Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n epl draft Wave function correlations and the AC conductivity of disorderedwires beyond the Mott-Berezinskii law
G. M. Falco , Andrei A. Fedorenko and Ilya A. Gruzberg Amsterdam University of Applied Studies, Weesperzijde 190, 1097 DZ, Amsterdam, the Netherlands Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France Ohio State University, Department of Physics, 191 W. Woodruff Ave, Columbus OH, 43210, USA
PACS – Localization effects (Anderson or weak localization)
PACS – Quantum wires
PACS – Single particle states
Abstract – In one-dimensional disordered wires electronic states are localized at any energy.Correlations of the states at close positive energies and the AC conductivity σ ( ω ) in the limit ofsmall frequency are described by the Mott-Berezinskii theory. We revisit the instanton approachto the statistics of wave functions and AC transport valid in the tails of the spectrum (largenegative energies). Applying our recent results on functional determinants, we calculate exactlythe integral over gaussian fluctuations around the exact two-instanton saddle point. We derivecorrelators of wave functions at different energies beyond the leading order in the energy difference.This allows us to calculate corrections to the Mott-Berezinskii law (the leading small frequencyasymptotic behavior of σ ( ω )) which approximate the exact result in a broad range of ω . Wecompare our results with the ones obtained for positive energies. Introduction. –
One-dimensional (1D) systems haveplayed an important role in developing the theory of co-herent quantum transport in disordered solids. Examplesinclude Anderson localization of non-interacting particlesin the presence of disorder [1], the Mott insulating phase ininteracting systems [2] and the recently discovered many-body localization which takes place in the middle of thespectrum of disordered interacting systems [3].In the absence of interactions and decoherence, electronsin a 1D wire are localized at any energy even by a weakrandom potential. Thus, 1D wires lack some features ofthe higher dimensional systems, such as mobility edges.On the other hand, 1D systems are amenable to powerfulnon-perturbative methods, such as the phase formalism,which provide access to spectral and localization proper-ties which are much more difficult to obtain in higher di-mensions [4]. For example, in 1D it is possible to calculateexactly the average density of states (DOS), the localiza-tion length and the Lyapunov exponent, quantities thatdescribe statistics of a single localized wave function.Less is known about the wave function correlations atdifferent energies and dynamical response functions, suchas the finite-frequency (AC) conductivity σ ( ω ). The math- In this paper we only consider systems in the unitary class. ematical description of such correlations is quite involvedand only provided asymptotic analytical expressions exactin the limit of small energy differences [5–7]. An impor-tant result of this kind is the behavior of the dissipativeAC conductivity Re σ ( ω ) at low frequencies expressed bythe Mott-Berezinskii (MB) formula [5, 8].According to the intuitive arguments by Mott, correla-tions of wave functions at close energies may be describedin terms of hybridization of localized states. The leadingmechanism for the AC conductivity is the resonant tunnel-ing between pairs of localized states with energies E − ω/ E + ω/ E . Mott argued thatthe conductivity is dominated by the