On the scaling properties of (2+1) directed polymers in the high temperature limit
OOn the scaling properties of (2+1) directed polymers in the high temperature limit
Victor Dotsenko a,b and Boris Klumov c a LPTMC, Sorbonne Universit´e, Paris, France b Landau Institute for Theoretical Physics, Moscow, Russia and c Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow, Russia (Dated: February 17, 2021)In this paper in terms of the replica method we consider the high temperature limit of (2+1)directed polymers in a random potential and propose an approach which allows to compute thescaling exponent θ of the free energy fluctuations as well as the left tail of its probability distributionfunction. It is argued that θ = 1 / different from the zero-temperature numerical valuewhich is close to 0 . PACS numbers: 05.20.-y 75.10.Nr 74.25.Qt 61.41.+e
I. INTRODUCTION
At present the statistical properties of one-dimensional directed polymers as well as the other systems belonging tothe so called KPZ universality class [1] are sufficiently well studied (for the reviews see e.g. [2–4]). In contrast to that,very little is achieved in the studies of the so called (2+1) model of directed polymers which describes the fluctuationsof an elastic string directed along the time axes which passes through a random medium in the three-dimensionalspace. Most of the information we have to date about such type of systems are due to numerical simulations [5–7]. Inparticular, it is rather convincingly established that at the zero-temperature and in the limit of large times t the freeenergy fluctuations of such directed polymers scale as t θ with the scaling exponent θ (cid:39) . θ is not equal to 1 / hightemperature limit allows to estimate the replica partition function in the limit of large number of replicas. This inturn, makes possible to derive the left tail asymptotics of the free energy fluctuations distribution function. Assumingthat this (unknown) distribution function is defined by only one energy scale one eventually finds that the time scalingexponent of the free energy fluctuations is θ = 1 /
2. This value is remarkably different from the zero temperaturenumerical results. If correct, this statement implies that unlike the (1 + 1) system (where it is rigorously proved that θ = 1 / T → T → ∞ ) in the two-dimensional case the free energy scaling exponent is non-universalbeing temperature dependent.In Section II we define the model and describe the general ideas of the present approach. In Section III we presentthe systematic mean-field method (hopefully valid in the limit of large number of replicas) which eventually can bereduced to the solution of two-dimensional one-particle non-local (integral) non-linear differential equation (25). Inthe high temperature limit this equation can be (numerically) solved providing the value of the exponent θ = 1 / II. THE MODEL AND THE REPLICAS APPROACH
We consider the model of directed polymers defined in terms of an elastic string described by the two-dimensionalvector φ ( τ ) ≡ (cid:0) φ x ( τ ) , φ y ( τ ) (cid:1) directed along the τ -axes within an interval [0 , t ] which passes through a random mediumdescribed by a random potential V ( φ , τ ). The energy of a given polymer’s trajectory φ ( τ ) is H [ φ ( τ ); V ] = (cid:90) t dτ (cid:26) (cid:2) ∂ τ φ ( τ ) (cid:3) + V [ φ ( τ ) , τ ] (cid:27) ; (1)Here the disorder potential V [ φ , τ ] is supposed to be Gaussian distributed with a zero mean V ( φ , τ ) = 0 and thecorrelation function V ( φ , τ ) V ( φ (cid:48) , τ (cid:48) ) = u δ ( τ − τ (cid:48) ) U ( φ − φ (cid:48) ) (2)The parameter u is the strength of the disorder and U ( φ ) is a smooth function characterized by the correlation length (cid:15) . For simplicity we take U ( φ ) = 12 π (cid:15) exp (cid:110) − φ (cid:15) (cid:111) (3) a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b One-dimensional, or the so called (1+1) version of this problem (when instead of the vector we have a scalar field φ ( τ )) with the δ -correlated random potential has been the focus of intense studies during past three decades [8–26]. Atpresent it is well established that the fluctuations of the free energy of this system are described by the Tracy-Widom(TW) distribution [27] and their typical value scale with time as t / .The general formulation of the considered (2+1) problem in terms of the replica approach looks quite similar to the(1+1) one. For a given realization of the random potential V ( φ , τ ) the partition function of the considered system(with fixed boundary conditions) is Z ( r , t ) = (cid:90) φ ( t )= r φ (0)= D φ ( τ ) exp (cid:8) − βH [ φ ( τ ) , V ] (cid:9) = exp (cid:8) − βF ( r , t ) (cid:9) (4)where β is the inverse temperature, F ( r , t ) is the free energy which is a random quantity and the integration is takenover all trajectories φ ( τ ) starting at (at τ = 0) and ending at a point r (at τ = t ). Note that this problem isequivalent to the KPZ equation [1] ∂ t F ( r , t ) = 12 β ∇ F ( r , t ) − (cid:16) ∇ F ( r , t ) (cid:17) + V ( r , t ) (5)which describe the time evolution of the two-dimensional manifold F ( r , t ) in a random potential V ( r , t ).For simplicity, in what follows we are going to consider the problem with the zero boundary conditions: φ ( τ = 0) = φ ( τ = t ) = . The free energy probability distribution function P ( F ) of this system can be studied in terms of theinteger moments of the above partition function, eq.(4): Z N ≡ Z ( N, t ) = N (cid:89) a =1 (cid:90) φ a ( t )=0 φ a (0)=0 D φ a ( τ ) (cid:32) exp (cid:110) − β N (cid:88) a =1 H [ φ a ( τ ) , V ] (cid:111)(cid:33) = (cid:90) + ∞−∞ dF P ( F ) exp (cid:8) − βN F (cid:9) (6)where ( ... ) denotes the averaging over the random potentials V [ φ , τ ] Performing this simple Gaussian averaging weget Z ( N, t ) = N (cid:89) a =1 (cid:90) φ a ( t )=0 φ a (0)=0 D φ a ( τ ) exp (cid:110) − βH N [ φ ( τ ) , φ ( τ ) , ... , φ N ( τ )] (cid:111) (7)where βH N [ φ ( τ ) , φ ( τ ) , ... , φ N ( τ )] = (cid:90) t dτ (cid:34) β N (cid:88) a =1 (cid:16) ∂ τ φ a ( τ ) (cid:17) − β u N (cid:88) a,b =1 U (cid:0) φ a ( τ ) − φ b ( τ ) (cid:1)(cid:35) ; (8)is the replica Hamiltonian which describes N elastic strings (cid:8) φ ( τ ) , φ ( τ ) , ... , φ N ( τ ) (cid:9) with the attractive interactions U (cid:0) φ a − φ b (cid:1) , eq.(3). To compute the replica partition function Z ( N, t ), eq.(7), one introduces the function:Ψ( r , r , ... r N ; t ) = N (cid:89) a =1 (cid:90) φ a ( t )= r a φ a (0)= D φ a ( τ ) exp (cid:110) − βH N [ φ ( τ ) , φ ( τ ) , ... , φ N ( τ )] (cid:111) (9)such that Z ( N, t ) = Ψ( r , r , ... r N ; t ) (cid:12)(cid:12) r a =0 (10)Here the spatial arguments of this function are N two-dimensional vectors { r a } . One can easily show thatΨ( r , r , ... r N ; t ) is the wave function of N quantum bosons defined by the imaginary time Schr¨odinger equation β ∂∂t Ψ( r , r , ... r N ; t ) = 12 N (cid:88) a =1 ∆ a Ψ( r , r , ... r N ; t ) + 12 β u N (cid:88) a,b =1 U ( r a − r b ) Ψ( r , r , ... r N ; t ) (11)where ∆ a is the two-dimensional Laplacian with respect to the coordinate r a . The corresponding eigenvalue equationfor the eigenfunctions ψ ( r , r , ... r N ), defined by the relationΨ( r , r , ... r N ; t ) = ψ ( r , r , ... r N ) exp (cid:8) − t E N (cid:9) (12)reads: − β E N ψ ( r , r , ... r N ) = N (cid:88) a =1 ∆ a ψ ( r , r , ... r N ) + β u N (cid:88) a,b =1 U ( r a − r b ) ψ ( r , r , ... r N ) (13)It is at this stage that we are facing the crucial difference of the considered problem with the corresponding (1+1)one. The general solution of the one-dimensional counterpart of eq.