Instanton theory for bosons in disordered speckle potential
aa r X i v : . [ c ond - m a t . d i s - nn ] M a y Instanton theory for bosons in disordered speckle potential
G. M. Falco , and Andrei A. Fedorenko Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, D-50937 K¨oln, Germany CNRS UMR5672 – Laboratoire de Physique de l’Ecole Normale Sup´erieure de Lyon, 46, All´ee d’Italie, 69007 Lyon, France (Dated: May 7, 2015)We study the tail of the spectrum for non-interacting bosons in a blue-detuned random specklepotential. Using an instanton approach we derive the asymptotic behavior of the density of states in d dimensions. The leading corrections resulting from fluctuations around the saddle point solutionare obtained by means of the Gel’fand-Yaglom method generalized to functional determinants withzero modes. We find a good agreement with the results of numerical simulations in one dimension.The effect of weak repulsive interactions in the Lifshitz tail is also discussed. I. INTRODUCTION
Effect of disorder on quantum systems has been at-tracted considerable interest in condensed matter physicsduring the last several decades. In recent years, it wasrealized that ultracold atomic gases in optical speckle po-tentials may serve as quantum simulators for diverse phe-nomena in disordered quantum systems [1–3]. Contraryto the experiments in condensed matter physics, the op-tical speckles allow one to create a controllable randompotentials acting on ultracold atoms [3, 4]. Many inter-esting features of Bose-Einstein condensates (BECs) indisordered speckle potentials have been addressed fromboth the experimental and the theoretical sides [5–9].These include inhibition of transport properties [6, 7, 10],fragmentation effects [5, 11, 12], frequency shifts [5, 13],damping of collective excitations [5, 12, 13] and Andersonlocalization [14–21]. Also, the superfluid-insulator tran-sition [22–30] and the transport of coherent matter waveshave been recently investigated from the theoretical pointof view for speckles in higher dimensions [31–36].When a coherent laser light is scattered from a roughsurface the partial waves passing through the differentparts of the surface acquire random phase shifts. Theinterference of these randomly phased waves produces aspeckle pattern which consists of the regions or grains oflight intensity with random magnitude, size and position.The local intensity of the speckles, I ( x ) = |E ( x ) | , isdetermined by the electric field E ( x ). To a very goodapproximation, the electric field E ( x ) can be viewed as acomplex Gaussian variable with finite correlation length ξ giving the typical size of the light intensity grains inthe speckle pattern. The distribution of intensity I ( x )across the speckle pattern follows a negative-exponential(or Rayleigh) law [13, 37], P [ I ] = exp [ − I/I ] /I , (1)where I = h I i is the mean intensity while the most prob-able intensity is zero. The speckle pattern shined on asample of atoms creates a random potential felt by theatoms provided that the wavelength of the laser light isslightly detuned from the atomic resonance. The poten-tial is proportional to the local light intensity, V ( x ) = α I ( x ), so that the non-interacting atoms in the speckle potential can be described by the Schr¨odinger equation (cid:20) − ¯ h m ∇ + α I ( x ) (cid:21) ψ ( x ) = E ψ ( x ) , (2)where m is the mass of atoms and ψ ( x ) is the singleparticle wave function. The constant α is proportionalto the inverse of the detuning ∆ between the laser andthe atomic resonance. The detuning can be either posi-tive (blue detuning) or negative (red detuning) [34]. Theblue-detuned case corresponds to a disordered potentialconsisting of a series of barriers bounded from below. Thered-detuned speckle produces a potential bounded fromabove and made of potential wells.The precise single particle spectrum of the speckle po-tential is unknown even in the 1D case despite the in-tense research activity in this field. It is widely believedthat the density of states (DOS) of blue-detuned repul-sive speckles is characterized by a usual Lifshitz tail forpotentials bounded from below [38]. However, manyexact results known for 1D random potentials [38, 39]cannot be directly applied to the speckle potential sinceit is correlated and non-Gaussian. In the previous pa-per [40] we investigated analytically and numerically the1D single-particle spectrum for the both red-detuned andblue-detuned speckle potentials. Since the speckle pat-tern is characterized, besides the mean intensity I , bythe correlation length ξ , this introduces a new energyscale E ξ = ¯ h / m ξ . We have shown that for dimen-sional reasons, the single-particle properties are deter-mined by the dimensionless parameter s = 2 m ξ αI / ¯ h for the both red and blue detuned speckle potentials.We identified different Lifshitz regimes controlled by thedimensionless parameter s which vary from the semiclas-sical limit | s | ≫ | s | ≪ E ξ theLifshitz tail does not depend on the precise form of theelectric field correlations since the DOS is determined bythe states localized in the regions with very small inten-sity whose size is much larger than ξ . In this regime theinstanton approach proposed in Ref. [41] for the poten-tial bounded from below predicts the