩= ⟨τ2y1−1 ⟩.

For the correlator (34) we obtainG(x2 −2t, x2; t) = (1 −β)−1⟨ηx2−2t ⟩(x2 odd) . (40)Here the connected correlation function in a region of uniform high density does not vanish.17

C. Correlation function inside the light coneFirst we note that in a region of uniform high density the result is again trivial. In sucha region particles on the odd sublattice are found everywhere with equal probability (theequal-time connected two-point function is 0) and since they move with the velocity of lightthe time-dependent connected two-point function does also vanish.If the profile is not uniform the calculation inside the light cone is non-trivial.

Withx1 = 2y1 −1 as above and x2 increasing beyond 2y1 + 2 −2t more and more contributionsfrom the r.h.s. of (36) are non-zero.

We evaluated G(x1, x2; t) for t = 1, 2, 3 inside the lightcone on the computer (using the software system Mathematica [14]) by calculating the exactform of (36) and then implementing the fusion rules (27) on the multi-point correlatorson the r.h.s. of (36).

First we noticed that in a region of uniform high (low) density (all⟨ηx ⟩=1(0)) one obtains G(x1, x2; t) = 1(0) and therefore Gc(x1, x2; t) = 0 as it should be.This observation is indeed a highly non-trivial test of the conjectured fusion rules (27) onwhich our calculation is based: Since eqs. (27) are supposed to be exact, the result of thecalculation of the time-dependent correlator must also be exactly 1 (0) if all ηx involved areset to 1 (0).

Any other result would have shown that the fusion rules do not hold. Secondlywe observed that by taking α = 1−β the exact general form of the correlator becomes fairlyobvious for arbitrary values of x1, x2 and t with x1 inside the light cone of x2.

We found bygeneralizing our result from t = 1, 2, 3 to arbitrary tG(x, x + 2y; t) =t+y−2Xk=0( 2t−2k!β2t−2−k(1 −β)k·(β⟨ηx+3−2t+2k ⟩+ (1 −β)⟨ηx+4−2t+2k ⟩)} +1 −t+y−2Xk=0 2t−2k!β2t−2−k(1 −β)k⟨ηx+2y ⟩. (41)This the main result of this section, valid for t < (x −1)/2 and −t + 2 ≤y ≤t −1.

Thefirst restriction is due to boundary effects, the second defines the interior of the light cone.In order to check this result we explicitly calculated G(x, x + 2y; t = 4) on the computer18

for arbitrary α and β using (36) and the fusion rules and found it in exact agreement withour conjecture when setting α = 1 −β. As a second, independent test we set ηx = 1(0)corresponding to the bulk value in the high density region (low density region) and indeedobtained Gc(x1, x2; 4) = 0 for arbitrary α and β.The choice α = 1 −β is not too restrictive as far as the physics is concerned: sincethis curve runs across the phase diagram it covers both the high density phase and the lowdensity phase and crosses the phase transition line at α = β = 1/2.

In what follows westudy G(x1, x2; t) in the low density phase along the curve β = 1 −α > 1/2 and on thephase transition line at β = 1/2.In the low density phase we focus on the boundary region with a non-uniform densityprofile. For β > 1/2, α = 1 −β the expression (23) for the density profile for large L gives⟨η2x−1 ⟩= ⟨η2x ⟩= ((1 −β)/β)L+2−2x and thereforeβ⟨ηx+3−2t+2k ⟩+ (1 −β)⟨ηx+4−2t+2k ⟩=1−ββL+1−x+2t−2k .

(42)Inserting this into (41) and introducing the incomplete β-functiont+y−2Xk=0 2t−2k!β2t−2−k(1 −β)k = Iβ(t −y, t + y −1)= 1 −I1−β(t + y, t −y) − 2t−2t−y−1!βt−y(1 −β)t+y−1(43)the correlation function (41) can conveniently be rewrittenG(x, x + 2y; t) =1−ββL−x I1−β(t −y, t + y) +β1−β2y−1 I1−β(t + y, t −y). (44)For large times t (such that |y|/t ≪1) the incomplete β-function has the asymptoticformI1−β(t + y, t −y) =(1 −34ξt) e1/ξtsξt4πt e−(y/ξr+t/ξt) e−y2/tβ > 12P( y√t) −stπξte−y2/tβ = 12 + (2ξt)−1/2(45)with19

ξ−1t= −ln (4β(1 −β)),ξ−1r= −ln 1−ββ. (46)and the probability integral P(u) = 1/√2πR u−∞exp (−t2/2)dt.

