Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Spectrum of the totally asymmetric simple exclusionprocess on a periodic lattice - bulk eigenvalues
Sylvain Prolhac
Laboratoire de Physique Th´eorique, IRSAMC, UPS, Universit´e de Toulouse, FranceLaboratoire de Physique Th´eorique, UMR 5152, Toulouse, CNRS, France
Abstract.
We consider the totally asymmetric simple exclusion process (TASEP)on a periodic one-dimensional lattice of L sites. Using Bethe ansatz, we derive para-metric formulas for the eigenvalues of its generator in the thermodynamic limit. Thisallows to study the curve delimiting the edge of the spectrum in the complex plane.A functional integration over the eigenstates leads to an expression for the density ofeigenvalues in the bulk of the spectrum. The density vanishes with an exponent 2 / PACS numbers:
Keywords:
TASEP, non-Hermitian operator, complex spectrum, Bethe ansatz,functional integration pectrum of periodic TASEP - bulk eigenvalues
1. Introduction
Markov processes [1] form a class of mathematical models much studied in relation withnon-equilibrium statistical physics. Their evolution in time is generated by an operator M , the Markov matrix, whose non-diagonal entries represent the rates at which thestate of the system changes from a given microstate to another.For processes verifying the detailed balance condition, which forbids probabilitycurrents between the different microstates at equilibrium, the operator M is realsymmetric, up to a similarity transformation. It implies that its eigenvalues are realnumbers. An example is the Ising model with e.g. Glauber dynamics. Processes thatdo not satisfy detailed balance, on the other hand, generally have a complex spectrum.This is the case for the asymmetric simple exclusion process (ASEP) [2, 3, 4, 5, 6, 7, 8, 9],which consists of classical hard-core particles hopping between nearest neighbour sitesof a lattice with a preferred direction.In one dimension, ASEP is known to be exactly solvable by means of Bethe ansatz.This has allowed exact calculations of the gap of the spectrum [10, 11, 12, 13, 14] and ofthe fluctuations of the current, both in the infinite line setting [15, 16, 17, 18, 19, 20] andon a finite lattice with either periodic [21, 22, 23, 24, 25, 26] or open [27, 28, 29] boundaryconditions. The exponents and scaling functions obtained in these articles are universal:they characterize not only driven diffusive systems [30, 31] far from equilibrium, to whichASEP belongs, but also interface growth models [32, 33, 34] and directed polymers in arandom medium [34, 35, 36, 37]. This forms the Kardar-Parisi-Zhang universality class[38, 39, 40].We focus in this article on the special case of ASEP with unidirectional hopping ofthe particles called totally asymmetric simple exclusion process (TASEP). We considerthe system with periodic boundary conditions, for which the number of microstates isfinite: the spectrum is then a discrete set of points in the complex plane. The aim of thepresent article is to obtain a large scale description of these points. We obtain explicitexpressions for the curve delimiting the edge of the spectrum in the complex plane.This curve is singular near the eigenvalue 0, with an imaginary part scaling as the realpart to the power 5 /
3. By a functional integration over the eigenstates, we also deriveexpressions for the density of eigenvalues. Near the eigenvalue 0, the density vanisheswith an exponent 2 / pectrum of periodic TASEP - bulk eigenvalues Figure 1.
Totally asymmetric simple exclusion process with N = 3 particles on aperiodic one-dimensional lattice with L = 16 sites. The particles hop to the nearestsite in the clockwise direction with rate 1 provided that site is empty.
2. Totally asymmetric exclusion process on a ring
We consider in this article the totally asymmetric simple exclusion process (TASEP)with N particles on a periodic one-dimensional lattice of L sites (see fig. 1). A siteis either empty or occupied by one particle. We call Ω the set of all microstates (or configurations ), which has cardinal | Ω | = (cid:0) LN (cid:1) . A particle at site i can hop to site i + 1if the latter is empty. The hopping rate for any particle is equal to 1, i.e. each particleallowed to move has a probability δt in a small time interval δt . In the following, wecall ρ = N/L the density of particles.
We call P t ( C ) the probability to observe the system in the microstate C at time t . Theprobabilities evolve in time by the master equation dP t ( C ) dt = X C ′ = C h w ( C ← C ′ ) P t ( C ′ ) − w ( C ′ ← C ) P t ( C ) i . (1)The rate w ( C ′ ← C ) is equal to 1 if it is possible to go from configuration C toconfiguration C ′ by moving one particle to the next site, and 0 otherwise.The master equation (1) can be conveniently written as a matrix equation bydefining the vector | P t i = P C∈ Ω P t ( C ) |Ci of the configuration space V with dimension | Ω | , where |Ci is the canonical vector of V corresponding to the configuration C . Then,calling M the matrix with non-diagonal entries hC| M |C ′ i = w ( C ← C ′ ) and diagonal hC| M |Ci = − P C ′ = C w ( C ′ ← C ), one has ddt | P t i = M | P t i , (2)which is formally solved in terms of the time evolution operator e tM as | P t i = e tM | P i .The graph of allowed transitions for TASEP presents an interesting cyclic structurewith period L : let us consider the observable X such that X ( C ) is the sum of the pectrum of periodic TASEP - bulk eigenvalues Figure 2.
Graph of all allowed transitions between the configurations of TASEP with N = 2 particles on a periodic lattice of L = 6 sites. positions of the particles (we take a fixed arbitrary site to be the origin of positions).We see that each time a particle hops to the next site, X ( C ) increases of 1 modulo L . It is then possible to split the configurations in L sectors according to the value of X modulo L . The only allowed transitions between configurations are then transitionsfrom configurations of a sector r to configurations of sector r + 1 (modulo L ), see fig. 2.We emphasize that this cyclic structure is not a consequence of the periodic boundaryconditions. Indeed, a similar cyclic structure exists for ASEP on an open segment of L sites connected to reservoirs of particles, with periodicity L + 1 instead of L . Equal time observables, such as the density profile of particles in the system or thenumber of clusters of consecutive particles can be extracted directly from the knowledgeof | P t i . Other observables, however, require the P t ( C ) for several values of the time. Amuch studied example is the current of particles and especially the fluctuations aroundits mean value.We define the observable Y t , which counts the total displacement of particles pectrum of periodic TASEP - bulk eigenvalues t . Starting with Y = 0, it is then updated by Y t → Y t + 1each time a particle hops anywhere in the system.The joint probability P t ( C , Y ) to observe the system in the configuration C with Y t = Y obeys the master equation dP t ( C , Y ) dt = X C ′ = C h w ( C ← C ′ ) P t ( C ′ , Y − − w ( C ′ ← C ) P t ( C , Y ) i . (3)It is convenient to introduce the quantity F t ( C ) = P ∞ Y = −∞ e γY P t ( C , Y ). It verifies thedeformed master equation [21] dF t ( C ) dt = X C ′ = C h e γ w ( C ← C ′ ) F t ( C ′ ) − w ( C ′ ← C ) F t ( C ) i . (4)Introducing the vector | F t i = P C∈ Ω F t ( C ) |Ci and a deformation M ( γ ) of the Markovmatrix, one has ddt | F t i = M ( γ ) | F t i . (5)For γ = 0, F t reduces to P t and M (0) = M . In the following, we will be interested inthe spectrum of the operators M and M ( γ ). It is well known that TASEP can be mapped to a model of growing interface [34]: foreach occupied site i of the system, we draw a portion of interface decreasing from height h i to h i +1 = h i − (1 − ρ ), and for each empty site i , we draw a portion of interfaceincreasing from height h i to h i +1 = h i + ρ . The interface obtained is continuous andperiodic, see fig. 3. The dynamics of TASEP then implies that parallelograms (squaresat half-filling N = L/
2) deposit on local minima of the interface with rate 1.Unlike TASEP, the set of microstates of the growth model is not finite since the totalheight is not bounded: there always exists a local minimum from which the interface cangrow. It is possible to identify any microstate of the growth model by the correspondingconfiguration of the exclusion process and the total current Y defined in section 2.2.Then, (3) can be interpreted as the master equation for the growth model, to which isassociated the infinite dimensional Markov matrix M = M (+) ⊗ S + M (0) ⊗ , (6)where M (0) and M (+) are respectively the diagonal and non-diagonal part of M , and S is the translation operator in Y space, S = P ∞ Y = −∞ | Y ih Y − | .We would like to diagonalize M in order to study its spectrum. We note that M commutes with S . This implies that the eigenvectors | Ψ i of M must be of the form | Ψ i = | ψ i ⊗ | φ ( θ ) i , (7)where | ψ i is a vector in configuration space and | φ ( θ ) i = 1 √ π ∞ X Y = −∞ e − i θ Y | Y i (8) pectrum of periodic TASEP - bulk eigenvalues S with eigenvalue e i θ . The eigenvalue equation for | Ψ i can bewritten E | Ψ i = [(e i θ M (+) + M (0) ) | ψ i ] ⊗ | φ ( θ ) i = [ M (i θ ) | ψ i ] ⊗ | φ ( θ ) i , (9)where M (i θ ) is the deformation of the Markov matrix introduced in section 2.2 and | ψ i and E an eigenvector and an eigenvalue of M (i θ ). We finally find that the spectrum of M is the reunion of the spectra of the M ( γ ) with | e γ | = 1. This spectrum is representedin fig. 4 for L = 8, N = 4.The spectrum of M can also be constructed from the finite spectra obtained bycounting the total current Y modulo KL with K a positive integer, and taking thelimit K → ∞ . The case K = 1 is the usual TASEP because of the cyclic structure ofthe graph of allowed transitions discussed at the end of section 2.1. The case K = N corresponds to TASEP with distinguishable particles, restricted to a subspace with agiven cyclic order of the particles since the particles cannot overtake each other.We call M ( K ) ( γ ) the corresponding deformed Markov matrix. The K | Ω | configurations arrange themselves in KL sectors according to the value of Y . Calling P r the projector on the r -th sector and M ( K ) r,r +1 ( γ ) = e γ P r +1 M ( K ) P r + P r M ( K ) P r , one has M ( K ) ( γ ) = KL X r =1 M ( K ) r,r +1 ( γ ) . (10)Introducing U = P KLr =1 e rγ P r , one finds U − M ( K ) ( γ ) U = KL − X r =1 M ( K ) r,r +1 + e KLγ P M ( K ) KL, P KL + P KL M ( K ) KL, P KL , (11)with M ( K ) r,r +1 = M ( K ) r,r +1 (0). This implies that the spectrum of M ( K ) ( γ ) is invariant underthe transformation γ → γ + 2i π/ ( KL ). Starting from a given eigenvalue of M ( K ) ( γ )and following the eigenvalue during the continuous change from γ to γ + 2i π/ ( KL ),one does not in general come back to the initial eigenvalue: we observe by numericaldiagonalization that one goes from one eigenvalue to the next one anticlockwise on thesame continuous curve in fig. 4. Typical eigenvalues of the Markov matrix M scale proportionally with L . Dividing allthe eigenvalues by L , the rescaled spectrum fills a region of the complex plane in thethermodynamic limit. We call e ( µ ) a parametrization of the curve at the edge of thisregion. An exact representation of this curve, (54), (55), is obtained in section 4. Weobserve that e ( µ ) is singular near e = 0, with the scalingRe e ≃ − / / π / | Im e | / . (12)The density of eigenvalues D ( e ) in the rescaled spectrum is also studied in section 5. Itgrows for large L as D ( e ) ∼ e Ls ( e ) . (13) pectrum of periodic TASEP - bulk eigenvalues .............................. ........................................ .......... Figure 3.
