Topological Phase Transitions and Edge-State Transfer in Time-Multiplexed Quantum Walks

TL;DR

The paper investigates topological phase transitions and edge-state behavior in a time-multiplexed nonunitary Floquet quantum walk with sublattice symmetry. It develops a tunable Floquet operator with gain/loss to compare unitary and nonunitary regimes, and employs non-Bloch band theory via the generalized Brillouin zone to restore a generalized BBC in the non-Hermitian case. A key finding is a transfer phenomenon where edge modes of different sublattice sectors can localize at the same boundary, driven by nonunitarity and diagnosed by complex spectral loops and non-Bloch winding numbers. The work provides a comprehensive framework for understanding non-Hermitian topological phases in Floquet QWs and guides experimental realization and control of edge states in these systems.

Abstract

We investigate the topological phase transitions and edge-state properties of a time-multiplexed nonunitary quantum walk with sublattice symmetry. By constructing a Floquet operator incorporating tunable gain and loss, we systematically analyze both unitary and nonunitary regimes. In the unitary case, the conventional bulk-boundary correspondence (BBC) is preserved, with edge modes localized at opposite boundaries as predicted by topological invariants. In contrast, the nonunitary regime exhibits non-Hermitian skin effects, leading to a breakdown of the conventional BBC. By applying non-Bloch band theory and generalized Brillouin zones, we restore a generalized BBC and reveal a transfer phenomenon, where edge modes with different sublattice symmetries can become localized at the same boundary. Furthermore, we demonstrate that the structure of the spectral loops in the complex quasienergy plane provides a clear signature for these transfer behaviors. Our findings deepen the understanding of nonunitary topological phases and offer valuable insights for the experimental realization and control of edge states in non-Hermitian quantum systems.

Figures (8)

• Figure 1: Schematic of a single step in the evolution of a nonunitary QW, initialized in the state $\left|\psi(t=0)\right\rangle = \left|n, H\right\rangle$. Circles denote lattice sites. Red (blue) arrows indicate the horizontal ($\left|H\right\rangle$) [vertical ($\left|V\right\rangle$)] polarization states, while green arrows represent superposition states of $\left|H\right\rangle$ and $\left|V\right\rangle$. The length of each arrow is proportional to the absolute values of the corresponding probability amplitude.
• Figure 2: Floquet quasienergy spectra as functions of $\theta_2$ for the unitary QWs with $\theta_1=0.2\pi$ and $\gamma=0$ under (a) PBCs and (b) FBCs ($N=60$). (c) Topological phase diagram in the coin parameter space $\left( \theta_1, \theta_2 \right)$. The distinct phases are labeled by the winding numbers $\left(\nu_0,\nu_\pi\right)$, and the topological phase boundaries for $E = 0$ and $E = \pi$ are marked by blue and purple dashed lines, respectively. The blue triangle, gray rectangle, and purple circle denote parameter sets with $(\theta_{1},\theta_{2})=(0.2\pi,-0.15\pi)$, $(0.2\pi,0.05\pi)$, and $(0.2\pi,0.15\pi)$, respectively.
• Figure 3: Spatial profiles of right eigenmodes in unitary QWs with $\gamma = 0$: (a) three arbitrarily chosen bulk states at $\left( \theta_1, \theta_2 \right) = (0.2\pi, -0.15\pi)$; (b)–(d) two degenerate $0$-modes; and (e)–(g) two degenerate $\pi$-modes. In (b)–(d) and (e)–(g) with $\theta_{1}=0.2\pi$, the columns from left to right correspond to $\theta_2=-0.15\pi$, $0.05\pi$, and $0.15\pi$, marked by the blue triangle, gray rectangle, and purple circle in Fig. \ref{['fig1']}(c), respectively.
• Figure 4: $T_{\tilde{\alpha}}(\eta)$ of the $\tilde{\alpha}$ modes as functions of $\theta_2$ with $\theta_{1}=0.2\pi$ for (a) $\tilde{\alpha}=0$ and (b) $\tilde{\alpha}=\pi$.
• Figure 5: Floquet quasienergy spectra as functions of $\theta_2$ for the nonunitary QWs with $\theta_1=0.2\pi$ and $\gamma=0.2$ under (a) PBCs and (b) FBCs ($N=60$). (c) GBZs parametrized by $\beta$ in the complex plane. (d) Non-Bloch topological phase diagram. The phases are labeled by the non-Bloch winding numbers $\left(\tilde{\nu}_0,\tilde{\nu}_\pi\right)$. The topological phase boundaries for non-Bloch quasienergy $0$ and $\pi$ are marked by blue and purple dashed lines, respectively.