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

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 $z=1/2$.