following form ofthe DOS tail ν ( E ) = A d ( E ) exp h − v d F d [ln ( I /E )] ( E ξ /E ) d/ i , (3)where v d is a constant which depends on the dimensionand F d and A d are some functions of the energy.The paper is organized as follows. In Sec. II we de-rive the replicated action for a particle in a blue-detunedspeckle potential. The saddle point solution of the classi-cal equation in d dimensions is discussed in Sec. III wherethe expressions for v d and F d are calculated. The fluc-tuations around the instanton solution are investigatedin Sec. IV. There, the prefactor A d is calculated usingthe Gel’fand-Yaglom (GY) method [42, 43] generalizedby Tarlie and McKane (TM) [44] to functional determi-nants with excluded zero modes. In order to illustrate thepower of this method, in Appendix C we reconsider theproblem of a particle in Gaussian uncorrelated disorderfor which there are some discrepancies in the existingliterature. In Sec. V we consider a weakly interactingBose gas in a speckle potential. The obtained results aresummarized in Conclusion. II. AVERAGING OVER DISORDER:REPLICATED ACTION
The DOS for a particle in a particular realization ofthe electric field E ( x ) can be related to the imaginarypart of the one-particle Green function ν ( E ) = − π Im G ( x, x ; E ) . (4)To average over different realizations of disorder potentialwe employ the replica trick [45]. We introduce N repli-cas of the original system and use the functional integralrepresentation G ( x, x ′ ; E ) = lim N → Z D φ φ ( x ) φ ( x ′ ) e − S [ ¯ φ ] (5)with the action S [ ¯ φ ] = 12 Z d d x (cid:26) ¯ h m (cid:0) ∇ ¯ φ ( x ) (cid:1) + [ V ( x ) − E ] ¯ φ ( x ) (cid:27) , (6)where ¯ φ ( x ) is an N -component scalar field. The disor-der potential is proportional to the local intensity of thespeckles pattern created by a laser, V ( x ) = α E ∗ ( x ) E ( x ),where E ( x ) is the electric field and we fix α = +1 in thecase of a blue-detuned speckle. To a very good approxi-mation, the electric field is a random complex Gaussianfield with zero mean and variance hE ∗ ( x ) E ( y ) i = G ( x − y ) , (7) where the function G ( x − y ) has the width of ξ and itsprecise form depends on the experimental setup [18, 32].Then the average of the disorder potential can be ex-pressed as exponential of the sum of loop diagrams: (cid:28) exp[ − Z d d x E ∗ ( x ) E ( x ) ¯ φ ( x )] (cid:29) = exp (cid:26) −
11! + 12! −
13! + ... (cid:27) . (8)In Eq. (8) the lines stand for the correlator (7) andthus have two distinct ends corresponding to E ∗ and E . Two lines can be connected only by ends of differ-ent types and the corresponding vertex carries a factorof ¯ φ ( x ) = P Nn =1 φ n ( x ). The loop diagram with j vertices in Eq. (8) has the combinatorial factor of ( j − j lines with distinct ends. Summing up the alldiagrams for an arbitrary function G ( x ) and field ¯ φ ( x )is a formidable task. However, there are several caseswhen one can solve this problem at least partially. Fora special class of correlation functions G ( x ) the sum ofthe diagrams can be rewritten as a ratio of functional de-terminants (see Appendix A). The summation of the dia-grams can be also performed within a variational methodwith Gaussian correlators and trial functions (see Ap-pendix B). These approximations can be used to showthat the low-energy tail of the DOS does not depend onthe precise form of the electric field correlator for E ≪ E ξ and it is completely determined by E ξ and I . This is incontrast to the Gaussian unbounded potential [45] wherethe presence of correlations changes the low-energy Lif-shitz tail of the DOS [46]. For E ≪ E ξ we can approxi-mate the electric field correlator by G ( x − y ) = I ξ d δ dξ ( x − y )where we have defined the regularized δ -function of width ξ such thatlim ξ → δ dξ ( x − y ) = δ d ( x − y ) and δ dξ (0) = 1 ξ d . As a result the loop diagram with j insertions of ¯ φ ( x )can be expressed as( − j j ! f f f f f f = Z d d xξ d ( − j j j ( I ξ d ) j ¯ φ j ( x ) . (9)By using the relation ∞ X j =1 ( − j j j ( I ξ d ) j ¯ φ j ( x ) = − ln (cid:20) I ξ d ) ¯ φ ( x ) (cid:21) , we can sum up all the diagrams. The averaged replicatedaction then reads S av = Z d d x n ¯ h m ( ∇ ¯ φ ( x )) − E ¯ φ ( x )+ ξ d ln (cid:2) ( I ξ d ) ¯ φ ( x ) (cid:3)o . (10) III. SADDLE POINT SOLUTION
The field theory (10) has a trivial vacuum state ¯ φ = 0.However, a perturbative expansion around it does notcontribute to the DOS at any finite order since the Greenfunction remains real in this approximation. FollowingRef. [41, 45] we assume that the functional integral (5) isdominated by a spherically symmetric saddle point fieldconfiguration. The integration over fluctuations aroundthis instanton solution brings a finite imaginary part tothe Green function G leading to a finite DOS.It is convenient to express the action (10) in terms ofthe rescaled quantities ¯ φ ( x ) = √ φ ( x/ξ ) / p E ξ ξ d , ˜ x = x/ξ and ˜ E = E/E ξ . By omitting the tildes on ¯ φ and x we arrive at S av = Z d d x n(cid:2) ∇ x ¯ φ ( x ) (cid:3) − ˜ E ¯ φ ( x ) + ln (cid:2) s ¯ φ ( x ) (cid:3)o . (11)The variational principle gives the following classicalequation of motion ∇ ¯ φ + ˜ E ¯ φ = ¯ φs − + ¯ φ . (12)Assuming that Eq. (12) has a solution of the form¯ φ cl ( x ) = ¯ n φ ( x ) (13)with ¯ n = 1, we rewrite it as ∇ φ + ˜ Eφ = φ s − + φ . (14)It is instructive to compare this equation with the cor-responding saddle point equation (C3) in the case of δ -correlated Gaussian disorder (see [45] and Appendix C).At variance with the Gaussian disorder, one cannot elim-inate the explicit energy dependence in Eq. (14) by anyvariable transformation. In the limit E →
