Improved field theoretical approach to noninteracting Brownian particles in a quenched random potential
IImproved field theoretical approach to noninteracting Brownian particles in aquenched random potential
Wonsang Lee and Joonhyun Yeo
1, 2, ∗ Department of Physics, Konkuk University, Seoul 05029, Korea School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea (Dated: June 12, 2020)We construct a dynamical field theory for noninteracting Brownian particles in the presence of aquenched Gaussian random potential. The main variable for the field theory is the density fluctu-ation which measures the difference between the local density and its average value. The averagedensity is spatially inhomogeneous for given realization of the random potential. It becomes uniformonly after averaged over the disorder configurations. We develop the diagrammatic perturbation the-ory for the density correlation function and calculate the zero-frequency component of the responsefunction exactly by summing all the diagrams contributing to it. From this exact result and thefluctuation dissipation relation, which holds in an equilibrium dynamics, we find that the connecteddensity correlation function always decays to zero in the long-time limit for all values of disorderstrength implying that the system always remains ergodic. This nonperturbative calculation relieson the simple diagrammatic structure of the present field theoretical scheme. We compare in detailour diagrammatic perturbation theory with the one used in a recent paper [B. Kim, M. Fuchs andV. Krakoviack, J. Stat. Mech. (2020) 023301], which uses the density fluctuation around the uniformaverage, and discuss the difference in the diagrammatic structures of the two formulations.
Keywords: Diffusion in random media, Brownian motion, Ergodicity breaking
I. INTRODUCTION
Dynamics of fluids in a quenched random environmenthas been studied in connection with many different re-search areas ranging from the structural glass transition[1–5] to biological [6, 7] and engineering [8–11] applica-tions. Theoretically the main focus has been on the pos-sible existence of an anomalous diffusion [12–22], whichhas been studied in connection with the spatial dimensionand the range of the random potential and the thermalnoise [23–25]. One of the main physical quantities is thelate time diffusion constant of a tagged particle. The cal-culation of effective transport properties in the presenceof disorder has also been extensively studied [26–32].All these studies are, however, based on the sin-gle particle picture. An alternative way is to use thefield theoretical approaches [33–44], which are based onthe Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD)-type dynamical field theory [45–47] for the stochasticequations governing collective variables such as the den-sity field. There are a number of advantages of using thefield theoretical formalism. One can for example extractphysical information from the symmetry property of theaction functional. For instance, the fluctuation dissipa-tion relation (FDR) of an equilibrium dynamics is ob-tained from the invariance of the action functional undertime-reversal transformations of fields [38, 48]. Anotherpoint is the availability of systematic techniques such asthe loop expansion and the diagrammatic resummationmethods in the field theoretical setting. In fact, therehave been a series of attempts [38–43] to obtain the mode ∗ [email protected] coupling theory (MCT) [49] of the structural glass tran-sition as a first-order self-consistent renormalized theoryfrom the field theoretical formulation of dense liquids orcolloids.There are nontrivial technical issues arising in the fieldtheoretical formalism. For colloidal fluids described byBrownian particles obeying overdamped Langevin equa-tions for the particle positions, an alternative stochasticequation for the microscopic density known as the Dean-Kawasaki (DK) equation [50, 51] can be obtained. Thefield theoretical formulation of the DK equation even inthe simplest case of noninteracting Brownian particlesturns out to be nontrivial [44]. Even in the absenceof interaction among particles, the field theory containsan interaction originated from the multiplicative noisein the DK equation. The multiplicative noise is also re-sponsible for the nontrivial form of the response func-tion [37, 38] which appears in the FDR. Recently, theeffect of quenched disorder on the field theoretical for-mulation of noninteracting Brownian particles was stud-ied [52]. The quenched disorder produces another formof interaction in addition to the one due to the mul-tiplicative noise. Various schemes of perturbation andself-consistent renormalization theories consistent withthe FDR were developed. It was shown that for somerenormalized perturbation scheme the system becomesnonergodic for strong disorder in the sense that the con-nected correlation function does not decay to zero in thelong time limit, while still exhibiting the normal diffusion[52].In this paper, we present an improved field theoret-ical formulation compared to that in Ref. [52] for thenoninteracting Brownian particles in a quenched randompotential with a Gaussian correlation. The main vari- a r X i v : . [ c ond - m a t . d i s - nn ] J un able in the field theory is the density fluctuation, whichmeasures the difference of the local density from its av-erage value, which is spatially inhomogeneous dependingon a given realization of the random potential. Only af-ter averaged over the disorder, a uniform average densityemerges. We present the field theoretical formulationin terms of the density fluctuation around its inhomo-geneous average value. In Ref. [52], the field theory forthe same problem was formulated in terms of the densityfluctuation with respect to the uniform average value.We find that although the two versions of the field the-ory differ only by the definition of the main variable andeventually give the same physical quantities, the detaileddiagrammatic perturbation theories are quite different.We show that within the formalism of Ref. [52], the per-turbation expansion in terms of the disorder strength pro-duces a multitude of unnecessary diagrams which eithercancel among themselves or vanish. In our version of thefield theory, however, the diagrammatic expansion has amuch simpler structure. As an example of the simplicity,we show that we can evaluate the zero-frequency limitor the time integral of the response function exactly bysumming all the diagrams contributing to it. This quan-tity gives, via the FDR, the value of the density-densitycorrelation function in the long time limit. From the ex-act calculation, we find that the system remains ergodicfor all values of the disorder strength in the sense thatthe connected density-density correlation function alwaysdecays to zero in the long time limit.This paper is organized as follows. In Sec. II, wepresent the MSRJD field theory for the noninteractingBrownian particles in the presence of the quenched ran-dom potential. In the next section, we compare the dia-grammatic perturbation theory of our formulation withthat used in Ref. [52]. In Sec. IV, we calculate the zerofrequency limit of the response function and discuss itsimplication on the ergodicity of the system. We concludein the following section with discussion. II. MSRJD FIELD THEORY FORNONINTERACTING BROWNIAN PARTICLESIN A QUENCHED RANDOM POTENTIAL