In terms of ξt the inequalityβ > 1/2 in the upper expression of the r.h.s. of (45) has to be understood as 1 ≪ξt<∼t.

Inthe lower expression we assume 1 ≪t<∼ξt. Note that the two length scales ξt and ξr are notindependent quantities but related through ξ−1t= ln cosh2(ξ−1r /2).

As β approaches 1/2, ξtand ξr diverge and are asymptotically related through ξt ≈4ξ2r.We define ξ = ξr and r = |y| = 1/2|x2 −x1| and insert (45) into (44). This gives thescaling form of the time-dependent correlation function in the scaling region of large 2ξ<∼t1/2G(x1, x2; t) = e−R/ξ e−r/ξs4ξ2πt e−t/(4ξ2) e−r2/t(47)whereR =L + 1 −x2 if 2y = x2 −x1 > 0L + 1 −x1 if 2y = x2 −x1 < 0(48)measures the distances of x2 or x1 from the boundary, depending on the sign of x2 −x1.G(x1, x2; t) is invariant under the scaling transformationR →λR,r →λr,ξ →λξt →λ2t.

(49)This is of the form corresponding to dynamical scaling with a dynamic critical exponentz = 2 and critical exponent x = 0.If ξ increases beyond the crossover length scale t1/2 the correlation function changes itsform. At the critical point β = 1/2 we have ⟨ηx ⟩= x/L and up to corrections of order 1/Lthe exact expression (41) for the correlation function givesG(x, x + 2y; t) = x + 2yL−2t + 2yLI 12(t −y, t + y −1)+2L122t−2 t+y−2Xk=0k 2t−2k!.

(50)20

Using2t+y−2Xk=0k 2t−2k!β2t−2−k(1 −β)k = 2(1 −β)(2t −2)Iβ(t −y, t + y −1)−2(t + y −1) 2t−2t−y−1!βt−y(1 −β)t+y−1(51)and the expansion (45) of I 12(t + y, t −y) we obtain with 2y = x2 −x1 and u = y/√tG(x1, x2; t) = x1 + x22L−tL122t−2 2t−2t−y−1!−yL1 −2I 12(t −y, t + y)≈x1 + x22L−√t√πLe−u2 + √πu(1 −2P(u)). .

(52)For finite distances y, t one has (up to corrections of order L−1 which we neglect) x1 = x2 = xand the correlation function at the critical point is space and time independent with anamplitude x/L depending on the relative position of x in the bulk. For large times, t ∝Land y/t ≪1, the correlation function gets an contribution order L−1/2.G(x1, x2; t) isinvariant under the scale transformations (49) with the length scale ξ replaced by the sizeof the system L.This result has a simple interpretation in terms of the constituent profiles discussed inthe preceding section.

For simplicity we consider y = 0. The operator ηxT tηx gives 1 ifx is in a region of high density both at times t0 and t0 + t. The probability that x is inthe high density region at time t0 is x/L.

This accounts for the constant x/L in (52) If weassume that the domain wall separating the low density region from the high density regionperforms a random walk of two lattice units per time step starting from its position x0 attime t0 then the resulting expectation value ⟨ηxT tηx ⟩will indeed be of the form (52).VI. SUMMARY AND COMPARISON WITH OTHER EXCLUSION MODELSLet us summarize our main results.

We obtained recursion rules (14) which allow theconstruction of the steady state of the model defined by (1) and (2). Moreover we found forarbitrary values α and β of the injection and absorption rates exact steady state expressions21

for the density-profile (18), the current (19), the equal-time n-point density correlationfunction (30) and the time dependent two-point function eqs. (37) - (41).§We made the following observations:(a) The phase diagram (fig.