0, the classicalsolution of Eq. (14) approaches the form φ ( x ) ≈ r a d ln (cid:16) s/ ˜ E (cid:17) / ˜ E L d (cid:16) x p ˜ E (cid:17) (15)in the region 0 ≤ x p ˜ E ≤ µ d and essentially vanisheselsewhere. The functions L ( t ) = cos t , L ( t ) = J ( t )and L ( t ) = sin t/t are the spherical Bessel functionsin d dimensions, µ d is the first zero of L d ( µ = π/ µ = 2 . µ = π ) and the constants are a d =1 , . , π in d = 1 , ,
3, respectively [41]. Substitut-ing this approximative solution for the saddle point intothe action (11) leads immediately to the DOS tail of theform [3, 40, 41] ν ( E ) = A d exp h − v d ( E ξ /E ) d ln ( I /E ) i , (16)with v d = µ dd π d/ / Γ( d/ S cl = Z dx n [ ∇ x φ ( x )] − U ( φ ) o , (17) of a particle moving in the potential U ( φ ) = ˜ Eφ ( x ) − ln (cid:2) sφ ( x ) (cid:3) with space coordinate φ and time x .Since the system is conservative the energy of the particle E = ˙ φ + U ( φ ) (18)is constant along any trajectory. The saddle point so-lution corresponds to the particle trajectory at E = 0.By using a simple variable transformation the action (17)can be rewritten as S cl = 2 z Z dz q z − ln(1 + s z ) − ˜ E, (19)where z corresponds to zero of the expression under theroot. For ˜ E → S cl ≈ π q / ˜ E ln (cid:16) s/ ˜ E (cid:17) in agreement with the DOS (16)in one dimension. IV. FLUCTUATIONS AROUND THE SADDLEPOINT
In this section we extend the instanton approach in or-der to calculate the dependence of the prefactor A d upon E . To this end we expand the action (10) around the sad-dle point ¯ φ = ¯ φ cl + ¯ φ ′ to second order in the fluctuationfields. The instanton contribution to the Green functionis then given by G ( x, x ′ ; E ) ∼ Z D φ ′ φ cl1 ( x ) φ cl1 ( x ′ ) × exp[ − Z d d xφ ′ α M αβ φ ′ β ] . (20)Assuming the saddle point solution of the form (13) theoperator M αβ can be diagonalized using the longitudinaland transverse projector operators in the replica space as M αβ = M L n α n β + M T ( δ αβ − n α n β ) . (21)The transverse and longitudinal operators can be writtenin the form M T,L = −∇ + U T,L + m (22)where we have defined the mass m = s − ˜ E such thatthe potentials U T ( r ) = s s φ − s, (23) U L ( r ) = s s φ − s − s φ (cid:0) s φ (cid:1) (24)vanish at infinity. Note that the transverse projector op-erator has ( N −
1) zero modes corresponding to invari-ance under O ( N ) rotations in the replica space while thelongitudinal operator has d zero modes corresponding totranslational invariance. In order to obtain a finite resultfrom the Gaussian integration in Eq. (20), the zero modesof the operators M T and M L have to be separated andintegrated out exactly without using the Gaussian ap-proximation. To that end one can perform transforma-tion to a collective coordinates x and ¯ n [45, 47]. Thisyields G ( x, x ′ ; E ) ∼ Z d d x d ¯ n J t φ ( x − x ) φ ( x ′ − x ) × Z D ¯ φ ′ exp[ − Z d d xφ ′ α M ′ αβ φ ′ β ] , (25)where J t is the Jacobian of the transformation to the col-lective coordinates x and ¯ n and the prime in M ′ αβ meansthat the zero modes have been omitted. The Jacobiancalculated to leading order in the energy ˜ E by expand-ing the model in the fields around the minimum is givenby [48] J t ∼ (cid:20)Z d d x (cid:0) ∇ ¯ φ (cid:1) (cid:21) d/ (cid:20)Z d d x ¯ φ (cid:21) ( N − / . (26)The functional integral in Eq. (25) contributes with Z D ¯ φ ′ exp (cid:20) − Z dxφ ′ α M ′ αβ φ ′ β (cid:21) = det ′ ˜ M L − / · det ′ ˜ M T − ( N − / (27)where det ′ stands for the product of all non-zero eigenval-ues including the continuous part of the spectrum. Theoperators M T and M L given by Eq. (22) are Schr¨odinger-like operators. Their spectrum turns out to be very sen-sitive to the precise form of the saddle point solution ofEq. (14), which is not known analytically even in onedimension. Unfortunately, the dependence of the eigen-values on the energy parameter ˜ E cannot be easily ex-tracted using simple scaling arguments as in the case ofGaussian uncorrelated disorder (see Appendix C). Nev-ertheless, there are methods which allow one to calculatethe functional determinants with excluded zero modeseven without knowing precisely the spectrum. A. One dimensional case
We start with the one dimensional case. The poten-tials U T and U L corresponding to the operators M T and M L obtained from numerical solution of the saddle pointequation in d = 1 are shown in Fig. 1. In the low en-ergy limit E →
0, we find that the potentials U T and U L approach asymptotically a square potential well ofwidth π q / ˜ E and depth s ∓ ˜ E (cid:16) / ln s ˜ E (cid:17) respectively.The operator M T has one zero mode corresponding tothe lowest symmetric state. The operator M L has theonly zero energy state corresponding to the lowest anti-symmetric state while its lowest symmetric state has anegative eigenvalue which gives a non-zero contributionto the imaginary part of the Green function. x x U T,L
30 20 10 0 10 20 300.60.50.40.30.20.10.0 - - -
