A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $\epsilon$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $\epsilon$, $P(\epsilon)\sim 1/\epsilon^{1+\alpha}$ with $\alpha\in(0,2)$. Our model serves as a generalization of 1D Lloyd's model, which corresponds to $\alpha=1$. First, we verify that the ensemble average $\left\langle -\ln G\right\rangle$ is proportional to the length of the wire $L$ for all values of $\alpha$, providing the localization length $\xi$ from $\left\langle-\ln G\right\rangle=2L/\xi$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $\alpha$ and $\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$ is proportional to $G^\beta$, for $G\to 0$, with $\beta\le\alpha/2$, in agreement to previous studies.

Of particular interest is the comparison between the one-dimensional (1D) Anderson model (1DAM) [44] and the 1D Lloyd's model, since the former represents the most prominent model of disordered wires [45]. Indeed, both models are described by the 1D tight-binding Hamiltonian: [ǫ n | n n | −ν n,n+1 | n n + 1 | −ν n,n−1 | n n − 1 | ] ; (2) where L is the length of the wire given as the total number of sites n, ǫ n are random on-site potentials, and ν n,m are the hopping integrals between nearest neighbors (which are set to a constant value ν n,n±1 = ν). However, while for the standard 1DAM (with white-noise on-site disorder ǫ n ǫ m = σ 2 δ nm and ǫ n = 0) the on-site potentials are characterized by a finite variance σ 2 = ǫ 2 n (in most cases the corresponding probability distribution function P (ǫ) is chosen as a box or a Gaussian distribution), in the Lloyd's model the variance σ 2 of the random on-site energies ǫ n diverges since they follow a Cauchy distribution.
It is also known that the eigenstates Ψ of the infinite 1DAM are exponentially localized around a site position n 0 [45]: where ξ is the eigenfunction localization length. Moreover, for weak disorder (σ 2 ≪ 1), the only relevant parameter for describing the statistical properties of the transmission of the finite 1DAM is the ratio L/ξ [46], a fact known as single parameter scaling. The above exponential localization of eigenfunctions makes the transmission or dimensionless conductance G exponentially small, i.e., [47] − ln G = 2L ξ ; (4) thus, this relation can be used to obtain the localization length. Remarkably, it has been shown that Eq. (4) is also valid for the 1D Lloyd's model [41] implying a single parameter scaling, see also [38]. It is also relevant to mention that studies of transport quantities through 1D wires with Lévy-type disorder, different from the 1D Lloyd's model, have been reported. For example, wires with scatterers randomly spaced along the wire according to a Lévy-type distribution were studied in Refs. [3,4,48,49]. Concerning the conductance of such wires, a prominent result reads that the corresponding probability distribution function P (G) is fully determined by the exponent α of the powerlaw decay of the Lévy-type distribution and the average (over disorder realizations) − ln G [48,49]; i.e., all other details of the disorder configuration are irrelevant. In this sense, P (G) shows universality. Moreover, this fact was already verified experimentally in microwave random waveguides [2] and tested numerically using the tightbinding model of Eq. (2) with ǫ n = 0 and off-diagonal Lévy-type disorder [50] (i.e., with ν n,m in Eq. (2) distributed according to a Lévy-type distribution).
It is important to point out that 1D tight-binding wires with power-law distributed random on-site potentials, characterized by power-laws different from α = 1 (which corresponds to the 1D Lloyd's model), have been scarcely studied; for a prominent exception see [41]. Thus, in this paper we undertake this task and study numerically the conductance though disordered wires defined as a generalization of the 1D Lloyd's model as follows. We shall study 1D wires described by the Hamiltonian of Eq. (2) having constant hopping integrals, ν n,n±1 = ν = 1, and random on-site potentials ǫ n which follow a Lévytype distribution with a long tail, like in Eq. (1) with 0 < α < 2. We name this setup the 1DAM with Lévytype on-site disorder. We note that when α = 1 we recover the 1D Lloyd's model. Therefore, in the following section we shall show that (i) the conductance distribution P (G) is fully determined by the power-law exponent α and the ensemble average − ln G ; (ii) for α ≤ 1 and − ln G ∼ 1, bimodal distributions for P (G) with peaks at G ∼ 0 and G ∼ 1 are obtained, revealing the coexistence of insulating and ballistic regimes; and (iii) the probability distribution P (ln G) is proportional to G β , for vanishing G, with β ≤ α/2.