tunneling events be-tween states that are separated by the optimal distance(Mott scale) L M ∼ | ln ω | . This leads to the low-frequencybehavior Re σ ( ω ) ∝ ω ln ω in 1D.Ivanov and co-authors [9] augmented Mott’s argumentsby combining them with assumptions about statistics ofthe localized wave functions and hybridization matrix ele-ments. As a result, the authors quantitatively reproducedthe asymptotic features of correlators found in [6, 7].Mott’s arguments can be put on a rigorous basis whenthe resonant states are deep in the Lifshits tails of thespectrum. Then one can apply the instanton approach,p-1. M. Falco, A. A. Fedorenko, and I. A. Gruzberg, EPL , 37004 (2017)where the dominant contribution to observables is givenby a saddle point of the action, and subleading contribu-tions come from Gaussain fluctuations about the saddlepoint. This method has been used to calculate the aver-age single-particle Green functions (GFs) and the DOS insystems with Gaussian white noise [10–14] and correlateddisorder [15], systems in magnetic field [16], the Lorentzmodel with repulsive scatterers [17], systems with boundednon-Gaussian disorder [18], and speckle potentials [19].The instanton approach was applied in refs. [20, 21]to the average two-particle dynamic correlation func-tion S ( ω, x ) and the AC conductivity σ ( ω ) for small ω .These observables can be written as path integrals whichare dominated by non-trivial two-instanton saddle points.The latter correspond to hybridized states in the Mott’squalitative picture. Both the saddle points and the fluc-tuations around them were found only approximately inref. [20], which led to some inconsistent results. Theauthors of ref. [21] found exact two-instanton solutions,but their treatment of fluctuations was still approximateand restricted only to small frequency ω , reproducing theasymptotic MB formula. Kirsch et al. [22] tried to putMott’s argument on a rigorous basis using an expansionin small density of the localizing potential wells, and wereable to derive the MB formula as well as asymptotic for-mulas for wave function correlators in the limit of smallenergy differences.Recently we have achieved a significant progress in ex-act calculations of functional determinants [23]. Our re-sults are particularly well suited for applications in in-stanton calculations. As we have shown, in this case somecomplicated factors exactly cancel. Using these results,in this paper we calculate exactly the Gaussian fluctua-tions around the non-trivial two-instanton saddle pointsand derive the AC conductivity and local DOS correla-tions applicable in a broad range of frequency ω , and thusgoing beyond the seminal MB law. Model. –
We consider the model of non-interactingelectrons in the presence of disorder in 1D H = − ~ m ∇ + V ( x ) , ∇ ≡ ddx , (1)where V ( x ) is a random white noise Gaussian potential: h V ( x ) i = 0 , h V ( x ) V ( x ′ ) i = γδ ( x − x ′ ) . (2)We will work in the tails of the spectrum at E < A E = ~ √ m | E | / γ ≫ . (3)It is convenient to switch to dimensionless quantities byintroducing units of length and time: λ E = ~ / p m | E | , τ E = ~ / | E | . (4)These units play a role analogous to the mean free pathand time, but unlike the case of E >
0, they do not dependon the disorder strength, only on | E | . Density of states. –