FIG. 1. One-dimensional potential wells U T (blue dashed)and U L (red solid) for s = 0 . E = 0 . When the spectrum of the Schr¨oedinger operator isknown analytically the zero mode can be explicitly ex-cluded from the product of eigenvalues. Therefore, thedeterminant can be calculated simply as an infinite prod-uct of non-zero eigenvalues. This is illustrated in Ap-pendix C 1 for the fluctuation operators M T and M L arising in the Gaussian disorder model [45]. For the bluedetuned speckle the determinants of M T and M L cannotbe calculated simply as a product of non-zero eigenvaluesbecause the spectrum cannot be found analytically. For-tunately, Gel’fand and Yaglom (GY) [42, 43] derived longago a general formula which allows one to calculate thefunctional determinant of a Schr¨odinger like operator atleast in one dimension without knowing any of its eigen-values. The GY method can be applied to an operatorof the form M = − d dx + U ( x ) + m , (28)which is defined on x ∈ [ − L, L ] for the wave functionssatisfying the boundary conditions u ( − L ) = u ( L ) = 0.The limit L → ∞ can be taken at the end of the calcu-lation. Since the well defined object is rather a ratioof two determinants than a single functional determi-nant itself it is convenient to introduce a free operator M free = − d dx + m . The GY theorem [42] states thatdet M det M free = u ( L ) u free ( L ) , (29)where u ( x ) and u free ( x ) are the solutions of the Cauchyproblems M u ( x ) = 0 and M free u free ( x ) = 0 (30)with the initial conditions: u ( − L ) = u free ( − L ) = 0 , u ′ ( − L ) = u ′ free ( − L ) = 1 . (31)Due to the presence of eigenfunctions with zero eigen-value, whose contributions to the determinant have to beexcluded, the GY formula (29) has to be slightly modifiedfor the operators M T and M L . A simple regularizationconsists of introducing an infinitesimal shift of the spec-trum by a small shift of the mass m . Then, the originaldeterminant with the excluded zero mode can be derivedby differentiating with respect to the mass. This methodis illustrated for the operators M T and M L of the Gaus-sian disorder model in Appendix C 2.In the case of the speckle potential it turns out to bemore convenient to use another regularization approachwhich has been recently proposed by Tarlie and McKane(TM) in Ref. [44]. It is based on the GY method gener-alized to an arbitrary boundary conditions by Forman inRef. [49]. The basic idea is to regularize the determinantby modifying the boundary conditions. This changes thezero eigenvalue to a nonzero one which can be estimatedto lowest order in the difference between the original andregularized boundary conditions. Assuming that the zeromode of the operator M is given by v ( x ) the ratio of thetwo determinants with excluded zero mode can be writ-ten as det ′ M det M free = − h v | v i v ′ ( − L ) v ′ ( L ) u ′ free ( − L ) u free ( L ) , (32)where we defined the scalar product h v | v i := Z L − L dx v ( x ) . (33)The TM formula is very useful because the zero mode ofthe operator M T is given by the classical solution φ ( x )while for the operator M L the zero mode is simply givenby its derivative φ ′ ( x ). This is true not only in the caseof the one dimensional speckle potential but also for theproblem with uncorrelated Gaussian disorder where theTM formula is shown to reproduce the correct result ofCardy (see Appendix C 3). By inserting the zero modesolutions in Eq. (32) the ratio of the two determinants M T and M L of Eq. (27) in one dimension can be rewrittenas det ′ M T det ′ M L = lim L →∞ h φ | φ ih φ ′ | φ ′ i φ ′′ ( − L ) φ ′′ ( L ) φ ′ ( − L ) φ ′ ( L ) , (34)because the contribution from the free operator cancelsout. Moreover, in one dimension, the derivatives of theclassical solution can be easily obtained from the firstorder differential equation (18) derived from the analogywith the particle in a conservative potential. Althoughthe function φ ( x ) is known only numerically, the limit ofEq. (34) can be calculated analytically by using that thissolution is regular at infinity. We find the exact relationdet ′ M T det ′ M L = − h φ | φ ih φ ′ | φ ′ i (cid:16) s − ˜ E (cid:17) . (35)When we substitute this result in the DOS (25) the ratioof two scalar products in Eq. (35) cancel exactly the Ja-cobian (26). Using the saddle point solution (15) in the E I - I - x FIG. 2. The one particle DOS for a blue-detuned specklepotential computed numerically for s = 1 (red squares) and s = 0 . asymptotic limit E → A ∼ ln (cid:16) s/ ˜ E (cid:17) ˜ E / s / . (36)By collecting all contributions, we find the tail of theDOS in one dimension ν = AξI (cid:18) I E (cid:19) / ln (cid:18) I E (cid:19) exp " − πs − / r I E f (cid:18) I E (cid:19) , (37)where A is a numerical constant and f ( y ) = 2 π z Z dz p z − ln(1 + yz ) − , (38)whose asymptotics for large y is f ( y ) = ln y . In orderto check this formula we have fitted the low energy tailof the DOS in 1D speckle potential which we computednumerically using the exact Hamiltonian diagonalizationin Ref. [40]. The result is shown in Fig. 2. B. Higher dimensions