1) shows two phases. In the low density phase (α < β) theaverage density is ρ = α/2 (in the limit L →∞) and in the high density phase one hasρ = 1 −β/2.

On the phase transition line α = β the average density is ρ = 1/2. (b) There is a shift in the average densities between the even sublattice and the odd sublat-tice (see eqs.

(24) - (26)). This shift is the current j.

In terms of the total average densityρ one finds j = 2ρ in the low density phase and j = 2(1 −ρ) in the high density phase. Onthe phase transition line (ρ = 1/2) the current is j = α (i.e.

j ̸= 1 = 2ρ = 2(1 −ρ)). (c) In the low density phase the density profile decays on both sublattices to its respectivebulk value exponentially with increasing distance from the boundary while in the high den-sity phase it increases exponentially to its respective bulk value with increasing distancefrom the origin.

At the phase transition line α = β the length scale ξ (23) associated withthe exponential shape of the profile diverges and the profile increases linearly on both sub-lattices. The average density and the first derivatives of the current w.r.t.

α and β have adiscontinuity at the phase transition line (in the thermodynamic limit L →∞). (d) The decay length ξ is identical with the correlation length of the connected two-pointdensity correlation function.

Outside the light cone this correlator is of the scaling formG(x1, x2; t = 0) = Arκ exp (−r/ξ) with exponent κ = 0.The amplitude A is space-dependent as a result of breaking of translational invariance (32) and (33). (e) The time-dependent two-point correlation function G(x1, x2; t) inside the light cone near the criticalline (47) has a form compatible with dynamical scaling with dynamic critical exponent z = 2and scaling dimension x = 0 (see the transformations (49)).

It contains the exponential§The expression (41) for the time-dependent two-point function inside the light cone was foundonly for α = 1 −β.22

exp (−r2/t) characteristic for local dynamical scaling, but the amplitude is again space-dependent due to breaking of translational invariance. The correlation function changes itsform when the correlation length increases beyond a crossover length scale of order t1/2.

Onthe critical line it becomes a space-dependent constant up to corrections of order L−1 if t islarge but finite, and up to corrections of order L−1/2 for times of order L.(f) From an analysis of the two-point correlation function we found that the density pro-file can be considered as a superposition of step-function type profiles with average den-sity ρ(even)1= α and ρ(odd)1= 0 up to some point x0 and average density ρ(even)2= 1 andρ(odd)2= 1 −β beyond this point up to the boundary. These densities are the sublatticedensities in the low density phase and high density phase respectively.

The probability offinding the “domain wall” at site x0 separating the two regions of high density and lowdensity decreases exponentially with increasing distance from the boundary (origin) in thelow density (high density) phase. On the phase transition line this probability is space inde-pendent and the domain wall can be found everywhere with the same probability.

Studyingthe time-dependent correlation function suggests that it performs a random walk around itsposition x0 at time t0.We conclude our discussion of the deterministic p/o-model with a brief comparison withother exclusion models and some conjectures for probabilistic exclusion models with paralleldynamics.In the deterministic p/p-model with a defect we found a phase diagram showing a lowdensity phase, a coexistence phase where a low density density region coexists with a highdensity region and a high density phase. The relation between the current and the sublatticedensities and the total density discussed in (b) for the low and high density phases is identicalwith that in the p/p-model in the respective phases.

This is a consequence of the bulkdynamics which are identical in both cases and could have been guessed.It is interesting to observe that there is also a correspondence between the coexistencephase of the p/p-model and the phase transition line here: In the p/p-model with defectstrength 1−q the quantity q is the hopping probability at a single link of the ring, say between23

sites L and 1 and therefore corresponds to an absorption of particles at site L and injectionof particles at site 1 with rate q. This injection and absorption is correlated because of theparticle number conservation.

In the coexistence phase the current is density-independentand one has j = q. In the p/o-model on the phase transition line discussed here particles arealso injected and absorbed with the same rate α, but uncorrelated.