The average DOS at the energy E is obtained from the average GF: ρ ( E ) = − απ Im G α (0; E ) , G α ( x − x ′ ; E ) = h G α ( x, x ′ ; E ) i ,G α ( x, x ′ ; E ) = h x | ( E + iα + − H ) − | x ′ i , (5)where α = ± x ) =( φ ( x ) , χ ( x )) T and its conjugate Φ † ( x ) = ( φ ∗ ( x ) , ¯ χ ( x )),where φ = φ x + iφ y is a complex bosonic field, and χ ,¯ χ are two fermionic fields. Upon rotation of the fieldsΦ → √− iα Φ in the complex plane [21], the average GFand the action in terms of dimensionless variables become G α ( x − x ′ ; E ) = − | E | γ Z D Φ χ ( x ) ¯ χ ( x ′ ) e −S , (6) S = A E Z dx h Φ † (cid:0) − ∇ (cid:1) Φ −
14 (Φ † Φ) i . (7)The corresponding classical equations of motion are (cid:16) − ∇ − | φ | (cid:17) φ = 0 , χ = 0 . (8)The trivial solution φ = 0 does not contribute to the imag-inary part of the GF and thus to the DOS. A nontrivial,one-instanton solution that we need is φ cl ( x − x , θ ) = e iθ ϕ ( x − x ) , ϕ ( x ) = 2 / cosh x. (9)The parameters x , θ describe translations in real spaceand rotations in the plane ( φ x , φ y ). The action of theone-instanton solution φ cl does not depend on x and θ : S ( E ) = 163 A E = 83 ~ √ m | E | / γ . (10)We now substitute φ = φ cl + ρ into the action and ex-pand to second order in fluctuations ρ : S = S + A E Z dx (cid:0) ρ x O d ρ x + ρ y O s ρ y + ¯ χO s χ (cid:1) , (11)where the fluctuation operators are O ν = −∇ + 1 − C ν / cosh ( x − x ) , (12)where ν = s, d (shallow and deep), C s = 1, and C d = 3.Gaussian integrals over ρ and χ give determinants of theoperators O ν . However, these operators have zero modesdue to the fact that any particular choice of x and θ breaks symmetries of the action. Whenever a brokensymmetry is described by a parameter ζ i , the function ψ i ( x ) = ∂ ζ i φ cl is a zero mode of one of the fluctua-tion operators. For instance, the lowest eigenvalue of O s , λ s = 0, is related to breaking the rotation invariance inthe ( φ x , φ y ) plane. The spectrum of O d starts with a neg-ative eigenvalue λ d <
0, followed by the zero eigenvaluep-2C conductivity of disordered wires λ d = 0 related to breaking the translation invariance. Thezero modes of O s and O d are ψ s ( x − x , θ ) = ∂ θ φ cl = ie iθ ϕ ( x − x ) , (13) ψ d ( x − x , θ ) = ∂ x φ cl = − e iθ ϕ ′ ( x − x ) . (14)We now separate the negative and zero modes and per-form the Gaussian integration of eq. (6) with the ac-tion (11). This yields, formally, G α ( x − x ′ ; E ) = − | E | γ e −S Λ s Λ d Λ d (cid:16) Det ′ A E O s Det ′′ A E O d (cid:17) / × ϕ ( x − x ) ϕ ( x ′ − x ) / h ϕ | ϕ i , (15)where h f | f i = R dxf ∗ ( x ) f ( x ), and Det ′ (Det ′′ ) standsfor a functional determinant with excluded zero (negativeand zero) eigenvalues. The contributions of the excludedmodes are denoted by Λ s , Λ d and Λ d . The last factor ineq. (15) comes from the fermionic integral in eq. (6) withthe action (11).The contributions Λ s and Λ d from the zero modes arecomputed by introducing the so-called collective coordi-nates [25–27]. When there are n zero modes ψ i , the col-lective coordinates are the parameters ζ i describing brokensymmetries, and the relevant contributionΛ ...n = 1 π n/ Z n Y i =1 dζ i q det h ψ i | ψ j i (16)involves the determinant of the n × n matrix whose ele-ments are overlaps of the zero modes h ψ i | ψ j i . In our casethis formula givesΛ s = 2 p π h ϕ | ϕ i , Λ d = 1 √ π p h ϕ ′ | ϕ ′ i Z dx . (17)The contribution Λ d from the negative mode can becomputed by an analytic continuation [21, 25]. We needto rotate the integration contour toward the saddle pointin the direction ∝ √− iα . At the saddle point the contourturns by − απ/ / α :Λ d = Z ∞−∞ dζ √ π e − λ d ζ → iα Z ∞ dζ √ π e λ d ζ = iα p − λ d . (18)Combining this contribution (18) with the ratio of deter-minants in eq. (15) we obtainΛ d (cid:16) Det ′ A E O s Det ′′ A E O d (cid:17) / = 12 (cid:16) Det ′ O s Det ′ O d (cid:17) / . (19)Here Det ′ O d <
