Topological characterization of phase transitions and critical edge states in one-dimensional non-Hermitian systems with sublattice symmetry

TL;DR

This work addresses how to characterize topology in 1D non-Hermitian systems with sublattice symmetry at both gapped and gapless transitions. It introduces two characteristic functions $P(z)=g(z)/f(z)$ and $Q(z)=f(z)g(z)$, derives winding numbers $W_1$ and $W_2$ from zeros/poles inside the generalized Brillouin zone (GBZ), and defines a bulk invariant $W=(W_1+W_2)/2$ together with edge counting $N_0=|W_1+W_2|=2|W|$, applicable to both Bloch and non-Bloch descriptions. The theory is applied to non-Hermitian SSH chains, showing how $W_1$, $W_2$, and $W$ classify gapped phases, gapless phase boundaries, and topological phase transitions, including transitions without bulk-gap closing and the emergence of critical edge states. The results reveal non-Hermitian-induced topological criticality (gSPT-like behavior) protected by sublattice symmetry, with bulk-edge correspondence $N_0=2|W|$ extending to gapless regimes and potentially guiding experimental realizations in photonic, acoustic, or electronic simulators.

Abstract

Critical edge states appear at the bulk gap closing points of topological transitions. Their emergence signify the existence of topologically nontrivial critical points, whose descriptions fall outside the scope of gapped topological matter. In this work, we reveal and characterize topological critical points and critical edge states in non-Hermitian systems. By applying the Cauchy's argument principle to two characteristic functions of a non-Hermitian Hamiltonian, we obtain a pair of winding numbers, whose combination yields a complete description of gapped and gapless topological phases in one-dimensional, two-band non-Hermitian systems with sublattice symmetry. Focusing on a broad class of non-Hermitian Su-Schrieffer-Heeger chains, we demonstrate the applicability of our theory for characterizing gapless symmetry-protected topological phases, topologically distinct critical points, phase transitions along non-Hermitian phase boundaries and their associated topological edge modes. Our findings not only generalize the concepts of topologically nontrivial critical points and critical edge modes to non-Hermitian setups, but also yield additional insights for analyzing topological transitions and bulk-edge correspondence in open systems.

Figures (11)

• Figure 1: Schematic diagrams of the theory and model. (a) and (b) illustrate the configurations of zeros (in purple dots) and poles (in green circles) of the characteristic functions $P(z)$ and $Q(z)$ [Eq. (\ref{['eq:PzQz']})] on the complex $z$ plane, with the GBZ given by the sinuous contours. Only zeros/poles inside the GBZ contribute to the winding numbers $W_1$ and $W_2$ [Eq. (\ref{['eq:W12']})], whose combination predicts the topological invariant $W=1$ [Eq. (\ref{['eq:W']})] and number of edge zero modes $N_0=2$ [Eq. (\ref{['eq:N0']})] for the case illustrated in (a)--(b). (c) shows NHSSH $\alpha$ chains with $\alpha=0,1,2$. In each case, the solid and dotted curves with arrows denote hopping amplitudes from left to right and right to left between sublattices A (cyan balls) and B (orange balls).
• Figure 2: Phase diagram, GBZ and zero/pole distributions of the NHSSH $(0,1)$ chain. (a) Topological phase diagram, with the values of $W$ shown explicitly in each phase. Phase boundaries are given by the gray solid lines. Along the dotted line, the symbols $\CIRCLE$, $\blacktriangleright$, $\blacklozenge$, $\blacktriangleleft$, and $\blacksquare$ highlight the points in the parameter space with $J=\sqrt{2}$ and $\gamma=0.5,1,1.3,\sqrt{3}$, and $2$, respectively. (b)--(f) show the BZ (in dashed rings), GBZ (in solid rings), zeros of $P(z)$ (in $\vartriangle$), poles of $P(z)$ (in $+$), zeros of $Q(z)$ (in $\triangledown$), and poles of $Q(z)$ (in $*$) for the five points in (a) from left to right on the complex $z$-plane, respectively.
• Figure 3: Spectra and edge states of the NHSSH $(0,1)$ chain with $J=\sqrt{2}$. (a) shows the absolute value of spectrum $|E|$ vs $\gamma$ under the OBC (in blue dots), with the red curve denoting the magnitude of spectral gap at $E=0$ under the PBC. (b)--(d) show the spectra at different $\gamma$ on the complex plane, with the shared color bar giving the inverse participation ratio of each eigenstate. (e) shows the probability distributions $P_{0{\rm L}}$ and $P_{0{\rm R}}$ of the two zero modes in (d) for a lattice with $10$ unit cells.
• Figure 4: Phase diagram, GBZ and zero/pole distributions of the NHSSH $(1,2)$ chain. (a) Topological phase diagram, with the values of $W$ shown explicitly in each phase. Phase boundaries are given by the gray solid lines. Along the dotted line, the symbols $\CIRCLE$, $\blacktriangleright$, $\blacklozenge$, $\blacktriangleleft$, and $\blacksquare$ highlight the points in the parameter space with $J=\sqrt{2}$ and $\gamma=0.5,1,1.3,\sqrt{3}$, and $2$, respectively. (b)--(f) show the BZ (in dashed rings), GBZ (in solid rings), zeros of $P(z)$ (in $\lozenge$ and $\vartriangle$ for the third order and first order ones), poles of $P(z)$ (in $+$), zeros of $Q(z)$ (in $\triangledown$), and poles of $Q(z)$ (in $*$) for the five exemplary points in (a) from left to right on the complex $z$-plane.
• Figure 5: Spectra and edge states of the NHSSH $(1,2)$ chain with $J=\sqrt{2}$. (a) shows the absolute value of spectrum $|E|$ vs $\gamma$ under the OBC (in blue dots), with the red curve denoting the magnitude of spectral gap at $E=0$ under the PBC. (b) and (c) show the spectra at $\gamma=1$ and $\sqrt{3}$ on the complex plane, with the shared color bar giving the inverse participation ratio of each eigenstate. (d) shows the probability distributions $P_{0{\rm L}}$ and $P_{0{\rm R}}$ of the two zero modes in (b) for a lattice with $10$ unit cells. The two edge modes in (c) have the same profiles $P_{0{\rm L}}$ and $P_{0{\rm R}}$.