Toward simulation of topological phenomena with one-, two-, and three-dimensional quantum walks

We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We focus mainly on the completion of a table for the protocols of the quantum walk that could simulate different families of the topological phases in one, two, and three dimensions. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences. To further enrich the simulation of the topological phenomena, we include step-dependent coins in the evolution operators of the quantum walks. This leads to step dependence of the simulated topological phenomena and their properties which introduces dynamicity as a feature of simulated topological phases and boundary states. This dynamicity provides the step number of the quantum walk as a means to control and engineer the numbers of topological phases and boundary states, their numbers, types, and even occurrences.

Figure 1

Quantum walk in two dimensions. Modification of the energy (only the positive branch) with $\alpha =\pi /3$ and $\gamma =\pi /4$ is shown for subsequent steps of $T=2$, 3, and 6 in (a) a quantum walk with only PHS and (b) quantum walks without the three symmetries. The structure of the simulated topological phenomena changes in dependence on the number of steps and from one protocol to another. The numbers of topological phases and boundary states (topological phase transitions) increase as quantum walks proceed. In the third step of the quantum walk with only PHS, we have Dirac cone boundary states; the same can be observed for the sixth step of the other protocol. Usually, each step contains both Dirac cones and Fermi arcs with characteristic flat-band boundary states.

Figure 2

Two-dimensional quantum walk with only PHS. The energy is plotted as a function of rotation angle and momenta for (a) $\alpha =\pm (2c+1)\pi /T$ and $\beta =\pm (2c+1)\pi /2T$, (b) $\alpha =\pm 2c\pi /T$ and $\beta =\pm (2c+1)\pi /2T$, (c) $\alpha =\pm 2c\pi /T$ and $\beta =\pm c\pi /T$, (d) $\alpha =\pm 2c\pi /T$ and $\beta =(6c\pm 1)\pi /3T$, and (e) $\alpha =\beta =(6c\pm 1)\pi /3T$. In (a) the energy bands are independent of $k_y$. If we fix $k_y$, the observable boundary state is a Fermi arc; for variation of $k_y$, we notice boundary states in the form of lines, which could be interpreted as flat-band boundary states. In (b), (d), and (e) we explore different ways that Fermi arc boundary states can be formed. In (c) we observe two types of boundary states: Dirac cone boundary states if we fix one of the momenta and let the other one vary and characteristically flat-band boundary states if both of the momenta are traversing the first Brillouin zone.

Figure 3

Two-dimensional quantum walk with only PHS. Modification of the energy and $\mathit{d}$ are shown with $T=2$ and $\alpha =\pi /3$. The $\mathit{d}$ surfaces are plotted by variation of the momenta through the first Brillouin zone. For $\beta =\pi /12$ and $\pi /2$, since the $\mathit{d}$ does not completely surround the origin, the Chern number is 0 (trivial phases). In the case of $\beta =\pi /4, \mathit{d}$ surrounds the origin, which indicates a nontrivial topological phase with Chern number $+1$. If $\beta =\pi /6, \pi /3$, and $2\pi /3$, energy bands close their gaps, resulting in boundary states. Their corresponding $\mathit{d}$ passes the origin. Chronologically, we observe energy bands change from gapped to gapless and the topological phase transitions from one to the other as they meet boundary states (the phase transition point).

Figure 4

Two-dimensional quantum walk without the three symmetries. Modification of the energy and $\mathit{d}$ are shown with $T=3, \alpha =\pi /3$, and $\gamma =\pi /4$. The $\mathit{d}$ surfaces are plotted as the momenta traverse the first Brillouin zone. For $\beta =0$ and $\pi /3$, we have two trivial phases with Chern number 0 since $\mathit{d}$ does not completely surround the origin. In the case of $\beta =\pi /2$, a nontrivial topological phase with Chern number $+1$ is observable. If we set $\beta =-\pi /4, \pi /4$, and $3\pi /4$, energy bands close their gaps and $\mathit{d}$ passes the origin, and so we have boundary states. Chronologically, we observe two trivial phases next to each other followed by a nontrivial topological phase. Therefore, the phase structure is different from those observed in PHS quantum walks.