0, and we need to choose the phase ac-cording to eq. (18).Calculation of the functional determinants can be doneexplicitly (see e.g. [19]) but in the subsequent study ofcorrelations of wave functions similar explicit calculations will be impossible. Thus, we use results of ref. [23] (gener-alizing those of [28, 29]), where we have shown that whenan n × n matrix Schr¨odinger operator O defined on theinterval x ∈ ( a, b ) with homogeneous boundary conditionshas n zero modes ψ i ( x ) (vectors with components ψ ij ( x ), i, j = 1 , .., n ), its functional determinant with excludedzero eigenvalues is equal toDet ′ O = ( − n det h ψ i | ψ j i det ψ ′ ij ( a )[det ψ ′ ij ( b )] ∗ . (20)This equation is formal and needs to be used in a ratio oftwo determinants. For the ratio in eq. (19) we obtainDet ′ O s Det ′ O d = h ϕ | ϕ ih ϕ ′ | ϕ ′ i lim R →∞ ϕ ′′ ( − R ) ϕ ′′ ( R ) ϕ ′ ( − R ) ϕ ′ ( R ) = − h ϕ | ϕ ih ϕ ′ | ϕ ′ i , (21)where in the limit it is sufficient to use the asymptoticform of the solution ϕ ( x ).When all the factors are combined, the overlaps of thezero modes cancel, as they should [23], and we get G α ( x − x ′ ; E ) = − iα | E | γ e −S x − x ′ sinh( x − x ′ ) . (22)For the DOS we get a well-known expression ρ ( E ) = 4 | E | πγ exp (cid:18) − ~ √ m | E | / γ (cid:19) . (23) Correlation functions. –
Statistics of localizedstates at two energies E , = E ∓ ~ ω/
2, are character-ized by the two-point local DOS correlation function [6] R ( ω, x ) = ρ − m D X kl δ ( E k − E ) δ ( E l − E ) × | ψ k (0) | | ψ l ( x ) | E , (24)and (the real part of) the dynamic correlation function [7] S ( ω, x ) = ρ − m Re D X kl δ ( E k − E ) δ ( E l − E ) × ψ k (0) ψ ∗ k ( x ) ψ l ( x ) ψ ∗ l (0) E . (25)Here ρ m = ρ ( E ) ρ ( E ). Functions (24) and (25) can becalculated from disorder averages of products of GFs: G α α ( x , x ′ ; x , x ′ ) = D Y a =1 G α a ( x a , x ′ a ; E a ) E (26)by means of R ( ω, x ) = − π ρ m X α ,α α α G α α (0 , , x, x ) , (27) S ( ω, x ) = − π ρ m Re X α ,α α α G α α (0 , x, x, . (28)p-3. M. Falco, A. A. Fedorenko, and I. A. Gruzberg, EPL , 37004 (2017)Assuming that both energies are in the tail of the spec-trum, E a <
0, let us introduce the following notation: E a = −| E | k a , k , = √ ± ¯ ω, ¯ ω = ~ ω/ | E | . (29)The supersymmetry representation for G α α requires twosuperfields Φ a = ( φ a , χ a ), a = 1 ,
2. After rotations of thefields in the complex plane we have G α α ( x , x ′ ; x , x ′ ) = | E | γ Z Y a D Φ a χ a ( x a ) ¯ χ a ( x ′ a ) e −S , S = A E Z dx h X a Φ † a (cid:0) k a − ∇ (cid:1) Φ a − (cid:16) X a Φ † a Φ a (cid:17) i . The bosonic saddle point equation is a two-componentnon-linear Schr¨odinger equation h k a − ∇ −
12 ( | φ | + | φ | ) i φ a = 0 , a = 1 , . (30)Equation (30) has the trivial solution ( φ , = 0), two one-instanton solutions ( φ = 0, φ = 0), and ( φ = 0, φ =0), and a two-instanton solution with ( φ , = 0). Thetrivial and the one-instanton solutions do not contributeto the functions R and S , which are determined by thetwo-instanton solutions.The exact two-instanton solutions found in ref. [21] canbe written as φ a, cl ( x − x ) = e iθ a ϕ a ( x − x ), where (cid:18) ϕ ( x ) ϕ ( x ) (cid:19) = (cid:18) cos θ ( x ) sin θ ( x )sin θ ( x ) − cos θ ( x ) (cid:19) (cid:18) ϕ L ( x ) ϕ R ( x ) (cid:19) , (31) ϕ L ( x ) = 4 k e − x e − x + e x , ϕ R ( x ) = 4 k e x e − x + e x , x , ( x ) = ( k + k ) x ± [ D − ln sin 2 θ ( x )] ,D = ln 2( k + k ) k − k , cot θ ( x ) = e ( k − k ) x + f . (32)These solutions contain four free parameters θ , θ , x , f ,so we expect to have four zero modes in the fluctuationspectrum. The parameter f determines the distance D = D + ln cosh f (33)between the left and right “instantons” ϕ L and ϕ R , whoseminimal value D ( ∼ ln(4 / ¯ ω ) for ¯ ω ≪
