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Non-analyticity in the distribution of conductances in quasi one dimensional wires

We show that the distribution P(g) of conductances g of a quasi one dimensional wire has non-analytic behavior in the insulating region, leading to a discontinuous derivative in the distribution near g=1. We give analytic expressions for the full distribution and extract an approximate scaling behavior valid for different strengths of disorder close to g=1.

It is well known that the probability distribution P (g) of the (dimensionless) conductances g of a disordered conductor is Gaussian in the deeply metallic regime and log-normal in the deeply insulating regime [1]. It has been proposed recently [2] that for a quasi one dimensional (1D) wire with mean dimensionless conductanceḡ ≪ 1, P (g) has a sharp cut-off beyond g = 1.
To be sure, in one dimension, when g ≤ 1, P (g) drops discontinuously to zero at g = 1. In higher dimensions the discontinuity will be smeared in some way, but it is difficult to say how much of it will survive. Using the saddle point method developed in [2], we are able to show here that nonanalytic behavior remains near g = 1 at least in quasi-1D systems. One should note that the exact results for the mean and variance of the conductance, obtained within the non-linear sigma model [3], do not give any clue on the abrupt change of P (g) near g = 1.
The existence of the sharp cutoff for allḡ ≪ 1 has recently been confirmed numerically, while the exact shape of the distribution very close to g = 1 seems to be different from a log-normal distribution [4,6,7,5,8].
In the present work, we focus on the insulating region in quasi 1D near |g − 1| ≪ 1, in order to understand better the nature of the unexpected sharp feature in the distribution. For simplicity, we will restrict our discussions to the unitary case of broken time reversal symmetry. We propose from a simple generalization of [2] that for large enough disorder the distribution has a non analyticity near g = 1, giving rise to large discontinuities in its derivatives. We obtain e.g. that in the insulating limit, the leading contribution to the distribution P (g) at |g − 1| ≪ 1, is given by where Φ is the error function, ν = cosh −1 1/(g − 1), ν 1 = cosh −1 2/g, and α = 1/ cosh 2 (3/2Γ) [9]. The disorder parameter Γ = ξ/L, where ξ is the localization length and L is the system size (Γ ≫ 1 would correspond to metals and Γ ≪ 1 corresponds to insulators). The term C is independent of g, but depends on Γ. The discontinuity is at g = 1 + α. For the case considered here, Γ ≪ 1 and α ∼ e −3/Γ . From (1) it follows that P ′ = dP (g)/dg has a discontinuity at g = 1 + α, with P ′ ∼ −e 2/Γ very large for g 1 + α (growing exponentially with increasing disorder) and |P ′ | ≪ 1 for g < 1 + α. Note that in the limit Γ → 0, the distribution would have an essential singularity at g = 1.
The above results are obtained assuming Γ ≪ 1, describing the insulating limit. As Γ increases, but still in the insulating regime, the dominant contribution remains the same with renormalized parameters, and our results remain qualitatively valid. In the metallic limit the singularity is absent, giving rise to the possibility that the singularity disappears across the crossover region (Γ ∼ 1/2), where our current approximations are not valid.
In addition to the sharp structure, our results suggest an approximate scaling behavior for different strengths of disorder near g ∼ 1. As is evident from (1), in the expression for the ratio P (g)/P (1), the dominant dependence on the disorder parameter Γ and on g − 1 appears only in the combination √ Γ(ν − 1/2Γ). Our numerical results agree with this approximate scaling. We also find that the distribution P (g) for g 1 is a slowly varying function of g, which is approximately constant close to g = 1. This leads to an exponential distribution for P (ln g) in this region, rather than the log-normal distribution proposed in [2], and agrees with various existing numerical results [6,4]. For Γ ≪ 1, the numerical results suggest that the exponential form crosses over to the log-normal form for g ≪ 1.
We now briefly discuss the details of the method which is based on a simple generalization of [2]. The probability distribution p(λ) of the N variables λ i , where λ i are related to the transmission eigenvalues T i of an N-channel quasi 1d wire by λ i = (1 − T i )/T i , satisfy the well known DMPK equation [10], whose solutions in the metallic and insulating regimes can be written in the general form [11] where Z = i dλ i exp[−βH] is a normalizing factor independent of λ i , H(λ) may be interpreted as the Hamiltonian function of N classical charges at positions λ i , and β = 2 for the unitary case. Since the dimensionless conductance is given by the probability distribution P (g) can be written as Following [2], we define a "Free energy" such that the distribution can be written as It has been shown in [2] that separating out the lowest eigenvalue λ 1 and treating the rest of the eigenvalues as a continuum beginning at λ 2 , the distribution can be obtained within a generalized saddle point approximation, given by where is the saddle point action, obtained from a saddle point free energy The insulating region in this model is given by x 2 ≫ 1, and also The saddle point free energy terms in this limit were calculated to be [12] .
