Quasi-resonant diffusion of wave packets in one-dimensional disordered mosaic lattices

To describe 1D non-interacting spinless particle systems, we use the standard single-chain tight-binding model

where $\psi_{n}$ and $\varepsilon_{n}$ are the wave function amplitude and the on-site potential at the $n$-th lattice site respectively. $E$ is the energy and $J$ is the coupling strength between nearest-neighbor sites. From now on, we will measure all energy scales in units of $J$ and set it equal to 1. In this study, we will investigate the transport and localization properties of a model which we call disordered mosaic lattice model. This model is defined by

where the inlay parameter $\kappa$ representing the period of the mosaic modulation is a fixed positive integer larger than 1 and $m$ is an integer running from 1 to $N$. Then the total number of sites $L$ is equal to $\kappa N$. The on-site potential $\beta_{n}$ at the $m\kappa$-th site is a random variable uniformly distributed in the interval $[-W,W]$, where $W$ is the strength of disorder. In all other sites, the on-site potential takes a constant value of $V_{0}$. In [43], the authors have studied a quasiperiodic mosaic lattice model, where $\beta_{n}$ is a quasiperiodic potential of Aubry-André-type [59]. Our model is different from that model in that $\beta_{n}$ is random instead of quasiperiodic. We note that if $\beta_{n}$ is a constant potential different from $V_{0}$, the model becomes perfectly periodic with period $\kappa$.

Following the procedure given in [58], we first assume that a monochromatic wave of energy $E$ is incident from the left side of the disordered region and define the amplitudes of the incident, reflected, and transmitted waves $A$, $B$, and $C$ by

where $q$ is related to $E$ by the free-space dispersion relation $E=2\cos q$. In the absence of dissipation, the law of energy conservation $|B|^{2}+|C|^{2}=|A|^{2}$ should be satisfied. In order to solve Eq. (1) numerically, we first fix $\psi_{L-1}$ to 1, then we obtain $C=\exp[-iq(L-1)]$ and $\psi_{L}=\exp(iq)$. Knowing the values of $\psi_{L}$ and $\psi_{L-1}$, we can solve Eq. (1) iteratively to obtain $\psi_{L-2}$, $\psi_{L-3}$, $\cdots$, $\psi_{2}$, $\psi_{1}$. Using the definition of $A$ and $B$ given in Eq. (7), we can express them in terms of $\psi_{1}$ and $\psi_{2}$:

We are interested in the case where the length of the disordered region is sufficiently large. Then the large portion of the incident wave power will be reflected. Therefore we focus on the behavior of the reflection coefficient $\tilde{r}(E)$ and the reflectance $\tilde{R}(E)$ defined by

Next we consider a Gaussian wave packet characterized by the spectrum

where $E_{0}$ is the central energy of the wave packet and $\sigma$ is its spectral width. $f(E)$ satisfies

The time-dependent reflection coefficient $r(t)$ of the incident Gaussian wave packet can be calculated using

where the range of integration is limited to $-2\leq E\leq 2$, since the reflection coefficient $\tilde{r}(E)$ can be defined only inside the band satisfying $E=2\cos q$. The disorder-averaged time-dependent reflectance $\left\langle R(t)\right\rangle$ is obtained from

where $\langle\cdots\rangle$ denotes averaging over a large number of different disorder configurations.

According to the previous theories of Anderson localization in the time domain [50, 52, 53, 54, 55, 58], the exponent characterizing the power-law decay of $\left\langle R(t)\right\rangle$ in the long-time limit provides an important information on the state of the system. Specifically, it has been established that $\left\langle R(t)\right\rangle$ decays as $t^{-2}$ in the localized regime, while it decays as $t^{-3/2}$ in the regime of classical diffusion. Finally, we point out that studying Anderson localization based on the measurements in the reflection geometry has substantial experimental advantages over more conventional approaches in the transmission geometry, in that the reflectance is often more easily measurable than the transmittance and there exist situations where measurements in the transmission mode are not possible. These advantages are especially relevant to the fields such as optics, acoustics, and seismology [60, 61, 62, 63].

In most of our calculations, we have set $V_{0}=0$ and computed $\langle R(t)\rangle$ by averaging over 10,000 distinct random configurations of $\varepsilon_{n}$. The step size in energy, $\Delta E$, is $10^{-3}$. We have chosen the system size $L$ to be sufficiently large so that the transmission is negligible and the system can be considered an effectively semi-infinite medium.