Mapping between periodic TASEP and an interface growth model at half-filling. - - - - - - - Figure 4.
Spectrum of the Markov matrix of TASEP with 4 particles on 8 sites. Theblack dots are the eigenvalues for undistinguishable particles, while the black + red(gray in printed version) dots are the eigenvalues for distinguishable particles. Thecurves are the eigenvalues of the corresponding interface growth model.
The quantity s ( e ) is computed exactly. For eigenvalues close to 0, but far from the edgeof the spectrum ( i.e. with | Im e | / ≪ | Re e | ), one finds in particular s ( e ) ≃ ξ ( − Re e ) / with ξ ≃ . . (14)The exponent 2 / / pectrum of periodic TASEP - bulk eigenvalues We consider the quantity Q ( t ) = 1 | Ω | tr e tM = 1 | Ω | X C∈ Ω hC| e tM |Ci . (15)Here hC| e tM |Ci is the probability that the system is in the microstate C at time t conditioned on the fact it was already in the microstate C at time 0. Since allconfigurations are equally probable in the stationary state of periodic TASEP [2], Q ( t )is simply the stationary probability that the system is in the same microstate at bothtimes 0 and t . We consider more generally (15) with M replaced by the deformation M ( γ ), anddefine f ( t ) = 1 L log tr e tM ( γ ) . (16)The quantity f ( t ) can be seen as the generating function of the cumulants of theeigenvalues of M ( γ ) (or more precisely the cumulants of the uniform probabilitydistribution on the set of the eigenvalues). The moments µ k = | Ω | − L − k tr M ( γ ) k andthe cumulants c k = L − k f ( k ) (0) of the eigenvalues are indeed related from (16) bylog (cid:16) ∞ X k =1 µ k t k k ! (cid:17) = ∞ X k =1 c k t k k ! . (17)The first cumulants are c = µ = f ′ (0), c = µ − µ = f ′′ (0) /L and c = µ − µ µ + 2 µ = f ′′′ (0) /L The moments and cumulants of the eigenvalues are independent of the deformation γ . Indeed, one can writetr M ( γ ) k = X C ,..., C k e γ P kj =1 {C j = C j +1 } hC | M |C ihC | M |C i . . . hC k | M |C i . (18)Because of the cyclic structure of allowed transitions explained at the end of section 2.1,only k -tuples of configurations such that P kj =1 {C j = C j +1 } is divisible by L contribute to(18). For k < L , it implies that tr M ( γ ) k cannot depend on γ . One has then f ( t ) = (cid:16) L log tr e tM + O ( t L ) (cid:17) . (19)It is possible to calculate directly the coefficients of the expansion near t = 0 of f ( t )by considering the case γ → −∞ , for which the matrix M ( γ ) becomes diagonal inconfiguration basis: hC| M ( γ ) |Ci is equal to minus the number m ( C ) of clusters ofconsecutive particles in the system. It implies f ( t ) = 1 L X C∈ Ω e − tm ( C ) = 1 L N X m =1 | Ω( m ) | e − tm . (20) pectrum of periodic TASEP - bulk eigenvalues | Ω( m ) | of configurations with m clusters can be calculated in thefollowing way: a configuration with m clusters for which the last site is occupied can bedescribed as a (0) ≥ b (1) > a (1) > b ( m ) > a ( m ) > A m +1 ( N + 1) A m ( L − N ) with A m ( r ) = ∞ X b (1) ,...,b ( m )=1 { b (1)+ ... + b ( m )= r } = I d z π z m − r − (1 − z ) m = (cid:18) r − m − (cid:19) . (21)From particle-hole symmetry, the number of configurations with m clusters for whichthe last site is empty is A m +1 ( L − N + 1) A m ( N ) This implies | Ω( m ) | = mLN ( L − N ) (cid:18) Nm (cid:19)(cid:18) L − Nm (cid:19) . (22)A saddle point approximation of the sum over m in (20) finally gives f ( t ) = ρ log (cid:16) (1 − ρ ) + 2 ρ e t + p ρ (1 − ρ )(e t − ρ e t (cid:17) + (1 − ρ ) log (cid:16) − (1 − ρ ) + 2(1 − ρ )e t + p ρ (1 − ρ )(e t − − ρ )e t (cid:17) , (23)which simplifies at half-filling to f ( t ) = log(1 + e − t/ ) . (24)The expressions (23) and (24) must be understood as an equality between Taylor series.In section 6, we consider again the quantity f ( t ) as a testing ground for the formulasderived from Bethe ansatz in section 3 for the eigenvalues of M in the thermodynamiclimit. We write the summation over the eigenvalues as an integral over a function η that index the eigenstates. After a saddle point calculation in the functional integral,we recover (23). t The total number of particles hopping during a finite time t is roughly proportionalto the average number of clusters of consecutive particles in the system. Fortypical configurations, this number scales proportionally with the system size in thethermodynamic limit at a finite density of particles. Because of the cyclic structurewith period L in the graph of allowed transitions described at the end of section 2.1, thequantity f ( t ) should have oscillations for finite times. The same reasoning also works forundistinguishable non-interacting particles. For distinguishable particles, on the otherhand, a similar argument shows that f ( t ) should show oscillations on the scale t ∼ L ,since the cyclic structure of the graph of allowed transitions has then period N L : allthe particles need to come back to their initial state.The oscillations of f ( t ) are observed for TASEP from numerical diagonalization,see fig. 5. For non-interacting particles, they are confirmed by a direct calculation inAppendix D, see fig. D2. In both cases, we observe that the oscillations of f ( t ) are not pectrum of periodic TASEP - bulk eigenvalues Figure 5.
Plots of L − log tr e tM ( γ ) as a function of t for a system at half-fillingwith L = 18 sites, obtained from numerical diagonalization of the (deformed) Markovmatrix. The different curves correspond to γ = −∞ , , . , . , . , . , .
5, frombottom to top. smooth: the function f ( t ) is defined piecewise. There exists in particular a time t ( γ )such that f ( t ) is analytic (and independent of γ ) for t between 0 and t ( γ ). We find t ( γ ) ≃ . − γ for undistinguishable free particles. For TASEP at half-filling, fig. 5seems to indicate that t (0) is slightly larger than 1.
3. Parametric formulas for the eigenvalues
In this section, we derive an exact parametric expression, (35), (36), for all theeigenvalues of the Markov matrix of TASEP.
