Duality in Power-Law Localization in Disordered One-Dimensional Systems

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, $1/r^a$. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of $a>0$. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops ($a<1$) and short-range hops ($a>1$), in which the wave function amplitude falls off algebraically with the same power $\gamma$ from the localization center.

Figure 1

Duality for $0<a<1$ and $1<a<2$ of the localization properties of the PLE model (blue circles) and the model of Refs. [20, 21] with diagonal disorder and deterministic sign-definite, long-range hopping (red triangles): (a) slope $k$ of the MFS extrapolated to $N\to \infty$; (b) exponent $\gamma$ of the algebraic localization of $|\psi {|}^2$. Dashed lines correspond to $\gamma =1/k=2a$ for $1<a<2$ or to $\gamma =4-2a$ for $0<a<1$. The error bars mainly stem from the extrapolation to infinite size systems.

Figure 2

Average of $\ln (|\psi (n){|}^2)$ for 1D chain of $N=2\times {10}^4$ sites vs logarithm of the distance $n$ from the localization center. (a) PLE model with random site positions and deterministic hopping function $1/r^a$ for $a=0.5$ and $a=1.5$. The duality Eq. (1) manifests itself in the same slope $-\gamma (a)$ of the curves at the tail. (b) Model of Ref. [21] on regular lattice with diagonal disorder and deterministic power-law hopping for $a=0$, $a=0.1$, and $a=1.9$. Notice a drastic difference between the exactly solvable [30] case $a=0$ and the two dual cases with $a$ close to zero and two ($a=0.1$ and $a=1.9$). At $a\to 0$, the eigenstates are algebraically localized with $\gamma \to 4$, while at $a=0$ they are “critically multifractal” with $f(\alpha )=\alpha /2$ for $0<\alpha <2$ and have a form of a sharp peak on the top of a background [31]. In contrast to the eigenstates of the standard Anderson model, where the background is exponentially small, the critically multifractal states are characterized by much stronger background $\sim N^{-2}$.

Figure 3

Dependence of $\gamma$ on the energy for the PLE model. (a) DOS for $a=0.5$. (b) DOS for $a=1.5$. (c) Average of $\ln (|\psi (n){|}^2)$. For all energies $E_{-}<E<E_{+}$, in both cases $a=0.5$ and $a=1.5$, the slope of the curves $-\gamma \approx -3$ is the same. The fraction of states beyond this energy interval, i.e., at small and large energies, is $\sim 3\%$ in both cases.

Figure 4

Multifractal spectrum $f(\alpha )$ (blue curve) obtained from the distribution function $P(|{\psi }_s(n){|}^2)$, extrapolated to $N\to \infty$, for (a) $a=0.5$; (b) $a=1$, and (c) $a=1.5$. We consider eigenstates at the maximum of the density of states. The inset of (a) shows that the inverse participation ratio $I_2$ does not depend on $N$, thus proving insulating behavior. Note that the concave feature in $f(\alpha )$ appears in all cases (see text). The red dashed line in each case shows the triangular $f_{\mathrm{tr}}(\alpha )$ with $k<1/2$ that corresponds to the same moments $I_q$ as $f(\alpha )$. The blue dashed line indicates that $f(0)=0$, which is confirmed by our numerics.