Our main goal is to study the dynamics of wave packet propagation in 1D disordered mosaic lattices with $\kappa\geq 2$. However, it is instructive to first consider the $\kappa=1$ case corresponding to the ordinary Anderson model to clarify the effects of the mosaic modulation. In a recent study, a numerical analysis of the time-dependent reflectance for the 1D Anderson model has been presented [58]. It has been reported that for any $W\in[1,4]$ and $E_{0}\in[-2+2\sigma,2-2\sigma]$, the time dependence of $\langle R(t)\rangle$ obeys an inverse-square law of the form $\langle R(t)\rangle\propto t^{-2}$ in the long-time limit. This implies that all the incident wave packets exhibit an Anderson localization behavior. It has also been shown that this behavior is independent of the spectral shape of the wave packet or its spectral width $\sigma$ as long as it is as small as 0.05 and 1. In the present work, we consider Gaussian wave packets with a fixed spectral width of $\sigma=0.05$. In Fig. 1, we show our calculations for the time decay of $\langle R(t)\rangle$ when $\kappa=1$, $W=2$, and the system size $L=1000$, which agree well with [58].

We are interested in exploring the transport behavior when the random mosaic modulation defined by Eq. (4) is introduced into the on-site potential of a lattice model. In Fig. 2, we set $\kappa=2$ and plot the time evolution of $\langle R(t)\rangle$ for various values of $E_{0}$ when $W=1$, 2, and 4. We find that for all values of $E_{0}$, $\langle R(t)\rangle$ shows a power-law decay of the form $\langle R(t)\rangle\propto t^{-\gamma}$ in the long-time limit. The exponent $\gamma$ is equal to 2 for almost all $E_{0}$ values except for a narrow region close to $E_{0}=0$. In contrast, the value of $\gamma$ at $E_{0}=0$ is equal to $3/2$ with a good approximation and is markedly different from the Anderson localization behavior shown for other $E_{0}$ values. In addition, the value of $\langle R(t)\rangle$ for $E_{0}=0$ is noticeably larger than those for other $E_{0}$ values when $\ln t$ is sufficiently large, as can be seen from the curves drawn with thick lines in Fig. 2. This behavior occurs for all values of the disorder parameter $W$ considered here, as long as the system length $L$ is sufficiently large such that the effect due to the leakage of the wave packet into the transmitted region is negligible. When the disorder is weak, we need to use a larger $L$ to satisfy such a condition.

In Fig. 3, we set $\kappa=3$ and plot the time evolution of $\langle R(t)\rangle$ for the same values of $E_{0}$ and $W$ as in Fig. 2. Similarly to the $\kappa=2$ case, the localization behavior takes place for most values of $E_{0}$, while the diffusive behavior appears only near a special value of $E_{0}$. In contrast to the previous case, however, the diffusive behavior with $\gamma=3/2$ is observed at $E_{0}=1$. In addition, we have verified numerically that a similar diffusive behavior also occurs at $E_{0}=-1$, though it has not been shown here explicitly.

Next, in Fig. 4, we consider the $\kappa=4$ case. The overall long-time behavior is similar to the $\kappa=2$ and 3 cases, but the diffusive behavior with $\gamma=3/2$ occurs at two different values $E_{0}=0$ and 1.4 ($\approx\sqrt{2}$) in the present case. In addition, a similar behavior is also observed at $E_{0}=-\sqrt{2}$. Since the diffusive long-time behavior appears only in a narrow range of $E_{0}$ values, it can be considered as a kind of quasi-resonance. Combining the results obtained for $\kappa=2$, 3, and 4, we deduce that the quasi-resonances occur symmetrically with respect to $E_{0}=0$ and their total number is equal to $\kappa-1$. We also find that this behavior is unaffected by the disorder strength when $W\gtrsim 1$.

In order to examine the quasi-resonant nature of the diffusive long-time behavior more clearly, we plot the exponent $\gamma$ obtained by fitting the numerical results in the long-time region where $5<\ln t<8$ versus $E_{0}$, when $W=2$, 3, 4, 5 and $\kappa=2$, 3, 4 in Fig. 5. We point out that there is a mirror symmetry in a statistical sense with respect to $E_{0}=0$, though the region with $E_{0}<0$ is not shown here. Numerical results for many $E_{0}$ values around the sharp dips at $E_{0}=0$, $1$, and $\sqrt{2}$ in addition to those shown in Figs. 2, 3, and 4 have been used in Fig. 5. As $E_{0}$ varies away from the values at the dips, $\gamma$ increases rapidly from 1.5 to 2. The half-widths (full widths at half maximum) of all the dips are similar and roughly equal to 0.15. In order to show the positions of the quasi-resonances more clearly, some values of $E_{0}$ versus $\gamma$ around the sharp dips are listed in Table 1.

In the rest of this subsection, we present some additional results which are useful in deducing a general formula for the quasi-resonant energy values. In Fig. 6, we plot $\langle R(t)\rangle$ versus $t$ at $E_{0}=0$ when $W=4$, $\kappa=2$, 3, 4, 5, and 6, and $V_{0}=0$. We find that the behavior of the incident wave packet at $E_{0}=0$ depends on the parity of $\kappa$. When $\kappa$ is odd, the wave packet exhibits Anderson localization, while, when $\kappa$ is even, it does a diffusive behavior.

