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A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions

Using a parent Hermitian tight-binding model on a bipartite lattice with chiral symmetry, we theoretically generate non-Hermitian models for free fermions with $p$ orbitals per unit cell satisfying a complex generalization of chiral symmetry. The $p$ complex energy bands in $k$ space are given by a common $k$-dependent real factor, determined by the bands of the parent model, multiplied by the $p$th roots of unity. When the parent model is the Su-Schrieffer-Heeger (SSH) model, the single-particle energy levels are the same as those of free parafermion solutions to Baxter's non-Hermitian clock model. This construction relies on fully unidirectional hopping to create Bloch Hamiltonians with the form of generalized permutation matrices, but we also describe the effect of partial unidirectional hopping. For fully bidirectional hopping, the Bloch Hamiltonians are Hermitian and may be separated into even and odd parity blocks with respect to inversion of the orbitals within the unit cell. Partially unidirectional hopping breaks the inversion symmetry and mixes the even and odd blocks, and the real energy spectrum evolves into a complex one as the degree of unidirectionality increases, with details determined by the topology of the parent model and the number of orbitals per unit cell, $p$. We describe this process in detail for $p=3$ and $p=4$ with the SSH model. We also apply our approach to graphene, and show that $AA$-stacked bilayer graphene evolves into a square root Hamiltonian of monolayer graphene with the introduction of unidirectional hopping. We show that higher-order exceptional points occur at edge states and solitons in the non-Hermitian SSH model, and at the Dirac point of non-Hermitian graphene.