Anderson localization in generalized discrete-time quantum walks

We study Anderson localization in a generalized discrete-time quantum walk—a unitary map related to a Floquet driven quantum lattice. It is controlled by a quantum coin matrix which depends on four angles with the meaning of potential and kinetic energy, and external and internal synthetic flux. Such quantum coins can be engineered with microwave pulses in qubit chains. The ordered case yields a two-band eigenvalue structure on the unit circle, which becomes completely flat in the limit of vanishing kinetic energy. Disorder in the external magnetic field does not impact localization. Disorder in all the remaining angles yields Anderson localization. In particular, kinetic-energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. Strong disorder in potential and internal magnetic field energies allows one to obtain analytical expressions for spectrally independent localization length, which is highly useful for various applications.

Figure 1

The schematics of the discrete-time quantum walk. The arrows indicate the directions of a single step transfer.

Figure 2

The dispersion relation $\omega \left(k\right)$ for different values of $\theta$: $\theta =0$ (no gap, black solid line), $\pi /4$ (finite gap, blue line), $\pi /2$ (flat bands, red lines). Here, $\phi ={\phi }_1=0$.

Figure 3

The dependence of the localization length on the frequency $\omega$ for disorder in $\phi$. From top to bottom at $\omega =\pi /2$: $W=\pi /20,\pi /5,\pi /2,\pi$. Here, $\theta =\pi /4$. The red vertical lines indicate the boundaries of the allowed bands (6).

Figure 4

The dependence of the localization length on the frequency $\omega$ for disorder in ${\phi }_2$. From top to bottom at $\omega =\pi /2$: $W=\pi /20,\pi /5,\pi /2,\pi$. Here, $\theta =\pi /4$. The red vertical lines indicate the boundaries of the allowed bands (6).

Figure 5

The localization length for weak disorder in ${\phi }_2$ near the band center $\omega =\pi /2$ ($\theta =\pi /4$). Here, $W=\pi /50$. Symbols are the results of computations; lines are guides for the eye. Inset: Same as main figure, but resolving the frequency dependence of the localization length on a logarithmic scale close to the band center.

Figure 6

The dependence of the localization length on the frequency $\omega$ for disorder in $\theta$. From top to bottom at $\omega =\pi /2$: $W=\pi /20,\pi /5,\pi /2,\pi$. Here, $\theta =\pi /4$. The red vertical lines indicate the boundaries of the allowed bands (6).

Figure 7

The numerically calculated dependence of the localization length $L_{\mathrm{loc}}$ on the characteristic frequency $\omega$ near the band center $\omega =\pi /2$ for disorder in $\theta$. The average angle ${\theta }_0=\pi /4$ and the strength of disorder $W=\pi /2$. Symbols: results of numerical computations; lines are a guide for the eye. Inset: Resolving the frequency dependence of the localization length on a logarithmic scale close to the band center. Symbols: results of numerical computations. The straight line is a linear fit of the data.

Figure 8

The numerically calculated dependence of the localization length on the frequency close to $\omega =0$ for disorder in $\theta$. The strength of disorder $W=\pi /2$, and the average angles ${\theta }_0=0$ (red circles) and ${\theta }_0=\pi /4$ (blue squares). Symbols: results of numerical computations; lines are a guide for the eye. Inset: Same as main figure, but resolving the frequency dependence of the localization length on a logarithmic scale close to $\omega =0$. The straight line is a linear fit of the data.

Figure 9

The localization length as a function of the disorder strength for disorder in $\phi$ for $\omega =\pi /2$ (band center). Solid lines: the analytical result (26); symbols: numerical computations. Here, ${\theta }_0=\pi /8,\pi /4,3\pi /8$, from top to bottom. The predicted scaling $L_{\mathrm{loc}}\sim 1/W^4$ is observed.

Figure 10

The localization length as a function of the disorder strength for disorder in $\phi$ for $\omega ={\theta }_0$ (band edge). Solid lines: the analytical result (27) with $\eta =1.36$; symbols: numerical computations. Here, ${\theta }_0=\pi /8,\pi /4,3\pi /8$, from top to bottom. The predicted scaling $L_{\mathrm{loc}}\sim 1/W^{2/3}$ is observed.

Figure 11

The mean localization length inside the band as a function of the disorder strength for disorder in ${\phi }_2$. Solid lines: the analytical result (29) with an additional fitting parameter $\eta =0.5$; symbols: numerical computations. Here, ${\theta }_0=\pi /8,\pi /4,3\pi /8$, from top to bottom. The predicted scaling $L_{\mathrm{loc}}\sim 1/W^2$ is nicely observed.

Figure 12

The numerically (symbols) and analytically (solid lines) calculated dependence of the coefficient $S=\eta \frac{12{cos}^2{\theta }_0}{W^2}$ [see Eq. (36)] on the strength of disorder $W$, with the numerical fitting parameter $\eta =0.22$. Here, ${\theta }_0=\pi /8,\pi /4,3\pi /8$, from top to bottom. The predicted scaling $L_{\mathrm{loc}}\sim 1/W^2$ is observed.

Figure 13

Localization length under maximal disorder in $\varphi$. Red solid lines are the analytical result (48); blue dots are the numerical result.

Figure 14

The inverse localization length $1/L_{\mathrm{loc}}(\omega =0)$ as a function of ${\theta }_0$ and $W$ according to Eq. (50). The inverse localization length strictly vanishes on the lines $\theta =0,\pi$ and $W=\pi$.