We consider N noninteracting Brownian particles mov-ing in d dimensions under the influence of an externalrandom potential Φ. The position X i ( t ) of the i -th par-ticle at time t is described by the overdamped Langevinequations, d X i ( t ) dt = − Γ ∇ Φ( X i ( t )) + η i ( t ) , (1)where η i ( t ) is the Gaussian white noise having zero meanand variance (cid:104) η µi ( t ) η νj ( t (cid:48) ) (cid:105) = 2 Dδ ij δ µν δ ( t − t (cid:48) ) (2)for i, j = 1 , . . . , N and µ, ν = 1 , . . . , d and D = Γ T withthe temperature T ( k B = 1). The quenched random potential Φ( x ) is selected from the Gaussian distributionwith zero mean and the varianceΦ( x )Φ( x ) = ∆( x − x ) , (3)where ∆ is a short-ranged function of the distance be-tween the two points. Here the overline indicates theaverage over the quenched random potential.In this paper, we study this system using the micro-scopic density field ρ ( x , t ) = N (cid:88) i =1 δ ( d ) ( x − X i ( t )) , (4)as a main dynamical variable. The above equation canthen be rewritten as a stochastic equation with a multi-plicative noise [50, 51] as ∂ρ ( x , t ) ∂t = Γ ∇ · { ρ ( x , t ) ∇ Φ( x ) } + D ∇ ρ ( x , t ) − ∇ · (cid:110) ξ ( x , t ) (cid:112) ρ ( x , t ) (cid:111) , (5)where (cid:104) ξ µ ( x , t ) ξ ν ( x (cid:48) , t (cid:48) ) (cid:105) = 2 Dδ µν δ ( t − t (cid:48) ) δ ( d ) ( x − x (cid:48) ) . (6)We note that in the above derivation, Ito’s discretizationconvention was used.Many physical quantities such as the intermediate scat-tering functions are obtained from the correlation func-tions of density fluctuation. For given realization of theexternal random potential, we consider the fluctuation δρ ( x ) around its average value (cid:104) ρ ( x , t ) (cid:105) , which we de-note by ρ Φ ( x ): δρ ( x , t ) ≡ ρ ( x , t ) − ρ Φ ( x ) . (7)In the presence of the external potential, ρ Φ ( x ) will beinhomogeneous and can be determined from the station-arity condition of Eq. (5) as ∇ · { ρ Φ ( x ) ∇ Φ( x ) } + D ∇ ρ Φ ( x ) = 0 (8)with the solution ρ Φ ( x ) = N e − Φ( x ) /T (cid:82) d d x e − Φ( x ) /T . (9)We can show that the mean and the variance of the de-nominator on the right hand side of Eq. (9) over theGaussian distribution of the random potential are bothproportional to the volume (see Appendix A). There-fore, in the thermodynamic limit, we can replace it byits disorder average value, (cid:82) d d x e − Φ( x ) /T = V e ∆( ) / (2 T ) with the volume V. This is consistent with the fact thatwhen averaged over the disorder realizations, we have ρ Φ ( x ) = ρ ≡ N/V . For future use, we rewrite Eq. (9)as ρ Φ ( x ) = ρ ∗ e − Φ( x ) /T , (10)where ρ ∗ ≡ ρ e − ∆( ) / (2 T ) . We shall use δρ defined inEq. (7) as the main variable for the MSRJD field theo-retical development below. An alternative way is to usethe density fluctuation around its uniform average as∆ ρ ( x , t ) ≡ ρ ( x , t ) − ρ . (11)The field theoretical formulation using ∆ ρ will be dis-cussed in the next section in detail.We write ρ = ρ Φ + δρ in Eq. (5) and transform it into a field theoretical setting using the MSRJD formalism[45–47], for which the generating functional is written aspath integrals over the density fluctuation δρ ( x , t ) andthe auxiliary response field ˆ ρ ( x , t ). For given externalpotential Φ, the average of an observable O ( δρ, ˆ ρ ) is thengiven by the functional integral (cid:104) O ( δρ, ˆ ρ ) (cid:105) = (cid:90) D δρ (cid:90) D ˆ ρ O ( δρ, ˆ ρ ) e S Φ [ δρ, ˆ ρ ] , (12)where S Φ [ δρ, ˆ ρ ] = (cid:90) d d x (cid:90) dt (cid:20) − i ˆ ρ ( x , t ) (cid:20) ∂δρ ( x , t ) ∂t − D ∇ δρ ( x , t ) − Γ ∇ · { δρ ( x , t ) ∇ Φ( x ) } (cid:21) + Dρ Φ ( x ) {∇ i ˆ ρ ( x , t ) } + Dδρ ( x , t ) {∇ i ˆ ρ ( x , t ) } (cid:21) . (13)The average over the disorder realization can be obtained by integrating it over the distribution P [Φ] = 1 N exp (cid:20) − (cid:90) d d x (cid:90) d d x (cid:48) Φ( x )∆ − ( x − x (cid:48) )Φ( x (cid:48) ) (cid:21) (14)for the external potential, where N is the normalization constant and ∆ − is the matrix inverse of ∆. Therefore wehave (cid:104) O ( δρ, ˆ ρ ) (cid:105) = (cid:90) D δρ (cid:90) D ˆ ρ (cid:90) [ D Φ] O ( δρ, ˆ ρ ) e S [ δρ, ˆ ρ, Φ] , (15)where the functional integral [ D Φ] contains the normalization factor 1 / N and the effective action S is given by S [ δρ, ˆ ρ, Φ] ≡ S Φ [ δρ, ˆ ρ ] − (cid:90) d d x (cid:90) d d x (cid:48) Φ( x )∆ − ( x − x (cid:48) )Φ( x (cid:48) ) . (16)If we expand Eq. (10) in powers of Φ, we can separate the effective action into the Gaussian and non-Gaussian partas S = S G + S v where S G = (cid:90) d d x (cid:90) dt (cid:20) − i ˆ ρ ( x , t ) (cid:26) ∂δρ ( x , t ) ∂t − D ∇ δρ ( x , t ) (cid:27) + Dρ ∗ ( ∇ i ˆ ρ ( x , t )) (cid:21) − (cid:90) d d x (cid:90) d d x (cid:48) Φ( x )∆ − ( x − x (cid:48) )Φ( x (cid:48) ) , (17)and S v = (cid:90) d d x (cid:90) dt (cid:34) Dδρ ( x , t )( ∇ i ˆ ρ ( x , t )) + Γ( i ˆ ρ ( x , t )) ∇ · { δρ ( x , t ) ∇ Φ( x ) } + Dρ ∗ ( ∇ i ˆ ρ ( x , t )) ∞ (cid:88) n =1 Φ n ( x ) n !( − T ) n (cid:35) . (18)In the absence of the random potential (Φ = 0 and∆ = 0), we recover the action studied in Ref. [44] for theBrownian gas. The first term in Eq. (18), which we referto as the noise vertex, comes from the multiplicative noiseand its effect has been studied in the field theoretical set-ting in the absence of disorder in Ref. [44]. The secondand third terms in Eq. (18) are the vertices arising fromthe quenched random potential. The second term is lin-ear in Φ, but the third term is an infinite series of termswhich originate from the expansion of ρ Φ . We denote these as the type- A and type- B n vertices ( n = 1 , , · · · ),respectively. Below we study the effects of these verticeson the perturbation theory in detail.The central quantities in this paper are the two-pointfunctions defined by G αβ ( x − x (cid:48) , t − t (cid:48) ) = (cid:104) ψ α ( x , t ) ψ β ( x (cid:48) , t (cid:48) ) (cid:105) S , (19)where ψ α and ψ β represent ˆ ρ or δρ and the average (cid:104)· · · (cid:105) S is with respect to the effective action S in Eqs. (17) and(18). (In the subscript, we use ρ instead of δρ for brevity.) FIG. 1. Diagrams for the bare propagators, i ˜ G ρ ˆ ρ ( q , t ), i ˜ G ρρ ( q , t ) and ˜ G ρρ ( q , t ) from left to right. Double lines arefor δρ and single lines for i ˆ ρ . q q q q FIG. 2. Diagrams for the noise vertex (left), type- A vertex(right) Their Fourier transforms are given by˜ G αβ ( q , t ) = (cid:90) d d x e − i q · x G αβ ( x , t ) . (20)We first note that due to causality G ˆ ρ ˆ ρ = 0.The bare propagators are obtained from the Gaussianaction in Eq. (17). These will be denoted by G ψ α ψ β andby the lines shown in Fig. 1. The double lines indicate δρ and the single lines are i ˆ ρ . The bare propagators aregiven by i ˜ G ρ ˆ ρ ( q , t ) = Θ( t ) e − D q t , (21) i ˜ G ρρ ( q , t ) = Θ( − t ) e D q t , (22)˜ G ρρ ( q , t ) = ρ ∗ e − Dq | t | , (23)where Θ( t ) is the step function. The bare propagatorinvolving Φ is G ( x − x (cid:48) ) = ∆( x − x (cid:48) ) of which Fouriertransform is given by˜∆( q ) = (cid:90) d d x e − i q · x ∆( x ) . (24)It will be denoted by dashed lines in diagrams, whichcarry momentum but not frequency.The nonlinear vertices in the action, Eq. (18) are rep-resented graphically as in Figs. 2 and 3. At each blackdot, the momentum conservation holds. The cubic noisevertex involves two i ˆ ρ fields and one δρ field and is shownon the left panel of Fig. 2. It contributes the factor of − D q · q and does not involve the random potential.The type- A vertex is shown on the right panel of Fig. 2.It gives the factor of Γ q · q . Finally, there are infi-nite number of vertices, labelled as type- B n , as shown inFig. 3, each of which carries the factor of − ρ ∗ D q · q n !( − T ) n . (25)The correlation functions are obtained by connectingthe lines in the vertices using the bare propagators. Be-fore going into this discussion, however, we first note thatthe ∆-dependent part in the average density ρ ∗ appear-ing in Eqs. (17), (18), (23) and (25) can be eliminated by q q ... k n k k FIG. 3. Diagram for the type- B n vertex. q q ... ... k n k k FIG. 4. Diagram representing the renormalized type- B n ver-tex. considering the renormalized type- B n vertices in the fol-lowing way. We first consider the type- B n +2 m vertex forgiven n, m = 1 , , · · · . If we connect the m pairs of dashedlines in the vertex as in Fig. 4, we end up with anothertype- B n vertex. Since there are ( n + 2 m )! / (2 m n ! m !) dis-tinct ways of doing this, this new type- B n vertex carriesa factor of − ρ ∗ D q · q ∆ m ( )2 m n ! m !( − T ) n +2 m . (26)Now if we sum all these contributions for m = 1 , , · · · and combine with the bare vertex in Eq. (25), which cor-responds to m = 0 case, we have − ρ ∗ D q · q n !( − T ) n ∞ (cid:88) m =0 ∆ m ( ) m !2 m ( − T ) m = − ρ D q · q n !( − T ) n . (27)We note that the effect of this renormalization is justthe use of ρ instead of ρ ∗ in the type- B n vertex. Thiskind of discussion also applies to the third term in theGaussian action in Eq. (17), which can be regarded asthe n = 0 case. If we sum over all these self-pairingcontributions from the type- B m vertices, we end up withthe renormalized term which again carries ρ instead of ρ ∗ . The renormalization of the Gaussian action thereforeresults in the renormalized bare propagator˜ G ρρ ( q , t ) = ρ e − Dq | t | . (28)with just ρ . Note that for this renormailzed bare prop-agator, we have ˜ G ρρ ( q , t ) = ρ [ i ˜ G ρ ˆ ρ ( q , t ) + i ˜ G ρρ ( q , t )],which we will use throughout this paper. In the pertur-bation theory developed below, we regard that the pair-ings of the dashed lines originating from a single vertexas done in Fig. 4 are already taken care of by using therenormalized type- B n vertex.We now investigate the diagrams contributing to thetwo-point functions, Eq. (19) in the perturbation the-ory. An interesting feature of the field theory constructed FIG. 5. Possible loops in the field theory. They all vanish dueto causality. from these propagators and vertices is that the numberof the single lines (ˆ ρ ) in every vertex is greater than orequal to that of the double lines ( δρ ). Since the corre-lation functions are obtained by pairing up these linesand since G ρ ˆ ρ = 0, every nonzero diagram should con-tain equal or more number of δρ than ˆ ρ . For ˜ G ρ ˆ ρ ( q , t )and ˜ G ˆ ρρ ( q , t ), any combinations of the noise vertex withtwo ˆ ρ ’s and a δρ and the type- B n vertices with two ˆ ρ ’sdo not contribute, since they all have more ˆ ρ ’s than δρ ’s.For this type of correlation function, therefore, only thetype- A vertex given in Fig. 2 contributes.For ˜ G ρρ ( q , t ), in addition to the type- A vertex, thetype- B n vertices may contribute, since the two external δρ ’s can make up for the two extra ˆ ρ ’s in that vertex.In terms of the number of ˆ ρ ’s and δρ ’s, the noise vertexmay appear to contribute to ˜ G ρρ ( q , t ). This is, however,not the case. Since there are only two external lines for˜ G ρρ ( q , t ) and since the noise vertex has three legs, theonly way for the noise vertex to contribute to ˜ G ρρ ( q , t ) isto form a closed loop. But the closed loops that appearin this field theory are all in the form of response loopsdescribed in Fig. 5, since every vertex has only one doubleline ( δρ ) at most. These loop diagrams, however, vanishdue to the causal structure of ˜ G ρ ˆ ρ ( q , t ) and ˜ G ρρ ( q , t ).The absence of closed loops is an important feature ofthe present field theory, which has already been noted inRef. [44] for the Brownian gas without disorder.For the functions G ρ ˆ ρ and G ˆ ρρ , as discussed already,only the type- A vertex is relevant. Since there are noloops with respect to the single and double solid lines,all the diagrams are tree diagrams with the connecteddashed lines representing the disorder strength ˜∆( q ). Wecan present a general recipe for constructing a genericdiagram contributing to, say G ˆ ρρ at an arbitrary orderas in Fig. 6. We first put 2 n dots between the externallines and make a single line by connecting them all withthe bare G ρρ ’s. We then connect n pairs of the dots usingthe dashed lines in all possible ways to generate a generaldiagram contributing to G ˆ ρρ .For G ρρ , by counting the number of ρ ’s and ˆ ρ ’s, wenote that the type- B n vertices can only appear once atmost in a diagram contributing to this function. we canclassify the diagrams contributing to G ρρ into two dis-tinct categories: (a) those in which the type- B n verticesis not used, and (b) those where it is used only once. Thestructures of these two kinds of diagrams are describedin Fig. 7. They are constructed in a similar way to theprevious case by connecting all possible pairs of dashed FIG. 6. Construction of a generic diagram contributing to G ρ ˆ ρ . There are even number of dots. Each diagram is gener-ated by connecting all possible n pairs of the dashed lines. ··· FIG. 7. Two possible structures of diagrams contributing to G ρρ : (a) diagrams without type- B n vertex (above) and (b)diagrams containing one type- B n vertex (below). As in Fig. 6,the dashed lines are paired up and connected in all possibleways. lines.It is now a straightforward matter to develop a pertur-bation expansion in powers of ∆. We have carried outthe calculation of G ρ ˆ ρ ( q , t ) and G ρρ ( q , t ) to the first or-der of ∆. We find that in the presence of disorder, thesefunctions decay much slower than the exponential decayexhibited by the bare functions. In fact, we find that G ρ ˆ ρ ( q , t ) and G ρρ ( q , t ) behave in the long time limit as t − d/ − and t − d/ , respectively. The latter behavior wasalready observed in Ref. [52]. In this paper, rather thancontinuing the perturbation theory calculation to higherorders, we focus on a nonperturbative calculation whichwill be presented in Sec. IV. III. FIELD THEORY FOR THE DENSITYFLUCTUATION AROUND THE UNIFORMAVERAGE VALUE