The deformed Markov matrix M ( γ ) of TASEP is equivalent by similarity transformationto minus the Hamiltonian of a ferromagnetic XXZ spin chain with anisotropy ∆ = ∞ and twisted boundary conditions [4]. The integrability of TASEP is a consequence ofthis. The eigenfunctions of M ( γ ) are given by the Bethe ansatz as [2, 3, 4] ψ ( x , . . . , x N ) = det h(cid:16) y k y j (cid:17) N − j e γx j (1 − y k ) x j i j,k =1 ,...,N , (25)provided the quantities y j , called Bethe roots, verify the Bethe equations(1 − y j ) L y Nj = ( − N − e − Lγ N Y k =1 y k . (26)The Bethe equations have many different solutions, corresponding to the variouseigenstates of M ( γ ). The eigenvalue of M ( γ ) corresponding to (25) is E = N X j =1 y j − y j . (27) pectrum of periodic TASEP - bulk eigenvalues Remark : the Bethe ansatz is usually written in terms of the variables z j = e i q j =e γ (1 − y j ) instead of the y j ’s. The eigenvectors are then linear combinations of planewaves with momenta q j . Remark 2 : proving the completeness of the Bethe ansatz for periodic TASEP is stillan open problem, although one observes numerically for small systems that it does giveall the eigenstates. The main difficulties consist in proving that the Bethe equations(26) have | Ω | solutions, and that all the eigenvectors generated form a basis of theconfiguration space of the model. Alternatively, the completeness would follow from adirect proof of the resolution of the identity, P | Ω | k =1 | ψ k ih ψ k | . For periodic TASEPin a discrete time setting with parallel update, such a proof was given by Povolotskyand Priezzhev in [41]. The Bethe equations (26) of TASEP have the particularity that the rhs does not dependspecifically on y j , but is instead a symmetric function of all the y k ’s. We write this rhs( − N − /B L . This allows to solve the Bethe equations in a parametric way, by firstsolving for each y j the polynomial equation(1 − y j ) L /y Nj = ( − N − /B L (28)as a function of B , and then solving a self-consistency equation for B . This ”decouplingproperty” was already used in [10, 11, 13, 14] for the calculation of the gap, and in[21, 22] for the calculation of the eigenvalue of M ( γ ) with largest real part. The sameproperty is also true for periodic TASEP in a discrete time setting with parallel update[41]. Taking the power 1 /L of the Bethe equations, there must exist numbers k j , j = 1 , . . . , N , integers if N is odd, half-integers if N is even, such that1 − y j y ρj = e πk j /L B , (29)with B a solution oflog B = γ + 1 L N X j =1 log y j . (30)We also define ℓ = − ρ log ρ − (1 − ρ ) log(1 − ρ ) . (31)The Bethe equations (29) involve a function g , defined as g : C \ R − → C \ (e i πρ [e ℓ , ∞ [ ∪ e − i πρ [e ℓ , ∞ [) y − yy ρ . (32)It turns out that this function is a bijection for 0 < ρ <
1, see Appendix A. This is akey point, as it allows to formally solve the Bethe equations as y j = g − (e πk j /L /B ) , (33) pectrum of periodic TASEP - bulk eigenvalues B is fixed by solving(30). We assumed that e πk j /L /B does not belong to the cut of g − . If this is not thecase for some eigenstate, a continuity argument in the parameter γ should still allow touse (33). Remark : there are exactly | Ω | ways to choose the k j ’s with the constraint 0 < k <. . . < k N ≤ L . We observe numerically on small systems that all | Ω | eigenstates arerecovered with this constraint. A similar argument was given in [13], using instead arewriting of the Bethe equations as a polynomial equation of degree L for the z j ’s: thenumber of ways to choose N roots from this polynomial equation is also equal to | Ω | . ϕ and ψ We define the rescaled eigenvalue e = E/L and the parameter b = log B . We introducethe functions ϕ ( z ) = g − ( z )1 − g − ( z ) and ψ ( z ) = log g − ( z ) . (34)In terms of ϕ and ψ , the parametric expression (27), (30) rewrites as e = 1 L n X j =1 ϕ (cid:16) e πkjL − b (cid:17) (35) b = γ + 1 L n X j =1 ψ (cid:16) e πkjL − b (cid:17) . (36)From (34) and (32), the functions ϕ and ψ verify the relationslog z + ρ log ϕ ( z ) + (1 − ρ ) log(1 + ϕ ( z )) = 0 (37)e ψ ( z ) + z e ρ ψ ( z ) = 1 . (38)They are also related by the two equations ϕ ( z ) = e ψ ( z ) − e ψ ( z ) (39) z ϕ ( z ) = e (1 − ρ ) ψ ( z ) . (40)The expansion near z = 0 of ϕ ( z ) and ψ ( z ) at arbitrary filling can be computed explicitlyfrom (34) and the observation [21] that for any meromorphic function h , h ( g − ( z )) canbe written as a contour integral. One has h ( g − ( z )) = I g (Γ) d w π h ( g − ( w )) w − z = I Γ d y π g ′ ( y ) h ( y ) g ( y ) − z , (41)where the contour Γ encloses g − ( z ) but none of the poles of h , and does not cross thecut R − of g . We will need to expand h ( g − ( z )) for small z . In the previous expression,it is possible to expand the integrand near z = 0 as long as | g ( y ) | > | z | for all y ∈ Γ.As shown in fig. 6, it is possible to find such a contour Γ only if | z | < e ℓ , otherwise thecontour would have to go through the cut. pectrum of periodic TASEP - bulk eigenvalues h ( y ) = log y and expand for small z inside the integral. Aftercomputing the residues at y = 1 (which is always inside the contour, see fig. 6), weobtain ψ ( z ) = ∞ X r =1 (cid:18) ρ rr (cid:19) ( − r z r ρ r . (42)Using (41) again for h ( y ) = y/ (1 − y ), one also finds ϕ ( z ) = z − − (1 − ρ ) ∞ X r =0 (cid:18) ρ rr (cid:19) ( − r z r r + 1 . (43)The first term takes into account the pole at y = 1 of y/ (1 − y ), so that the contour Γencloses both poles of the integrand y = 1 and y = g − ( z ).At half filling, the summation over r in (43) and (42) can be done explicitly. Onefinds ϕ ( z ) = −
12 + √ z z (44) ψ ( z ) = − z/
2) = − (cid:16) z r z (cid:17) . (45)The expressions (42) and (43) rely on the assumption that the solution b of (36) is suchthat Re b > − ℓ with ℓ defined in (31), otherwise the expansion of g − for small argumentwould not be convergent. The condition is satisfied if γ is a large enough real positivenumber, in which case b ≃ γ . Besides, when γ = 0, numerical checks on small systemseem to indicate that (36) always has a solution inside the radius of convergence of theseries in e − b for all the eigenstates, except the one with eigenvalue 0. For the latter, itseems that (36) does not have a solution b ∈ C , although formally b = −∞ is a solutionof (36) for which (35) gives e = 0. We emphasize that for all other solutions of (36),even the ones for which the eigenvalue is close to 0 such as the gap, equation (36) seemsto have a solution. For large L , both (35) and (36) become independent of the detailed structure of the k j ’s,and only retain information about the density profile of the k j ’s. For each eigenstate,we introduce a function η such that L η ( u )d u is the number of k j in the interval[ L u, L ( u + d u )]. The function η then obeys the normalization ρ = Z d u η ( u ) , (46)while the eigenvalue (35) becomes e = Z d u η ( u ) ϕ (e πu − b ) , (47)and the equation for the parameter b (36) rewrites b = γ + Z d u η ( u ) ψ (e πu − b ) . (48) pectrum of periodic TASEP - bulk eigenvalues - - - - - - - - Figure 6.
Curves of the points y such that | g ( y ) | = | z | for ρ = 1 / z = 0 .
99 e ℓ (left) and z = 1 .
01 e ℓ (right). The function g is defined in eq. (32) and ℓ is given by(31). The gray area corresponds to | g ( y ) | < | z | and the white area to | g ( y ) | > | z | . Thered half line is the cut R − of the function g . In sections 5 and 6, we will need to count the number of eigenstates corresponding to agiven function η in order to sum over the eigenvalues. Since the number of ways to place L η ( u )d u k j ’s in any interval L d u is (cid:0) L d uL η ( u )d u (cid:1) , Stirling’s formula implies that the totalnumber of eigenstates corresponding to η is Ω[ η ] ∼ e Ls , where we defined an ”entropyper site” s = − Z d u [ η ( u ) log η ( u ) + (1 − η ( u )) log(1 − η ( u ))] . (49)
4. Edge of the spectrum
In the thermodynamic limit, the rescaled eigenvalues e = E/L fill a bounded domainin the complex plane, see fig. 9. We study in this section the boundary of this domain,called in the following edge of the spectrum.We observe numerically that the eigenvalues located at the edge of the spectrumcorrespond to eigenstates for which the k j ’s are consecutive numbers. There are L such possibilities, that we index with an integer m between 0 and L −
1. We write k j = m + j − ( N + 1) /
2. The eigenvalue with largest real part corresponds to m = 0 for γ >
0. For γ = 0, however, we remind that there is no solution to (36) when m = 0. Wewill see that it corresponds to a singular point for the curve at the edge of the spectrum.In the limit L → ∞ , the corresponding density profile η ( u ) of the k j ’s is the functionwith period 1 such that η ( u ) = (cid:12)(cid:12)(cid:12) µ − ρ < u < µ + ρ , (50) pectrum of periodic TASEP - bulk eigenvalues µ = m/L . With this choice of η , writing explicitly the dependency in µ of the eigenvalue and of the parameter b , one finds e ( µ ) = Z ρ/ − ρ/ d u ϕ (e π ( u + µ ) − b ( µ ) ) (51)and b ( µ ) = Z ρ/ − ρ/ d u ψ (e π ( u + µ ) − b ( µ ) ) . (52)At half-filling, a nice parametric representation of the curve e ( µ ) can be written byreplacing b ( µ ) by a new variable d ( µ ) defined by d ( µ ) = − i arccos (cid:16) e πµ − b ( µ ) (cid:17) . (53)Taking the derivative with respect to µ of the equation for b ( µ ) and calculating explicitlythe integrals, one obtains e ( µ ) = − tanh d ( µ ) − d ( µ )2i π (54) d ′ ( µ ) = π d ( µ ) tanh d ( µ ) . (55)The initial condition for the differential equation (55) depends on the value of γ . For γ = 0, since e (0) = 0 (largest eigenvalue of a Markov matrix), then one must have d (0) = 0, which corresponds to b (0) = − log 2. This is a singular point for the differentialequation (55), which is related to the fact that b = − log 2 corresponds to the border ofthe region where the expansion of ϕ ( z ) and ψ ( z ) for small z in section 3 is convergent.Expanding the differential equation at second order near µ = 0 leads to 3 solutions.Inserting them in the equation for e ( µ ), one finds that only the solution d ( µ ) = e − π/ (3 π µ ) / + π µ O (cid:0) µ / (cid:1) (56)gives Re e ( µ ) < µ positive or negative. One has e ( µ ) = − i πµ π/ / π / µ /
10 + O (cid:0) µ / (cid:1) . (57)Studying the stability of the solutions of the differential equation near d = 0, oneobserves that taking as initial condition d (0) = ǫ e i θ with 0 < ǫ ≪ − π < θ < − π/ e ( µ ) is singular near the origin with a power5 / e ( µ ) obtained from (54) and (55) is plottedalong with the full spectrum of TASEP for L = 18, N = 9. The agreement is alreadyvery good with the large L limit. pectrum of periodic TASEP - bulk eigenvalues Figure 7.
Solutions of the differential equation (55) for small µ with initial condition d (0) = 10 − e i θ , for θ = π/ , π/ , . . . , π/
30. The initial values d (0) arerepresented on the circle. The arrows indicate the direction of the increment given bythe derivative at µ = 0, d ′ ( µ ) ≃ π /d (0) . The curves starting on the circle representthe solution of the differential equation (55) solved numerically, which is essentiallyundistinguishable from the solution of d ′ ( µ ) = π /d ( µ ) for the values of d shown.The dotted lines represent the 3 solutions of (55) unstable under a perturbation near d = 0. An alternative expression to (55) can be written by solving explicitly the differentialequation for d in terms of a dilogarithm function, as d ( µ ) − π
24 + d ( µ ) log (1 + e − d ( µ ) ) −
12 Li ( − e − d ( µ ) ) = π µ − π (cid:16) j µ − k(cid:17) . (58)For µ in a neighbourhood of 0, the rhs of the previous equation is equal to π µ . Thesecond term in the rhs is however needed since e − d ( µ ) crosses the cut [1 , ∞ ) of thedilogarithm at µ = 1 /
5, see fig. 8. Indeed, inserting d (1 /
5) = − log((1 + √ / − i π/ (cid:16) √ (cid:17) = − π − log (cid:16) − √ (cid:17) , (59)one observes that the equation (58) is verified, with e − d (1 / = (3 + √ / >
1. For µ = − /
5, the solution is also explicit: usingLi (cid:16) − √ (cid:17) = π − log (cid:16) √ (cid:17) , (60)one finds d ( − /
5) = log((1 + √ / − i π/
2. Here however, e − d ( − / = (3 − √ / < pectrum of periodic TASEP - bulk eigenvalues - - - - - - - - Figure 8.
Curve in the complex plane of d ( µ ) (left) and − e − d ( µ ) (right), from anumerical resolution of the differential equation (55). In both graphs, the two dots arethe explicit values − e − d ( ± / = (3 ± √ /
2. In the graph on the left, the outer, redcurve encloses the domain for which Re b > − log 2, for which the expansions (42) and(43) hold. In the graph on the right, the thick, red line correspond to the cut of thedilogarithm [1 , ∞ ). - - - - - - - - - - Figure 9.
Spectrum of TASEP with N = 9 particles on L = 18 sites. The smallblack dots are the eigenvalues E of the Markov matrix divided by L . The black curveis the edge of the spectrum in the thermodynamic limit at half-filling. The red (grayin printed version) curve corresponds to the asymptotic expression (12) of the edge ofthe spectrum near the origin. The two big blue dots correspond to the explicit values E/L = − ± π (cid:0) √ − log √ (cid:1) . The first order correction to the edge from eq. (61)is plotted in gray. pectrum of periodic TASEP - bulk eigenvalues L (half-filling) We observe in fig. 9 that the edge of the spectrum shows L small peaks. These peaksare a 1 /L correction to the leading behaviour (54), (55). They are a consequence of theconstraint that all the integers k j are different. This phenomenon does not happen fornon-interacting particles, see fig D1.In order to study this correction, one must go back to the exact expressions (35) and(36). Numerically, one observes that the k j that contribute to the edge are such that allthe k j ’s are consecutive except at most one of them. We will write k j = µL + j − ( N +1) / j = 1 , . . . , N − k N = ( µ + ν ) L . On the scale studied here, µ can only take thevalues 1 /L , 2 /L , . . . , 1. The parameter ν verifies 0 ≤ ν ≤ − ρ . We focus again on thehalf-filled case ρ = 1 / L , one recovers (54), (55) using again the change of variable(53). Writing e = e ( µ + 1 / (2 L )) + δe ( µ, ν ), one finds at the end of the calculation δe ( µ, ν ) = tanh d ( µ )2 d ( µ ) L (cid:18) d ( µ ) q − e πν cosh d ( µ )i e πν sinh d ( µ ) − arcsinh h i e πν cosh d ( µ ) i(cid:19) , (61)where d ( µ ) is the solution of (55).From (58) and the relation Li ( − e z ) + Li ( − e − z ) = − π / − z /
2, the function d ( µ )verifies the symmetry relation d ( − µ ) = − d ( µ ), where · denotes complex conjugation.This implies δe ( − µ, / − ν ) = δe ( µ, ν ). The latter symmetry is a consequence of theterm +1 / (2 L ) in the definition of δe . The first-order correction (61) is plotted in fig. 9along with the exact spectrum for L = 18, N = 9.
5. Density of eigenstates
The total number of eigenstates for TASEP is | Ω | ∼ e ℓL , with ℓ defined in terms of thedensity of particles ρ in (31). In the bulk of the spectrum, the number of eigenstateswith a rescaled eigenvalue E/L close to a given e is expected to be of the form e Ls ( e ) .The function s ( e ) is studied in this section. η The density of eigenstates near the rescaled eigenvalue e can be formally defined by thefunctional integral D ( e ) = Z D η { e [ η ]= e } { R d u η ( u )= ρ } e Ls [ η ] . (62)We write explicitly the dependency in η of e , b and s in this section. We want tomaximize for ηs [ η ] + λ (cid:16) Z d u η ( u ) − ρ (cid:17) + Re(2 ω ( e [ η ] − e )) . (63)It gives an optimal function η ∗ , which depends on the two Lagrange multipliers λ ∈ R and ω ∈ C . Those must then be set such that the constraints R d u η ( u ) = ρ and pectrum of periodic TASEP - bulk eigenvalues e [ η ∗ ] = e are satisfied, and we can finally write (13) with s ( e ) = s [ η ∗ ].For given values of the Lagrange multipliers, writing the variation of s [ η ], b [ η ] and e [ η ] for a small variation δη of η and using (46), (47) and (48), we find δs = − Z d u δη ( u ) log η ( u )1 − η ( u ) , (64) δb = R d u δη ( u ) ψ (e πu − b [ η ] )1 + π R d u η ( u ) ∂ u ψ (e πu − b [ η ] ) , (65)and δe = Z d u δη ( u ) h ϕ (e πu − b [ η ] ) + a [ η ] ψ (e πu − b [ η ] ) i . (66)We have defined a [ η ] = − π R d u η ( u ) ∂ u ϕ (e πu − b [ η ] )1 + π R d u η ( u ) ∂ u ψ (e πu − b [ η ] ) . (67)It implies that the optimal function η ∗ verifies − log η ∗ ( u )1 − η ∗ ( u ) + λ + Re[2 ω ϕ (e πu − b [ η ∗ ] ) + 2 a [ η ∗ ] ω ψ (e πu − b [ η ∗ ] )] . (68)The optimal function is then equal to η ∗ ( u ) = (cid:16) − λ − ω ϕ ( e πu − b [ η ∗ ] ) + a [ η ∗ ] ω ψ ( e πu − b [ η ∗ ] ) ] (cid:17) − . (69)This is a real function that satisfies 0 < η ∗ ( u ) < u . The expression (69)is reminiscent of a Fermi-Dirac distribution. On the other hand, the correspondingexpression for undistinguishable non-interacting particles, (D.5), resembles a Bose-Einstein distributions: allowing some momenta to be equal gives a term − For given values of the Lagrange multipliers λ and ω , the expression (69) is completelyexplicit except for the two unknown complex quantities a [ η ∗ ] and b [ η ∗ ], that must bedetermined self-consistently from (67) and (48). It is possible to simplify the problema little by noticing that we can replace a [ η ∗ ] and b [ η ∗ ] by two real quantities α and β .Indeed, the definitions (67) and (69) implyIm( a [ η ∗ ] ω ) = Im h − ω π Z d u η ∗ ( u ) ∂ u (cid:16) ϕ (e πu − b ) + aψ (e πu − b ) (cid:17)i = 14 π Z d u ( η ∗ ) ′ ( u )1 − η ∗ ( u ) = 0 . (70)We define α = a [ η ∗ ] ω ∈ R . The imaginary part of b [ η ∗ ] can be eliminated by a shift of u and a redefinition of η : we introduce σ such that σ (e πu ) = η ∗ (cid:16) u + Im( b )2 π (cid:17) . (71) pectrum of periodic TASEP - bulk eigenvalues β = Re( b [ η ∗ ]), one has σ ( z ) = (cid:16) − λ − ωϕ (e − β z )] − α Re[ ψ (e − β z )] (cid:17) − . (72)It is not possible to eliminate the quantity β by changing the contour of integration,since e.g. Re[ ψ (e − β z )] is not an analytic function of z . It can also be seen by writing2 Re[ ψ (e − β z )] = ψ (e − β z ) + ψ (e − β z − ) for | z | = 1.The quantities s = s [ η ∗ ], β , α can be rewritten in terms of σ as s = − π I d zz σ ( z ) log σ ( z ) + (1 − σ ( z )) log(1 − σ ( z )) , (73) β = Re γ + Re h π I d zz σ ( z ) ψ (e − β z ) i , (74)and α = − π I d z σ ( z ) ∂ z (cid:16) ωϕ (e − β z ) + αψ (e − β z ) (cid:17) . (75)All the contour integrals are over the circle of radius 1 and center 0 in the complexplane. Similarly, the constraints for ρ and e [ η ∗ ] give ρ = 12i π I d zz σ ( z ) , (76)and e = 12i π I d zz σ ( z ) ϕ (e − β z ) . (77)For given ρ and e , one has to solve (76) and (77) in order to obtain λ and ω in termsof them. Like for (75) and (74), it does not seem that these equations can be solvedanalytically in general. At half-filling, however, it is possible to show that λ = Re ω , (78)leaving only the equations for α , β and ω to be solved numerically. Indeed, for ρ = 1 / ϕ ( − z ) = − − ϕ ( z ) and ψ ( − z ) = − ψ ( z ). Setting λ = Re ω in(72) then implies σ ( z ) + σ ( − z ) = 1, from which (76) follows at half-filling.We note that if ω ∈ R then e ∈ R . This is a consequence of (77), σ ( − z ) = σ ( z ) andIm[ ϕ ( − z )] = − Im[ ϕ ( z )]. By the definition (63) of the Lagrange multiplier ω , it impliesthat s is also the density of eigenvalues with a given real part when ω is real.The maximum of s is located at e = − ρ (1 − ρ ), s = ℓ . It corresponds to ω = 0, σ ( z ) = ρ , λ = log[ ρ/ (1 − ρ )], α = 0, β = Re γ . Unlike free particles, the spectrum is notsymmetric with respect to the maximum of s .In fig. 10, e is plotted for various values of ω at half-filling, along with the optimalfunction σ (e πu ). In fig. 11, s is plotted as a function of e ∈ R and compared withthe density of real part of eigenvalues obtained from numerical diagonalization of theMarkov matrix M for N = 9, L = 18. The agreement is not very good for eigenvaluesclose to the edges. This is caused by the ”arches” at distance ∼ /L of the edge of thespectrum, which still contribute much for N = 9, L = 18, see fig. 9. pectrum of periodic TASEP - bulk eigenvalues - - - - - - - - - - Figure 10.
