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Non-Hermitian systems exhibiting topological properties are attractinggrowing interest. In this work, we propose an algorithm for solving the groundstate of a non-Hermitian system in the matrix product state (MPS) formalismbased on a companion Hermitian Hamiltonian. If the eigenvalues of thenon-Hermitian system are known, the companion Hermitian Hamiltonian can bedirectly constructed and solved using Hermitian variational methods. When theeigenvalues are unknown, a gradient descent along with the companion HermitianHamiltonian yields both the ground state eigenenergy and the eigenstate. Withthe variational principle as a solid foundation, our algorithm ensuresconvergence and provides results in excellent agreement with the exactsolutions of the non-Hermitian Su-Schrieffer-Heeger (nH-SSH) model as well asits interacting extension. The approach we present avoids solving anynon-Hermitian matrix and overcomes numerical instabilities commonly encounteredin large non-Hermitian systems.

Variational Matrix Product State Approach for Non-Hermitian System Based on a Companion Hermitian Hamiltonian