Topology and edge modes surviving criticality in non-Hermitian Floquet systems

Abstract

The discovery of critical points that can host quantized nonlocal order parameters and degenerate edge modes relocate the study of symmetry-protected topological phases (SPTs) to gapless regions. In this letter, we reveal gapless SPTs (gSPTs) in systems tuned out-of-equilibrium by periodic drivings and non-Hermitian couplings. Focusing on one-dimensional models with sublattice symmetry, we introduce winding numbers by applying the Cauchy's argument principle to generalized Brillouin zone (GBZ), yielding unified topological characterizations and bulk-edge correspondence in both gapped phases and at gapless critical points. The theory is demonstrated in a broad class of Floquet bipartite lattices, unveiling unique topological criticality of non-Hermitian Floquet origin. Our findings identify gSPTs in driven open systems and uncover robust topological edge modes at phase transitions beyond equilibrium.

Figures (6)

• Figure 1: Schematics of theory and model. (a) and (b) show the zero/pole locations of $f_{1}(z)$ and $f_{2}(z)$ on complex plane, with the wiggled contour showing the FGBZ. Orange dots (cyan circles) denote the zeros (poles) of $f_{1,2}(z)$. The spectrum related to (a), (b) is sketched in (c). The two bands are touched at both $E=0,\pi$. The critical $0$ ($\pi$) edge modes are depicted by the bells hung at $E=0$ ($\pi$), with their profiles illustrated at the bottom (top) of (c). (d) shows the NHSSH model under periodic quenches. The red (blue) balls at the center of each chain denote sublattice A (B) of unit cell $n$.
• Figure 2: Phase diagram of PQNHSSH $\alpha$-chain. Hopping amplitudes over $\alpha'$ unit cells are $J_{\alpha'}^{{\rm L(R)}}=J-(+)\gamma$. Regions of distinct colors are gapped phases, with their topological indices $(w_{0},w_{\pi})$ shown explicitly. Red solid (dotted) lines have $(w_{0},w_{\pi})=(\alpha,0)$, with bulk gap closes at $E=0$ ($\pi$). Red dash-dotted lines have $(w_{0},w_{\pi})=(\alpha',0)$, with bulk gap closes at $E=\pi$. Red dashed lines have $(w_{0},w_{\pi})=(\alpha,\alpha'-\alpha)$, with bulk gap closes at $E=0$.
• Figure 3: Spectra of PQNHSSH $\alpha$-chain at criticality with $(\alpha,\alpha')=(1,2)$, shown in terms of the Floquet operator's eigenvalue $F\equiv e^{-iE}$. The color of each point records the inverse participation ratio (IPR) of related state in a lattice with $1000$ unit cells. System parameters are $J_{1}^{{\rm L}}=J_{1}^{{\rm R}}=J_{1}$, $(J_{2}^{{\rm L}},J_{2}^{{\rm R}})=(J-\gamma,J+\gamma)$, $J=3\pi/4$ for all panels, and $J_{1}=0.32\pi$ ($0.68\pi$) for (a)--(c) ((d)--(f)). $\gamma=\sqrt{J^{2}-(\pi-J_{1})^{2}}$ for (a,e), $\sqrt{J^{2}-J_{1}^{2}}$ for (b,d), $\sqrt{J^{2}+{\rm acosh}^{2}(1/\cos J_{1})}$ for (c) and $\sqrt{J^{2}+{\rm acosh}^{2}(-1/\cos J_{1})}$ for (f).
• Figure 4: Floquet spectra along critical lines. The color of each point records the IPR of related state in a lattice with $1000$ unit cells. Numbers of $0$ and $\pi$ edge modes are given in each panel. In (a) ((d)), parameters are varied along the dash-dotted to dotted (dashed to solid) lines in Fig. \ref{['fig:PD']} for $\gamma=\pi/4\rightarrow 3\pi/4$. In (b)--(c) ((e)--(f)), parameters are varied along the dotted (solid) line in Fig. \ref{['fig:PD']} for $\gamma=3\pi/4\rightarrow\pi$.
• Figure 5: Central charge of PQNHSSH $\alpha$-chain with $(\alpha,\alpha')=(1,2)$. System parameters are $(J_{2}^{{\rm L}},J_{2}^{{\rm R}})=(J-\gamma,J+\gamma)$ with $J=3\pi/4$. The fitting curve $S(L,l)\sim2c\ln[\sin(\pi l/L)]/3$ is used for bipartite EE at half-filling with total system size $L=1000$ and varied subsystem sizes $l$.