1) plays the role ofthe Mott scale L M . The action of a two-instanton solutiondoes not depend on the free parameters: S = S ( E ) + S ( E ) = 163 ( A E + A E ) . (34)We now introduce the fluctuation fields ρ x = ( ρ x , ρ x ), ρ y = ( ρ y , ρ y ), χ † = ( ¯ χ , ¯ χ ), and expand the actionaround the saddle point: S = S + A E Z dx [ ρ x O x ρ x + ρ y O y ρ y + χ † O y χ ] , (35) where the fluctuation 2 × O y = −∇ −
12 ( ϕ + ϕ ) + (cid:18) k k (cid:19) ,O x = O y − (cid:18) ϕ ϕ ϕ ϕ ϕ ϕ (cid:19) . (36)The operator O x has two negative modes ψ x and ψ x (whose explicit form we do not need) and two zero modes ψ x ( x ) = (cid:18) ϕ ′ ( x ) ϕ ′ ( x ) (cid:19) , ψ x ( x ) = (cid:18) ∂ f ϕ ( x ) ∂ f ϕ ( x ) (cid:19) , (37)related to the translation invariance and to changing thedistance D . The operator O y has two zero modes ψ y ( x ) = (cid:18) ϕ ( x )0 (cid:19) , ψ y ( x ) = (cid:18) ϕ ( x ) (cid:19) , (38)related to the rotation invariance with respect to θ a .A formal Gaussian integration gives G α α = | E | γ e −S Λ x Λ x Λ y (cid:16) Det ′ A E O y Det ′′ A E O x (cid:17) / × Y a ϕ a ( x a − x ) ϕ a ( x ′ a − x ) h ϕ a | ϕ a i . (39)Equation (16) gives the contributions of zero modes:Λ x = 1 π Z df Z dx q h ψ x | ψ x ih ψ x | ψ x i − |h ψ x | ψ x i| , Λ y = 4 π p h ϕ | ϕ i h ϕ | ϕ i . (40)Similarly to eq. (19) we now haveΛ x (cid:16) Det ′ A E O y Det ′′ A E O x (cid:17) / = − α α (cid:16) Det ′ O y Det ′ O x (cid:17) / . (41)To compute this determinant ratio we use eq. (20):Det ′ O y Det ′ O x = h ϕ | ϕ ih ϕ | ϕ ih ψ x | ψ x ih ψ x | ψ x i − |h ψ x | ψ x i| F (¯ ω ) , (42)where F (¯ ω ) is easily computed from the asymptotic be-havior of the solutions (31) at infinity: F (¯ ω ) = lim R →∞ ϕ ′′ ( R ) ∂ f ϕ ′ ( R ) − ϕ ′′ ( R ) ∂ f ϕ ′ ( R ) ϕ ′ ( R ) ϕ ′ ( R )= k + k √ ω + √ − ¯ ω . (43)Using eqs. (23), (29), and (34), we have | E | γ e −S = π ρ m k k = π ρ m (1 − ¯ ω ) . (44)Collecting all factors we arrive at G α α = − α α π ρ m Y (¯ ω ) Q, Y (¯ ω ) = √ ω + √ − ¯ ω − ¯ ω ) ,Q ( x , x ′ ; x , x ′ ) = 12 Z Z df dx ϕ ( x − x ) ϕ ( x ′ − x ) × ϕ ( x − x ) ϕ ( x ′ − x ) . (45)p-4C conductivity of disordered wires Fig. 1: The dynamic S and the local DOS R correlation func-tions. Solid curves show the numerically exact correlators eval-uated for three values of ¯ ω . The dashed curves near the originare plots of the exact equation (48), and the ones near the Mottscale L M are plots of the approximations (59) and (61). Substituting this into eqs. (27) and (28), we finally obtain R ( ω, x ) = Y (¯ ω ) Q R ( x ) , S ( ω, x ) = Y (¯ ω ) Q S ( x ) , (46) Q R ( x ) = Q (0 , , x, x ) , Q S ( x ) = Q (0 , x, x, . (47)Equations (45)-(47) allow us to establish some exactproperties of the functions R and S such as R ( ω, x = 0) = S ( ω, x = 0) = 1 /
