Disorder-induced two-body localized state in interacting quantum walks

We observe the onset of nonergodicity from ballistic propagation of a two-body bound state in an interacting discrete-time quantum walk (DTQW) due to time-dependent disorder in the interaction. The effect of the disorder on the two-body state can be interpreted as Anderson-type localization in a projected DTQW, but without saturation of entanglement entropy. We characterize the two-body localization in terms of the rate of growth of entanglement entropy and compute the localization length. We find indications for two distinct growth laws for the entanglement: A logarithmic law in the delocalized phase and a double-logarithmic law in the localized phase. We discuss similarities with many-body localization.

Figure 1

Dispersion in the manifold $X_1-X_2=0$ vs quasimomentum $p$. The parameters correspond to the unbiased coin $\rho =1/2$ (red) and fully biased coins $\rho =1$ (blue, Dirac cones) and $\rho =0$ (magenta, flat bands at $\mu +\varphi =0,\pm \pi$).

Figure 2

Two noninteracting unbiased Hadamard walks [$\mu \left(t\right)=0$]. The entanglement entropy $S_A$ for various partitionings $A$ and probability distribution of the quantum walk at step 15 (inset). Particles 1 and 2 remain unentangled (red line). We denote the position and spin of particle $i$ by $X_i$ and ${\zeta }_i$, respectively, and the total set of degrees of freedom is $X_1,{\zeta }_1,X_2,{\zeta }_2$. The initial condition, $|\mathrm{\Psi }\left(0\right)\rangle =|x_0\rangle \otimes |+\rangle$, is the same for both particles. At odd (even) time steps the even (odd) sites are unoccupied, and the joint probability distribution is symmetric about the antidiagonal.

Figure 3

Collision of two interacting unbiased Hadamard walks [$\mu \left(t\right)=3\pi /2, W=0$]. The entanglement entropy $S_A$ for various partitionings $A$. Particles 1 and 2 remain unentangled until the collision at $t=5$ (red line), which also breaks the symmetry between the particles. The initial condition, $|{\mathrm{\Psi }}^{\left(2\right)}\left(0\right)\rangle =|x_{-5},+\rangle \otimes |x_5,+\rangle$, contains the particles evenly separated by ten sites, but with the same spin state $|+\rangle$.

Figure 4

Bound state in interacting unbiased Hadamard walks [$\mu \left(t\right)=3\pi /2, W=0$]. Shown are the probability distribution at $t=100$ (top left) and $t=150$ (top right) and the entanglement entropy (bottom). The initial condition is $|{\mathrm{\Psi }}^{\left(2\right)}\left(0\right)\rangle =|x_0,-\rangle \otimes |x_0,-\rangle$. The symmetry of the initial condition imposes a similar symmetry on the partitioning, meaning that the entropy $S_A$ for $A=X_2$ and $A={\zeta }_2$ is the same as shown for particle 1. The entropy partitioned between the two particles (red line with pentagons) grows as $\sim {log}_{10}\left(t\right)$ at large steps $t$.

Figure 5

Entanglement entropy vs disorder strength $W$. (a) $W=\pi /30$. (b) $W=\pi /6$. (c) $W=\pi /2$. (d) $W=\pi$. The legend is the same as in Fig. 3.

Figure 6

Entanglement entropy with maximal disorder $W=\pi$. The entanglement grows slowly, in accordance with the double-logarithmic law (31). A straight line in this plot corresponds to a double-logarithmic law as the $x$ axis is scaled by ${log}_{10}\left[{log}_{10}\left(t\right)\right]$. The legend is the same as in Fig. 3.

Figure 7

Localization length $\xi$ vs quasienergy $\varphi$ without disorder. In the numerical evaluation we use $x={10}^2$ (red) and $x={10}^4$ (blue). The dashed line is the analytical result from Eq. (32). We take ${\alpha }_n={\beta }_n\equiv \varepsilon$. States in the allowed bands are delocalized, identified numerically by $\xi \gg x$ and analytically by ${\mathrm{\Lambda }}_0=0$.

Figure 8

Localization length $\xi$ vs quasienergy $\varphi$ with correlated disorder. From top to bottom at $\varepsilon +\varphi =0, W=\pi /30$ (red), $\pi /6$ (blue), $\pi /2$ (orange), and $\pi$ (magenta). We take ${\alpha }_n={\beta }_n\equiv \varepsilon$, and $x={10}^6$. The bound states of the two DTQWs become localized as a result of disorder, with a finite nonzero $\xi$.

Figure 9

Localization length $\xi$ vs quasienergy $\varphi$ with uncorrelated disorder. From top to bottom at $\varepsilon +\varphi =0, W=\pi /30$ (red), $\pi /6$ (blue), $\pi /2$ (orange), and $\pi$ (magenta). Here $x={10}^6$, and we take ${\alpha }_n$ and ${\beta }_n$ to be uncorrelated. As a result of the uncorrelated disorder, the bound states become more strongly localized compared to the effect of correlated disorder ${\alpha }_n={\beta }_n$ (Fig. 8); in particular, there is no distinct enhancement of $\xi$ near the band centers $\varepsilon +\varphi =0,\pm \pi$ that arises from strong correlation in the disorder in Fig. 8.

Figure 10

Localized bound state with maximal disorder $W=\pi$. (a) The probability distribution at $t=100$ (left) and $t=150$ (right). (b) The density on the antidiagonal $X_1-X_2=0$ at $t=100$ (left) and $t=150$ (right). The solid black line is $\sim e^{-X/\xi }$, where $\xi \approx 2.886$ is the prediction from the transfer matrix approach, which agrees nicely with the localization length obtained from direct two-body propagation (green triangles). Due to the chessboard symmetry even and odd sites $X$ along the antidiagonal are empty depending on the parity of $t$, and the empty sites are left unplotted. The initial condition is $|{\mathrm{\Psi }}^{\left(2\right)}\left(0\right)\rangle =|x_0,+\rangle \otimes |x_0,+\rangle$.

Figure 11

Spin-dependent interaction in an unbiased two-body Hadamard walk ($W=\pi$). Shown are the probability distribution at $t=100$ (top left) and $t=150$ (top right) and the entanglement entropy (bottom). The legend is the same as in Fig. 3. The bound state disappears. The initial condition is $|{\mathrm{\Psi }}^{\left(2\right)}\left(0\right)\rangle =|x_0,+\rangle \otimes |x_0,+\rangle$.