(13) is given by the Bethe ansatz wave functionwhich is valid only for U ( x ) = δ ( x ) and which is based on the exact two-particle wave functions ( N = 2) solutionexhibiting finite value energy E N =2 . It is this fundamental property of (1+1) problem which eventually allows toderive the Tracy-Widom distribution for the free energy fluctuation.The situation in the (2+1) case, eq.(13), is much more complicated. First of all, in two dimensions there exists no finite two-particle solution for U ( r ) = δ ( r ). One can easily construct an approximate ground state solution of thetwo-particle problem for the finite-size function U ( r ), eq.(3), but then one finds that in the limit (cid:15) → U ( r )turns into the δ -function) the ground state energy of this solution E N =2 → −∞ . In other words, in two dimensions(unlike one-dimensional case) one can not consider the problem with δ -correlated random potential. We have to studythe system with finite size function U ( r ) and the value of its spatial size (cid:15) must explicitly enter into the final results.Second, one can easily demonstrate that in the two-dimensional case the construction of the N -particle wavefunction `a la Bethe ansatz structure based on the approximate two-particle solution (for finite (cid:15) ) doesn’t work. Sothat, unlike one-dimensional case, here even the ground state energy E N as well as N -particle ground state wavefunction ψ ( r , r , ... r N ) of eq.(13) are not known. All that makes the perspective to find the exact solution of the(2+1) problem rather doubtful.Here we would like to propose somewhat different strategy which, at least in the high-temperature limit, makespossible to estimate the replica partition function Z ( N, t ) at N (cid:29) III. MEAN FIELD APPROACH
First, to simplify notations, let us eliminate the parameter (cid:15) of the correlation function U (cid:0) r (cid:1) . Redefining r → (cid:15) r (14)and E N = 1 (cid:15) ˜ E N , (15)instead of eq.(13) we get − β ˜ E N ψ ( r , r , ... r N ) = N (cid:88) a =1 ∆ a ψ ( r , r , ... r N ) + β u N (cid:88) a,b =1 U ( r a − r b ) ψ ( r , r , ... r N ) (16)where U ( r ) = 12 π exp (cid:26) − r (cid:27) (17)It is evident that in a general case eq.(16) can not be solved. However in the limit of large number of particles, N (cid:29)
1, one hopefully can use the standard trick of the mean field approximation, in which the N -particle wavefunction ψ ( r , r , ... r N ) factorizes into the product of N one-particle functions, namely ψ ( r , r , ... r N ) (cid:39) N (cid:89) a =1 ψ ( r a ) (18)Substituting this into eq.(16), we obtain − β ˜ E N N (cid:89) a =1 ψ ( r a ) = N (cid:88) a =1 ∆ a ψ ( r a ) N (cid:89) b (cid:54) = a ψ ( r b ) + β u N (cid:88) a (cid:54) = b U ( r a − r b ) ψ ( r a ) ψ ( r b ) N (cid:89) c (cid:54) = a,b ψ ( r c ) + 12 π β uN N (cid:89) a =1 ψ ( r a ) (19)Introducing notations ˜ E N = − N β λ (20) β u N = 12 κ (21)and integrating eq.(19) over all r , r , ... r N (taking into account that (cid:82) d r ∆ ψ ( r ) = 0) we get κ ( N − C − (cid:90) d r (cid:90) d r U ( r − r ) ψ ( r ) ψ ( r ) = λ N − π κ (22)where C = (cid:90) d r ψ ( r ) (23)Now, integrating eq.(19) over r , r , ... r N we obtain12 λN ψ ( r ) = ∆ ψ ( r ) + κ (cid:16) − N (cid:17) C − ψ ( r ) (cid:90) d r U ( r − r ) ψ ( r )+ 12 (cid:16) − N (cid:17) ψ ( r ) κ ( N − C − (cid:90) d r (cid:90) d r U ( r − r ) ψ ( r ) ψ ( r ) + 14 π κ ψ ( r ) (24)Substituting here eq.