Finally, in Fig. 7, we compare the long-time scaling behavior at $E_{0}=0$ and 0.2, when $V_{0}=0$ and $V_{0}=0.2$ for $\kappa=2$, 4, and 6. A diffusive behavior is observed at $E_{0}=0$ when $V_{0}=0$, while it is found at $E_{0}=V_{0}$ when $V_{0}=0.2$. Therefore the quasi-resonance energy is found to depend on the value of $V_{0}$ as well as $\kappa$.

The numerical results presented above clearly show that a wave packet propagating inside a disordered mosaic lattice displays a diffusive behavior for some special values of the wave packet’s central energy in a quasi-resonant manner, although it exhibits Anderson localization for all the other values of the central energy. In other words, mosaic modulation of the lattice potential causes the usual exponential Anderson localization to be destroyed at special discrete values of the energy dependent on the modulation period. Thus we find that the periodic mosaic modulation is a feature that can induce a new type of delocaliztion in low-dimensional disordered systems.

From our numerical results given in the previous subsection, we can easily deduce that the central energy at the quasi-resonances, $E_{R}$, is given precisely by

The number of $E_{R}$ is equal to $\kappa-1$. $E_{R}$ is distributed symmetrically with respect to $V_{0}$ and $E_{R}=V_{0}$ is included only when $\kappa$ is even. For several small values of $\kappa$, we obtain

The analytical formula for the quasi-resonance energy $E_{R}$ appears to agree with the formula obtained for the resonance energy [e.g., Eq. (16) of Ref. 29] where the delocalization occurs in 1D binary random $N$-mer models with $N=\kappa$. However, this agreement is only coincidental and superficial. In the binary random $N$-mer model, the resonance energy has been obtained from the condition that the transmittance through a single $N$-mer is unity, and therefore the transmittance at the resonance energies is identically equal to 1 and the corresponding states are completely extended [27, 28, 29, 30]. In contrast, at our quasi-resonance energies, the transmittance is not equal to 1 and the states are not extended but critical states, as will be explained in Sec. III.3. In the binary random $N$-mer model of Ref. 29, the on-site potential can take one of the two values 0 and $V_{0}$. The problem becomes nonrandom and trivial if $V_{0}=0$ and so we need to exclude that case. In order to have extended states, it is also necessary to have the condition that $|E_{R}|\leq 2$, which gives another constraint for $V_{0}$ [29]. On the contrary, in our model, $V_{0}$ is just a background on-site potential and can take any arbitrary value including zero. We also point out that the cosine form for the resonance energy of the $N$-mer model arises only when the on-site potentials at all the $N$ sites of an $N$-mer are the same. Even if they are not uniform, there can still exist resonance energies, but they will not be given by a simple cosine form.

We pay close attention to the results that the dependence of $\gamma$ on $E_{0}$ shown in Fig. 5 is insensitive to the strength of disorder $W$. In our model, the random potential is present only at periodically spaced sites $n=\kappa m$ ($m=1,2,3,\cdots$). In order for the numerical results to be insensitive to the disorder strength, it is natural to assume that the wave-function amplitudes take very small values close to zero at those sites. That is, the wave function has nodes there. This implies that the eigenfunctions have a periodicity with the wavelength $\lambda$ satisfying

For such eigenfunctions, the energy eigenvalues should be very close to the values obtained in the disorder-free case, which are given by $E=V_{0}+2\cos q$. Since the wave vector $q$ is related to $\lambda$ by $q=2\pi/\lambda$, we can obtain Eq. (14) in a straightforward way. This argument strongly suggests that the states at the energies $E_{R}$ are rather insensitive to the disorder and therefore are not standard exponentially localized states. Our conjecture that the wave functions at the quasi-resonance energies have nodes at $n=\kappa m$ will be confirmed later in Fig. 9 in Sec. III.3.

It is highly instructive to consider a related periodic model, which we may call periodic mosaic lattice model, defined by Eq. (4), but with $\beta_{n}$ replaced by a constant $\beta$. Since this model is strictly periodic with a period $\kappa$, it contains alternating forbidden and allowed bands. In Fig. 8, we show its band structure when $\kappa=4$, $\beta=10$, and $V_{0}=0$. The gray-colored region denotes the forbidden bands. The dispersion relation between $E$ and $(4/\pi)q$ is plotted in red inside the allowed bands. We have found that the energy values at the upper edges of the allowed bands are precisely given by Eq. (14) for all values of $\kappa$, $\beta$, and $V_{0}$. As the potential strength $\beta$ increases to large values, the widths of the allowed bands become very small. In the large $\beta$ limit, the allowed bands consist of infinitesimally narrow regions at the energies given by Eq. (14). If we compare this limiting case with the disordered mosaic lattice model for large $W$, we find that the forbidden bands and the narrow allowed bands at $E_{R}$ in the periodic case are replaced respectively by exponentially localized states and diffusive states in the random case.