On the left, graph of e ( ω ) for fixed values of | ω | (black), and fixed valuesof arg ω (gray), obtained from (77) after solving numerically the system (74), (75) athalf-filling (left). The different curves correspond to | ω | = 0 . , . , , , , ω = 0 , π/ , π/ , . . . , π/
20. The outer, red curveis the edge of the spectrum, computed numerically from (54), which is recovered from(77) in the limit | ω | → ∞ . On the right, optimal function σ (e πu ) plotted as a functionof u for ω = 8 e i π/ . - - - - - - - Figure 11.
Number of eigenvalues with real part
L e of the Markov matrix M ofTASEP at half-filling, plotted as a function of e . The thick black curve correspondsto the expressions (73), (77) parametrized by ω ranging from −
50 to 60, where thequantities α and β are solved numerically using equations (75) and (74) for each value ofthe parametrization ω . The thick red (gray in printed version) curve is the asymptotics(14). The histograms correspond to the density of real part of eigenvalues obtainedfrom numerical diagonalization of M for the finite system with N = 9 particles on L = 18 sites. Because of the logarithmic corrections in L of L − log | Ω | ≃ log 2, where | Ω | = (cid:0) LL/ (cid:1) is the total number of microstates, we shift the height of the histogramsso that their maximum is log 2. pectrum of periodic TASEP - bulk eigenvalues ρ = 1 / ) We consider the limit e → s ( e ) at half-filling for the Markov matrix (deformation γ = 0). It corresponds to | ω | → ∞ with arg ω → | ω | → ∞ , the optimal function σ (e πu ) approaches 1 if u < u < u and0 if u < u < u + 1. The relation σ ( − z ) = 1 − σ ( z ) at half-filling and the normalizationcondition (76) imply that u = u + 1 /
2. Furthermore, if we consider a scaling such thatarg ω → | ω | → ∞ , one has u = − / u = 1 /
4, which is the same as whatwe had in section 4 for the edge of the spectrum near the eigenvalue 0.In Appendix B, we compute explicitly the large ω limit of the integrals (73), (74),(75) and (77) with λ given by (78). We find two different regimes, depending on therespective scaling between the real and imaginary part of e .The first regime, e → | Im e | / / Re e →
0, corresponds to the central partof the spectrum, far from the edge. We find β ≃ − log 2 + δβ | ω | − / , where δβ is thesolution of 16 π = Z ∞ d x Im[(1 + 2i πx ) / ]1 + e (2 δβ )3 / Im[(1+2i πx ) / ] . (79)The real part of the eigenvalue e and the ”entropy” s are equal toRe e = 4 √ δβ / | ω | / h π − Z ∞ d x Im[(1 + 2i πx ) / ]1 + e (2 δβ )3 / Im[(1+2i πx ) / ] i (80)and s = − | ω | Re e . (81)Solving (79) numerically, one finds δβ ≃ . e ≃− . | ω | − / , s ≃ . | ω | − / and (14).The second regime, e → e and Im e related by (12), corresponds to theedge of the spectrum. In this regime, one finds s ( e ) ≃ / π / / / ( − Re e ) / r / / π / | Im e | / Re e . (82)The crossover between the two regimes corresponds to | Im e | / / ( − Re e ) convergingto a constant different from the 2 / / π / /
10 characteristic of the edge. Explicitexpressions are given in Appendix B.
6. Trace of the time evolution operator
In this section, we study the quantity f ( t ) defined in eq. (16). This is another applicationof the formulas (47), (48) derived in section 3 for the eigenvalues. pectrum of periodic TASEP - bulk eigenvalues η As in section 5 for the density of eigenvalues, one can write the summation over alleigenvalues as tr e tM = X { k ,...,k N } e tE ( k ,...,k N ) ≃ Z D η { R d u η ( u )= ρ } e L ( s [ η ]+ te [ η ]) . (83)If the functional integral is dominated by the contribution of an optimal function η ∗ ,one finds from (16) f ( t ) = s [ η ∗ ] + te [ η ∗ ]. The function η ∗ generally depends on t . Thenormalization (46) of η is enforced by the Lagrange multiplier λ (cid:16) − ρ + Z d u η ( u ) (cid:17) . (84)The change in s , b and e from a variation δη of η is still given by (64), (65) and (66).The optimal function then verifies − log η ∗ ( u )1 − η ∗ ( u ) + λ + t (cid:16) ϕ (e πu − b [ η ∗ ] ) + a [ η ∗ ] ψ (e πu − b [ η ∗ ] ) (cid:17) , (85)with a [ η ] still given by (67). We obtain η ∗ ( u ) = (cid:16) − λ − t ( ϕ ( e πu − b [ η ∗ ] ) + a [ η ∗ ] ψ ( e πu − b [ η ∗ ] )) (cid:17) − . (86) Remark : unlike section 5, the optimal function η ∗ ( u ) is not a real function. This meansthat the saddle point of the functional integral (83) lies in the complex plane. Thefunction η ∗ of (86) can be recovered from the function η ∗ of (69) by the choice ω = t and ω = 0, where · denotes complex conjugation, and ω and ω must be thought of asindependent variables. The optimal function η ∗ ( u ) is an analytic function of z = e πu − b [ η ∗ ] . One defines σ ( z ) = η ∗ ( u ) = (cid:16) − λ − t ( ϕ ( z )+ a [ η ∗ ] ψ ( z )) (cid:17) − . (87)Under the assumption that the contours of integration can be freely deformed from | z | = e − Re b [ η ∗ ] to something independent of b [ η ∗ ], we recover the equations (73) and (76)for s ≡ s [ η ∗ ] and ρ , but with σ now given by (87) instead of (72). The equations for thequantities e ≡ e [ η ∗ ] and a = a [ η ∗ ] become e = 12i π I d zz σ ( z ) ϕ ( z ) (88)and a = − I d z π σ ( z ) ∂ z ( ϕ ( z ) + a ψ ( z )) . (89)Eq. (87) implies ϕ ( z ) + aψ ( z ) = − λt − t log 1 − σ ( z ) σ ( z ) . (90) pectrum of periodic TASEP - bulk eigenvalues - - - - - - - - - - - Figure 12.
On the left, the singularities of the function σ for t = 5 (poles and cuts)are drawn in red (gray in printed version). The circle of radius r t (94) represents apossible contour of integration. On the right, the image by σ of this contour is drawnfor t between 1 (inner curve) to 5 (outer curve). Combining this with (89) gives a = I d z πt ∂ z log(1 − σ ( z )) = wt , (91)where w ∈ Z is the winding number of 1 − σ ( z ) around the origin. We assume in thefollowing that w = 0, hence a = 0 and σ ( z ) = (cid:16) − λ − tϕ ( z ) (cid:17) − . (92)The property a = 0 is compatible, at least for small times, with numerical solutions of(76) with a = 0, see fig. 12.Unlike section 5, the expression (92) for σ ( z ) allows to calculate explicitly theLagrange multiplier λ , the eigenvalue e , the ”entropy” s and the function f ( t ) = s + t e .Details are given in Appendix C. In the end, we recover (23). σ ( z )In the previous subsection, we used a deformation of the contour of integration inorder to eliminate completely the parameter b . We implicitly assumed that this waspossible without crossing singularities of σ ( z ). We come back to this issue here, withthe expression (92) for σ ( z ) resulting from the assumption a = 0.We focus on the half-filled case. From (44) and (C.7), the singularities of thefunction σ consist in an essential singularity at z = 0, two cuts starting at ± ± z k , k ∈ N , with z k = 2i t p t + 4(2 k + 1) π . (93)For t = 0, we observe that all the poles are located at the origin. For times t < t ( γ ),where t ( γ ) is the first non-analyticity of f discussed at the end of section 2.5, the contour pectrum of periodic TASEP - bulk eigenvalues r t = 1 + t/ √ t + 4 π (94)is a possible contour.We observed in section 2.5 that the function f ( t ) is defined piecewise. The non-analyticity of f ( t ) should be the sign of the presence of several saddle points competingin the functional integral (83). This is similar to what happens in the direct calculationsof f ( t ) for free particles in Appendix D, with a simple integral instead of a functionalintegral. It is not completely clear, however, how several saddle points emerge fromthe functional integral (83) for TASEP. It might be due to a change in the contour ofintegration at t = t ( γ ), with a new contour that does not enclose all the poles of σ .There could also be a transition in eq. (89) from the solution a = 0 to another valuedue to a change in the winding number of 1 − σ ( z ) around 0.
7. Conclusion
Parametric expressions can be derived for all the eigenvalues of TASEP using Betheansatz. These expressions allow a study of large scale properties of the spectrum in thethermodynamic limit, in particular the curve marking the edge of the spectrum, thedensity of eigenvalues in the bulk of the spectrum and the generating function of thecumulants of the eigenvalues.A natural extension of the present work would be to analyse the structure of theeigenvalues closer to the origin. Of particular interest are eigenvalues with a real partscaling as L − / , which control the relaxation to the stationary state. Another goalwould be to obtain asymptotic expressions for the scalar product between an eigenstatecharacterized by a density η of k j ’s as in sections 5 and 6 and a microstate characterizedby a density profile of particles. This would allow to study physical quantities moreinteresting than the trace of the time evolution operator.Another very interesting extension would be the case of the asymmetric simpleexclusion process with partial asymmetry, where particles are allowed to hop in bothdirections, with an asymmetry parameter controlling the bias. It would be nice if itwere possible to derive parametric expressions for the eigenvalues analogous to (35) and(36). A good starting point seems to be the quantum Wronskian formulation of theBethe equations [42, 26], where a parameter analogous to the parameter B we used hereexists. It would allow to study how the spectrum changes at the transitions betweenequilibrium and non-equilibrium.Finally, it would be nice to understand how the approach used here to study thespectrum of TASEP relates to thermodynamic Bethe ansatz [43]. The latter followsfrom the observation that, in the thermodynamic limit, Bethe roots accumulate on acurve in the complex plane. The density of Bethe roots along this curve can usually beshown to be the solution of a nonlinear integral equation. We would like to understand pectrum of periodic TASEP - bulk eigenvalues Acknowledgements
I thank B. Derrida for several very helpful discussions. I also thank D. Mukamel for hiswarm welcome at the Weizmann Institute of Science, where early stages of this workwere done.
