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.
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.
The simulation of interacting many-body quantum systems is a central chal- lenge in condensed matter physics due to the exponential growth of the Hilbert space with system size. In this project, we implement a Neural Network Quantum State (NQS) ansatz based on a Complex Restricted Boltzmann Machine (RBM) to solve for the ground state of the 1D Transverse Field Ising Model (TFIM). Using Variational Monte Carlo (VMC) optimization via Stochastic Reconfiguration (SR), we successfully train the network to approximate the ground state wave function across the paramagnetic-ferromagnetic phase transition. We benchmark our results against Exact Diagonalization (ED) for small systems (N = 12), demonstrating high accuracy in ground state energy estimation. Furthermore, we investigate quantum correlations and order parameters, specifically the transverse magnetization mx and longitudinal spin-spin correlations Czz(r), highlighting the RBM’s ability to capture long-range order and critical behavior near the quantum critical point hx/J = 1.
Non-Hermitian (NH) quantum systems have emerged as a powerful framework for describing open quantum systems, nonequilibrium 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.
Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. Based on anecdotal evidence, it has been claimed repeatedly in the literature that neural network quantum state simulations are insensitive to the sign problem. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving exactly how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify a number of new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. Our results suggest that great care needs to be taken with neural network quantum state based variational Monte Carlo when the system under study is highly frustrated.
Non-Hermitian (NH) quantum many-body systems exhibit a rich array of physical phenomena, including NH skin effects and exceptional points, that remain largely inaccessible to existing numerical techniques. In this Letter, we investigate the application of variational Monte Carlo and neural network wave function representations to examine their ground-state properties. Due to the breakdown of the Rayleigh-Ritz variational principle in NH 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. Leveraging this tool, we probe and report accurate NH physical observables for a two-dimensional transverse-field Ising model with a complex longitudinal field, spanning both parity-time symmetric and broken phases. Lastly, we show, through extensive numerical evidence, that our method offers a scalable and flexible computational tool to investigate NH quantum many-body systems, beyond the reach of conventional numerical techniques such as the density-matrix renormalization group algorithm.
In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a Z2 symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a U(1) symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.
We investigate nonstabilizerness, also known as ``magic,'' to understand criticality and exceptional points in non-Hermitian quantum many-body systems. Our focus is on parity-time (PT)-symmetric spin chains, specifically the non-Hermitian transverse-field Ising and XX models. We calculate stabilizer Rényi entropies in their ground states using non-Hermitian matrix product state methods. Our findings show that magic exhibits unique and model-specific signs of phase transitions. In the Ising chain, it peaks near the regular Hermitian-like critical line but disappears across exceptional points. In contrast, in the XX chain, it reaches its maximum at the exceptional line where PT symmetry is broken. Finite-size scaling reveals that these effects become more pronounced with larger systems, highlighting nonstabilizerness as a sensitive marker for both quantum criticality and non-Hermitian spectral degeneracies. We also investigate magic in momentum space for the XX model analytically and find that it reaches a minimum around exceptional points. Our results indicate that magic takes extremal values at the exceptional points and serves as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter.