Biorthogonal Neural Network Approach to Two-Dimensional Non-Hermitian Systems

Abstract: Non-Hermitian quantum many-body systems exhibit a rich array of physical phenomena, including non-Hermitian skin effects and exceptional points, that remain largely inaccessible to existing numerical techniques. In this work, we investigate the application of variational Monte Carlo and neural network wavefunction representations to examine their ground-state (the eigenstate with the smallest real part energy) properties. Due to the breakdown of the Rayleigh-Ritz variational principle in non-Hermitian settings, we develop a self-consistent symmetric optimization framework based on variance minimization with a dynamically updated energy estimate. Our approach respects the biorthogonal structure of left and right eigenstates, and is further strengthened by exploiting system symmetries and pseudo-Hermiticity. Tested on a two-dimensional non-Hermitian transverse field Ising model endowed with a complex longitudinal field, our method achieves high accuracy across both parity-time symmetric and broken phases. Moreover, we propose novel optimization routines that address the challenges posed by exceptional points and provide reliable convergence to the ground state in regimes where standard variational techniques fail. Lastly, we show, through extensive numerical evidence, that our method offers a scalable and flexible computational tool to investigate non-Hermitian quantum many-body systems, beyond the reach of conventional numerical techniques such as the density-matrix renormalization group algorithm.