Appendix A. Bijection g The function g is defined in (32) on the whole complex plane minus the negative realaxis ( −∞ , y = r e i θ with r > − π < θ < π . Then g ( y ) is divergent inthe limit of small and large values of r . One has g ( y ) ≃ r − ρ e − i ρ θ for r → g ( y ) ≃ − r − ρ e i(1 − ρ ) θ for r → ∞ . (A.2)This implies arg g ( y ) ∈ ( − ρπ, ρπ ) for small r and arg g ( y ) ∈ ( − π, − ρπ ) ∪ ( ρπ, π ] forlarge r . The image of a point on the cut of g is g ( − r ± i 0 ± ) = e ± i πρ rr ρ . (A.3)The derivative of g ( y ) with respect to θ at this point verifies ∂ θ gg ( − r ± i 0 ± ) = i(1 − ρ ) r + 1 h r − ρ − ρ i . (A.4)This implies that the curves { g ( r e i θ ) , θ ∈ ( − π, π ) } join the cuts in the image spaceorthogonally (which already follows from the local holomorphicity of g , as the image ofthe orthogonality of any circle of center 0 with the negative real axis), from one side orthe other depending on whether r is smaller or larger than ρ/ (1 − ρ ).When y spans ( −∞ , ± i πρ [e ℓ , ∞ ) ∪ e − i πρ [e ℓ , ∞ ),with ℓ defined in equation (31). The bijective nature of g is clearly seen in fig. A1 andfig. A2 where the functions g and its inverse g − are represented. Appendix B. Density of eigenvalues close to the origin
In this appendix, we perform the calculations related to the limit e → | ω | → ∞ ,arg ω → α (75), β (74), e (77) and s (73). Theintegrals must be decomposed as a sum of 4 terms. Up to terms exponentially small in ǫ − ≫
1, one has Z d u F [ σ (e πu ) , e πu ] ≃ Z − ǫ − + ǫ d u F [1 , e πu ] + Z − ǫ + ǫ d u F [0 , e πu ] (B.1) pectrum of periodic TASEP - bulk eigenvalues - -
15 0 15 30 - - - - - - - - - - Figure A1.
Deformation of a grid by the function g (32), with ρ = 1 /
3. On theleft is a grid in polar coordinates, with angles θ regularly spaced of 2 π/
10 and radii0 . . . .
02, 0 .
05, 0 .
2, 0 .
5, 2, 5, 10, 20, 40 coloured from blue for smallradius to red for large radius. The thick red line corresponds to the cut of the function g . On the right, the image by the function g of the previous grid is drawn, using thesame colors for a curve in the initial grid and its image by g . The almost semi-circularcurves on the left part of the complex plane are red, the ones on the right are blue.The two thick red lines correspond to the image of the cut by g . - - - - - - - - - Figure A2.
Deformation of a grid by the inverse function g − (32), with ρ = 1 / θ regularly spaced of 2 π/
15 andradii 0 . ℓ , 0 .
25 e ℓ , 0 . ℓ , 0 .
75 e ℓ , e ℓ , 1 . ℓ , 1 . ℓ coloured from blue for small radiusto red for large radius, with ℓ defined in (31). The two thick red lines correspond tothe cuts of the function g − . On the right is represented the image by the function g − of the previous grid, using the same colors for a curve in the initial grid and itsimage by g − . The almost circular shapes of large radius and small radius around 0are red. The small almost circular shapes around 1 are blue. pectrum of periodic TASEP - bulk eigenvalues α − r/ β + log 2 Re e Im e sc < − . r / . r / − . r / . χr − c +4 / . r / = = = < c <
13 0 . χ r − c +1 / . r / − . r / . χr − c +4 / . r / < c < √ χr c √ χ r − c − / / χ / πr − c ) / − / χ / / πr − c ) / π / / χ / r (1+ c ) / Table B1.
Various scalings of the parameters needed for the calculation of the densityof eigenvalues close to the eigenvalue 0, after writing ω = r + i χr c . Exact expressionsfor the numerical constants are given in section 5.3 and Appendix B.1. + Z − − ǫ − + ǫ d u F [ σ (e πu ) , e πu ] + Z − ǫ + ǫ d u F [ σ (e πu ) , e πu ] . After a little rewriting, one obtains Z d u F [ σ (e πu ) , e πu ] ≃ Z − d u F [1 , e πu ] + Z d u F [0 , e πu ] (B.2)+ Z − ǫ d u (cid:16) F [ σ (i e πu ) , i e πu ] + F [ σ ( − i e πu ) , − i e πu ] − F [1 , i e πu ] − F [0 , − i e πu ] (cid:17) + Z ǫ d u (cid:16) F [ σ (i e πu ) , i e πu ] + F [ σ ( − i e πu ) , − i e πu ] − F [0 , i e πu ] − F [1 , − i e πu ] (cid:17) . For the four quantities α , β , e and s , the terms with F [0 , e πu ] vanishes, as well as theterms with F [1 , e πu ] for s .In order to continue the calculations, several scalings need to be considered for Im ω when | ω | → ∞ . We write ω = r + i χr c with r > χ > c <
1, and will take the limit r → ∞ . The case χ < c = − − − /
2, 0, 1 /
6, 1 /
5, 1 /
4, 1 /
3, 1 /
2, 2 /
3, 3 /
4, 4 / r = 100 , , . . . , r → ∞ . For all the scalings studied, therelative errors in the numerical coefficients of α , β , Re e , Im e and s obtained from theBST algorithm were smaller than 10 − compared to the exact values. Appendix B.1. Scaling c = 0 . Writing α = r/ δα/r / , β = − log 2 + δβ/r / and expanding the equations (74)and (75) respectively up to order r − and r − / with ǫ ∼ r − / in (B.2) give afterstraightforward, but rather tedious calculations (79) and δα h π − Z ∞ d x Im[(1 + 2i πx ) − / ]1 + e (2 δβ )3 / Im[(1+2i πx ) / ] i (B.3) pectrum of periodic TASEP - bulk eigenvalues δβ h π − Z ∞ d x Im[(1 + 2i πx ) / ]1 + e (2 δβ )3 / Im[(1+2i πx ) / ] i + √ δβ χ Z ∞ d x Re[(1 + 2i πx ) / ] Re[(1 + 2i πx ) − / ] e (2 δβ )3 / Im[(1+2i πx ) / ] (cid:16) (2 δβ )3 / Im[(1+2i πx ) / ] (cid:17) . We used the expansion Z / − / d u ψ (2 e πu − ǫ ) = − log 2 + ǫ − √ π ǫ / + 2 √ π ǫ / + O (cid:0) ǫ / (cid:1) . (B.4)We observe that the equations (79) and (B.3) for δβ and δα decouple in the scaling c = 0,unlike the original equations (74) and (75) for β and α .Expanding the equation (77) for e , we find at leading order in r (80) andIm e = − δβ χr / Z ∞ d x [Re[(1 + 2i πx ) / ]] e (2 δβ )3 / Im[(1+2i πx ) / ] (cid:16) (2 δβ )3 / Im[(1+2i πx ) / ] (cid:17) , (B.5)while the equation (73) for s gives (81). Numerically, one has δα ≃ − . . χ and Im e ≃ − . χ r − / . Appendix B.2. Scaling c = 1 / . Writing α = r/ δα r / , β = − log 2 + δβ r − / and expanding the equations for β (74)and α (75) respectively up to order r − and r with ǫ ∼ r − / in (B.2) give after againlong but straightforward calculations13 π = Z ∞ d x h Im[(1 + 2i πx ) / ]1 + e Φ + ( x ) + Im[(1 + 2i πx ) / ]1 + e Φ − ( x ) i (B.6)and δα (cid:16) π − Z ∞ d x h Im[(1 + 2i πx ) − / ]1 + e Φ + ( x ) + Im[(1 + 2i πx ) − / ]1 + e Φ − ( x ) i(cid:17) = − χ Z ∞ d x h Re[(1 + 2i πx ) − / ]1 + e Φ + ( x ) − Re[(1 + 2i πx ) − / ]1 + e Φ − ( x ) i , (B.7)with the definitionΦ ± ( x ) = (2 δβ ) / πx ) / ] − p δβ δα Im[(1 + 2i πx ) / ] ± p δβ χ Re[(1 + 2i πx ) / ] . (B.8)We observe that the equations for δα and δβ are now coupled, unlike in the scaling c = 0.Similar calculations with the equations for e (77) and s (73) give at leading order in r Re e = 2 √ δβ / r / (cid:16) π − Z ∞ d x h Im[(1 + 2i πx ) / ]1 + e Φ + ( x ) + Im[(1 + 2i πx ) / ]1 + e Φ − ( x ) i(cid:17) , (B.9)Im e = √ δβ / r Z ∞ d x h Re[( δβ + 2i πx ) / ]1 + e Φ + ( x ) − Re[(1 + 2i πx ) / ]1 + e Φ − ( x ) i , (B.10) pectrum of periodic TASEP - bulk eigenvalues s = − r Re e + 3 χ r / Im e . (B.11)In the limit χ →
0, we see that δβ , Re e and s converge to their value in the scaling c = 0, while δα ≃ . χ and Im e ≃ − . χ/r , with the same numericalconstants as in the scaling c = 0.The limit χ → ∞ is a bit more complicated. We check that δα ≃ √ χ/
12 and δβ ≃ √ χ/ δα and δβ , we observe thatΦ + ( χu ) > u >
0, while Φ − ( χu ) < u < / (4 π ) and Φ − ( χu ) > χ to the integral, while the second term ofthe integrands contributes only for v/χ < / (4 π ), with e Φ − ( v ) →
0. Performingexplicitly the integrals then shows that (B.6) and (B.7) are indeed verified. We alsochecked numerically that the solutions of (B.6) and (B.7) for finite χ seem to grow as δα ≃ √ χ/