Figure 5

One-dimensional quantum walk with only PHS. Modification of the energy (the positive branch) is plotted as a function of momentum and rotation angle $\alpha$ for the two cases (a) $\beta =\pi /3$ and (b) $\beta =(\alpha +\pi )/3$. In (a) each step contains only one type of boundary state (Dirac cone or Fermi arc). In (b) we observe the simulation of the Dirac cone and Fermi arc boundary states together in each step. There are two types of Dirac cone boundary states: In the first type, gapless energy bands are observed at $k_x=0$ and $\pm \pi$ while for the second type observation additionally happens at $k_x=\pm \pi /2$. As the step number increases, the size of the topological phases decreases. Therefore, the numbers of topological phases and boundary states (topological phase transitions) increase in dependence on the number of steps.

Figure 6

One-dimensional quantum walk with only PHS. The energy (top panels) and positive branch of the group velocity (bottom panels) are plotted for (a) $T=6$ and (b) $T=8$ with $\beta =(\alpha +\pi )/3$. As we scan the rotation angle $\alpha$ through the limited range, the protocol starts with the first type of Dirac cone boundary state (orange line) followed by a series of topological phases (blue lines) and confronted with a Fermi arc boundary state (red line). After the Fermi arc, another set of topological phases emerges (purple lines). Finally, we reach the second type of Dirac cone boundary state (black line). Comparing (a) with (b), we see that the positions of the topological phases and boundary states change in dependence on the number of steps. In the bottom panels of (a) and (b) we observe that around the first type of Dirac cone boundary state, the group velocity approximates to approximately $\pm 1$, while for the second type of Dirac cone, it approximates to approximately $\pm 2$. The characteristic behavior of the group velocity around the gapless point of each boundary state differs significantly. In contrast, for all of these boundary states, the sign of the group velocity swaps from positive to negative or vice versa at the gapless point of the energy bands.

Figure 7

One-dimensional quantum walk with only CHS. Modification of the energy (the positive branch) is plotted as a function of momentum and rotation angle $\alpha$ for the two cases (a) $\beta =\pi /3$ and (b) $\beta =(\alpha +\pi )/3$. (a) In the early stages of the walk, only Fermi arc boundary states are observable and an increment in step number results in simulation of both the Dirac cone and Fermi arc boundary states together. Only the first type of Dirac cones is present for boundary states. Contrary to PHS protocol, the boundary states form for arbitrary values of $k_x$. (b) If the rotation angles are linearly related to each other, simulation of the Fermi arc and the first type of Dirac cone boundary states in each step is guaranteed.

Figure 8

One-dimensional quantum walk with only CHS. The energy (top panels) and positive branch of group velocity (bottom panels) are plotted for (a) $T=6$ and (b) $T=8$ with $\beta =(\alpha +\pi )/3$. As we scan the rotation angle $\alpha$ through the limited range, we start with a Fermi arc boundary state (orange line) which is followed by topological phases (blue lines) until we reach another Fermi arc boundary state (red line). By further increasing the rotation angle, we first come across topological phases (purple lines) and finally meet the first type of Dirac cone boundary state (black line). In the bottom panels of (a) and (b), the group velocity for Fermi arcs is nonlinearly modified as we scan the first Brillouin zone for $k_x$, while it is linearly modified for Dirac cones. The swapping of the group velocity's sign to the opposite happens at gapless points of energy bands. (c) Modification of $\mathit{n}$ (blue curves) as $k_x$ traverses the first Brillouin zone in a plane orthogonal to $\mathit{A}$ for $T=6$ with $\beta =(\alpha +\pi )/3$. The closed loops correspond to nontrivial topological phases in which the clockwise winding direction has $w=-1$ while counterclockwise has $w=1$. If the loop is not closed and does not cross the origin, it indicates a trivial phase with $w=0$. For the boundary states, $\mathit{n}$ passes the origin.

Figure 9

One-dimensional quantum walk with only CHS: (a) energy and (b) positive branch of the group velocity with $\alpha =\beta =\pi /3$ for five subsequent steps. The formation of Fermi arcs happens for arbitrary values of $k_x$, which changes in dependence on the number of steps.