The GY method can be extended to determinants ofthe Schr¨odinger-like operators in d dimensions in the caseof radially symmetric potentials [50]. Due to the radialsymmetry of the operators, their eigenfunctions can bedecomposed into a product of radial parts and hyper-spherical harmonicsΨ ( r, ϑ ) = 1 r d − ψ ℓ ( r ) Y ℓ ( ϑ ) . (39)The radial parts ψ ℓ ( r ) are then solutions to the radialequations M ( ℓ ) T,L ψ ℓ ( r ) := − d dr + (cid:0) ℓ + d − (cid:1) (cid:0) ℓ + d − (cid:1) r ++ U T,L ( r ) + m (cid:1) ψ ℓ ( r ) = λ ψ ℓ ( r ) . (40)The radial eigenfunctions ψ ℓ ( r ) come with a degeneracyfactor of deg ( ℓ ; d ) = (2 ℓ + d − ℓ + d − ℓ !( d − . (41)The determinant of a radially separable operator can becalculated by combining the determinants for each par-tial wave ℓ with the weights given by the degeneracy fac-tor (41) as followsln det ′ M T,L det M free = ∞ X ℓ =0 deg ( ℓ ; d ) ln det ′ M ( ℓ ) T,L det M ( ℓ )free , (42)where the free operators have been defined as M ( ℓ )free = − d dr + (cid:0) ℓ + d − (cid:1) (cid:0) ℓ + d − (cid:1) r + m . (43)In order to compute the determinants of the partial oper-ators in Eq. (42) one can use the GY method for those de-terminants that have no zero modes and the MT methodfor those that have them. The only partial operatorswhich have zero modes are M (0) T and M (1) L . All otherdeterminants can be computed using the GY formuladet M ( ℓ ) T,L det M ( ℓ )free = lim R →∞ u ( ℓ ) ( R ) u ( ℓ )free ( R ) , (44)where u ( ℓ ) and u ( ℓ )free are the solution of the followingCauchy problems M ( ℓ ) u ( ℓ ) ( r ) = 0 , u ( ℓ ) ∼ r ℓ + ( d − , r → , (45) M ( ℓ )free u ( ℓ )free ( r ) = 0 , u ( ℓ )free ∼ r ℓ + ( d − , r → . (46)The determinants of the operators M (0) T and M (1) L withexcluded zero modes are given by the generalization ofEq. (32) which readsdet ′ M det M free = − lim R →∞ h v | v i v ′ ( R ) u free ( R ) . (47)Here we have used that the zero modes v ( r ) of the op-erators M (0) T and M (1) L and the solution u free ( r ) of theCauchy problem (46) have the same behavior at r → ℓ in Eq. (42) diverges for d ≥ /ℓ . This divergence is a general property offunctional determinants in d ≥ ℓ in Eq. (42). Following Ref. [50] we use thediagrammatic representation of the determinants M T,L explained in Appendix A and given by Eq. (A2) with
PSfrag replacements ℓ ˜ E l n ( d e t ′ M T / d e t ′ M L ) − A T + A L FIG. 3. The logarithm of the ratio of determinants M T and M L with subtracted diagrams A T,L computed numerically for s = 0 . d = 2 as a function of ˜ E using Eq. (48). The solidblue line is a fit by 3 . / p ˜ E + 2 . E + 10 . d = 2, ˜ E = 0 . s = 0 . ℓ . The solidblue line is the asymptotic behavior 162085 / ( ℓ + 21 . . V ( r ) = U T,L ( r ). For 2 ≤ d < A which is linear in V . For d = 4 one hasalso subtract the diagram A which is quadratic in V .Subtracting the divergent diagram from the partial de-terminants we obtainln det ′ M T,L det M free = A + ∞ X ℓ =0 deg( ℓ ; d ) " ln det ′ M ( ℓ ) T,L det M ( ℓ )free ∅ ( V ) , (48)where the symbol [ ... ] ∅ ( V ) means that the parts of order V have been subtracted. The sum over ℓ in Eq. (48)becomes finite because the UV divergences have been ac-cumulated in the regularized Feynman diagram A . Theexplicit form of the terms which have to be subtracted fora general potential V ( x ) is given by the expansion [50]ln det M ( ℓ ) det M ( ℓ )free = ∞ Z dr r V ( r ) K ν ( mr ) I ν ( mr ) − ∞ Z dr r V ( r ) K ν ( mr ) r Z dr ′ r ′ × V ( r ′ ) I ν ( mr ′ ) + O ( V ) , (49)where I ν and K ν are the Bessel function with ν ≡ l + d − ℓ in Eq. (48) converges asymp-totically as 1 /ℓ in d = 2 and as 1 /ℓ in d = 3. As anexample, this sum is shown as a function of ˜ E for d = 2and s = 0 . A T − A L = Z q q + m Z d d x [ U T ( x ) − U L ( x )] . (50)Neglecting a power-law correction resulting from the Ja-cobian (26) we arrive at A d ( E ) ∼ exp " γ d (cid:18) I /E ln( I /E ) (cid:19) ( d − / (51)with UV cutoff-dependent coefficients γ = 1 .
89 ln(Λ /s )and γ = 14 . V. WEAKLY INTERACTING BOSE GAS INSPECKLE POTENTIAL
We now consider the effect of weak repulsive interac-tion on the bosons in the Lifshitz tail. We restrict ourconsideration to the three dimensional system in diluteregime. The corresponding Hamiltonian has the well-known form [55] H = Z d xψ † ( x ) (cid:18) − ¯ h m ∇ + V ( x ) − µ (cid:19) ψ ( x )+ g Z d d x ( ψ † ( x ) ψ ( x )) , (52)where ψ ( x ) is the secondary quantized wave function, V ( x ) - random potential and µ - the chemical poten-tial. The positive coupling constant is given by g =4 π ¯ h a s /m , where a is the scattering length and we as-sume a low concentration of bosons n , such that na s ≪ E is domi-nated by the optimal wells of width R with the energy E ( R ) = − ¯ h / (2 m R ) which decreases with shrinkingof R . The density of optimal wells, which can be foundfrom the corresponding instanton solution, is n w ( R ) ∼ e −L /R /R , where L = ¯ h / ( m γ ) is the so-called Larkinlength related to the strength of disorder γ [55]. In thepresence of weak repulsive interactions the positive re-pulsion energy per particle grows with decreasing R as E r ( R ) = 3 g N ( R ) / (4 πR ), where N ( R ) = n/n w ( R ) isthe typical number of bosons in optimal wells. Since theboth energies E ( R ) and E r ( R ) have opposite behaviorwith respect to decreasing R one has to optimize the to-tal energy in order to find the size of the optimal wellrenormalized by interactions. This yields with the loga-rithmic precision R ( n ) = L / ln( n c /n ). Then the relation between the chemical potential and the density is givenby [55] µ ( n ) = − ¯ h m R ( n ) ≈ − E (cid:16) ln n c n (cid:17) , (53)where E = ¯ h / ( m L ) and n c = (3 L a s ) − .In the case of the speckle potential both the energycorresponding to the optimal well E ( R ) = ¯ h / (2 m R )and the positive repulsion energy decay with growing thesize of the typical well. Thus, there is no competition be-tween the disorder and interactions so that we have noneed for optimization of the total energy. Neglecting thekinetic energy and using Eq. (16) to estimate the den-sity of the optimal wells we obtain with the logarithmicprecision µ ( n ) ≈ E ξ (cid:18) v ln I E ξ (cid:19) / (cid:16) ln n n (cid:17) − / , (54)where n = (6 ξ a s ) − and v = 4 π /