12 and δβ ≃ √ χ/ χ .Similar calculations lead to Re e ≃ − / / χ / / (5 πr / ) and Im e ≃− / χ / / (3 / πr ) in the limit χ → ∞ . Using (B.11), it implies s = 0 on the scale χ / . Going beyond that requires some more work: one has to take into account thecontribution of the integrals near v = 3 χ/ (4 π ), making the change of variables v =3 χ/ (4 π ) + χ − / u . Writing δα = √ χ/
12 + χ − δα and δβ = √ χ/ χ − δβ , one findsat the end of the calculation δα ≃ − π / δβ →
0, Re e ≃ − / / χ / / (5 πr / ) − π/ (2 / / χ / r / ) and Im e ≃ − / χ / / (3 / πr ) + π/ (2 / / χ / r ). It finally leadsto s ≃ π/ (2 / / χ / r / ). Appendix B.3. Scalings c < and < c < / . The crossover scaling c = 0 is surrounded by the 2 regimes c < < c < / c = 0 allow to compute thequantities α , β , Re e , Im e and s at leading order in r .We observe that the results found in the regime c < χ → c = 0. Similarly, the results found in the regime 0 < c < / χ → ∞ in the scaling c = 0.Since the regimes c < < c < / δα , which is just an intermediate quantity needed for the calculations, one can for allpurpose consider this as a unique regime, c < /
3: the quantities of interest e and s then depend in a simple way on r , χ and c in the whole regime. Appendix B.4. Scaling / < c < . The regime 1 / < c < u = ± / d /r − c + d /r − c ) + . . . + d m /r m (1 − c ) + v/r ( c +1) / inthe integrals, where the constants d j must be such that the argument of the exponentialin σ ( u ) is of order r . Using a similar rewriting to (B.2), but with ± / pectrum of periodic TASEP - bulk eigenvalues ± / d /r − c + d /r − c ) + . . . + d m /r m (1 − c ) , the same kind of calculations as in theother scalings can in principle be done.We checked only the specific case c = 2 /
3. There, making the change of variables u = ± / d /r / + d /r / + v/r / in the integrals, we find that d must be solutionof the equation(3 χ − πd )Re[( δβ + 2i πd ) / ] = 2( δβ − δα )Im[( δβ + 2i πd ) / ] , (B.12)while d has a rather complicated (but completely explicit) expression in terms of χ , δα , δβ and d . The equation for β at order r − / then givesRe[( δβ + 2i πd ) / ] = 0 , (B.13)while the equation for α at order r / leads to3 χ − πd = 2i( δβ − δα ) . (B.14)Gathering the last 3 equations, one finds d = 3 χ/ (4 π ), δα = √ χ/
12 and δβ = √ χ/ d then gives d = − χ / (10 √ π ). From the equation for e , one obtainsRe e = − / / χ / / (5 πr / ) and Im e = − / χ / / (3 / πr / ), while the equation for s leads to s = π/ (2 / / χ / r / ). We observe that the numerical constants are thesame as in the limit χ → ∞ of the scaling c = 1 /
3. We conjecture that this is the casefor all the scaling 1 / < c < Appendix C. Explicit calculations for the generating function f ( t )In this appendix, we calculate explicitly the contour integrals in (76), (88) and (73),with σ ( z ) given by (92). In order to do this, we first prove two useful formulas, (C.1)and (C.3) for the exponential of the functions ψ (42) and ϕ (43). Appendix C.1. Formula for e xψ ( z ) From (34), one has e x ψ ( z ) = ( g − ( z )) x . Using (41) with h ( y ) = y x , the expansion near z = 0 leads toe x ψ ( z ) = x ∞ X r =0 (cid:18) ρ r + xr (cid:19) ( − r z r ρ r + x . (C.1) Appendix C.2. Formula for e − xϕ ( z ) Expanding the exponential in e − xϕ ( z ) and using (40), one findse − xϕ ( z ) = 1 + ∞ X k =1 ( − z ) − k x k k ! e k (1 − ρ ) ψ ( z ) . (C.2)Eq. (C.1) then leads toe − xϕ ( z ) = 1 + (1 − ρ ) ∞ X k =1 x k ( k − ∞ X r =0 (cid:18) ρ r + (1 − ρ ) kr (cid:19) ( − r − k z r − k ρ r + (1 − ρ ) k . (C.3) pectrum of periodic TASEP - bulk eigenvalues I d z π e − xϕ ( z ) z = ρ + (1 − ρ )e x . (C.4) Appendix C.3. Exact expression for the Lagrange multiplier λ The Lagrange multiplier λ is fixed by the normalization condition (76) with σ given by(92). One has ρ = ∞ X r =0 ( − r (e − λ ) r I d z π e − rtϕ ( z ) z . (C.5)Using (C.4) with x = rt implies ρ = ρ − λ + 1 − ρ t − λ . (C.6)Solving for λ finally gives λ = log 2 ρ e t − ρ + p ρ (1 − ρ )(e t − , (C.7)which simplifies at half filling to λ = t/
2. The sign + is chosen in front of the squareroot for continuity at t = 0, for which one has λ = log[ ρ/ (1 − ρ )]. Appendix C.4. Exact expression for the eigenvalue e The expression (88) for the eigenvalue can be made completely explicit. From (92), onehas e = ∞ X r =0 ( − r (e − λ ) r I d z π e − rtϕ ( z ) ϕ ( z ) z = I d z π ϕ ( z ) z + ∂ t h ∞ X r =1 ( − r − (e − λ ) r r I d z π e − rtϕ ( z ) z i . (C.8)Using (C.4) to compute the residue, one finds e = − − ρ t − λ = 1 − p ρ (1 − ρ )(e t − t − . (C.9) Appendix C.5. Exact expression for s After a little rewriting, the expression (73) for s becomes s = I d z πz h (1 − σ ( z ))( λ + tϕ ( z )) + log (1 + e − λ − tϕ ( z ) ) i . (C.10)Using (76), (88) and (92), one has s = λ (1 − ρ ) − t (1 − ρ ) − te + ∞ X r =1 ( − r − e rλ r I d z π e − rtϕ ( z ) z . (C.11)Using (C.4) to compute the residue, one finds s = − te + ρ log(1 + e − λ ) + (1 − ρ ) log(1 + e λ − t ) . (C.12) pectrum of periodic TASEP - bulk eigenvalues Appendix D. Free particles
In this appendix, we study a system of N = ρL non-interacting particles hopping to thenearest site on the right with rate 1 on a periodic lattice of L sites. Unlike TASEP, thereis no exclusion constraint. We will consider both the case of distinguishable particlesand the case of undistinguishable particles.Similarly to TASEP, we call M ( γ ) the deformation of the Markov matrix whichcounts the current of particles. The action of M ( γ ) on a microstate with particles atpositions x , . . . , x N is M ( γ ) | x , . . . , x N i = N X j =1 (e γ | . . . , x j + 1 , . . . i − | . . . , x j , . . . i ) . (D.1)The x j ’s need not be distinct. For distinguishable particles, the j -th element of | x , . . . , x N i is the position of the j -th particle, and the total number of microstatesis | Ω dfree | = L N . For undistinguishable particles, the positions are kept ordered as x ≤ . . . ≤ x N , and the number of configurations is then | Ω ufree | = (cid:0) L + N − N (cid:1) . In bothcases, the eigenvectors of M ( γ ) are products of plane waves, and the eigenvalues are ofthe form E = N X j =1 (e γ − πk j /L − , (D.2)where each k j is an integer between 1 and L . For distinguishable particles, there is nofurther restriction on the k j ’s. For undistinguishable particles the k j ’s must be ordered, k ≤ . . . ≤ k N .As in the case of TASEP, we define a density of eigenvalues D ( e ) as in (62) anda quantity f ( t ) as in (16). We calculate in this appendix D ( e ) in the thermodynamiclimit in the case of undistinguishable particles, and f ( t ) for both distinguishable andundistinguishable particles. Appendix D.1. Density of eigenvalues for undistinguishable particles
Defining the density profile η ( u ) of the k j ’s as in section 3.4, the rescaled eigenvalue e = E/L is expressed in terms of η as e [ η ] = Z d u η ( u )(e γ − πu − . (D.3)The number Ω[ η ] of ways to place L η ( u ) du k j ’s in an interval of length L du is (cid:0) L du + L η ( u ) du − L η ( u ) du (cid:1) . Stirling’s formula implies Ω[ η ] ∼ e Ls , where the ”entropy” s [ η ] is s [ η ] = Z d u [ − η ( u ) log η ( u ) + (1 + η ( u )) log(1 + η ( u ))] . (D.4)The difference with eq. (49) for TASEP comes from the fact that several k j ’s can beequal for non-interacting particles. pectrum of periodic TASEP - bulk eigenvalues - - - - - - - - - - - - Figure D1.
