Disorder-free localization in quantum walks

The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder-free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved moments. When the interaction is weak, the spread of the walker shows subdiffusive behavior having downscaled ballistic tails in the evolving probability distribution at intermediate timescales. However, as the interaction gets stronger the walker gets completely localized in total absence of disorder in both lattice and initial state. Using a matrix-product-state ansatz, we investigate the relaxation and entanglement dynamics of the on-site spins due to their coupling with the quantum walker. Surprisingly, we find that, even in the delocalized regime, entanglement growth and relaxation occur slowly, unlike majority of the other models displaying a localization transition.

Figure 1

Schematic representation of the first 3 steps of the quantum walk model employed in the paper. The quantum walker (blue-dark balls) interacts with the spin chain (gray-light balls) in each step such that the spin rotates by an angle $\varphi$ when the walker is on that site and has no dynamics otherwise. The starting point of the walk is indicated by 0 and a balanced walk is depicted.

Figure 2

Quasienergy $E$ spectrum and the inverse participation ratio $\widehat{\text{IPR}}$ of the corresponding eigenstates with respect to the interaction parameter $\varphi$ for the quantum walker stepping with ${\widehat{W}}_M$. For a given $E$, the inverse participation values are averaged over degenerate cases and normalized by the number of spins, $N=18$. After the critical spin rotation angle $\varphi =\pi /4$, the band gaps close and the quasienergy eigenstates become localized.

Figure 3

(a) Probability distribution of the walk in the position space after 400 steps and (b) its variance in position space as a function of step number for the initial state $\left|{\mathrm{\Psi }}_0\right.〉$. (c) Inverse participation ratio (IPR) and (d) variance are shown as a function of the spin rotation angle $\varphi$, at different time steps. IPR is normalized by the lattice size $N=801$. The walker is exponentially localized for $\varphi >\pi /4$.

Figure 4

(a) Entanglement entropy between two halves of the spin chain as a function of time for different values of $\varphi$ increasing from bottom to top curve and (b) as a function of the angle $\varphi$ at different time steps. (c) Expectation values of $z$ component and (d) $y$ component of local spins along the chain at time $t=100$ for different values of $\varphi$ increasing from bottom to top curve.

Figure 5

Entanglement entropy between bipartite partitions of the chain at the $n\mathrm{th}$ site, $S_n$, as a function of time.

Figure 6

(a) The logarithmic $A$ and linear $B$ contributions to the growth of the entanglement entropy $S$ [see Eq. (A10)]. (b) Entanglement entropy between bipartite partitions of the chain at the $n\mathrm{th}$ site, $S_n$, as a function of the angle $\varphi$ (b) after 100 time steps.

Figure 7

(a) Entanglement entropy as a function of time for a lattice with $N=15$ for different values of $\varphi$. (Partitioning of the system is done in half, i.e., $7+8=15$, and only spins on the left are traced out for these calculations.) For $\varphi >\pi /4, S$ grows as $\propto logt$ in the second decade of the evolution before the systems eventually thermalize. (b) Entanglement entropy per site $S/N$ as a function of system size $N$ after thermalization at $t={10}^5$ approaches a constant indicating a volume law.

Figure 8

Effect of the symmetry-breaking field in Eq. (B1) for the interaction angles $\varphi =\pi /8$ (upper panel) and $\varphi =3\pi /8$ (lower panel). Panels (a) and (d) show the probability distribution. Time behavior of the variance in position space is shown in panels (b) and (e). Time behavior of the entanglement entropy is shown in panels (c) and (f).