Symmetries, topological phases, and bound states in the one-dimensional quantum walk

Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.

Figure 1

The discrete-time quantum walk. A spin-$1/2$ particle starting from a site of a discrete lattice undergoes alternating spin rotations $R$ and spin-$z$ dependent unitary shifts $S$. The first few time steps are shown representing the effect of interference.

Figure 2

Dispersion relations of the 1D quantum walk. Continuous line shows a typical gapped phase, with $\theta =\pm \pi /4$. The two gapless dispersion relations are $\theta =0$ (slashed) and $\pm \pi$ (dotted). In both cases, the gaps at $E=0$ and $\pm \pi$ are closed.

Figure 3

Parameter space of the one-dimensional simple quantum walk. The only parameter is the coin rotation angle $\theta$. The parameter space consists of two gapped domains, with gaps around both $E=0$ and $\pi$. These are separated by the gapless points $\theta =0$ and $\pi$.

Figure 4

Time dependence of the probability distribution (color coding corresponding to $|{\Psi }_{x,\uparrow }{|}^2+{\left|{\Psi }_{x,\downarrow }\right|}^2$) of a walker on a lattice consisting of $N=40$ sites with periodic boundary conditions. The coin operator is taken as $R\left[\theta \right(x\left)\right]$, with $\theta \left(x\right)={\theta }_A$ for $10<x<31$ and $\theta \left(x\right)={\theta }_B$ otherwise. In each case, we start the walker localized on the boundary between the domains of ${\theta }_A$ and ${\theta }_B$, on site $x=10$, with spin up. In cases (a) and (b), the walk is homogeneous: in (a), ${\theta }_A={\theta }_B=0.4\pi$ and in (b), ${\theta }_A={\theta }_B=0.2\pi$. In (c), an inhomogenous system is considered, with a sharp boundary between two bulks with the same topology: ${\theta }_A=0.2\pi$ and ${\theta }_B=0.4\pi$. In (d), ${\theta }_A=-0.2\pi$ and ${\theta }_B=0.4\pi$; here, the two domains have different topology. Accordingly, a significant part of the wave function of the walker gets trapped at the interface.

Figure 5

Phase map of a 1D quantum walk with partially cut links. The various gapped domains (different shadings) have a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ invariant associated with them, which is indicated as a pair of numbers $(Q_0,Q_{\pi })$. Separating these domains are the lines where the gap at $E=0$ closes (continuous line), or the gap at $E=\pi$ closes (dotted line). The vertical (horizontal) slashed lines denote the parameters corresponding to reflective coins (cut links). To find the number of edge states at an edge with cut links, a the point corresponding to the quantum walk (circle) is connected to the horizontal slashed line representing the boundary conditions. In the example shown, there is a single bound state with $E=\pi$.

Figure 6

Time steps for a walker started from an edge. Continuous (dotted) circles and lines correspond to the sites and links of the bulk (boundary). The time steps are broken down to four successive operations, as in Eq. (33), each occurring in $1/4$ time. Continuous (dotted) circles and lines correspond to the sites and links of the bulk (boundary). For simplicity, the bulk is taken with $\theta =\pi /2$ and $\varphi =0$: a simple quantum walk. The boundary has $\theta =\pi /2$ and cut links: $\varphi =\pi /2$. If the reflection is on a cut link (a), there is a protected midgap edge state with energy $\pi$. If the reflection happens on a reflective coin (b), during two timesteps, the walker acquires a phase of ($-$1). Superpositions of the states at $t=0$ and 1 with a relative phase of $i$ ($-i$) are therefore stationary states with energy $-\pi /2$ ($\pi /2$) not protected by PHS.

Figure 7

Two successive time steps of a quantum walk, with a walker started in bulk A (a), at a sharp boundary (b), or in bulk B (c). Each time step is broken down to its four stages, given by the four factors in $U_2\left(k\right)=S_{\uparrow }R(0,\pi )S_{\downarrow }R(\pi /2,-\pi /2)$, with $R({\theta }_A,{\theta }_B)$ as defined in Eq. (21), with $x\in A\leftrightarrow x<1$. In each case, the walker returns to its initial site after two time steps. In the bulk, during the two time steps, a phase factor of ($-$1) is acquired by the walker, showing that stationary states (superpositions of the states at $t=0$ and 1 with relative phase $\pm i$) have quasienergy $\mp \pi$. At the boundary, this factor is (+1), therefore even and odd superpositions of the states at $t=0$ and 1 are stationary states with energy $0,\pi$. These are at the topologically protected midgap states.