Since 1/2|F ′′ | is exponentially large in the insulating regime, the saddle point solution is given by putting the coefficient of the term in the action S equal to zero, namely g = F ′ , which then gives As pointed out in [2], the saddlepoint solution is valid only for g < 1 since cosh x 1 ≥ 1, while the g > 1 region is determined by the boundary values of x 1 = 0 and x 2 = 2/πΓ. While this is a good approximation on both sides of g = 1, the fact that the boundary of the saddle point solution is at g = 1 makes it possible that the approximation is not accurate enough very close to g = 1. We will show below that at the next level of approximation suggested in [2], the region close to g = 1 can be better described by saddle point solutions valid on both sides. This improved approximation immediately leads to the non-analyticity mentioned before. It was shown in [13] that while separating out the lowest eigenvalue gave qualitatively good results both in the metallic and insulating regimes, it is important to separate out one additional eigenvalue to obtain good agreement with numerical and available exact results in the insulating and crossover regimes because the separation between the eigenvalues becomes large. The extension is straightforward, and the distribution P (g) now has an additional integral over λ 3 , which is the beginning of the continuum that represents the rest of the eigenvalues. It is clear that the insulating limit is characterized by x 3 ≫ x 2 ≫ 1 for typical values of x i , and that F ′ in this limit will now be given by [13] F ′′ now has the same expression as (12), with x 2 replaced by x 3 , while F 0 is given by (10) with x 1 , x 2 replaced by x 2 , x 3 plus additional terms involving x 1 The saddle point solutions for x 3 and x 2 are now given by x 3sp = 3/2Γ, which is independent of g and obtained again from g = F ′ . Since x 2 ≫ 1, the additional term in F ′ is exponentially small, and usually negligible. However, close to g = 1, this is the term that allows the saddle point solution to be valid on both sides of g = 1. The fluctuation correction to the x 2 integral contributes an additional factor e 2x2sp , and we are left with the final integral We have used finite limits in the integral because the saddle point solution of x 2 must be real and also we require that x 3sp > x 2sp . This gives x 1min = Re[cosh −1 1/(g − α)], α = 1/ cosh 2 (3/2Γ), and x 1max = cosh −1 2/g. The crucial point is that the restriction x 1min is zero for all g ≥ 1 + α, but becomes finite for g < 1 + α, with a discontinuous slope. It is the universal nature of this constraint which will eventually lead to the sharp changes in P (g) at g = 1 + α.
In the limit g → 1 and Γ ≪ 1, the upper limit of the integral x 1max ≪ 1. We can therefore neglect the x 2 1 term in (17). Changing variables to x 2 and rewriting ln x 1 , we obtain a simple integral with ν 1 as defined after (1), and x 2max = ν for g > 1 + α, but 3/2Γ for g < 1 + α. The integral immediately leads to Eq. (1). It is easy to see from (1) that the first derivative has a discontinuity at g = 1 + α and is large, of order e 2/Γ for g > 1 + α. Figure 1 shows the sharp changes at g = 1 + α according to Eq. (1). Since ν 1 in this region is only weakly dependent on g, this also leads to an approximate scaling behavior of P (g) as a function of z = √ Γ(ν − 1/2Γ). While the g-dependent contribution comes from the x 2 integral where x 2sp < x 3sp , there is a contribution to P (g) from integral over x 2 > x 3sp . The x 3 integral in this case is given by the boundary value at x 3 = x 2 , and the fluctuation correction to the x 2 integral becomes of order unity. The resulting integral leads to a g independent but Γ dependent term giving In the above, we considered the case Γ ≪ 1, where x 3 ≫ x 2 ≫ 1 and only the leading order terms were kept in the free energy. For weaker disorder but still in the insulating side, the next leading order terms will become important. First of all, the ln x 2 term arising from F 0 (x 2 , x 3 ) in (15) will start to contribute. Second, the next leading order contributions to F 0 will contribute some more x 2 and ln x 2 terms. In general, as long as the limit x 3 ≫ x 2 ≫ 1 remains valid, and the perturbative expansion of the free energy remains a good approximation, the integral (18) can be generalized to be of the form where a, b and p are parameters which in principle can be calculated within our framework by keeping higher order terms in the free energy. For p = 0 we get back (1) with renormalized parameters. For finite integer p the integral can again be