3. The asymptoticbehavior (54) holds for n ≪ n and E ξ ≪ I and is anagreement with Ref. [56]. VI. CONCLUSION
We have studied the low energy behavior of the DOSfor non-interacting bosons in a d dimensional blue de-tuned laser speckle potential. We have shown that for E ≤ E ξ the precise form of the electric field correla-tor does not affect the asymptotic behavior. Using aninstanton approach we have found the saddle point solu-tion which gives the leading exponential behavior. Inte-grating out the Gaussian fluctuation around this solutionwe have expressed the prefactor in the form of ratio oftwo functional determinants. In one dimension we cal-culated the ratio of functional determinants exactly us-ing the generalized GY method which allows one to takeinto account not only the discrete part of the spectrumof fluctuation operators but also the continuous one. Inhigher dimensions the corresponding ratio diverges thathas been overlooked in most of the previous work on theinstanton approach to the DOS of disordered systems.Using the partial wave decomposition we can separatethe UV divergences to a regularized one-loop Feynmandiagram and obtain a finite result for the DOS tail in d >
1. In Appendix C 4 we show that this method givesa correct result for the case of Gaussian uncorrelated dis-order. We also discussed the effect of weak repulsion in-teractions in the DOS tail. In contrast to the Gaussianunbounded disorder the interactions and disorder do notcompete and the optimal wells are not renormalized byinteractions that leads to a different dependence of thechemical potential on the bosons density.
ACKNOWLEDGMENTS
We would like to thank Thomas Nattermann, ValeryPokrovsky and Boris Shklovskii for useful discussions.AAF acknowledges support by ANR grants 13-JS04-0005-01 (ArtiQ) and 2010-BLANC-041902 (IsoTop).
Appendix A: Functional determinant representation
The diagrams in Eq. (A2) can not be summed up foran arbitrary distribution of the electric fields includingthose that appear in real experiments. In this appendixwe show that the sum can be performed for a specialchoice of the Gaussian distribution of the electric fieldswith zero mean and variance G ( x ) = I d/ Γ (cid:0) − d (cid:1) (cid:18) xξ (cid:19) − d K d/ − (cid:18) xξ (cid:19) , (A1)where K ν ( x ) is the modified Bessel function. A similarcorrelator appears in the problem of the Bragg glass stud-ied in Ref. [57]. For d = 1 the variance (A1) reduces to G ( x ) = I e −| x | /ξ . This is particular interesting becausethe asymptotic behavior of the DOS does not depend onthe precise form of correlations in disorder but it is de-termined by the lower energy states which spread overdistances larger than the disorder correlation length ξ .Therefore, in order to study the lowest order correctionsto the asymptotic tail due to presence of correlations onecan use Eq. (A1) as a reasonable approximation for thevariance of the electric field.The starting point is the following diagrammatic rep-resentation of the ratio of two functional determinantsln (cid:18) det( −∇ + V ( x ) + m )det( −∇ + m ) (cid:19) = ∞ X n =1 ( − n +1 n A n = −
12 + 13 − ... (A2)In the diagrams shown in Eq. (A2) the dots correspondto the potential V ( x ) and the lines stand for the Green’sfunction C ( x ) = m d − (2 π ) d/ K d/ − ( m | x | )( m | x | ) d/ − , (A3)which satisfies the equation (cid:2) −∇ + m (cid:3) C ( x ) = δ ( x ) . (A4)If we separate the combinatorial factors from the dia-grams in Eq. (8) we obtain the same series as in Eq. (A2).Thus, one can formally rewrite the sum of the diagrams inEq. (8) as a ratio of two functional determinants with themass m = 1 /ξ and the potential V ( x ) = I ¯ φ ( x ) /C (0)resulting from a random Gaussian electric field with zeromean and variance G ( x ) = I C ( x ) /C (0). Note thatdivergency of C (0) when d ≥ d ≥
2. Here we restrict ourselves to the case d = 1 and obtain S ( E ) = Z dx (cid:20) ¯ h m ( ∇ ¯ φ ( x )) − E ¯ φ ( x ) (cid:21) + ln (cid:18) det( −∇ + ( I /ξ ) ¯ φ ( x ) + 1 /ξ )det( −∇ + 1 /ξ ) (cid:19) . (A5)In the limit ξ → ∇ -operator in thedeterminants of Eq. (A5) and recover Eq. (10) using thatln det = Tr ln. The corresponding saddle point equation¯ h m ∇ ¯ φ ( x ) + E ¯ φ ( x ) = δδ ¯ φ ( x ) ln det (cid:2) −∇ + ( I /ξ ) ¯ φ ( x ) + 1 /ξ (cid:3) (A6)has the form of a gap equation known in relativistic quan-tum field theory [51]. Appendix B: Variational method with Gaussian trialfunctions