3, independently of ω . This propertycan be established only if the functional determinants arecalculated exactly for all ¯ ω . We also find R ( ω, x → ∞ ) = 1as expected from the clustering property of the correlatorat large distances. Finally, in the limit ¯ ω → R (0 , x ) = S (0 , x ) = Q ( x ) ≡ x coth x − x . (48)This function describes the peak near x = 0 shown in fig. 1by a black dashed curve.Numerical evaluation of eq. (45) gives the plots of thefunctions R and S shown in fig. 1 for three values of ¯ ω .The features at the Mott scale (a step in R and a negativebump in S ) have widths that are independent of ¯ ω for¯ ω ≪
1, see fig. 2 which shows the position L M of thenegative peak of S ( ω, x ) and its width at half maximum∆ L M as functions of ¯ ω .Analytical expressions for Q C ( C = R, S ) can be ob-tained for ¯ ω ≪
1. In this case one can neglect the pref-actor Y (¯ ω ), and represent the corrections to the centralpeak and the features at the Mott scale similar to ref. [9]: Q C ( x ) = Q ( x ) + δQ C ( x ) + Q CM ( x ) . (49)When ¯ ω ≪
1, in the leading order the mixing angle θ ( x )in eq. (31) becomes constant (cot θ = e f ), and ϕ L,R ( x )become instantons (9) separated by D ≫ ϕ L ( x ) = ϕ ( x + D/ , ˜ ϕ R ( x ) = ϕ ( x − D/ . (50)We substitute these approximate solutions to eq. (45), anddenote the result by ˜ Q . Then the integral over x becomes Fig. 2: The Mott scale L M as a function of ¯ ω . The red dotsare numerically exctracted minima of S (¯ ω, x ). The solid line isln(4 / ¯ ω ), and the dashed line is D − ln 2 + 1 ≈ ln(2 e/ ¯ ω ). Inset:the width of the negative peak of S (¯ ω, x ) as a function of ¯ ω . elementary, and after changing the integration variable f to D we obtain for x > δ ˜ Q R ( x ) = − Z ∞ D dD w ( D − D ) I ( x, D ) , (51)˜ Q RM ( x ) = Z ∞ D dD w ( D − D ) Q ( x − D ) , (52) δ ˜ Q S ( x ) = Z ∞ D dD w ( D − D ) I ( x, D ) , (53)˜ Q SM ( x ) = − Z ∞ D dD w ( D − D ) Q ( x − D ) , (54)where the tilde indicates that the approximation (50) hasbeen used, and we have defined the functions I ( x, D ) = 2 D coth D − x coth x cosh 2 D − cosh 2 x , w ( D ) = e − D √ e D − ,w ( D ) = 2 − e − D √ − e − D , w ( D ) = 2 p − e − D . (55)For D ≫ x ≪ D the function I ( x, D ) can be wellapproximated by I ( x, D ) ≈ e − D ( D − x coth x ) , (56)and with this approximation we have δ ˜ Q R ( x ) ≈ − e − D ( D − x coth x + 5 / − ln 2) , (57) δ ˜ Q S ( x ) ≈ e − D ( D − x coth x + 4 / − ln 2) . (58)The function w ( D ) approaches its asymptotic value of1 very rapidly. Replacing w by 1 gives the approximation˜ Q RM ( x ) ≈ I ( D + x ) + I ( D − x ) , (59) I ( x ) = 1 − coth x x x . (60)p-5. M. Falco, A. A. Fedorenko, and I. A. Gruzberg, EPL , 37004 (2017) Fig. 3: The ratio Σ(¯ ω ) / Σ MB (¯ ω ) as a function of ln ¯ ω (mainpanel) and ¯ ω (inset). The exact formula (63) is used for theblue solid curve, the approximation (65) for the green dashedcurve, and the sum of (65) and (67) for the red dashed curve. The function w ( D ) is peaked at D = 0. A reasonableapproximation is to replace it by a delta function δ ( D ) / Q SM ( x ) ≈ − (cid:2) Q ( x − D ) + Q ( x + D ) (cid:3) . (61)The approximations (59) and (61) are shown in fig. 1 bydashed lines for ¯ ω = 10 − . AC conductivity. –
The real part of the AC conduc-tivity σ ( ω ) is obtained from S ( ω, x ) asRe σ ( ω ) = π ~ e ρ m m | E | ) / ω Σ(¯ ω ) , (62)Σ(¯ ω ) = − Y (¯ ω ) Z dx x Q S ( x ) . (63)This expression is valid for any E in the tail, but arbitrary0 ¯ ω <
