Longitudinal-Field-Driven Transition in a non-integrable Non-Hermitian Transverse-Field Ising Chain via RBMs

Understanding the behavior of interacting quantum many-body systems in external fields is a central problem in condensed matter physics. While the standard transverse-field Ising model is exactly solvable, introducing a longitudinal magnetic field and non-Hermitian parameters (such as complex transverse fields representing gain and loss) destroys its free-fermion structure, rendering the system non-integrable. This work investigates the ground-state properties and quantum critical behavior of this non-Hermitian, non-integrable transverse-field Ising chain.

Because exact diagonalization is limited to small system sizes due to the exponential growth of the Hilbert space, the authors employ real-valued Restricted Boltzmann Machines (RBMs) optimized via Variational Monte Carlo (VMC) sampling. For PT-symmetric Hamiltonians in the unbroken phase, the ground-state optimization can be formulated using a single real-valued neural network ansatz. This approach avoids the complex optimization of independent left and right eigenvectors while accurately capturing non-local many-body correlations and magnetic observables.

Spectral analysis of finite chains reveals exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce, marking spontaneous parity-time (PT) symmetry breaking. Benchmark comparisons with exact diagonalization confirm that the RBM approach accurately reproduces ground-state energies, magnetization, and spin-spin correlations. Furthermore, mapping the system in the longitudinal field and imaginary transverse field parameter space demonstrates a qualitative reversal in how the longitudinal field affects the EP regions around a critical non-Hermitian threshold, successfully tracking the emergence of magnetic order and quantum criticality.