The current is density-independent, j = α as in the p/p-model. The profile is build by constituent profiles with aregion of low density up to some x0 and a high density region beyond that point.

x0 can beanywhere with same probability. This suggests that also the profile in the coexistence phaseof the p/p-model with defect is build by such constituent profiles, with the distinction thatthere x0 cannot be anywhere (because of particle number conservation) but the probabilityp(x0) of finding the domain wall is centered around some point R0.

We believe that similarphases occur also in probabilistic p/p-models with a defect and in probabilistic p/o-models.In such models particles do not always move if the neighbouring site was empty as here butcan stay with some non-zero probability even in the bulk. For mixed models (defect andsome uncorrelated injection and absorption) we expect correspondingly a softening of thedistribution p(x0).All the features summarized under (c) (except the implied anisotropy between the evenand odd sublattices) are in common with the phase transition from the low density phaseAI to the high density phase BI in the s/o-model [5].

This suggests that also the correlationfunctions are qualitatively the same (in the thermodynamic limit L →∞) near the phasetransition line. In the s/o-model however, the transition line α = β extends only up toα = β = 1/2 as opposed to α = β = 1 here.Correspondingly, in our model there isno maximal current phase with a power behaviour of the density profile and no phasescorresponding to the phases AII or BII (for α > 1/2 or β > 1/2) of [5] where the shape ofthe density profile is determined by a product of a power law behaviour with an exponentialdecay.We believe that phase transitions to such phases cannot occur in our model because ofthe deterministic nature of the dynamics.

In [5] we argued that these phases result from24

an “overfeeding” of the system with particles: The system reaches its maximal transportcapacity at density 1/2. In order to obtain average density 1/2 at the origin particles haveto be injected with rate 1/2.

If particles are injected at a higher rate they block each otherrather than moving into the bulk and cause a phase transition. Here this cannot happen.The system reaches its maximal transport capacity also at average density 1/2 but here thiscorresponds to a completely filled even sublattice and an empty odd sublattice.

An averagedensity of 1 on the even sublattice at the origin can only occur if particles are injected atrate 1. Thus there can be no “overfeeding”.

The deterministic hopping rules imply thatparticles injected at the origin move away with velocity of light, therefore there can be nomutual blockage near the origin. (Similar arguments can be used for a discussion of thedependence of the phase transitions on the absorption rate β by exchanging particles withholes and studying the injection of holes at the boundary).This discussion naturally leads to the conjecture that the probabilistic p/o-model (whichhas not yet been studied) will have a phase diagram similar to that of the s/o-model witha phase transition line at α = β up to some value α0 where the profile is constant on bothsublattices and additional phase transition lines at the values of α and β corresponding toan overfeeding of particles at the origin and holes at the boundary respectively.AcknowledgmentsIt is a pleasure to thank E. Domany and D. Mukamel for interesting dicussions.

Financialsupport by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.APPENDIX A: MAPPING TO A TWO-DIMENSIONAL VERTEX MODELFollowing the idea of ref. [6] we want to show how the exclusion process defined byrelations (1) and (2) is related to a two-dimensional vertex model.

The discussion is partlysimilarly to that in [1] which we repeat for the convenience of the reader unfamiliar with25

this mapping. The mapping of the boundary conditions to vertices in the vertex model isdifferent from [1].Consider a 4-vertex model on a diagonal square lattice defined as follows: Place an up-or down-pointing arrow on each link of the lattice and assign a non-zero Boltzmann weightto each of the vertices shown in figure 1.

(All other configurations of arrows around anintersection of two lines, i.e., all other vertices, are forbidden in the bulk.) The partitionfunction is the sum of the products of Boltzmann weights of a lattice configuration takenover all allowed configurations.❅❅❅❅❅✒✒■■❅❅❅❅❅✠✠❘❘❅❅❅❅❅✒✒❘❘❅❅❅❅❅✠✒❘■a1a2b2c2FIG.

2. Allowed bulk vertex configurations in the four-vertex model.

Up-pointing arrows corre-spond to particles, down-pointing arrows represent vacant sites. In the dynamical interpretationof the model the Boltzmann weights give the transition probability of the state represented by thepair of arrows below the vertex to that above the vertex.In the transfer matrix formalism up- and down-pointing arrows in each row of a diagonalsquare lattice built by M of these vertices represent the state of the system at some giventime t. Corresponding to the M vertices there are L = 2M sites in each row representedby the links of the diagonal lattice.