Before going to the nonperturbative calculation, westudy in this section the field theory using the densityfluctuation ∆ ρ ( x , t ) around the uniform average density ρ , which is defined in Eq. (11). This is the formalismused in Ref. [52]. We present a detailed comparison of thefield theory with respect to this variable with the one de-veloped in the previous section. We will show that bothformalisms give the same correlation functions and physi-cal quantities as expected. Nevertheless, we find it usefulto compare these two formalisms in detail for general un-derstanding of the diagrmmatics involved in these fieldtheories and for the nonperturbative calculation whichwill be done in Sec. IV. In fact, we find that the pertur-bative field theory using ∆ ρ is much more complicatedthan that for δρ .For given realization of the disorder potential, the ther-mal average (cid:104) ∆ ρ ( x , t ) (cid:105) = ρ Φ ( x ) − ρ ≡ δρ Φ ( x ) does notvanish. As we have seen in Eq. (10), only after aver-aged over the disorder potential, this quantity vanishes, (cid:104) ∆ ρ ( x , t ) (cid:105) = 0. Higher order moments of ∆ ρ ’s are re-lated to those of δρ ’s, since we can write ∆ ρ ( x , t ) = δρ ( x , t ) + δρ Φ ( x ). For example, we have [5, 52] (cid:104) ∆ ρ ( x , t )∆ ρ ( x (cid:48) , t (cid:48) ) (cid:105) = (cid:104) δρ ( x , t ) δρ ( x (cid:48) , t (cid:48) ) (cid:105) + δρ Φ ( x ) δρ Φ ( x (cid:48) ) , (29)where we can evaluate explicitly the last term as δρ Φ ( x ) δρ Φ ( x (cid:48) ) = ρ (cid:16) e ∆( x − x (cid:48) ) /T − (cid:17) . (30)The first term on the right hand side of Eq. (29) is just G ρρ ( x − x (cid:48) , t − t (cid:48) ) defined in the previous section. Therefore,the correlation function on the left hand side of Eq. (29) differs from G ρρ ( x − x (cid:48) , t − t (cid:48) ) by the time-independentquantity given by Eq. (30).We write ρ = ρ + ∆ ρ in Eq. (5) and again transform it into a field theory as done before. The average of anobservable O (∆ ρ, ˆ ρ ) is then given by (cid:104) O (∆ ρ, ˆ ρ ) (cid:105) = (cid:90) D ∆ ρ (cid:90) D ˆ ρ (cid:90) [ D Φ] O (∆ ρ, ˆ ρ ) e S ∆ [∆ ρ, ˆ ρ, Φ] , (31)where S ∆ = S G ∆ + S v ∆ with S G ∆ = (cid:90) d d x (cid:90) dt (cid:104) − i ˆ ρ ( x , t ) { ∂ ∆ ρ ( x , t ) ∂t − D ∇ ∆ ρ ( x , t ) } + Dρ ( ∇ i ˆ ρ ( x , t )) (cid:105) − (cid:90) d d x (cid:90) d d x (cid:48) Φ( x )∆ − ( x − x (cid:48) )Φ( x (cid:48) )(32)and S v ∆ = (cid:90) d d x (cid:90) dt (cid:104) D ∆ ρ ( x , t )( ∇ i ˆ ρ ( x , t )) + Γ( i ˆ ρ ( x , t )) ∇ · { (∆ ρ ( x , t ) + ρ ) ∇ Φ( x ) } (cid:105) . (33)The Gaussian part of the action S G ∆ takes the similarform to S G in Eq. (17). However, unlike S v , S v ∆ does notcontain an infinite number of terms, and takes a muchsimpler form. Here we regard the last term in Eq. (33),which is quadratic in fields as a part of vertices. As wewill see below, due to the unusual nature of this vertex,the perturbation expansion for this action is much morecomplicated than the corresponding scheme for δρ de-spite the apparent simplicity of the action. We note thatthe functional integral over Φ in Eq. (31) can actually becarried out to yield the effective action depending only on∆ ρ and ˆ ρ . This effective action has been used in Ref. [52].For the present discussion, we find it more convenient toconsider the Φ-dependent action as in Eq. (33). We willpair up the dashed lines in the perturbation expansionas before, which will produce the same diagrams as themethod used in Ref. [52].We now develop the perturbation theory for the two-point function F αβ ( x − x (cid:48) , t − t (cid:48) ) = (cid:104) ψ α ( x , t ) ψ β ( x (cid:48) , t (cid:48) ) (cid:105) S ∆ (34)evaluated with respect to S ∆ for the variables ψ α , ψ β rep-resenting ˆ ρ or ∆ ρ (denoted by ρ in the subscript again).Since the Gaussian part S G ∆ is identical to S G in Eq. (17)except for ∆ ρ and ρ playing the roles of δρ and ρ ∗ , re- q FIG. 8. Diagram for the type- A vertex which exists only inthe field theory for ∆ ρ . spectively, the bare propagators F αβ take the same formas the renormalized G αβ . The noise vertex and the type- A vertex in S v ∆ have the same structure as before with ∆ ρ taking the place of δρ . Instead of the the type- B n vertex,the action contains a new vertex as shown in Fig. 8 whichcomes from the last term in Eq. (33). We denote it bythe type- A vertex. This carries the factor of − ρ Γ q .The type- A vertex contains just one ˆ ρ field. Thereforethe only way it can contribute to F ρρ is to appear atmost twice in a diagram. Otherwise, we would have toomany ˆ ρ ’s to make a nonzero diagram. We can thereforeclassify the diagrams contributing to ˜ F ρρ ( q , t − t (cid:48) ) intothree distinct categories depending on the number of thenew vertices in a diagram: (A) no type- A vertex, (B)one type- A vertex, and (C) two type- A vertices. Thediagrams in category (A) are identical to the diagramsin (a) for G ρρ discussed in the previous section.For diagrams in category (C), one cannot form a path q q FIG. 9. Combination of the noise vertex and the type- A vertex which is equivalent to the renormalized type- B vertex. q q FIG. 10. Combination of the noise, type- A and type- A ver-tices which is equivalent to the renornalized type- B n vertex. from an external leg at time t to the other one at time t (cid:48) using only the solid (single or double) lines. The ex-ternal legs have to be connected by a dashed line, whichdoes not carry the time in it. This means that those incategory (C) give a constant contribution independentof t − t (cid:48) (but still a function of q ). We shall explicitlyshow below that the contributions from the diagrams incategory (B) reproduce those from the diagrams in (b)for G ρρ and that those in category (C) correspond to thetime-independent term, Eq. (30), for F ρρ .We investigate diagrams in category (B) in more de-tail. The diagrams in (B) must contain one noise vertexalong with one type- A vertex. Figure 