1. It can be evaluated with a various degree ofaccuracy. For ¯ ω ≪ Q S by ˜ Q S . If we ne-glect Y (¯ ω ) in front of the integral in eq. (63), then neglect δ ˜ Q S , and use eq. (61) for ˜ Q SM , we get an approximationfor Σ(¯ ω ): Σ(¯ ω ) ≈ D ≈ ln ω ≡ Σ MB (¯ ω ) . (64)The last expression is what is called the Mott formula inref. [21]. Deviations of the numerically exact Σ(¯ ω ) fromΣ MB (¯ ω ) are demonstrated in fig. 3, where the solid curveis the plot of the ratio Σ / Σ MB .A much better approximation is obtained if we keep Y (¯ ω ), substitute eqs. (53) and (54) into (63), and performthe x integration. This gives Σ(¯ ω ) ≈ Σ (¯ ω )+Σ (¯ ω ), whereΣ (¯ ω ) = 2 Y (¯ ω ) Z ∞ D dD w ( D − D ) D = Y (¯ ω ) (cid:2) ( D − ln 2 + 1) + 1 − π / (cid:3) , (65)Σ (¯ ω ) = − Y (¯ ω )6 Z ∞ D dD w ( D − D ) D ( D + π )sinh D . (66) The last integral can be evaluated after we approximatesinh D ≈ e D /
2, and gives a long expression whose leadingterms areΣ (¯ ω ) ≈ − Y (¯ ω ) e − D D (cid:2) D + (16 / − D + 52 / π / −
16 ln 2 + 6 ln (cid:3) . (67)The ratios Σ / Σ MB and (Σ +Σ ) / Σ MB are shown in fig. 3as the green and red dashed curves. We see that Σ + Σ well approximates the exact result in a wide range of ¯ ω . Discussion and conclusions. –
All qualitative fea-tures of the functions R and S that we found, includingthe shape of the central peak and the features at the Mottscale, are in agreement with ref. [22], see their eq. (4.20)and the list that follows. This is expected, since this paperdeals with states deep in the tails of the spectrum, and sodo we. However, when we compare our results with thoseobtained in the regime of large positive E in refs. [6, 7](conveniently summarized in tables I and II in [9]) we seemany differences.The values of the correlators R and S and their deriva-tives at x = 0 differ between the two cases, but are notexpected to be universal. The decay of the central peak Q ( x ) has a different rate in the exponential, as well asa different power-law prefactor. The difference of the fea-tures at the Mott scale is more drastic: while in our casethe behavior around L M is exponential (see eqs. (59) and(61)), it is an error function and a Gaussian for positive E . We attribute these differences to a different nature ofthe localized wave functions. The localized states in theoptimal fluctuations of the disorder potential are simplerthan the Anderson-localized states at E ≫
0. In particu-lar, they do not have log-normally distributed tails, which,according to ref. [9], affect the hybridization of wave func-tions at E ≫
0. In spite of this “non-universality” of thestructure of the correlators R and S at the Mott scale, thebehavior of the AC conductivity σ ( ω ) at low frequenciesis still universal, and is given by the MB formula.In conclusion, we have calculated exactly the Gaussianfluctuations around two-instanton saddle points of thefunctional integral for the disorder average of two-pointGreen functions in a one-dimensional wire in the tail of thespectrum. This allowed us to derive the local DOS and dy-namic correlation functions and the AC conductivity be-yond the Mott-Berezinskii law, that is, for a broad rangeof frequencies. Unlike other approaches, our method canbe applied to quasi-one dimensional systems [30]. We alsohope that it can be useful for systems with non-Gaussiandisorder-like random speckle potential [19,31], and for sys-tems with weak interactions [32, 33]. ∗ ∗ ∗ IG is grateful to D. Khmelnitskii for stimulating and il-luminating discussions. AAF acknowledges support fromthe French Agence Nationale de la Recherche throughp-6C conductivity of disordered wiresGrants No. ANR-12-BS04-0007 (SemiTopo), No. ANR-13-JS04-0005 (ArtiQ), and No. ANR-14-ACHN-0031(TopoDyn). IG was supported by the NSF Grant No.DMR-1508255.
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