Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials

TL;DR

The paper analyzes a dimerized Hatano-Nelson chain with staggered imaginary potentials to resolve how bulk topology manifests under non-Hermitian conditions. By solving both periodic and open-boundary problems analytically, it establishes a rigorous bulk-boundary correspondence where periodic winding numbers dictate the number of singular values that vanish in open systems, corresponding to edge modes that become exact zero modes only in the semi-infinite limit. It uncovers a rich exceptional-point structure, including a dense-EP phase in the PT-symmetric regime that yields hyper-ballistic transport, and shows how entanglement spectra reveal topological features via protected eigenvalues at the entanglement gap. The results clarify why eigenvalues in finite NH systems need not reflect bulk-edge features and provide concrete, testable predictions for experiments on NH lattices and quantum simulators.

Abstract

It is well-known that the standard bulk-boundary correspondence does not hold for non-Hermitian systems in which also new phenomena such as exceptional points do occur. Here we study, mostly by analytical means, a paradigmatic one-dimensional non-Hermitian model with dimerization, asymmetric hopping, and imaginary staggered potentials. We present analytical solutions for the singular-value and the eigensystem of this model with both open and closed boundary conditions. We explicitly demonstrate that the proper bulk-boundary correspondence is between topological winding numbers in the periodic case and singular values, {\it not eigenvalues}, in the open case. These protected singular values are connected to hidden edge modes which only become exact zero-energy eigenmodes in the semi-infinite chain limit. We also show that a non-trivial topology leads to protected eigenvalues in the entanglement spectrum. In the $\mathcal{PT}$-symmetric case, we find that the model has a so far overlooked phase where exceptional points become dense in the thermodynamic limit. This phase shows unusual hyper-ballistic transport properties with a dynamical critical exponent $z=1/2$.

Figures (15)

• Figure 1: The considered minimal model has a two-site unit cell, asymmetric intra- ($V_L,V_R$) and inter-cell ($W_L,W_R$) hoppings as well as complex staggered potentials $\pm iu$.
• Figure 2: The non-Hermitian gap $\Delta$, see Eq. \ref{['NHgap']}, as a function of the model parameters in the thermodynamic limit for periodic boundary conditions. In panel (a), the zero field (sublattice symmetric) case, $u=0$, is shown for $V_R=W_R=1$ with two winding numbers $(\nu_1,\nu_2)$ characterizing each phase. In panel (b), we set $u=0.5$, therefore breaking the sublattice symmetry and leaving only a single winding number $I_1$ labeling the emerging phases. In panel (c), we consider the $\mathcal{PT}$-symmetric case $V_L=V_R=V$, $W_L=W_R=W$ and $u=1$. The regions in blue constitute a gapless phase where exceptional points are dense. The central "diamond" is a complex (anti-$\mathcal{PT}$) phase. The remaining phases are $\mathcal{PT}$-unbroken phases, either trivial or topological.
• Figure 3: Scaling of the gap \ref{['NHgap']} in the phase where exceptional points become dense for $V/W=(1-\sqrt{5})/2$ (black circles) and $V/W=(1+\sqrt{5})/2$ (red squares). The lines are fits $\Delta\sim N^{-1/2}$, i.e., the dynamical critical exponent is $z=1/2$.
• Figure 4: Loci of exceptional points $\left(\sqrt{\delta}\right)_\text{EP}$, see Eq. \ref{['gapclosingLodd1']}, for $N=41$ and different rescaled fields $\bar{u}$.
• Figure 5: In the upper panel, solutions of the transcendental equation (\ref{['trans2']}) for different fields $\bar{u}$ and $N=40$ are shown. In the middle panel, we fix $\bar{u}=1/2$ and consider different lattice sizes. In the bottom panel, we show the exceptional points associated with (\ref{['trans2']}).