One can also sum up all diagrams in Eq. (8) for aparticular class of functions ¯ φ ( x ) that can be used tofind variationally an approximative instanton solution bymeans of the trial function method. Let us assume thatthe one dimensional electric field correlator has the form G ( x − y ) = I ξδ ξ ( x − y ) with δ ξ ( x ) = 1 ξ e − πx /ξ . (B1)For the trial function we consider φ ( x ) = ¯ nφ ( x ) with¯ n = 1 and φ ( x ) = √ Ce − πx / a . (B2)By substituting the trial function into the potential partof the action (8) we find that the diagram with n electricfield correlators reads f f f f f f = ( n − I n C n Z dx ...dx n exp (cid:20) − πx a − π ( x − x ) ξ ... − π ( x n − − x n ) ξ − πx n a − π ( x n − x ) ξ (cid:21) . (B3)Upon making the variable rescaling x i → ξx i /π , the in-tegral in Eq. (B3) can be rewritten as ξ n π n Z dx ...dx n exp − n X i,j =1 x i A ij x j = ξ n π n/ [det A n ] − / , (B4)where we have introduced the matrix A n = ǫ − ... − − ǫ − ... − ǫ ... ... ... ... ... ... ... ... ǫ − − ... − ǫ , with ǫ = ξ/a . The determinant of A n can then be calcu-lated, and we getdet A n = n − X m =0 Q mi =0 ( n − i )2 m (2 m + 1)!!( m + 1)! ǫ m = 4 sinh h n arcsinh ǫ i . (B5)Therefore, the action evaluated using the trial func-tion (B2) can be written as S ( E ) = Z dx (cid:20) ¯ h m ( φ ′′ ( x )) − Eφ ( x ) (cid:21) − ∞ X n =1 ( − n n ( I ξC ) n ( √ π ) n (cid:2) n arcsinh ǫ (cid:3) . (B6)When E ∼ /a ≪ E ξ = 1 / (cid:0) ξ (cid:1) we can approximatesinh (cid:2) n arcsinh ǫ (cid:3) ≈ nǫ/
2. As a result the second lineof Eq. (B6) is simplified to − Li [ − I ξC/ √ π ] /ǫ . This ex-pression can be also derived by applying the trial functionmethod directly to the action (10). This means that theaction (10) obtained in the limit of uncorrelated specklepotential properly describes the DOS in the presence ofcorrelations for E ≪ E ξ . For these low laying states thefinite range correlations play no role because the typicalwidth of the wave functions a is much larger than thecorrelation length of disorder ξ . Appendix C: The Lifshitz tail for a particle inGaussian uncorrelated disorder
In order to illustrate the power of the GY and MTmethods, we reconsider here the problem of a particlein uncorrelated Gaussian disorder. There existed a dis-agreement in the literature on the preexponential factorin the asymptotic behavior of the DOS in the tail of theband [45, 53, 58]. The two points that have not beensufficiently discussed in these works are the contributionof the continuous part of the spectrum to the functionaldeterminants and divergence of functional determinantsin d ≥ S av = Z d d x (cid:8) ( ∇ ¯ φ ( x )) − E ¯ φ ( x ) − ( γ/ ( ¯ φ ( x )) (cid:9) (C1)and we look for the asymptotic behavior of the DOS inthe limit E → −∞ . The saddle point solution to the action (C1) has the form¯ φ cl = ( − E ) / γ f ( √− Ex ) ¯ n, (C2)where f satisfies the dimensionless equation of motion ∇ f − f = − f . (C3)The action (C1) evaluated at the saddle point behaves as S cl ∼ ( − E ) − d/ /γ . By repeating the calculations (25)-(27) we find the DOS tail ν ( E ) ∼ ( − E ) / − d / d/ s det ′ M T det ′ M L e − S cl , (C4)where the regularized determinants of the fluctuation op-erators need to be calculated. The transverse and longi-tudinal operators derived by expansion around the saddlepoint solution have the form M T = −∇ − E (cid:20) − f (cid:16) √− Ex (cid:17) (cid:21) (C5) M L = −∇ − E (cid:20) − f (cid:16) √− Ex (cid:17) (cid:21) , (C6)where the energy E is assumed to be large and negative.
1. Brute force method in d = 1 The one dimensional case is interesting for testing theGY method because the ratio of determinants of the op-erators (C5) and (C6) can also be calculated directly fromthe product of their eigenvalues. The saddle point solu-tion of Eq. (C3) is f ( x ) = 4 √ x ) (C7)and the operators (C5)-(C6) can be rewritten in the formof P¨oschl-Teller operators [43, 51] M m,j = − ∂ ∂x + | E | h m − j ( j + 1) sech (cid:16) √− E x (cid:17)i , (C8)where j takes integer values. The operators (C5) and(C6) correspond to the case M T = M , and M L = M , . The spectrum of the P¨oschl-Teller operator M m,j contains a discrete part which have j bound states ε ℓ = | E | (cid:0) m − ℓ (cid:1) ( ℓ = 1 , ..., j ) and the continuous part ε ( k ) = | E | (cid:0) k + m (cid:1) with the density of states ν j ( k ) = L π − π j X ℓ =1 ℓℓ + k , (C9)0which has been regularized by putting the system in abox of size L . This yieldslog det M m,j det M m, = j X ℓ =1 log (cid:2) | E | (cid:0) m − ℓ (cid:1)(cid:3) + Z ∞−∞ dk ν j ( k ) log (cid:2) | E | (cid:0) k + m (cid:1)(cid:3) , (C10)where the free particle contribution L/ π has been can-celed. From this formula the functional determinants ofthe operators (C5)-(C6) can be determined straightfor-wardly. The operator M T = M , has just one discretezero eigenvalue. Excluding this zero mode, the only con-tribution comes from the continuous spectrum given bythe second line in Eq. (C10),det ′ M T det M free = 14 | E | , (C11)where M free := M m, . The operator M L = M , has twodiscrete eigenvalues: a negative eigenvalue ( ℓ = 2) givinga finite imaginary part of the Green function (4), anda zero eigenvalue ( ℓ = 1). The negative mode and thecontinuous spectrum contribute withdet ′ M L det M free = − | E | . (C12)We obtainlim N → det ′ M L − / · det ′ M T − ( N − / = i √ , (C13)which is independent of E .