9 shows the sim-plest vertex structure appearing in diagrams that belongto (B). We note that the external legs of this diagramhave the same structure as the type- B vertex used inthe previous section. In the diagram in Fig. 9, the noiseand the type- A vertices give the factors, − D q · q and − ρ Γ( q + q ) , respectively. Since one of the time inte-grals gives (cid:90) ∞ dt i ˜ F ρ ˆ ρ ( q + q , t ) = 1 D ( q + q ) , (35)we end up with the overall factor of ρ Γ q · q . FromEq. (27), we can see that this is exactly the factor for therenormalized type- B vertex. Therefore, we can concludethat the diagram in Fig. 9 plays exactly the same roleas the renormalized type- B vertex. In general, we canshow that the diagram shown in Fig. 10 which consistsof one noise vertex, one type- A vertex and n − A vertices is equivalent to the renormalizd type- B n vertexshown in Fig. 4. The detailed proof for the equivalenceis presented in Appendix B. This can be regarded as oneof the advantages in using the field theory for δρ over thecorresponding one using ∆ ρ . A complicated combinationof vertices in the field theory for ∆ ρ can be representedas a single vertex in the corresponding formalism for δρ .We now consider the diagrams obtained by connect-ing the dashed lines in the diagram in Fig. 10. We re-call that for the type- B n vertex, connecting the dashed q q FIG. 11. Diagrams obtained by connecting the dashed lines inFig. 10. These kinds of diagrams vanish due to the incomingzero momentum. lines within a single vertex results in its renormalization.But, as we have seen above, the diagram in Fig. 10 isalready equivalent to the renormalized type- B n vertex.We therefore expect that the diagrams obtained by pair-ing up the dashed lines in Fig. 10 all vanish, which wewill demonstrate below. This means that the pertur-bation expansion for the ∆ ρ -field theory generates nu-merous unnecessary diagrams which as a whole give avanishing contribution. We first note that when the tworightmost dashed lines are paired as in Fig. 11, the mo-mentum flowing through the ˜ F ρ ˆ ρ propagator right nextto the loop must be zero. Then, since the type- A vertexinvolves the dot product between the zero momentumvector and another one, such a diagram vanishes. Whenthere is no such loop, a diagram does not vanish on itsown in general. The diagrams in Fig. 12 show such ex-amples. For these diagrams, when we perform the timeintegrals for the bare propagators, we obtain, apart fromthe common factor of ( ρ D/T ) q · q , (cid:90) d d q (cid:48) (2 π ) d ( q + q ) · q (cid:48) ( q + q ) ( q + q + q (cid:48) ) · q (cid:48) ( q + q + q (cid:48) ) ˜∆( q (cid:48) ) , (36)and (cid:90) d d q (cid:48) (2 π ) d ( q + q ) · q (cid:48) ( q + q ) ( q + q + q (cid:48) ) · ( q + q )( q + q + q (cid:48) ) ˜∆( q (cid:48) ) , (37)respectively. Here the momentum q (cid:48) flows through thedashed line loop. We find that the individual diagramsdo not vanish, but the sum does, due to the rotational in-variance of the integration over q (cid:48) . We expect that for thehigher order vertices equivalent to the type- B n , a similarcancellation must occur when the dashed lines are pairedwithin the vertex. This cancellation is a generic featureof the field theory involving ∆ ρ . We stress again that inthe field theory for δρ , these kinds of unnecessary dia-grams do not arise. These self-loops in these vertices arehandled by the simple renormalization of the B n vertexfrom the outset.As mentioned above, the diagrams in category (C)are responsible for the time-independent part given inEq. (30). This part arises only in this formalism, since ∆ ρ is not a density fluctuation around its own average value,but around the uniform value. We again find that manydiagrams in the category (C) cancel each other and donot contribute at all. We note that a generic diagram in q q q q FIG. 12. Diagrams obtained by connecting the dashed linesin Fig. 10. These diagrams cancel each other.FIG. 13. Typical diagram in category (C). Note that theremust be a gap in the middle. An example of six dashed lines isshown. For a general case, each disconnected part can containan arbitrary number of dashed lines. All possible pairs of thedashed lines are to be connected. (C) can be described schematically as in Fig. 13. There isa gap in the middle between the two type- A vertices andwe can naturally split the diagram into two disconnectedparts. We then have to connect all possible pairs of thedashed lines. We find that a set of diagrams containing adashed line connection within a disconnected part gives avanishing contribution. An example is shown in Fig. 14.We can easily see that these diagrams cancel each other,as the diagrammatic structure is essentially the same asthose in Fig. 10. The nonvanishing diagrams are thosewith the dashed lines on the left part are connected tothose on the right part. As we show in detail in AppendixC, such a diagram with 2 n dashed lines gives ρ n ! T n (cid:90) n (cid:89) j =1 (cid:104) d d q j (2 π ) d ˜∆( q j ) (cid:105) (2 π ) d δ ( d ) ( q − n (cid:88) i =1 q i ) , (38)where q is the momentum flowing through the diagramin Fig. 13. This is exactly the Fourier transformation ofthe right hand side of Eq. (30) at O (∆ n ).In this section we have shown that, although the fieldtheory for the density fluctuation around the uniformaverage is equivalent to that for δρ studied in the previoussection, its perturbation expansion generates numerousdiagrams most of which vanish or cancel each other. Thiscould make the calculation of the correlation functionsunnecessarily complicated if done in this formalism. IV. THE PHYSICAL RESPONSE FUNCTIONAND THE FLUCTUATION-DISSIPATIONRELATION
In this section, we further explore the usefulness ofthe field theory for δρ developed in Sec. II in the con-text of the FDR, which is obeyed by the equilibrium dy-namics described by the Langevin equations, Eqs. (1)and (5). The FDR, which comes from the time-reversal FIG. 14. Diagrams in category (C) that cancel each other.Compare this with Fig. 12. symmetry of the equilibrium state, provides the relation-ship between the correlation and the response functions.The MSRJD formalism is known to be suited for study-ing the correlation and response of a system describedby Langevin equations as the hatted variable arising inthe field theory naturally provides the expression for theresponse function. However, for the Langevin equationin Eq. (5) given in terms of the density variable, theresponse function does not take the simple form suchas (cid:104) ρ ( x , t ) i ˆ ρ ( x (cid:48) , t (cid:48) ) (cid:105) . It is well known that due to themultiplicative nature of the noise in Eq. (5), the physi-cal response to an external perturbation coupled to thedensity variable takes a more complicated form given by[37, 38, 41, 52] R ( x − x (cid:48) , t − t (cid:48) ) = − Γ (cid:104) ρ ( x , t ) ∇ (cid:48) · ( ρ ( x (cid:48) , t (cid:48) ) ∇ (cid:48) i ˆ ρ ( x (cid:48) , t (cid:48) )) (cid:105) , (39)which we refer to as the physical response function in thefollowing. The FDR relates R to the correlation function C via − ∂∂t C ( x , t ) = T R ( x , t ) − T R ( x , − t ) , (40)where C ( x − x (cid:48) , t − t (cid:48) ) = (cid:104) ρ ( x , t ) ρ ( x (cid:48) , t (cid:48) ) (cid:105) (41)is the density-density correlation function. If we use ρ ( x , t ) = ρ Φ ( x ) + δρ ( x , t ), this is related to our corre-lation function as C ( x − x (cid:48) , t − t (cid:48) ) = ρ Φ ( x ) ρ Φ ( x (cid:48) )+ G ρρ ( x − x (cid:48) , t − t (cid:48) ) . (42)The physical response function in our field theory for δρ is obtained by simply replacing ρ ( x , t ) in Eq. (39) by δρ ( x , t ) + ρ Φ ( x ). The average is then given with respectto the effective action S in Eqs. (17) and (18). We firstnote that both (cid:104) ρ Φ ( x ) ∇ (cid:48) · ( δρ ( x (cid:48) , t (cid:48) ) ∇ (cid:48) i ˆ ρ ( x (cid:48) , t (cid:48) )) (cid:105) S and (cid:104) δρ ( x , t ) ∇ (cid:48) · ( δρ ( x (cid:48) , t (cid:48) ) ∇ (cid:48) i ˆ ρ ( x (cid:48) , t (cid:48) )) (cid:105) S are equal to zero inthis field theory. The former is essentially G ρ ˆ ρ evaluatedat the same space time point. By connecting the endpoints in the generic diagram in Fig. 6, we find that weare left with the loops described in Fig. 5, which vanishdue to causality. The latter quantity must involve onenoise vertex and can contain an arbitrary number of the ··· FIG. 15. Generic diagram contributing to the physical re-sponse function R . All possible pairs of the dashed lines areto be connected. type- A vertices. But, since there is no vertex with two δρ fields, the loops that appear in the perturbation expan-sion of this quantity are again all in the form of Fig. 5.Therefore, Eq. (39) reduces to R ( x − x (cid:48) , t − t (cid:48) ) = − Γ (cid:104) δρ ( x , t ) ∇ (cid:48) · ( ρ Φ ( x (cid:48) ) ∇ (cid:48) i ˆ ρ ( x (cid:48) , t (cid:48) )) (cid:105) S . (43)In the absence of disorder, ρ Φ = ρ , and the physicalresponse function reduces to ρ Γ ∇ (cid:48) iG fρ ˆ ρ ( x − x (cid:48) , t − t (cid:48) )[44], where the superscript f denotes the case where Φ isset to zero. The effect of disorder in our field theory onthe physical response function is reflected simply in theappearance of the inhomogeneous average density ρ Φ .On the other hand, in the field theory using ∆ ρ , thephysical response function takes a more involved form.By substituting ρ with ρ + ∆ ρ in Eq. (39), we have R ( x − x (cid:48) , t − t (cid:48) ) = − ρ Γ ∇ (cid:48) iF ρ ˆ ρ ( x − x (cid:48) , t − t (cid:48) ) − Γ (cid:104) ∆ ρ ( x , t ) ∇ (cid:48) · (∆ ρ ( x (cid:48) , t (cid:48) ) ∇ (cid:48) i ˆ ρ ( x (cid:48) , t (cid:48) )) (cid:105) S ∆ . (44)Evaluating the physical response function in this caseinvolves the calculation of two correlation functions withrespect to the effective action S ∆ , one of which is a three-point function. The three-point function is nonvanishingin this case due to the presence of the type- A vertex.In the absence of disorder, the three-point function inEq. (44) vanishes, and we have only the first term andhave the same result as before: ρ Γ ∇ (cid:48) iF fρ ˆ ρ ( x − x (cid:48) , t − t (cid:48) )(note that F fρ ˆ ρ = G fρ ˆ ρ ). The effect of disorder in this caseis encoded in the three-point function as well as in F ρ ˆ ρ both of which have to be evaluated explicitly.The simple structure of the physical response function,Eq. (43), in our field theory for δρ enables us to do a non-perturbative calculation. If we expand ρ Φ in Eq. (43) inpowers of Φ, we find that a generic diagram contributingto R takes the form depicted in Fig. 15, where all pos-sible pairs of dashed lines are to be connected. On theright end point ( x (cid:48) , t (cid:48) ), we have multiple Φ( x (cid:48) )’s comingfrom the expansion of ρ Φ along with ˆ ρ field. In the mid-dle, we have dashed lines from a collection of the type- A vertices reaching up to ( x , t ) where δρ lies. The contribu-tion from this kind of diagram with total n dashed linesto the Fourier transform ˜ R ( q , t − t (cid:48) ) can be written as (cid:90) n (cid:89) i =1 (cid:104) d d k i (2 π ) d ˜Φ( k i ) (cid:105) Ξ n ( q ; k , k , · · · , k n ; t − t (cid:48) ) (45)for some vertex function Ξ n with the condition (cid:80) ni =1 k i =0. We note that, by construction, Ξ n is symmetric under the permutation of n momenta, ( k , k , · · · , k n ). Theresponse function R can then be obtained by connectingall possible pairs of Φ’s and by summing over all n =0 , , , · · · .Despite the simple diagrammatic structure, it is still adifficult task to find a general nonperturbative expressionfor Ξ n and thus for ˜ R ( q , t − t (cid:48) ). But we can make aprogress if we focus on the zero frequency component ofthe response function,˜ R ( q , ω = 0) = (cid:90) ∞−∞ dt ˜ R ( q , t ) , (46)where the time integral is actually from 0 to ∞ due tothe causal nature of the response function. This quantityprovides an important physical insight on the long-timebehavior of the correlation function, since it is equal to( ˜ C ( q , − ˜ C ( q , ∞ )) /T as can be obtained from the FDR,Eq. (40). In order to calculate this, we define˜Ξ n ( q ; k , k , · · · , k n ; ω = 0) ≡ (cid:90) ∞−∞ dt Ξ n ( q ; k , k , · · · , k n ; t ) . (47)We evaluate this in detail in Appendix D. The result isquite simple as it gives just a constant˜Ξ n ( q ; k , · · · , k n ; ω = 0) = − ρ ∗ ( − T ) n +1 n ! . (48)We now connect all possible pairs of Φ’s in Eq. (45) andsum these contributions over n = 0 , , , · · · . It is clearthat only the terms with even n = 2 p survive. We there-fore have˜ R ( q , ω = 0) = ∞ (cid:88) p =0 (2 p )!2 p p ! (cid:90) p (cid:89) i =1 d d k i (2 π ) d ˜∆( k i ) (49) × ˜Ξ p ( q ; k , − k , k , − k , · · · k p , − k p ; ω = 0)with the understanding that p = 0 term is given justby Ξ = ρ ∗ /T . The factor (2 p )! / (2 p p !) accounts for thenumber of possible ways to form all possible pairs of Φ’s.Using Eq. (48), we finally have˜ R ( q , ω = 0) = ρ ∗ T ∞ (cid:88) p =0 p ! (cid:40)(cid:90) d d k (2 π ) d ˜∆( k )2 T (cid:41) p = ρ ∗ T e ∆( ) / T = ρ T . (50)As mentioned above, this nonperturbative result onthe response function has an implication on the long-timebehavior of the density-density correlation function. Attime t = 0, the particles are at equilibrium with respectto a given realization of the external potential Φ( x ) attemperature T . The density-density correlation functionmust be equal to the static one given by (cid:104) ρ ( x ) ρ ( x (cid:48) ) (cid:105) st = δ ( x − x (cid:48) ) ρ Φ ( x ) + ρ Φ ( x ) ρ Φ ( x (cid:48) ) , (51)0with ρ Φ given in Eq. (10). Averaging over the disorderconfigurations, we therefore have C ( x − x (cid:48) ,
