Exceptional-point-constrained locking of boundary-sensitive topological transitions in non-Hermitian lattices

Abstract

Point-gap topology under periodic boundary conditions and line-gap topology under open boundary conditions are generally inequivalent in non-Hermitian systems. We show that, in chiral non-Hermitian lattices, these two boundary-sensitive topological transitions become locked when the parameter sweep is confined to an exceptional-point (EP)-constrained manifold, such that the Bloch spectrum remains pinned to a zero-energy degeneracy throughout the evolution. In an extended non-Hermitian Su-Schrieffer-Heeger chain, this locking can be established analytically in a tractable limit, where the EP-constrained manifolds and the corresponding PBC and OBC transition boundaries are obtained in closed form, and it persists away from this limit when the generalized Brillouin zone is determined numerically. Outside the EP-constrained manifold, the two transitions generally decouple, even in the presence of isolated EPs or Hermitian degeneracies. We further show that the same mechanism survives in a four-band spinful extension with branch-resolved generalized Brillouin zones, including branch-imbalanced regimes. These results identify EP-constrained band evolution as a simple organizing principle for boundary-sensitive topology in chiral non-Hermitian systems and suggest a useful route for diagnosing non-Bloch topological transitions from periodic-boundary spectral evolution when such spectral information can be accessed in photonic, circuit, and cold-atom platforms.

Figures (6)

• Figure 1: (a) Schematic of the extended non-Hermitian SSH chain. Each unit cell contains two sublattices, $A$ and $B$. The intracell hoppings are nonreciprocal ($t_1\neq t_2$), whereas the intercell hoppings are reciprocal and staggered as $t_3\pm\delta$. (b) Schematic illustration of a point gap in the complex-energy plane: the spectrum avoids the reference energy $E_{\rm b}$ (red dot). (c) Schematic illustration of a line gap in the complex-energy plane: the spectrum avoids a reference line (red line).
• Figure 2: Analytically solvable limit $\delta=t_3=0.5$. (a) Phase diagram in the $(t_1,t_2)$ plane. The solid lines are the EP-constrained manifolds, $|t_1|=2|t_3|$ or $|t_2|=2|t_3|$. The dotted line marks the PBC point-gap topological transition, $|t_1|=|t_2|$, and the dashed line marks the OBC real-line-gap transition, $|t_3|=\sqrt{|t_1t_2|}/2$. Stars indicate the representative parameter points used in (b)--(e). (b)--(e) Complex-energy spectra under PBC (open circles) and OBC (solid circles) with system size $N=50$ for (b) $(t_1,t_2)=(0.6,1.0)$, (c) $(t_1,t_2)=(1.0,1.0)$, (d) $(t_1,t_2)=(1.3,1.0)$, and (e) $(t_1,t_2)=(0.8,1.25)$.
• Figure 3: Locking between the OBC real-line-gap transition and the PBC point-gap transition away from the solvable limit. Here $\delta=0.5t_3$ and $t_2=2t_3$, so that the PBC spectrum remains pinned to a zero-energy degeneracy throughout the sweep. (a) Real parts of the PBC spectrum versus $t_1/t_3$, showing a real-part degeneracy at $t_1=2t_3$. (b) Real parts of the OBC spectrum, where the real line gap closes and reopens at the same point; zero-energy boundary modes are present for $t_1<2t_3$ but absent for $t_1>2t_3$. (c)--(e) Complex spectra at $t_1=1.6t_3$, $2.0t_3$, and $2.4t_3$, respectively. The PBC point-gap winding changes from $\nu_{\rm PBC}=-1$ to $\nu_{\rm PBC}=+1$ across $t_1=2t_3$, while the OBC real-line-gap transition and disappearance of zero-energy boundary modes occur at the same point. (f)--(h) Corresponding evolution of $\nu_{\rm OBC}$, $\bar{\kappa}$, and $\nu_{\rm PBC}$, all changing at $t_1=2t_3$.
• Figure 4: Spectral and topological evolution outside the EP-constrained manifold for $\delta=0.5t_3$. (a)--(d) Results along the sweep $t_2=t_1+0.25\,t_3\sin[(t_1/t_3-1)\pi]$: (a) real parts of the PBC spectrum, (b) real parts of the OBC spectrum ($N=50$), (c) OBC real-line-gap invariant $\nu_{\rm OBC}$, and (d) PBC point-gap winding number $\nu_{\rm PBC}$. (e)--(h) Results for the sweep with fixed $t_2=2.2t_3$: (e) real parts of the PBC spectrum, (f) real parts of the OBC spectrum, (g) $\nu_{\rm OBC}$, and (h) $\nu_{\rm PBC}$. All quantities are plotted as functions of $t_1/t_3$.
• Figure 5: Four-band spinful extension of the non-Hermitian SSH model. (a) Schematic of the spinful non-Hermitian SSH ladder, showing the intracell and intercell hoppings together with the intracell and intercell spin-flip couplings $u$ and $v$. (b) Branch-resolved generalized Brillouin zone (GBZ) loops for the two branches at $t_1=0.4t_3$. (c) Real parts of the PBC spectrum and (d) real parts of the OBC spectrum as functions of $t_1/t_3$. (e) OBC real-line-gap invariant $\nu_{\rm OBC}$. (f) PBC point-gap winding number $\nu_{\rm PBC}$. (g) Branch-resolved average skin exponents $\bar{\kappa}_1$ and $\bar{\kappa}_2$. (h)--(j) OBC eigenstate profiles at $t_1=0.4t_3$, $1.7t_3$, and $3.6t_3$, respectively. Bulk skin modes are shown in gray, while the two zero-energy boundary modes are highlighted in blue and red. Unless otherwise specified, the parameters are $t_2=3.4t_3$, $\delta=-0.5t_3$, $u=2t_3$, and $v=0.6t_3$.