Many-Body Neural Network Wavefunction for a Non-Hermitian Ising Chain

Non-Hermitian (NH) quantum systems have emerged as a powerful framework for describing open quantum systems, non-equilibrium dynamics, and engineered quantum optical materials. However, solving the ground-state properties of NH systems is challenging due to the exponential scaling of the Hilbert space, and exotic phenomena such as the emergence of exceptional points. Another challenge arises from the limitations of traditional methods like exact diagonalization (ED). For the past decade, neural networks (NNs) have shown promise in approximating many-body wavefunctions, yet their application to NH systems remains largely unexplored. In this paper, we explore different NN architectures to investigate the ground-state properties of a parity-time-symmetric, one-dimensional NH, transverse field Ising model with a complex spectrum by employing a recurrent neural network (RNN), a restricted Boltzmann machine~(RBM), and a multilayer perceptron (MLP). We construct the NN-based many-body wavefunctions and validate our approach by recovering the ground-state properties of the model for small system sizes, finding excellent agreement with ED. Furthermore, for larger system sizes, we demonstrate that the RNN outperforms both the RBM and MLP. However, we show that the accuracy of the RBM and MLP can be significantly improved through transfer learning, allowing them to perform comparably to the RNN for larger system sizes. These results highlight the potential of neural network-based approaches–particularly for accurately capturing the low-energy physics of NH quantum systems in case of both weak and strong non-Hermiticity.

💡 Research Summary

This paper investigates the ground‑state properties of a parity‑time (PT) symmetric non‑Hermitian (NH) transverse‑field Ising chain using neural‑network quantum states (NQS). The authors consider three distinct neural‑network architectures—recurrent neural networks (RNNs), restricted Boltzmann machines (RBMs), and multilayer perceptrons (MLPs)—and embed them within a variational Monte‑Carlo (VMC) framework that is adapted to the bi‑orthogonal formalism required for NH systems.

The model studied is a one‑dimensional spin‑½ Ising chain with nearest‑neighbour coupling J=1 and a staggered complex transverse field g=η+iξ applied to sublattice A (g) and its complex conjugate g* to sublattice B. The Hamiltonian respects PT symmetry, leading to a phase diagram where an unbroken PT region (real spectrum) is separated from a broken PT region (complex spectrum) by exceptional points (EPs). The authors analytically discuss the emergence of EPs, verify them numerically by tracking square‑root singularities in the spectrum and by measuring the maximal overlap between eigenvectors, which reaches unity at EPs.

In NH quantum mechanics the notion of a ground state is ambiguous because eigenvalues are complex. The authors adopt the convention of minimizing the real part of the energy expectation value, using the bi‑orthogonal expression ⟨Ψ₀ᴸ|O|Ψ₀ᴿ⟩/⟨Ψ₀ᴸ|Ψ₀ᴿ⟩. They show that the local energy can be computed with either the left or right eigenstate, simplifying the VMC implementation to a single neural network.

For the NQS, the RNN treats the spin configuration as a sequence, naturally capturing long‑range correlations and complex phases. The RBM uses a visible‑hidden bipartite graph with complex weights, while the MLP stacks fully‑connected layers with complex parameters. All three networks are trained by stochastic gradient descent (Adam) on the variational energy, with gradients obtained via the standard VMC estimator.

Benchmarking proceeds in two stages. First, for small chains (N=4,6,8) the authors compare the neural‑network energies and correlation functions to exact diagonalization (ED). All three architectures reproduce the ED results with errors below 10⁻⁴, correctly capturing the EP‑induced changes in the wave‑function phase. Second, for larger systems (N≈20–40) the RNN outperforms the RBM and MLP in terms of convergence speed and final energy error (ΔE_real ≈ 5×10⁻⁴). However, by employing transfer learning—initializing the RBM and MLP with parameters trained on a smaller system—their performance improves dramatically, reaching accuracy comparable to the RNN (ΔE_real < 10⁻³) and reducing training time.

The authors also analyze the behavior of the networks near EPs. The RNN’s recurrent structure allows it to adapt more readily to the rapid phase variation of the wavefunction, whereas the RBM and MLP exhibit larger fluctuations unless aided by transfer learning. The study confirms that minimizing the real part of the energy yields a physically meaningful ground state, which in this model remains purely real despite the non‑Hermitian terms.

In conclusion, the paper demonstrates that (i) neural‑network variational methods can be successfully extended to NH many‑body problems, (ii) RNNs provide the most robust representation for complex‑valued wavefunctions, and (iii) transfer learning enables traditional architectures (RBM, MLP) to achieve comparable accuracy on larger systems. The work opens the door to applying NQS techniques to higher‑dimensional NH models, to exploring NH topological phases, and to integrating these methods with quantum‑hardware simulators.