Genuine localization transition in a long-range hopping model

We introduce and study a banded random matrix model describing sparse, long-range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and a mapping to a long-range epidemics model, we establish the phase diagram of the model. A genuine localization transition, with well defined mobility edges, appears as the hopping rate decreases slower than ${\ell }^{-2}$, where $\ell$ is the distance. Correspondingly, the decay of the localized states evolves from a standard exponential shape to a stretched exponential and finally to a $exp(-C{ln}^{\kappa }\ell )$ behavior, with $\kappa >1$.

Figure 1

(a) A qualitative sketch of the phase diagram of the RSM in the regime $\mu \in (0,1)$, in the parameter plane of the disorder strength $\beta$ and the energy $\lambda$. (b) The analogous phase diagram for the BBRM ensemble. It differs from (a) in that ${\beta }_c=\infty$: mobility edges always for any large $\beta$.

Figure 2

(a) The absolute value of matrix elements of PBRM (line) and BBRM (dots) on one line of the matrix,as a function of the distance from the diagonal. For both matrices, we used the parameters $\mu =0.5,\beta =0.1$. The red dashed line depicts the standard deviation common to both. Many elements of the BBRM are too small to be drawn on the plot. (b) A schematic color plot of the matrix elements' amplitudes of a broadly distributed banded random matrix. Off diagonal, the matrix is quasisparse. (c) The same plot for the PBRM. The hopping elements have narrowly distributed amplitudes.

Figure 3

(a) The DOS of the randomized sparse matrix with parameters $\mu =0.5$, $\beta =1,2$, and 5, and matrix sizes $N=2^7$ and $2^{13}$. (b) DOS of the BBRM for $\mu =0.5$ and various $\beta$. The curve $\beta =20$ is compared to the prediction (11), plotted in thick curve. In all numerical data, lighter colors represent a small system size $N=2^7$ and the color black represents a large system size $N=2^{13}$.

Figure 4

An illustration of the block-diagonalization procedure. (a) Depicts a diagonal block of size ${\ell }_{k+1}\times {\ell }_{k+1}$, which is divided in into block matrices of size ${\ell }_k\times {\ell }_k$. We pick two blocks of sites labeled by $m$ and $n$, respectively. The diagonal ${\ell }_k$ blocks are banded matrices (represented as squares with gradient color). The choice of the scales [Eq. (14)] ensures that the off-diagonal block has a few large elements $Q_{mn}\sim O\left(1\right)$ (represented as a dot) while the typical elements are negligibly small. (b) Depicts the matrix ${\tilde{Q}}_{\alpha \gamma }^{\left(k\right)}$ [Eq. (12)] after diagonalizing the ${\ell }_k$ diagonal blocks, so the latter become diagonal. The unitary transformations smear out the quasisparseness of $Q_{mn}$ far from the diagonal, and the transformed elements ${\tilde{Q}}_{\alpha \gamma }^{\left(k\right)}$ are no longer broadly distributed.

Figure 5

The IPR of the eigenstates of the RSM, plotted for parameters $\mu =0.5$, $\beta =1,2$, and 5, for matrix sizes $N=2^7,\cdots ,2^{13}.$ The numerical protocol is identical to that of Fig. 6. We recall the for ideally extended (localized) states, the $y$ value is 1 (0, respectively). In panel (b), the estimated position of mobility edges ${\lambda }_c=\pm 1.0\left(2\right)$ is also indicated. In all plots, lighter colors represent smaller system sizes.

Figure 6

Measure of the IPR of BBRM eigenstates as function of eigenvalue $\lambda$, at $\mu =0.5$ and with various sizes of matrices, $N=2^7,\cdots ,2^{14}$ (darker colors indicate larger systems). According to Eq. (21), extended (localized) states would have $y$ coordinate $-\overline{ln{P}_2}/lnN\to 1$ ($\to 0$, respectively) when $N\to \infty$. (a) $\beta =5$. The dashed lines indicate the mobility edges, estimated at ${\lambda }_c\approx -0.65\left(5\right)$ and ${\lambda }_c\approx 0.70\left(5\right)$. (b) The same measure for $\beta =1$. As $N$ increases (darker color), the eigenstates with $\lambda \in (-3,3)$ become more and more delocalized.

Figure 7

(a) Numerical measure of ratio $\chi$ [Eq. (24)] for the BBRM model with $\mu =0.5$, $\beta \in (1,10)$ and of sizes $N=2^7$ (light color) to $2^{14}$ (dark color). Eigenvalues in $(0.5,1.5)$ are binned into five bins of equal width. More than ${10}^5$ different gaps are averaged over for each data point. (b) A zoom-in to the critical regime. We locate the critical value of the rescaled gap ratio observable ${\chi }_c\approx 0.28\left(2\right)$.

Figure 8

An illustration of the dynamics of the epidemics dynamics model of [13] in $d=1$. Initially ($t=0$), the only occupied site at $n=0$ infects other sites at a rate that decays as Eq. (25). Later, $t>0$, infected sites go on to infect remaining sites.

Figure 9

Illustration of models involved in the mapping described in Sec. 5a. (a) In the FPP model, a waiting time ${\tau }_{mn}$ is assigned to each pair. The first-passage time $T_n$ is a minimum of all paths, given by Eq. (26). (b) In the polymer model, which is a finite-temperature extension of the FPP model, we sum over all paths connecting two points 0 and $n$, ranging from the direct path (blue, dashed line) and detoured paths with loops, for example, the one in black. (c) The amplitude at site $n$ of (strong-disorder) BBRM eigenstates localized around 0 is in turn related to the polymer partition function by (31).

Figure 10

Numerical measure of the decay of the RSM model (a) and the BBRM (b) model in the $\mu >1$ regime. The dots are the numerical data, and the thin lines represent the prediction (32). For all measures $N=2^{11}$, and the $N/4$ states in the center of the spectrum of every realization are measured. (a) From top to bottom: $\mu =2.2,1.8,1.5,1.2$. For all data, $\beta =0.1$, $N=2^{11}$. (b) From top to bottom: $\mu =2.5,1.8,1.5,1.2$, $\beta =0.2,0.2,0.3,0.3$.

Figure 11

Representative samples of the long-range percolation random graph $G$ [see Eq. (33)]. The lattice is represented by the horizontal line, and the links by the semicircles. The lattices have sites $i=1,\cdots ,1024$, with open boundary condition. (a) When $\mu >1$, the graph is divided by the vertical lines (cuts) into intervals. (b) When $\mu <1$, no cut is present.