In order to better understand the nature of the states at the quasi-resonance energies, we digress briefly from the study of the time-dependent reflectance to consider the spectral properties of a finite disordered mosaic lattice. Specifically we calculate the participation ratio $P(E_{k})$ for the $k$-th eigenstate with the energy eigenvalue $E_{k}$, which is defined by [64]

where $\psi_{n}^{(k)}$ is the value of the $k$-th eigenfunction at the site $n$ ($n=1,2,\cdots,L$). For a finite lattice, $P(E_{k})$ gives approximately the number of sites over which the $k$-th eigenfunction is extended. To study the large-$L$ scaling behavior of the participation ratio in a disordered system, it is convenient to introduce a double-averaged quantity $\langle P(E)\rangle$, where $P(E)$ is obtained by averaging over all eigenstates within a narrow interval $\Delta E$ around $E$ and $\langle\cdots\rangle$ denotes averaging over a large number of independent disorder configurations. When $L$ is sufficiently large, we find a power-law scaling behavior of the form

with the scaling exponent $x$. It has been well-established that for extended states, the scaling exponent $x$ is equal to 1 and $\langle P(E)\rangle$ increases linearly with increasing the lattice size $L$. On the contrary, for (exponentially) localized states, the exponent $x$ is zero, that is, $\langle P(E)\rangle$ does not depend on $L$ and converges to a constant value as $L\to\infty$. For critical states at the boundary between extended and localized states, the exponent should be in the intermediate range of $0<x<1$. Thus the finite-size scaling analysis of $\langle P(E)\rangle$ provides a very useful information about the nature of the states.

In Figs. 9(a) and 9(b), we show the results of numerical calculations of $\langle P(E)\rangle$ obtained when $\kappa=3$, $\Delta E=0.1$, $W=1$, 2, and the number of disorder configurations is 1000. In Fig. 9(a), we find that the average participation ratio for $L=1000$ has pronounced peaks at the quasi-resonance energies $E=\pm 1$, while it remains small at other energies. We have confirmed that for the value of $E$ away from $\pm 1$, $\langle P(E)\rangle$ approaches a constant as $L$ increases to large values, implying that $x$ is zero and the states are exponentially localized. In contrast, at $E=\pm 1$, $x$ is neither 0 nor 1 as demonstrated for $E=1$ in Fig. 9(b). From fitting the data to Eq. (24), we obtain $x\approx 0.25$ for $W=1$ and $x\approx 0.23$ for $W=2$. These results clearly demonstrate that the states at the quasi-resonance energies are neither extended nor exponentially localized states. We notice that they closely resemble the critical states observed at Anderson localisation-delocalisation and quantum Hall plateau transitions [3, 65]. They are totally different from the resonant states appearing in the binary random $N$-mer model, where such states have been found to be completely extended.

In Figs. 9(c) and 9(d), we illustrate the spatial distributions of the wave function amplitudes obtained for a single disorder configuration when $\kappa=3$, $L=1000$, $W=5$, and $E=0$, 1. When the energy is away from the quasi-resonance energies, we find that the wave function is strongly localized as in Fig. 9(c). At the quasi-resonance energies, however, the wave function has a distinct spatial distribution with several disjointed occupied regions as in Fig. 9(d). This is a unique characteristic frequently observed in critical or multifractal states. We also find that the wave function amplitudes are very close to zero at all the sites satisfying $n=m\kappa$ ($m=1,2,\cdots$) as shown in the inset of Fig. 9(d), which is fully consistent with our conjecture that there are wave function nodes at such sites. This node structure is absent for localized wave functions and occurs only at the quasi-resonance energies. It leads to the behavior that when $W$ is sufficiently large, the quasi-resonant states are insensitive to the strength of disorder.

From separate calculations for the eigenfunctions of the periodic mosaic model, we have confirmed that they also have nodes at $n=m\kappa$ ($m=1,2,\cdots$) only for the energies given by Eq. (14). Let us suppose that starting from the periodic mosaic model, we turn on the weak disorder at the sites $n=m\kappa$ ($m=1,2,\cdots$) and increase its strength gradually. The special node structure of the wave function we have discussed so far will be maintained at the quasi-resonance energies regardless of the strength of disorder. Therefore the node structure of the wave functions is a crucial feature that connects the periodic and disordered mosaic lattice models and induces the critical states at the quasi-resonance energies. A more systematic and comprehensive analysis about the nature of the quasi-resonant states in the framework of the spectral problem will be presented elsewhere.