(22), redefining ψ ( r ) → C ψ ( r ) and neglecting terms of order N − we get the following non-linearmean-field equation for the one-particle function ψ ( r ):∆ ψ ( r ) − λ ψ ( r ) + κ ψ ( r ) (cid:90) d r (cid:48) U ( r − r (cid:48) ) ψ ( r (cid:48) ) = 0 (25)where the function ψ ( r ) is normalized, (cid:90) d r ψ ( r ) = 1 (26)and the function U ( r ) is given in eq.(17). Further strategy is in the following. For given values of the parameters λ (the energy, eq.(20)) and κ (the interaction parameter, eq.(21)) we have to find smooth non-negative solution ofeq.(25) such that ψ ( r → ∞ ) →
0. Next, substituting this solution into the constraint (26) we can find λ as a functionof κ , which eventually gives us the dependence of the ground state energy, eqs.(15) and (20), on the replica parameter N . First, let us demonstrate how this strategy works in the well studied one-dimensional case. A. The example of (1 + 1) system
The one-dimensional version of eqs.(25)-(26) reads ψ (cid:48)(cid:48) ( x ) − λ ψ ( x ) + κ ψ ( x ) (cid:90) dx (cid:48) U ( x − x (cid:48) ) ψ ( x (cid:48) ) = 0 (27)where (cid:90) + ∞−∞ dx ψ ( x ) = 1 (28)and U ( x ) = 1 √ π exp (cid:26) − x (cid:27) (29)Redefining ψ ( x ) = λκ φ (cid:0) √ λ x ) (30)and denoting √ λ x = z , instead of eqs.(27)-(28) we get φ (cid:48)(cid:48) ( z ) − φ ( z ) + φ ( z ) (cid:90) dz (cid:48) U λ ( z − z (cid:48) ) φ ( z (cid:48) ) = 0 (31)where the function φ ( z ) satisfy the constraint √ λκ (cid:90) + ∞−∞ dz φ ( z ) = 1 (32)and U λ ( z ) = 1 √ πλ exp (cid:26) − λ x (cid:27) (33)According to eq.(32), λ ( κ ) = I − κ (34)where I = (cid:90) + ∞−∞ dz φ ( z ) (35)In the high temperature limit both κ = ∝ β uN → λ →
0. Thus, in this limit, accordingto the definition (33), lim β → U λ ( z ) → δ ( z ) (36)so that eq.(31) reduces to φ (cid:48)(cid:48) ( z ) − φ ( z ) + φ ( z ) = 0 (37)One can easily check (numerically) that this equation has an instanton-like solution with φ (0) (cid:39) . , φ (cid:48) (0) = 0 and φ ( z → ∞ ) → FIG. 1: Instanton solution of eq.(37) I = (cid:90) + ∞−∞ dz φ ( z ) (cid:39) .
00 (38)Thus, according to eqs.(20), (21), (34) and (38) we find E N (cid:39) − β u N ∝ − N (39)This result, except for the numerical prefactor, perfectly fits with the exact value of ground state energy − β u N of the one-dimensional N -particle boson system (see e.g. [4]) and correspondingly provide the well known value ofthe free energy scaling exponent θ = 1 / B. (2 + 1) directed polymers The situation in the (2+1) case is more complicated. Assuming radial symmetry of the function ψ ( r ) = ψ ( | r | ) ≡ ψ ( r )eqs.(25)-(26) take the form ψ (cid:48)(cid:48) ( r ) + 1 r ψ (cid:48) ( r ) − λψ ( r ) + κ ψ ( r ) (cid:90) d r (cid:48) U ( | r − r (cid:48) | ) ψ ( r (cid:48) ) = 0 (40)2 π (cid:90) ∞ dr r ψ ( r ) = 1 (41)Redefining ψ ( r ) = λκ φ (cid:0) √ λ r ) (42)and denoting √ λ r = z , instead of eqs.(40)-(41) we get φ (cid:48)(cid:48) ( z ) + 1 z φ (cid:48) ( z ) − φ ( z ) + φ ( z ) (cid:90) d z (cid:48) U λ ( | z − z (cid:48) | ) φ ( z (cid:48) ) = 0 (43)2 π (cid:90) + ∞−∞ dz z φ ( z ) = κ (44)where U λ ( | z | ) = 12 πλ exp (cid:26) − λ | z | (cid:27) (45)The main difference with the one-dimensional case is that now in the limit λ → κ , eq.(44) remainsfinite. Indeed, according to eq.(45), lim λ → U λ ( | z | ) = δ ( z ) (46)In this case eq.(43) reduces to φ (cid:48)(cid:48) ( z ) + 1 z φ (cid:48) ( z ) − φ ( z ) + φ ( z ) = 0 (47)This equation has an instanton-like solution with φ (0) (cid:39) . , φ (cid:48) (0) = 0 and φ ( z → ∞ ) → λ = 0 κ ( λ = 0) ≡ κ (cid:39) .