II. RESULTS AND DISCUSSION
Since we are interested in the conductance statistics of the 1DAM with Lévy-type on-site disorder we have to define first the scattering setup we shall use: We open the isolated samples described above by attaching two semi-infinite single channel leads to the border sites at opposite sides of the 1D wires. Each lead is also described by a 1D semi-infinite tight-binding Hamiltonian. Using the Heidelberg approach [51] we can write the transmission amplitude through the disordered wires as t = −2i sin(k) W T (E − H eff ) −1 W, where k = arccos(E/2) is the wave vector supported in the leads and H eff is an effective non-hermitian Hamiltonian given by H eff = H − e ik WW T . Here, W is a L × 1 vector that specifies the positions of the attached leads to the wire. In our setup, all elements of W are equal to zero except W 11 and W L1 which we set to unity (i.e., the leads are attached to the wire with a strength equal to the inter-site hopping amplitudes: ν = 1). Also, we have fixed the energy at E = 0 in all our calculations, although the same conclusions are obtained for E = 0. Then, within a scattering approach to the electronic transport, we compute the dimensionless conductance as [52] First, we present in Fig. 1(a) the ensemble average − ln G as a function of L for the 1DAM with Lévy-type disorder for several values of α. It is clear from this figure that − ln G ∝ L for all the values of α we consider here. Therefore, we can extract the localization length ξ by fitting the curves − ln G vs. L with Eq. (4); see dashed lines in Fig. 1(a). This behavior should be contrasted to the case of 1D wires with off-diagonal Lévy-type disorder [53] which shows the dependence − ln G ∝ L 1/2 when α = 1/2 at E = 0 [50]. Also, we have confirmed that the cumulants (− ln G) k obey a linear relation with the wire length [41,54], i.e., where the coefficients c k , with c 1 ≡ ξ −1 , characterize the Lyapunov exponent of a generic 1D tight-binding wire with on-site disorder. We have verified the above relation, Eq. (5), for k = 1, 2, and 3; as an example in Fig. 1(b) we present the results for (− ln G) 2 as a  5), which can be used to extract the higher order coefficient c 2 . Now, in Fig. 2 we show different conductance distributions P (G) for the 1DAM with Lévy-type on-site disorder for fixed values of − ln G ; note that fixed − ln G means fixed ratio L/ξ. Several values of α are reported in each panel. We can observe that for fixed − ln G , by increasing α the conductance distribution evolves towards the P (G) corresponding to the 1DAM with white noise disorder, P WN (G), as expected. The curves for P WN (G) are included as a reference in all panels of Fig. 2 as red dashed lines [55]. In fact, P (G) already corresponds to P WN (G) once α = 2.
We recall that for 1D tight-binding wires with offdiagonal Lévy-type disorder P (G) is fully determined by the exponent α and the average − ln G [50]. It is there- fore pertinent to ask whether this property also holds for diagonal Lévy-type disorder. Thus, in Fig. 3 we show P (G) for the 1DAM with Lévy-type on-site disorder for several values of α, where each panel corresponds to a fixed value of − ln G . For each combination of − ln G and α we present two histograms (in red and black) corresponding to wires with on-site random potentials {ǫ n } characterized by two different density distributions [57], but with the same exponent α of their corresponding power-law tails. We can see from Fig. 3 that for each value of α the histograms (in red and black) fall on the top of each other, which is an evidence that the conductance distribution P (G) for the 1DAM with Lévy-type on-site disorder is invariant once α and − ln G are fixed; i.e., P (G) displays a universal statistics.
Moreover, we want to emphasize the coexistence of insulating and ballistic regimes characterized, respectively, by the two prominent peaks of P (G) at G = 0 and G = 1. This behavior, which is more evident for − ln G ∼ 1 and α ≤ 1 (see Figs. 2 and 3), is not observed in 1D wires with white-noise disorder (see for example the red dashed curves in Fig. 2). This coexistence of opposite transport regimes has been already reported in systems with anomalously localized states: 1D wires with obstacles randomly spaced according to Lévy-type density distribution [48,50] as well as in the so-called random-mass Dirac model [58].
Finally, we study the behavior of the tail of the distribution P (ln G). Thus, using the same data of Fig. 3, in Fig. 4 we plot P (ln G). As expected, since P (G) is determined by α and − ln G , we can see that P (ln G) is invariant once those two quantities (α and − ln G ) are fixed (red and black histograms fall on top of each other). Moreover, from Fig. 4 we can deduce a powerlaw behavior: for G → 0 when α < 2. For α = 2, P (ln G) displays a lognormal tail (not shown here), expected for 1D systems in the presence of Anderson localization. Actually, the behavior (6) was already anticipated in [41] as P (G) ∼ G −(2−λ)/2 for G → 0 with λ < α; which in our study translates as P (ln G) ∝ G λ/2 (since P (ln G) = GP (G)) with λ/2 ≡ β ≤ α/2. Indeed, we have validated the last inequality in Fig. 5 where we report the exponent β obtained from power-law fittings of the tails of the histograms of P (ln G). In addition, we have observed that the value of β depends on the particular value of − ln G characterizing the corresponding histogram of P (ln G). Also, from Fig. 5 we note that β ≈ α/2 as the value of − ln G decreases.

III. CONCLUSIONS
In this work we have studied the conductance G through a generalization of Lloyd's model in one dimension: We consider one-dimensional (1D) tight-binding wires with on-site disorder following a Lévy-type distribution, see Eq. (1), characterized by the exponent α of the power-law decay. We have verified that different cumulants of the variable ln G decrease linearly with the length wire L. In particular, we were able to extract the eigenfunction localization length ξ from − ln G = 2L/ξ. Then, we have shown some evidence that the probability distribution function P (G) is invariant, i.e., fully determined, once α and − ln G are fixed; in agreement with other Lévy-disordered wire models [2,[48][49][50]. We have also reported the coexistence of insulating and ballistic regimes, evidenced by peaks in P (G) at G = 0 and G = 1; these peaks are most prominent and commensurate for − ln G ∼ 1 and α ≤ 1. Additionally we have shown that P (ln G) develops power-law tails for G → 0, characterized by the power-law β (also invariant for fixed α and − ln G ) which, in turn, is bounded from above by α/2. This upper bound of β implies that the smaller the value of α the larger the probability to find vanishing conductance values in our Lévy-disordered wires. and ρ2(ǫ) = α (1 + ǫ) 1+α , where Γ is the Euler gamma function.