Longitudinal-Field-Driven Transition in a non-integrable Non-Hermitian Transverse-Field Ising Chain via RBMs
We investigate the ground-state properties and quantum critical behavior of a non-Hermitian transverse-field Ising chain subjected to longitudinal and complex transverse magnetic fields. To address this interacting many-body problem, we employ real-valued neural quantum states based on Restricted Boltzmann Machines (RBMs), optimized using Variational Monte Carlo (VMC) sampling. Spectral analysis of finite chains reveals exceptional points associated with spontaneous parity-time (PT) symmetry breaking. A real-valued RBM framework is developed to reconstruct the ground-state eigenstates of the non-Hermitian Hamiltonian. Benchmark comparisons with exact diagonalization demonstrate that the RBM approach accurately reproduces the ground-state energy, magnetization, and spin-spin correlations. Extending the analysis to larger system sizes, we identify a non-Hermitian quantum phase transition characterized by PT-symmetry breaking and the emergence of magnetic order. Our results establish real-valued neural quantum states as an efficient and scalable framework for investigating critical phenomena in interacting non-Hermitian quantum systems.