Thouless quantum walks in topological flat bands

TL;DR

The paper addresses realizing quantum walks under non-Abelian gauge structure using Thouless pumping on topological flat-band lattices. It introduces Thouless holonomic quantum walks (ThQWs) built from degenerate flat bands with holonomic coin and shift operations, implemented via pumping cycles. The key result is that the geometric displacement is quantized by the first Chern number $C_1$, with the walk exhibiting parity and time-reversal symmetry breaking and a continuum Weyl-like dynamics; a photonic Lieb-chain implementation is proposed. The work opens pathways to robust topological quantum transport and programmable quantum-state engineering, with potential extensions to higher dimensions and larger coin spaces.

Abstract

Non-Abelian gauge symmetries are cornerstones of modern theoretical physics, underlying fundamental interactions and the geometric structure of quantum mechanics. However, their potential to control quantum coherence, entangle- ment, and transport in engineered quantum systems remains to a large extent unexplored. In this work, we propose utilizing non-Abelian Thouless pumping to realize one-dimensional discrete-time quantum walks on topological lattices char- acterized by degenerate flat bands. Through carefully designed pumping cycles, we implement different classes of holonomic coin and shift operators. This frame- work allows for the construction of quantum walks that encode the topological and geometric properties of the underlying system. Remarkably, the resulting evolution exhibits parity symmetry breaking and gives rise to a dynamical pro- cess governed by a Weyl-like equation, highlighting the deep connection between parity and time-reversal symmetry breaking in the system.

Figures (4)

• Figure 1: Inception of Thouless quantum walk. Essential elements underlying the construction of Thouless quantum walk. In orange the discrete time quantum walk, whose compositions of coin and shift operators upon the level states induce quantum superposition that allow to overcome the mean walker’s path length beyond classical random walks. In blue flat band lattices, whose destructive interference grants the existence of degenerate orthogonal spatially compact states associated to the non-dispersive bands which mimic the coin levels of a quantum walk. In green non-Abelian Thouless pumping, whose cycles grant both conditional transport and geometrical unitary superposition of coin levels mimicking coin and shift operators of a quantum walk.
• Figure 2: Photonic implementation of a sample uni-directional Thouless quantum walk. (a) Flat band lattice profile with the unit cell coloured in grey. (b) Orthogonal symmetric $|p_n\rangle$ and antisymmetric $|q_n\rangle$ states with non-zero amplitudes coloured in dark blue. (c) Pumping cycles $\mathcal{C}_{\mathcal{T}}^{+ s}$ (red) and $\mathcal{C}_{\mathcal{T}}^{- s}$ (green) in the parameter space. The blue circles indicate the initial point. (d) Same as (c) for pumping cycles $\mathcal{C}_{\mathcal{R}}(\frac{\pi}{4})$ (red) and $\mathcal{C}_{\mathcal{R}}(\frac{\pi}{6})$ (green). (e) Illustration of three unit cells of the pumped lattice implementing a time-unit formed by a coin $\mathcal{C}_{\mathcal{R}}(\frac{\pi}{6})$ and a shift $\mathcal{C}_{\mathcal{T}}^{--}$. (f) Same as (e) with coin $\mathcal{C}_{\mathcal{R}}(\frac{\pi}{4})$ followed by a shift $\mathcal{C}_{\mathcal{T}}^{++}$. (g) Three steps propagation of the ThQWs generated by a step $\mathcal{C}_{\mathcal{T}}^{--} \mathcal{C}_{\mathcal{R}}(\frac{\pi}{6})$ from a single-cell excitation $|\Psi(z_0)\rangle = \frac{1}{\sqrt{5}} |p_0\rangle + \frac{2}{\sqrt{5}} |q_0\rangle$. In both cycles $\mathcal{C}_{\mathcal{T}}$ in (c) and $\mathcal{C}_{\mathcal{R}}$ in (d) we set $\lambda \mathcal{J}= 200$. The green lines indicate the time-steps $z_t$, while the blue ones separate the coin for the shift. The right panel shows the field's intensity, while the left panel shows the trace of the displacement matrix $D$ (red) and the center of mass (blue). (h) Same as (g) for a ThQWs generated by a step $\mathcal{C}_{\mathcal{T}}^{++} \mathcal{C}_{\mathcal{R}}(\frac{\pi}{4})$. (i) Variance $\sigma^2$ for $\mathcal{C}_{\mathcal{T}}^{--} \mathcal{C}_{\mathcal{R}}(\frac{\pi}{6})$ (orange) and $\mathcal{C}_{\mathcal{T}}^{++} \mathcal{C}_{\mathcal{R}}(\frac{\pi}{4})$ (blue) from $|\Psi(z_0)\rangle = \frac{1}{\sqrt{2}} |p_0\rangle + \frac{1}{\sqrt{2}} |q_0\rangle)$. The squares and circles indicate the $\sigma^2$ at integer multiples of $\lambda$ The dashed black line guides the eye, indicating $\sigma^2=t^2$.
• Figure 3: Parity breaking in Thouless quantum walk. Floquet quasi-energies for the quantum walks $C_{\mathcal{R}} C_{\mathcal{T}}^{++}$ (a) and $C_{\mathcal{R}} C_{\mathcal{T}}^{--}$ (b) for different values of the coin angle $\theta \in[0,\pi/2]$ (c) Example of propagation along composite cycles starting from a symmetric single-cell excitation $|\Psi(z_0)\rangle = ( |p_0\rangle + |q_0\rangle)/\sqrt{2}$. The pattern is generated by means of a two stage procedure. In the first stage we perform the $C_{\mathcal{R}} C_{\mathcal{T}}^{++}$ for $n\geq 0$ and $C_{\mathcal{R}} C_{\mathcal{T}}^{--}$ for $n\leq 0$ and suitably overlapping the two cycles for $n=0$. In the second stage which start at $t=100\lambda$, we apply time-reversal symmetry and we exchange $|p_n\rangle$ and $|q_n\rangle$ while keeping the chirality $\chi$ of the walks.
• Figure 4: Photonic implementation of bi-directional Thouless quantum walk. (a) Illustration of three unit cells of the lattice pumped by cycle $\mathcal{C}_{\mathcal{R}}$ with $\theta=\pi/4$ (marked by blue lines), cycle $\mathcal{C}_{\mathcal{T}}^{++}$ (marked by cyan lines) and cycle $\mathcal{C}_{\mathcal{T}}^{--}$ (marked by green lines). (b) Two steps propagation of the walk from a single-cell excitation. The green lines indicate the time-units $z_s$, while the blue ones separate the coin $\mathcal{C}_{\mathcal{R}}$ and shift $\mathcal{C}_{\mathcal{T}}^{\xi s}$. (c) Schematic illustrations of the two-lobes curve in the parameter space representing a two-cycle walk $W_M(A\cdot B)$. The red and green lobes represent cycle $A$ and $B$. (d) Three lobes curve representing the three-cycle walk $W_M(A\cdot B\cdot C)$. The red, green and blue lobes represent cycle $A$, $B$ and $C$ respectively. The two zooms schematically indicate the cycle order $A\rightarrow B \rightarrow C$ (blue) and $A\rightarrow C \rightarrow B$ (magenta) around the junction point of the three cycles.