Abstract
The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.
Highlights
Since the seminal paper by Kane and Mele [1], noninteracting topological phases of matter have attracted growing interest
1.0 b see that the polarization reproduces the parity of the winding number and that it is equivalent to χ. To show that this result is not limited to the case where the polarization is quantized, we introduce a term in the Hamiltonian, H = κσz, to break sublattice symmetry
More correctly the entanglement occupancy spectrum (EOS), it was shown that the line-gapped phases reproduced the topology of the energy spectrum, while the EOS of point-gapped phases was thought to be featureless [35]
Summary
Since the seminal paper by Kane and Mele [1], noninteracting topological phases of matter have attracted growing interest. Note that an invariant called “biorthogonal polarization” was introduced for non-Hermitian systems [25,26] It is, not a generalization of the Hermitian polarization, as it is calculated using only the topological zero-energy states. There are several issues, most importantly how to choose the “ground state” for which the reduced density matrix is computed, and how to properly define the reduced density matrix itself This was first studied by Herviou et al [34,35], where they generalize the equivalence between the entanglement spectrum and the EOS to noninteracting non-Hermitian systems. They showed numerically that the EOS correctly reproduced the topology of the periodic boundary condition system for line-gapped phases. Some of the more technical details are moved to the appendices
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