Phase rotation symmetry and the topology of oriented scattering networks

We investigate the topological properties of dynamical states evolving on periodic oriented graphs. This evolution, which encodes the scattering processes occurring at the nodes of the graph, is described by a single-step global operator, in the spirit of the Ho-Chalker model. When the successive scattering events follow a cyclic sequence, the corresponding scattering network can be equivalently described by a discrete time-periodic unitary evolution, in line with Floquet systems. Such systems may present anomalous topological phases where all the first Chern numbers are vanishing, but where protected edge states appear in a finite geometry. To investigate the origin of such anomalous phases, we introduce the phase rotation symmetry, a generalization of usual symmetries which only occurs in unitary systems (as opposed to Hamiltonian systems). Equipped with this new tool, we explore a possible explanation of the pervasiveness of anomalous phases in scattering network models, and we define bulk topological invariants suited to both equivalent descriptions of the network model, which fully capture the topology of the system. We finally show that the two invariants coincide, again through a phase rotation symmetry arising from the particular structure of the network model.

Figure 1

Example of an oriented network. The direction of propagation along the links is represented by an arrow. The nodes represent the unitary scattering events between incoming and outgoing amplitudes. Due to the unitarity of scattering events, the number of incoming links is equal to the number of outgoing links at each node.

Figure 2

Phase spectrum. Illustration of a phase spectrum with four bands and four gaps.

Figure 3

The phase rotation. On the level of the spectrum, a phase rotation of angle $\zeta$ maps eigenvalues ${\mathrm{e}}^{-\mathrm{i}\varepsilon }$ to ${\mathrm{e}}^{-\mathrm{i}(\varepsilon +\zeta )}$.

Figure 4

Examples of phase rotation invariant spectra. The spectra are invariant under a rotation of $2\pi /M$, with (a) $M=2$ and [(b) and (c)] $M=3$. In all cases, a fundamental domain $F$ for the symmetry can be chosen. For our purposes, the most convenient choice is an interval of length $2\pi /M$ with both ends lying in a spectral gap. In each case, a possible choice of fundamental domain is represented in purple.

Figure 5

L-lattice. (a) The L-lattice as a square Bravais lattice with basis vectors $e_x$ and $e_y$, with its four inequivalent links and two inequivalent nodes. A unit cell is enhanced in red and detailed in (b).

Figure 6

Two possible unit cells of the L-lattice.

Figure 7

Phase diagram and loops configurations of the L-Lattice. (Do not confuse with the phase spectrum of Figs. 4 and 9.) The L-lattice hosts two gapped phases with a transition at $\theta =\pi /4$. When varying $\theta$, each of these phases can be continuously deformed into lattices of clockwise ($\theta =0$) or anticlockwise ($\theta =\pi /2$) loops, that both satisfy a phase rotation symmetry with one band in the fundamental domain.

Figure 8

Interfaces of the L-lattice. We consider interfaces between the two phases of the L-lattice in a cylinder geometry (the system has periodic boundary conditions in the $x$ direction and is infinite in the $y$ direction). This allows one to (i) avoid potential ambiguities due to the relative character of the invariant and (ii) confirm that the existence of chiral edge states is indeed due to the bulk topology, and not merely from the oriented nature of the links. Remarkably, the two chiral edge states (one at each interface) are found to have different group velocities, which is consistent with the simple intuitive sketch in (a) where one of the two channels (in red) can flow easily rather than the other one (in blue) is forced to propagate in pilgrimage, resulting in a decreasing of its velocity along the $y$ axis compared to that of the other boundary state. (a) Interfaces between two networks with respectively $\theta =0$ and $\pi /2$. The system in periodic in both direction and finite in the $x$ direction. At the two interfaces, edge states with different velocity, in sign and amplitude, arise. (b) Eigenvalues of the corresponding Ho-Chalker evolution operator with $\theta \approx 0$ and $\approx \pi /2$ for clarity. Bulk states are represented in green, the fast boundary state in red, and the slow boundary state in blue. The code used to compute the phase spectra and the topological invariants is available at Ref. [22].

Figure 9

Relation between the spectra of $\mathcal{S}$ and $U_{\text{F}}^{\left(n\right)}$. The spectrum of $\mathcal{S}$ restricted to a fundamental domain $F$ of the phase rotation constraint corresponding to the structure constraint is in direct correspondence with the (full) spectra of all blocks $U_{\text{F}}^{\left(n\right)}$ of the repeated evolution operator ${\mathcal{S}}^s$. By phase rotation symmetry $D$, the first Chern number on the fundamental domain $F$ is zero, and thus the two bands of $F$ are opposite first Chern numbers.

Figure 10

Oriented Kagome lattice. (a) Oriented Kagome lattice with six inequivalent links and three inequivalent nodes per unit cell enhanced in red and detailed in (b).

Figure 11

Examples of loops configurations in the oriented Kagome lattice (a) and (b) Loops configurations that display a vanishing first Chern number phase. The phase spectra (d) and (e) for these configurations in a ribbon geometry confirm this result, and reveal that (a) corresponds to a trivial topological phase (with no edge state in the gaps), whereas (b) corresponds to an anomalous topological one (with one chiral edge state per edge in each gap). The configuration (c) displays no loop so that only the first Chern number on a fundamental domain is constrained to vanish. The phase spectrum on a strip geometry (f) confirms this result.

Figure 12

Construction of the interpolation of a stepwise evolution. The evolution is constituted of $s$ time-ordered step operators $U_j$. A time $t_j$ is attributed to each step and a continuous map is systematically built to interpolate between two successive steps. It follows a continuous map $t\to U\left(t\right)$ from $\text{Id}$ to $U_{\text{F}}^{\left(1\right)}$.