The configuration of arrows in the next row above(represented by the upper arrows of the same vertices) then corresponds to the state ofthe system at an intermediate time t′ = t + 1/2, and the configuration after a full timestep t′′ = t + 1 corresponds to the arrangement of arrows two rows above. Therefore eachvertex represents a local transition from the state given by the lower two arrows of a vertexrepresenting the configuration on sites j and j + 1 at time t to the state defined by theupper two arrows representing the configuration at sites j and j + 1 at time t + 1/2.

The26

correspondence of the vertex language to the particle picture used in the introduction canbe understood by considering up-pointing arrows as particles occupying the respective sitesof the chain while down-pointing arrows represent vacant sites, i.e., holes.The diagonal-to-diagonal transfer matrix T acting on a chain of L sites (L even) of thevertex model with vertex weights a1, . .

. , c1 as shown in fig.

2 is then defined by [1,15]T =L/2Yj=1T2j−1 ·L/2Yj=1T2j = T odd T even . (A1)The matrices Tj act nontrivially on sites j and j + 1 in the chain, on all other sites theyact as unit operator.

All matrices Tj and Tj′ with |j −j′| ̸= 1 commute. (The differencej −j′ is understood to be mod L).

For an explicit representation of the transfer matrixwe choose a spin-1/2 tensor basis where the Pauli-matrix σzj acting on site j of the chain isdiagonal and spin down at site j represents a particle (up-pointing arrow) and spin up a hole(down-pointing arrow). In this basis τj = 12(1 −σzj ) is the projection operator on particleson site j, σj = 12(1 + σzj ) is the projector on holes and s±j = 12(σxj ± iσyj ) (σx,y,z being thePauli matrices) create (s−j ) and annihilate (s+j ) particles respectively.The bulk dynamics of our model is encoded in the transfer matrix by choosing the vertexweights asa1 = a2 = b2 = c1 = 1(A2)In the bulk this leads toTj = 1 + s+j s−j+1 −τjσj+1 =1000011 0000 0000 1j,j+1.

(A3)In the particle language the matrices Tj describe the local transition probabilities of particlesmoving from site j to site j + 1 represented by the corresponding vertices. If sites j andj + 1 are both empty or occupied, they remain as they are under the action of Tj.

The sameholds for a hole on site j and a particle on site j +1, corresponding to the diagonal elements27

of Tj, representing vertices a1, a2 and c1. If there is a particle on site j and a hole on sitej + 1, the particle will move with probability one to site j + 1.

This accounts for vertex b2.As discussed in the introduction we assume open boundary conditions with injection ofparticles on site 1 and absorption of particles on site L. This allows for the additional verticesshown in fig. 3 together with vertex weights corresponding to the respective probabilities ofcreating and annihilating particles.❅❅❅❅❅✒✒■■❅❅❅❅❅✠✒❘❘❅❅❅❅❅✒✠❘❘❅❅❅❅❅✒✒❘■❅❅❅❅❅✒✒❘❘βαβ(1 −α)α(1 −β)(1 −α)(1 −β)FIG.

3. Additional vertex configurations allowed at the boundary and their Boltzmann weights.The left arrows of these vertices describe the particle configuration at the boundary site L of thesystem while the right arrows define the particle configurations at the origin (site 1).In a two-dimensional lattice (fig.

4) we consider the half-vertices at the left boundary asthe right arms of the vertices shown in fig. 2 and fig.

3 and the half-vertices at the rightboundary as their left arms. Thus the left arrows define the particle configuration on site Land the right arrows are considered as site 1.

Vertices a1, a2 and b2 have a different weightat the boundary: a′1 = 1 −β, a′2 = 1 −α, b′2 = αβ. Note that vertex b2 at the boundarydescribes simultaneous absorption of a particle at site L and creation of a particle at site 1.28

❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒■■■■■■■■■■■■12345678910 11 12012345FIG. 4.