On the left, spectrum of the Markov matrix M for a system of N = 6non-interacting undistinguishable particles on L = 12 sites. The black dots are theeigenvalues rescaled by a factor 1 /L . The black circle is the edge of the spectrum inthe thermodynamic limit with ρ = 1 /
2. The red (gray in printed version) curve is theparabolic approximation of the circle near the origin. On the right, number of rescaledeigenvalues with a given real part e for non-interacting undistinguishable particles athalf-filling, as a function of e . The black curve corresponds to the expressions (D.4),(D.3) parametrized by ω ranging from − N = 9 particles on L = 18 sites. The histograms are shifted so that their maximum is(log 2 + 3 log 3) / ≃ L − log | Ω ufree | , with | Ω ufree | the total number of microstates. We introduce the two Lagrange multipliers λ ∈ R and ω ∈ C as in (63). Theoptimal function η ∗ ( u ) that maximizes (63) with s [ η ] given by (D.4) is η ∗ ( u ) = (cid:16) − − λ − ω (e γ − πu − (cid:17) − . (D.5)Solving numerically (46) for several values of ω allows to plot s ( e ), see fig. D1. We areinterested in the limit e →
0, which corresponds to ω → ∞ , γ = 0. We will only treatthe case e ∈ R , for which ω ∈ R . We first expand the denominator and the exponentialin the expression (69) of η ∗ , as η ∗ ( u ) = ∞ X k =1 ∞ X j =0 k j j ! e k ( λ − ω ) ( ω e − πu + ω e πu ) j , (D.6)where · denotes complex conjugation. The integral over u can then be performed in(46), (D.3) and (D.4). After summing over j , we find ρ = ∞ X k =1 e k ( λ − ω ) I (2 kω ) , (D.7) e = − ρ + ∞ X k =1 e k ( λ − ω ) I (2 kω ) , (D.8) pectrum of periodic TASEP - bulk eigenvalues s = − ρλ − ωe + ∞ X k =1 e k ( λ − ω ) k I (2 kω ) . (D.9)In the previous expressions, I and I are modified Bessel functions of the first kind.Taking the asymptotics of the Bessel functions for large argument and summing over j gives √ πρ ≃ Li / (e λ ) √ ω + Li / (e λ )8(2 ω ) / + 9 Li / (e λ )128(2 ω ) / , (D.10) √ π ( e + ρ ) ≃ Li / (e λ ) √ ω − / (e λ )8(2 ω ) / −
15 Li / (e λ )128(2 ω ) / , (D.11)and √ π ( s + ρλ + 2 ωe ) ≃ Li / (e λ ) √ ω + Li / (e λ )8(2 ω ) / + 9 Li / (e λ )128(2 ω ) / . (D.12)In the limit ω → ∞ , one has λ → − . After expanding the polylogarithms, we finallyobtain s ( e ) ≃ ζ (3 / / ( − e ) / π / − π / ( − e ) / ρ ζ (3 / / + (cid:16) π ρ ζ (3 / + ζ (1 / ρ ζ (3 / − ζ (5 / ζ (3 / (cid:17) e , (D.13)where ζ is Riemann zeta function. We observe that s ( e ) vanishes for e = 0 with anexponent 1 /
3. This exponent should not be confused with the exponent 1 / E ≃ − π P Nj =1 k j /L for eigenvalues that do not scale proportionally with L . Appendix D.2. Function f ( t ) for distinguishable particles From the definition (16) and the expression (D.2) for the eigenvalues, one has f ( t ) = ρ log (cid:16) L X k =1 e t (e γ − πk/L − (cid:17) . (D.14)For finite times, the sum becomes an integral in the large L limit. After a rewriting asa contour integral, one finds f ( t ) − ρ log L = ρ log (cid:16) I d zz e t (e γ z − (cid:17) = − ρ t . (D.15)We are also interested in f ( t ) for times t of order L . Expanding the exponential in(D.14) leads to f ( τ L ) = ρ log (cid:16) L X k =1 ∞ X j =0 j X m =0 (cid:18) jm (cid:19) ( − j − m L j τ j e mγ e − πkm/L j ! (cid:17) . (D.16) pectrum of periodic TASEP - bulk eigenvalues - - - Figure D2.
Logarithm of the trace of the time evolution operator for non-interactingparticles. On the left, graph of f ( τ L ) /L as a function of τ for distinguishable particles,with f defined in (16). In black are exact computations for ρ = 1 / γ = 0 with L = 20(upper curve) and L = 100 (lower curve). The thick, red curve corresponds to the large L limit (D.22). On the right, graph of f ( t ) as a function of t for undistinguishableparticles, with f defined in (16). In black are exact computations for ρ = 1 / γ = 0with L = 20 (lower curve) and L = 80 (upper curve). The thick, red curve correspondsto the large L limit (D.37). Exchanging the order of the summations over j and m allows to perform the summationover j . One finds f ( τ L ) = ρ log (cid:16) L X k =1 ∞ X m =0 L m τ m e − τL e mγ e − πkm/L m ! (cid:17) . (D.17)The summation over k is then done with the help of L X k =1 e − πkm/L = L ∞ X r =0 δ m,rL , (D.18)which leads to f ( τ L ) − ρ log L = ρ log (cid:16) ∞ X r =0 L rL τ rL e − τL e rγL ( rL )! (cid:17) . (D.19)Using Stirling’s formula for ( rL )! (except for the term r = 0) and extracting the leadingterm of the sum, one has f ( τ L ) L ≃ max r ∈ N ρ ( − τ + r (1 + log(e γ τ /r ))) − { r ( τ ) ≥ } ρ log(2 πr ( τ ) L )2 L − { r ( τ ) ≥ } r ( τ ) L , (D.20)with the convention r log r = 0 for r = 0. In the second and third terms, r ( τ ) is the r corresponding to the maximum in the first term. Defining τ = 0 and for r ∈ N ∗ e γ τ r = e − r r ( r − r − , (D.21)one finally finds for τ r ≤ τ ≤ τ r +1 lim L →∞ f ( τ L ) L = − ρ τ + ρ r + ρ r log e γ τr . (D.22) pectrum of periodic TASEP - bulk eigenvalues r , one has e γ τ r ≃ r − /
2, hence for large τ lim L →∞ f ( τ L ) L ≃ ρ (e γ − τ − ρ (e γ τ − [e γ τ ]) γ τ , (D.23)where [e γ τ ] is the integer closest to e γ τ . Appendix D.3. Function f ( t ) for undistinguishable particles From the definition (16) and the expression (D.2) for the eigenvalues, one has f ( t ) = 1 L log (cid:16) X ≤ k ≤ ... ≤ k N ≤ L N Y i =1 e t (e γ − πki/L − (cid:17) . (D.24)Expanding the exponential leads to f ( t ) = 1 L log (cid:16) X ≤ k ≤ ... ≤ k N ≤ L N Y i =1 ∞ X j =0 j X m =0 (cid:18) jm (cid:19) ( − j − m t j e mγ e − πk i m/L j ! (cid:17) . (D.25)Exchanging the order of the summations over j and m allows to perform the summationover j . One finds f ( t ) = 1 L log (cid:16) X ≤ k ≤ ... ≤ k N ≤ L N Y i =1 ∞ X m =0 t m e − t e mγ e − πk i m/L m ! (cid:17) . (D.26)So far, the calculation parallels the one for distinguishable particles in the scaling t ∼ L .In order to perform the summation over the k i , we first use the relation S N ( L ) = X ≤ k ≤ ... ≤ k N ≤ L N Y i =1 f ( k i ) = I d z πz N +1 exp h ∞ X a =1 L X k =1 z a f ( k ) a a i , (D.27)The contour integral is over a contour enclosing 0. Eq. (D.27) can be proved byconsidering the formal series in z ∞ X N =0 z N S N ( L ) = L Y k =1 − zf ( k ) . (D.28)Eq. (D.27) gives f ( t ) = 1 L log (cid:16) I d z π e − Nt z N +1 exp h ∞ X a =1 z a a L X k =1 ∞ X m ,...,m a =0 a Y i =1 t m i e m i γ e − πkm i /L m i ! i(cid:17) . (D.29)Using (D.18) to sum over k leads to f ( t ) = 1 L log (cid:16) I d z π e − Nt z N +1 exp h L ∞ X a =1 z a a ∞ X r =0 ∞ X m ,...,m a =0 δ rL, a P i =1 m i a Y i =1 t m i e m i γ m i ! i(cid:17) . (D.30)The multinomial sum over the m i ’s can be performed. One has f ( t ) = 1 L log (cid:16) I d z π e − Nt z N +1 exp h L ∞ X a =1 z a a ∞ X r =0 ( a e γ t ) rL ( rL )! i(cid:17) . (D.31) pectrum of periodic TASEP - bulk eigenvalues a can be done explicitly. For r ≥
1, it gives a polylogarithm. Onefinds f ( t ) = 1 L log (cid:16) I d z π e − Nt z N +1 (1 − z ) L exp h L ∞ X r =1 (e γ t ) rL ( rL )! Li − rL ( z ) i(cid:17) . (D.32)We deform the contour of integration so that it encloses the negative real axis. In thethermodynamic limit, it is then possible to use the asymptoticsLi − n ( z ) ≃ Γ( n + 1)( − log z ) − n − (D.33)for large n . It leads to f ( t ) ≃ L log (cid:16) I d z π e − Nt z N +1 (1 − z ) L exp h ∞ X r =1 r (cid:16) − e γ t log z (cid:17) rL i(cid:17) . (D.34)Summing explicitly over r gives f ( t ) ≃ L log (cid:16) I d z π e − Nt z N +1 (1 − z ) L (cid:16) − (cid:16) − e γ t log z (cid:17) L (cid:17) (cid:17) . (D.35)Expanding the last factor of the denominator, we finally obtain f ( t ) ≃ L log (cid:16) ∞ X r =0 I d z π e − Nt (cid:16) − e γ t log z (cid:17) rL z N +1 (1 − z ) L (cid:17) . (D.36)The thermodynamic limit of f ( t ) is extracted by calculating the saddle point z r of thecontour integral. One finds f ( t ) ≃ max r ∈ N (cid:16) − ρ t − ρ log z r − log(1 − z r ) + r log (cid:16) − e γ t log z r (cid:17)(cid:17) , (D.37)where z r verifies the equation r log z r = z r − z r − ρ . (D.38)One has z = ρ/ (1 + ρ ). For ρ = 1 /
2, we find numerically z ≃ . z ≃ . z ≃ . t r , r ∈ N such that when t r < t < t r +1 , the saddlepoint that dominates (D.37) is z r . The value of t r is determined by continuity of f ( t ).One has t = 0, and for ρ = 1 /
2, e γ t = 1 . γ t = 3 . γ t = 5 . r , z r decreases to 0 as z r ≃ e − r/ρ . This implies, for t r < t < t r +1 , f ( t ) ≃ − ρt + r + r log ρ e γ tr , (D.39)with t r given for large r by ρ e γ t r ≃ e − r r ( r − r − ≃ r − . (D.40)For large times, we finally obtain f ( t ) ≃ ρ (e γ − t − ( ρ e γ t − [ ρ e γ t ]) ρ e γ t , (D.41)where [ ρ e γ t ] is the integer closest to ρ e γ t . This expression is very similar to the onefound for distinguishable particles on the scale t ∼ L . pectrum of periodic TASEP - bulk eigenvalues References [1] F. Spitzer. Interaction of Markov processes.
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