evaluated simply in terms of the error function. For non-integer values of p, the integral is more complicated but it is clear that the qualitative features of the results will remain the same. We expect that our results will remain qualitatively valid near the crossover region on the insulating side (Γ 1/2). In order to check some of our approximate expressions, we have calculated P (g) numerically for a quasi 1D wire. Recent techniques [4] allow us to obtain reasonable statistics for N = 5. Unfortunately, it was not possible to obtain enough data for either Γ ≪ 1 or 1 < g < 1.01. Nevertheless, (19) should be approximately valid in the region numerically accessible to us and the data can be used to check if the approximate scaling expected from (19) holds qualitatively. In Fig. 2, circles and squares show the numerical results for different values of average conductanceḡ. We use p = 2, Γ e = aΓ and b as fitting parameters to fit the data for differentḡ using (19) (for our range of parameters, the constant C of (1) turns out to be negligible). Agreement with the data is good for g − 1 < 0.15. In Fig. 3 the same data are plotted as a function of the scaling variable s = exp[− √ Γ e (ν − b/2Γ e ] shifted by s(1 + α), taking into account the fact that the singularities occur at g = 1 + α(Γ e ). Again the scaling is quite good. We have checked that the scaling fails for larger values of Γ e , which is expected because it is close to the crossover regime (Γ ∼ 1/2) where our perturbative expansion of the free energy starts to become invalid.
A non-analyticity in P (g) was obtained in [4] within Random Matrix Theory (RMT), based on the assumption that contributions from only the smallest two eigenvalues are important in the insulating regime. All derivatives of P (g) in such a model diverge at g = 1. However, a two-eigenvalue calculation within DMPK (obtained by neglecting the continuum in our model) gives rise to an essential singularity at g = 1, so that there is no discontinuity in the derivatives at that point. Therefore, it is not simple to see if the trivial discontinuity in the derivative mentioned in the introduction will persist beyond 1D. We emphasize that our present results for quasi 1D are fundamentally different from both of the above models: whereas all derivatives of P(g) diverge for the RMT model and vanish for the two-eigenvalue DMPK model, they are finite for our quasi 1D system. Also, in contrast to the above cases, the singularity for the quasi 1D system is not exactly at g = 1, but is shifted as a function of disorder. Finally, the RMT results do not satisfy the approximate scaling of Fig. 3.
We note that there is no phase transition in quasi 1D. It is therefore important to ask if a non-analyticity in P (g) in the absence of a true phase transition violates any fundamental principle. Our framework can be considered as an electrostatic problem in one dimension involving charges with repulsive interactions and confinement potentials. Our free energy and the density of charges are all analytic, ensuring that e.g. the total energy will be an analytic function of disorder. However, the conductance is a complicated function of the charge distribution, and the non-analyticity appears only in the distribution of the conductances where the charges have to satisfy certain constraints. There is no restriction on the analyticity of such a distribution.
In summary, we have shown that the distribution of conductances in quasi 1D systems in the insulating regime has a non-analytic behavior near g = 1. The non-analyticity gives rise to very sharp structures close to g = 1, with finite discontinuities in its derivatives. The presence of similar structures in higher dimensions in numerically studied systems [6,7,8] gives rise to the possibility that such non-analyticity might be present in the conductance distribution in higher dimensions as well, having important consequences for the Anderson transition. Whether the non-analyticity discussed here disappears abruptly at some critical value of Γ, or smoothly, as Γ is increased beyond 1/2 remains to be clarified. It is conceivable that an abrupt disappearance of the non-analyticity is a signature of the Anderson transition.
KAM is grateful for support from the Alexander von Humboldt foundation during his visit to U. Karlsruhe. Work of PW was supported in part by SFB 195 of the Deutsche Forschungsgemeinschaft (DFG). AGM was supported by the Emmy-Noether program of the DFG under Grant No. Bu 1107/2-1. VAG is grateful for support from CONACYT, México and hospitality at the IPCMS, France.