Configuration of particles (up-pointing arrows) on a lattice of length L = 12 in space(horizontal) direction M = 2t = 12 between times t = 0 and t = 5 + 1/2 (vertical direction).Down-pointing arrows denoting vacant sites have been omitted from the drawing. At time t = 0the even sublattice is filled and the odd sublattice empty.

Particles are injected at site 1 after timest = 0 and t = 4. At the boundary (site 12) particles get stuck at times t = 1 and t = 2 and areabsorbed at times t = 0, 3, 4, 5.With this convention TL(α, β) acting on sites L and 1 corresponding to the vertex weightsshown in fig.

3 is given byTL(α, β) = 1 + α(s−1 −σ1) + β(s+L −τL) + αβ(s+L −τL)(s−1 −σ1)=1 −α 0β(1 −α)0α1αββ00 (1 −α)(1 −β)000α(1 −β)1 −βL,1. (A4)The transfer matrix T = T(α, β) acts parallel first on all even-odd pairs of sites (2j, 2j+1)29

including the boundary pair (L,1), then on all odd-even pairs. Thus in the first half timestep T even shifts particles from the even sublattice to the odd sublattice (so far it was notoccupied) and then, in the second half step, T odd moves particles from the odd sublattice tothe even sublattice again.

As a result, we expect an asymmetry in the average occupationof the even and odd sublattice which is related to the particle current. In a model withtransfer matrix ˜T = T oddT even the asymmetry will be reversed, but there will be no essentialdifference in the physical properties of these two systems.A possible configuration of particles in a 12x12 lattice is shown fig.

4. Note that thepresence of particles at site x = 11 and times t = 2, 3 imply the existence of particles on theleft edge of their light cones as long as they move in a region where the even sublattice isfully occupied, i.e.

they move with velocity of light (two lattice units per time step) to theleft. A particle on an even lattice site at some (integer) time t always implies the existenceof a particle on the right edge of its light cone up to the boundary.The model has a particle hole symmetry.

We denote by |x1, x2, . .

. , xN⟩= s−x1s−x2 .

. .

s−xN| ⟩the N-particle state with particles on sites x1, . .

. , xN (| ⟩is the state with all spins upcorresponding to no particle).

The parity operator P reflects particles with respect to thecenter of the chain located between sites x = L/2 and x = L/2+1 and the charge conjugationoperator C = QLj=1 σxj interchanges particles and holes and therefore turns a N-particle stateinto a state with L −N particles. One finds(CP) T(α, β) (CP) = T(β, α) .

(A5)In the bulk the particle current is conserved and can be obtained from the commutatorsof τ2x and τ2x−1 with T. These relations play a crucial role in the construction of the steadystate and the computation of the time-dependent correlation function. Defining the currentoperators jeven2xand jodd2x−1 byjeven2x= τ2xσ2x+1(1 ≤x ≤L/2 −1)jodd2x−1 = (1 −σ2x−2σ2x−1)(1 −τ2xτ2x+1) (2 ≤x ≤L/2 −1)(A6)30

a straightforward calculation yields (x ̸= L/2):[T, τ2x−1] = T (τ2x−1 −(1 −σ2x−2σ2x−1)τ2xτ2x+1)= jodd2x−1 −jeven2x−2[T, τ2x]= T (σ2x−2σ2x−1(1 −τ2xτ2x+1) −σ2x)= jeven2x−jodd2x−1 . (A7)Current conservation implies that the expectation values of the current operators jeven2xand jodd2x−1 do not depend on x, ⟨jeven2x ⟩= ⟨jodd2x−1⟩= const = j.Note that the cases α, β = 0, 1 are trivial.

If α = 0 no particles are injected and thesteady state is | ⟩. If α = 1 then in each time step a particle is injected and therefore theeven sublattice fully occupied.

Particles on the odd sublattice are randomly distributedwith average density 1 −β. As discussed above they move with velocity of light everywhere.Therefore the connected time-dependent two-point function on the odd sublattice is 0 excepton the left edge of the (forward) light cone.31

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