00 (48)At non-zero λ (cid:28)
1, for κ > κ numerical solution of eqs.(43)-(44) demonstrate perfect linear dependence (see Fig.3) λ ( κ ) = γ ( κ − κ ) (49)with γ (cid:39) .
050 (50)On the other hand, for κ < κ eqs.(43)-(44) have no non-negative solutions (such that ψ ( r → ∞ ) →
0) for any λ . IV. FREE ENERGY SCALING
Substituting eqs (15), (20) and (21) into eq.(49) for the mean-field ground state energy of considered two-dimensional N -particle boson system we find E N (cid:39) − γ β(cid:15) N (cid:0) β uN − κ (cid:1) (51) FIG. 2: Instanton solution of eq.(47)
FIG. 3: Dependence of λ on ( κ − κ ), eq.(49) where (cid:15) is the correlation size of the random potential, eq.(3), and the numerical (approximate) values of the factors γ and κ are given in eqs.(48) and (50). Note that the above result is valid only for N > κ β u (cid:29) t → ∞ for the replica partition function we find the followingestimate Z ( N, t ) ∼ exp (cid:26) γ β(cid:15) N (cid:0) β uN − κ (cid:1) t (cid:27) (53)Correspondingly, we see that in the high-temperature limit and at large N eq.(6) reads (cid:90) + ∞−∞ dF P ( F ) exp (cid:8) − βN F (cid:9) ∼ exp (cid:26) γu (cid:15) ( βN ) t − γκ β (cid:15) βN t (cid:27) (54)The above relation shows that the total free energy can be split into two parts F = F + ˜ F (55)where F = γκ β (cid:15) t (56)and ˜ F is the fluctuating contribution described by the distribution function ˜ P (cid:0) ˜ F (cid:1) which according to eq.(54) is definedby the relation (cid:90) + ∞−∞ d ˜ F ˜ P (cid:0) ˜ F (cid:1) exp (cid:8) − βN ˜ F (cid:9) ∼ exp (cid:26) γu (cid:15) t ( βN ) (cid:27) (57)In the case the above relation would be valid for any N we would find that ˜ P (cid:0) ˜ F (cid:1) is just simple Gaussian distributionfunction. In fact, as eq.(57) is valid only for N > N ∗ (cid:29)
1, eq.(52), it gives us only the left tail of this distribution:˜ P (cid:0) ˜ F → −∞ (cid:1) ∼ exp (cid:26) − (cid:15) γ u t ˜ F (cid:27) (58)Note that this asymptotics sets in at | ˜ F | (cid:29) ( βN ∗ ) γut/(cid:15) ∼ κ γt/ ( β (cid:15) ) ∼ F so that here the linear part of the freeenergy F is negligible compared to the fluctuating part ˜ F . Now if we assume that the entire (unknown) distributionfunction ˜ P (cid:0) ˜ F (cid:1) is defined by only one free energy scale, the result (58) tells us that in the considered high-temperaturelimit the typical value of the free energy fluctuations scale as˜ F ∼ √ γ u(cid:15) t / (59)where γ (cid:39) .
05, eq.(50), and u and (cid:15) are the strength and the correlation length of the random potential, eqs.(2)-(3).The above eqs.(58)-(59) constitute the main results of the present research. V. CONCLUSIONS
Regardless of their somewhat ”trivial” form, the results presented in this paper, eqs. (58) and (59), imply coupleof rather non-trivial conclusions.First of all, the fact that in the high temperature limit the free energy time scaling exponent θ = 1 / . β / ) and depends only on the parameters of the disorder.It should be stressed however, that all the results presented in this paper are based on pure heuristic mean-field ansatz, eq.(18). It looks quite reasonable in the limit of large number of replicas (which according to eq.(52)corresponds to the high-temperature limit). Moreover, it works very well in the one-dimensional case (see SectionIII.A). But unfortunately, ”reasonable” does not always guarantee that it is correct... In view of that, numericalsimulations of the considered system at finite (or high) temperatures would be extremely helpful. Acknowledgments
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