2. Regularized Gel’fand-Yaglom formula
The same result can also be obtained using the GYmethod (29). The solution u m,j of the correspondingCauchy problem (30)-(31) for the P¨oschl-Teller opera-tors M m,j (C8) can be found analytically. For the sakeof compactness, we show here the formula for | E | = 1, u m,j ( L ) = 1(1 + j − m ) × − P mj ( Y ) Q mj ( − Y ) + P mj ( − Y ) Q mj ( Y ) P mj +1 ( − Y ) Q mj ( − Y ) − P mj ( − Y ) Q mj +1 ( − Y ) , (C14)where we defined Y := tanh L and P mj ( x ) and Q mj ( x ) arethe Legendre functions. The solution (C14) for the freeoperator M free = M m, reduces to u m, ( L ) = sinh [2 mL ] m . (C15)The ratio of the two determinants is then given bydet M m,j det M m, = lim L →∞ u m,j ( L ) u m, ( L ) . (C16) In order to exclude the zero modes, we apply a shift ofthe mass m → √ m + δm that givesdet (cid:0) M m,j + δm (cid:1) det ( M m, + δm ) ∼ δm det ′ M m,j det M m, , δm → . Restoring the dependence on E we obtaindet ′ M ,j det M , = ( − j +1 j ( j + 1) | E | (C17)in agreement with Eq. (C11)-(C12) for j = 1 and j = 2respectively.
3. McKane-Tarlie formula
We now apply the MT method. First, we need thezero modes of the operators M T = M , and M L = M , . They are given by v , ( x ) = | E | f ( √− Ex ) and v , ( x ) = | E | / f ′ ( √− Ex ), respectively. Then, accord-ing to Eq. (34), the ratio of the determinants with ex-cluded zero modes is given bydet ′ M T det ′ M L = lim L →∞ h v , | v , ih v , | v , i v ′ , ( − L ) v ′ , ( L ) v ′ , ( − L ) v ′ , ( L ) = − , (C18)where we used h v , | v , i = 2 | E | / , (C19) h v , | v , i = 23 | E | / , (C20) v ′ , ( −∞ ) v ′ , ( −∞ ) = − v ′ , ( ∞ ) v ′ , ( ∞ ) = √− E. (C21)
4. Gel’fand-Yaglom method generalized to radialoperators for d > The radial parts of the eigenfunctions of the opera-tors (C5) and (C6) satisfy Eq. (40) with the mass m = | E | and the potentials U T ( r ) = − | E | f (cid:16) √− Er (cid:17) , (C22) U L ( r ) = − | E | f (cid:16) √− Er (cid:17) . (C23)Scaling analysis shows that the solutions of the corre-sponding Cauchy problems (45) and (46) have the form u ( ℓ ) = | E | g ( ℓ ) ( | E | / r ) , (C24) u ( ℓ )free = | E | g ( ℓ )free ( | E | / r ) . (C25)There is no zero modes for ℓ > ℓ > R →∞ g ( ℓ ) ( | E | / R ) /g ( ℓ )free ( | E | / R ) , (C26)1which does not depend on E . Thus, all the ratios ofthe partial determinants with ℓ > E dependance of the full ratio of the functionaldeterminants.The operators M (0) T and M (1) L have a zero eigenvalueso that to exclude it we apply the MT method. Thecorresponding ratios of the partial determinants can becalculated using Eq. (47). The scaling behavior of thezero mode eigenfunction v ( r ) is again given by v ( r ) = | E | g ( | E | / r ) . (C27)This yields h v | v i ∼ | E | / , (C28)lim R →∞ v ′ ( R ) u free ( R ) = | E | / lim R →∞ g ′ ( R ) g free ( R ) , (C29)where the last limit is expected to be finite. Thus, thelogarithms of the determinant ratios are given (up to anenergy independent constants) byln det ′ M T det M free = deg (0; d ) ln det ′ M (0) T det M (0)free ∼ ln | E | − (C30)andln det ′ M L det M free = deg (1; d ) ln det ′ M (1) L det M (1)free ∼ d ln | E | − . (C31)Above we assumed that the ratios of the determinantsare finite and the sum over ℓ is converging. However, weknow that this sum diverges for d = 2 and d = 3. We haveto subtract from each ratio of the partial determinantsthe term resulting from the partial wave decompositionof the diverging diagram A . After that the regularizeddiagram A has to be added to the action as shown inEq. (48). The terms needed to be subtracted are given by Eq.(49) and read ∞ Z dr r U T,L ( r ) K ν ( | E | / r ) I ν ( | E | / r ) (C32)where ν ≡ l + d − U T,L are given by Eqs. (C22)and (C23). It is easy to see that the terms (C32) do notdepend on E . The bare diagram A can be written as A T,L = Z d d x U T,L ( | x | ) lim y → x C ( x − y ) , (C33)where C ( x ) is given by Eq. (A3). The expression (C33)diverges and has to be regularized, e.g. by the UV cutoffΛ as follows A T,L = Z | q | < Λ q + m Z d d x U T,L ( | x | ) , (C34)where the last integral behaves as ( − E ) − d/ . CombiningEqs. (C30)-(C31) and (C34) we obtain r det ′ M T det ′ M L ∼ i | E | d − e ( A T − A L ) / , (C35)where the factor i comes from the negative eigenvalueof the partial operator M (0) L . Inserting Eq. (C35) intoEq. (C4) gives the DOS tail for E → −∞ . 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