Topological Floquet Green's function zeros

Abstract

Motivated by recent advances in digital quantum emulation using noisy intermediate-scale quantum (NISQ) devices and an increased interest in topological Green's function zeros in condensed matter systems, we here study Green's function zeros in topological Floquet systems. We concentrate on interacting Kitaev-like Floquet chains (or equivalently transverse field Ising circuits) and introduce Floquet Green's-function-based topological invariants for the corresponding symmetry class BDI. In the vicinity of special points in the free fermion phase diagram and using tailor-made interactions which lead to the Floquet version of symmetric mass generation, we analytically calculate both edge and bulk Green's functions. Just as in the case of continuum time evolution, topological bands of Green's function zeros may also contribute to the topological invariant. However, contrary to the case of continuum time evolution, Floquet Green's functions can have zeros even in the absence of interactions. Finally, we also discuss an implementation of this Floquet system in a digital quantum emulator: We present a circuit which encodes the interaction under consideration and pinpoint the observables carrying information about the topological Green's function boundary zeros.

Figures (8)

• Figure 1: a) Illustration of the system and notational convention: in particular $g$ and $J$ describe the free fermion Floquet evolution (matchgate circuit) while $w$ adds interactions. b) Free fermion Floquet phase diagram. The labelling $( \nu_0 , \nu_\pi)$ corresponds to the topological free fermion invariant Eq. \ref{['eq:TopoFree']} and sets the number $\vert \nu_0\vert$ and $\vert \nu_\pi\vert$ of Majorana zero modes and Majorana $\pi$-modes at a given edge, respectively ThakurathiDutta2013CardosoMitra2025. Crosses (circles) indicate the points (regimes) in parameter space for calculations of Sec. \ref{['sec:FKFloquetEdge']} (\ref{['sec:FKFloquetBulk']}).
• Figure 2: Poles, zeros and winding of the free fermion Green's function Eqs. \ref{['eq:FreeFermionGFs']}, \ref{['eq:Ueff']}. a), c) Poles (solid) and zeros (dotted) of $\mathbf G_{\rm R}(\Omega, p)$ for periodic boundary conditions. Note the vicinity to the gap closing at $\Omega = 0$ and $\Omega = \pi$ in panels a) and c) respectively. b), d) Schematic spectrum for open boundary conditions. e), f) Parametric plot of the in-plane vector $(n_x(p),n_y(p))$, Eq. \ref{['eq:n']} entering the Green's function and its winding, Eq. \ref{['eq:WindingFree']}. Figures (a,b,e) are plotted for $(J,g) = (0.7,0.8)$ and (c,d,f) at $(J,g) = (0.7,0.4)$.
• Figure 3: a) Classification of Floquet SPTs (reproduced from Ref. PotterVishwanath2016): Symmetry preserving interactions fold the non-interacting $\mathbb Z \times \mathbb Z$ classification down to $\mathbb Z_8 \times \mathbb Z_4$ (shaded green). b)Illustration of the interaction for models 1 and 2 corresponding to $N_f = 8$ Kitaev chains (blue crosses in a). The complex $c_{{\mathbf{j}}, a}$ fermions entering the interaction term, Eq. \ref{['eq:FK']} are illustrated as green ellipses and composed of Majorana fermions on adjacent wires. c) Illustration of the interaction model 3 (red cross in a) corresponding to $N_f = 4$ wires. The complex $c_{{\mathbf{j}}, a}$ fermions entering the interaction term, Eq. \ref{['eq:FK']} are composed of linear combinations of Majorana Eq. \ref{['eq:MZMMPM']} from adjacent wires.
• Figure 4: Infinite temperature [panels a), b)] and zero temperature [c)-d)]. Green's functions of edge zero modes [a), c), e), g)] and $\pi$ modes [b), d), f)]. a)-f) correspond to Floquet Green's functions while g) is the Fourier transformed continuous time Green's function for the same parameters as e). e)[f)] is a cross-cut of c)[d)] at $w = 0.01$. Red lines in panels a)-d) delineate vanishing Re$[\mathbf G_{\rm R}(\Omega)]$, while the color plot is the spectral function $-{\rm Im}\left[G_R(\Omega)\right]/\pi$. All figures are plotted for finite broadening $\Omega \rightarrow \Omega + i \Gamma$, $\Gamma = 0.05, \lambda = 1$.
• Figure 5: Graphical illustration of the two interacting Floquet cycles, Eqs. \ref{['eq:FloquetCycles']}