0) = δ ( x − x (cid:48) ) ρ + ρ Φ ( x ) ρ Φ ( x (cid:48) ) . (52)The nonperturbative result, Eq. (50), together with theFDR, Eq. (40) implies that C ( x − x (cid:48) , ∞ ) = ρ Φ ( x ) ρ Φ ( x (cid:48) ) (53)or G ρρ ( x − x (cid:48) , t ) goes to zero in the long time limit. Thismeans that the system remains ergodic and there is no er-godicity breaking transition for all values of the disorderstrength. V. DISCUSSION AND CONCLUSION
We have constructed the MSRJD dynamical fieldtheory for the noninteracting Brownian particles in aquenched Gaussian random potential. We have set upthe diagrammatic perturbation scheme for the connecteddensity-density correlation function. The main variablefor our field theory is the density fluctuation around thenonuniform average value. The diagrammatic structuresare compared in detail with those in the field theory usingthe density fluctuation around the uniform average value,which is used in Ref. [52]. It is shown that the latter gen-erates many unnecessary diagrams which either vanish orcancel among themselves. Using our field theory, we wereable to evaluate the zero-frequency component of the re-sponse function exactly by summing all the diagrams.According to the FDR, our result implies that the con-nected density-density correlation function decays to zeroin the long time limit and the system remains ergodic forall values of the disorder strength.Our finding is in contrast to that was found in Ref. [52].In that paper, various renormalized perturbation schemesfor the density correlation function, which are consistentwith the FDR, are presented. In one of the schemes, theconnected density correlation function does not decay tozero in the long-time limit, but approaches a finite valuewhen the disorder strength exceeds some critical value.This signals an ergodic-nonergodic transition. This is, however, in contrast to our exact result, since if the tran-sition exists, the right hand side of Eq. (50) would givea different value at the transition. These renormalizedperturbation theories basically correspond to replacingthe bare correlation functions in some perturbation ex-pansion scheme with the renormalized ones. These inturn produce various types of self-consistent equationsfor the renormalized correlation function. In terms of theFeynman diagrams, a renormalized perturbation theorycorresponds to the partial resummation of a particularinfinite subset of diagrams contributing to the densitycorrelation function. Our exact result suggests that theergodic-nonergodic transition seen in Ref. [52] might bean artifact of the partial resummation of the diagrams.The situation is very reminiscent of the dynamical tran-sition predicted by the MCT of supercooled liquids [49],which can also be regarded as a partial resummation ofdiagrams for the full theory of supercooled liquids. It isexpected that the sharp transition will be smeared outwhen other effects such as activated hopping are includedand that the system remains ergodic [34, 35, 53].In the present work, we have only considered the zero-frequency component of the response function. In orderto find more useful information on the transport proper-ties of the system it will be necessary to find a reliablescheme to calculate the full time dependence of the cor-relation and response functions. We have tried to imple-ment various renormalized perturbation schemes includ-ing those presented in Ref. [52] for the density correlationfunction. We have also encountered the same problemsas in Ref. [52] such as spurious instabilities when we tryto find solutions to self-consistent equations. We believethat in order to further improve the renormalized pertur-bation theory for this system, one needs to consider therenormalization of the vertices as well as the propagatorsand to find self-consistent equations for these quantities.This is left for future work. For this kind of calculationsand also for an eventual generalization to the systemof interacting Brownian particles in a quenched randompotential, we believe that the perturbation scheme pre-sented in this work would provide a convenient startingpoint.
Appendix A: Properties of Gaussian disorder average
We consider the disorder average of the denominator on the right hand side of Eq. (9). We have (cid:90) d d x e − Φ( x ) /T = (cid:90) d d x e ∆( ) / (2 T ) = V e ∆( ) / (2 T ) , (A1)1where V is the volume. We calculate the variance of this quantity as (cid:20)(cid:90) d d x e − Φ( x ) /T − V e ∆( ) / (2 T ) (cid:21) = (cid:90) d d x (cid:90) d d x (cid:48) e (∆( )+∆( x − x (cid:48) )) /T − V e ∆( ) /T = V e ∆( ) /T (cid:90) d d x ( e ∆( x ) /T − . (A2)If we assume ∆( x ) is a short-ranged function, then the integral gives a finite quantity. Therefore the disorder averageand the variance of (cid:82) d d x e − Φ( x ) /T are both proportional to V and we can replace it by its average value in thethermodynamic limit. Appendix B: Evaluation of the diagram in Fig. 10
We consider the case where there are n dahsed lines in Fig. 10 carrying the momenta k , k , · · · k n . Figure 10 canbe represented in the action as (cid:90) dt (cid:90) d d q (2 π ) d (cid:90) d d q (2 π ) d (cid:90) n (cid:89) i =1 d d k i (2 π ) d (2 π ) d δ ( d ) ( q + q − n (cid:88) i =1 k i ) × ( − D q · q )Λ n ( k , k , · · · , k n ) i ˆ ρ ( − q , t ) i ˆ ρ ( − q , t ) n (cid:89) i =1 ˜Φ( k i ) (B1)for some vertex function Λ n . In order to evaluate Λ n , we consider the case where the momenta going in through thedashed lines from right to left in Fig. 10 are given by k , k , · · · k n . For convenience, we use the shorthand notationfor the bare propagator as g ( q , t ) ≡ i ˜ G ρ ˆ ρ ( q , t ) = i ˜ F ρ ˆ ρ ( q , t ) = Θ( t ) e − D q t , (B2)Then, for this particular configuration of the momenta, the contribution to Λ n is given by (cid:90) ∞−∞ n (cid:89) i =1 dt i {− ρ Γ k · k } g ( k , t − t ) {− Γ k · ( k + k ) } g ( k + k , t − t ) × · · · × {− Γ k n − · ( k + · · · + k n − ) } g ( k + · · · + k n − , t n − t n − ) × {− Γ k n · ( k + · · · + k n ) } g ( k + · · · + k n , t − t n ) , (B3)where t i denotes the time when the momentum k i is coming in for i = 1 , , · · · , n . We make the change of variables, s i = t i +1 − t i for i = 1 , · · · , n , ( t n +1 = t ) with the unit Jacobian. Then the time integral over s i is from 0 to ∞ dueto the step function in the bare propagator and can be evaluated explicitly as ρ n (cid:89) i =1 (cid:26) − Γ k i · ( k + k + · · · k i ) (cid:90) ∞ ds i e − D ( k + k + ··· k i ) s i (cid:27) = ρ ( − T ) n n (cid:89) i =1 k i · ( k + k + · · · k i )( k + k + · · · k i ) , (B4)As can be seen from Eq. (B1), Λ n can be symmetrized with respect to the permutation of its arguments. Therefore,we have Λ n ( k , k , · · · , k n ) = 1 n ! ρ ( − T ) n O n ( k , k , · · · , k n ) , (B5)where O n ( k , k , · · · k n ) ≡ (cid:88) P n (cid:89) i =1 k i · ( k + k + · · · k i )( k + k + · · · k i ) (B6)with the summation over the permutations P of ( k , k , · · · k n ).We now prove by induction that O n = 1 for all n = 1 , , · · · . It is trivial to see that O = 1 and that O ( k , k ) = k · ( k + k )( k + k ) k k + k · ( k + k )( k + k ) k k = 1 . (B7)2 q k n k n − k m +1 k m k t t n − m t n − m − t t FIG. 16. Diagram in Fig. 15 with the mometum labels. This is evaluated in Appendix D
Now we suppose that O n − = 1. We note that, compared to O n − , O n contains one additional factor which containsall the k i ’s. Therefore, we can write O n ( k , k , · · · , k n ) = n (cid:88) i =1 k i · ( k + k + · · · k n )( k + k + · · · k n ) O n − ( k , · · · , k i − , k i +1 , · · · , k n ) , (B8)and from the assumption we conclude that O n ( k , k , · · · , k n ) = n (cid:88) i =1 k i · ( k + k + · · · k n )( k + k + · · · k n ) = 1 . (B9)Therefore, from Eq. (B5), we have Λ n = ρ / ( n !( − T ) n ). By using this value in Eq. (B1) and comparing it with thelast term in Eq. (18), we find that the diagram in Fig. 10 is equal to the renormalized type- B n vertex, which carries ρ instead of ρ ∗ . Appendix C: Evaluation of the diagram in Fig. 13
We first note that both disconnected parts in Fig. 13 have the same structure as the diagram in Fig. 10 treated inAppendix B except that at the end points we have the ∆ ρ fields instead of the noise vertex. For the disconnected parton the left hand side, if we denote the external momentum coming out of the left end point by q and the momentacoming in the n -dashed lines by k i , i = 1 , , · · · , n , we can write this part as (cid:90) n (cid:89) i =1 d d k i (2 π ) d (2 π ) d δ ( d ) ( q − n (cid:88) i =1 k i )Λ n ( k , k , · · · , k n ) n (cid:89) i =1 ˜Φ( k i ) (C1)with the same vertex function Λ n as considered in Eq. (B1). As expected, there is no time dependence. We representthe part on the right hand side in the same way using the momenta q (cid:48) and k (cid:48) i , i = 1 , , · · · , n . We then connectall possible pairs of the dashed lines. This produces the disorder correlation ˜∆( k i ) with the delta function enforcing k i + k (cid:48) i = 0. Therefore, along with the overall delta function (2 π ) d δ ( d ) ( q + q (cid:48) ), we have n ! (cid:90) n (cid:89) i =1 (cid:18) d d k i (2 π ) d ˜∆( k i ) (cid:19) Λ n ( k , k , · · · k n )(2 π ) d δ ( d ) ( q − n (cid:88) i =1 k i ) , (C2)where n ! denotes the number of different ways of connecting the dashed lines. If we use the result Λ n = ρ / ( n !( − T ) n )obtained in Appendix B, we find that this is exactly Eq. (38). Appendix D: Evaluation of the vertex function ˜Ξ n ( q ; k , k , · · · , k n ; ω = 0) We consider the diagram in Fig. 15 with total n dashed lines. Among the n dashed lines, we consider the situationdescribed in Fig. 16, where there are m ( m ≤ n ) dashed lines at the right external source at time t (cid:48) and n − m dashed lines coming from a series of the type- A vertices reaching at the left external source at time t . We denotethe m momenta coming in through the dashed lines at the right end point by k , k , · · · k m , and those coming in attimes t , t , · · · , t n − m through the remaining dashed lines by k m +1 , k m +2 , · · · k n from right to left, respectively. For˜ R ( q , t − t (cid:48) ), the momentum coming out of the left external source at time t is q and the momentum conservation gives3the condition that that (cid:80) ni =1 k i = 0. For this particular configuration of { k i } , the contribution from this diagram toΞ n ( q ; k , k , · · · , k n ; t − t (cid:48) ) in Eq. (45) is − ρ ∗ Γ m !( − T ) m (cid:90) ∞−∞ n − m (cid:89) i =1 dt i (cid:110) − q · ( q + m (cid:88) j =1 k j ) (cid:111) g (cid:16) q + m (cid:88) j =1 k j , t − t (cid:48) (cid:17) × (cid:110) − Γ k m +1 · ( q + m +1 (cid:88) j =1 k j ) (cid:111) g (cid:16) q + m +1 (cid:88) j =1 k j , t − t (cid:17)(cid:110) − Γ k m +2 · ( q + m +2 (cid:88) j =1 k j ) (cid:111) g (cid:16) q + m +2 (cid:88) j =1 k j , t − t (cid:17) × · · · × (cid:110) − Γ k n · ( q + n (cid:88) j =1 k j ) (cid:111) g (cid:16) q + n (cid:88) j =1 k j , t − t n − m (cid:17) (D1)with the bare propagator g given in Eq. (B2). In general it is difficult to evaluate the time integrals to get a closedform. But, if we focus on the time integral over τ = t − t (cid:48) of the above quantity to get the zero-frequency limit of theresponse function from Eq. (47), we can make a progress. By changing the integration variables from t , · · · , t n − m and τ to s = t − t (cid:48) , s = t − t , · · · , s n − m = t n − m − t n − m − and s n − m +1 = t − t n − m , with the unit Jacobian, wecan explicitly evaluate all the time integrals as we have done in Appendix B. The result is − ρ ∗ m !( − T ) n +1 q · ( q + k + · · · + k m )( q + k + · · · + k m ) n (cid:89) i = m +1 k i · ( q + k + · · · + k i )( q + k + · · · k i ) . (D2)As can be seen from Eq. (45), we have to symmetrize this quantity over all the permutations of ( k , · · · , k n ). Wealso have to consider all possible cases of m = 0 , , · · · , n for given n . Therefore we can express the quantity inEq. (47) as ˜Ξ n ( q ; k , k , · · · , k n ; ω = 0) = − ρ ∗ ( − T ) n +1 n ! n (cid:88) m =0 O nm ( q ; k , k , · · · , k n ) , (D3)where O nm ( q ; k , k , · · · , k n ) ≡ (cid:88) P (cid:40) m ! q · ( q + k + · · · + k m )( q + k + · · · + k m ) n (cid:89) i = m +1 k i · ( q + k + · · · + k i )( q + k + · · · + k i ) (cid:41) , (D4)where (cid:80) P indicates the sum over all possible permutations of ( k , k , · · · , k n ). We will prove below that (cid:80) nm =0 O nm =1 for all n = 0 , , , · · · by mathematical induction. First we note that O = 1 and O ( q ; k ) = k · ( q + k )( q + k ) , O ( q ; k ) = q · ( q + k )( q + k ) , (D5)thus O + O = 1. Now we suppose that (cid:80) n − m =0 O n − ,m = 1. Using the method similar to the one used to deriveEq. (B8), we obtain O nm ( q ; k , k , · · · , k n ) = n (cid:88) i =1 (cid:110) k i · ( q + k + · · · + k n )( q + k + · · · + k n ) O n − ,m ( q ; k , · · · , k i − , k i +1 , · · · , k n ) (cid:111) , (D6)for m ≤ n −
1. We therefore find that n (cid:88) m =0 O nm ( q ; k , · · · , k n ) = n − (cid:88) m =0 O nm ( q ; k , · · · , k n ) + O nn ( q ; k , · · · , k n )= n − (cid:88) m =0 n (cid:88) i =1 (cid:110) k i · ( q + k + · · · + k n )( q + k + · · · k n ) O n − ,m ( q ; k , · · · , k i − , k i +1 , · · · , k n ) (cid:111) + O nn ( q , { k , · · · k n } )= n (cid:88) i =1 k i · ( q + k + · · · + k n )( q + k + · · · + k n ) + q · ( q + k + · · · + k n )( q + k + · · · + k n ) = 1 . (D7)4From Eq. (D3), we finally have ˜Ξ n ( q ; k , · · · , k n ; ω = 0) = − ρ ∗ ( − T ) n +1 n ! (D8)regardless of its arguments. ACKNOWLEDGMENTS
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