⚛️ quantum physics

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

This paper demonstrates that real-valued neural quantum states based on Restricted Boltzmann Machines can efficiently and accurately characterize the ground-state properties and PT-symmetry-breaking quantum phase transitions in a non-integrable, non-Hermitian transverse-field Ising chain subjected to longitudinal and complex transverse magnetic fields.

Original authors: M. E. Ateuafack, M. R. Kamsap, J. T. Diffo, L. C. Fai, M. N. Jipdi

Published 2026-08-18

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CC BY 4.0

Original authors: M. E. Ateuafack, M. R. Kamsap, J. T. Diffo, L. C. Fai, M. N. Jipdi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, scientists often study how tiny particles, like atoms or electrons, interact with one another to form larger structures. A common way to model these interactions is by imagining a chain of tiny magnets, called spins, that can point in different directions. In a standard setup, these spins are influenced by a magnetic field pushing them sideways, creating a tug-of-war between order and chaos. This simple model has been a cornerstone of physics for decades, helping researchers understand how materials change their properties, such as becoming magnetic or superconducting. However, real-world systems are rarely so simple. They often face additional forces, like a magnetic field pushing from the front, which complicates the interactions and makes the math incredibly difficult to solve. Furthermore, many modern experiments involve systems that are not perfectly isolated; they exchange energy with their surroundings, a situation that requires a different kind of mathematical description known as non-Hermitian physics. In these systems, energy levels can become complex numbers, leading to strange behaviors where the usual rules of symmetry break down. Understanding how these complex, interacting systems behave is a major challenge because the number of possible states grows so fast that even the most powerful supercomputers struggle to keep up.

A team of researchers has taken a significant step forward in solving this puzzle by applying a new kind of computational tool to a specific, difficult problem. They focused on a chain of quantum spins that is subjected to both a sideways magnetic field and a forward-pushing field, while also allowing the system to lose or gain energy in a controlled way. This combination makes the system "non-integrable," meaning it cannot be solved with standard mathematical tricks, and "non-Hermitian," meaning its energy properties are more exotic than usual. To tackle this, the researchers turned to artificial intelligence, specifically a type of neural network called a Restricted Boltzmann Machine. Think of this network as a flexible, digital brain that learns to guess the most likely arrangement of the spins by adjusting its internal settings over and over again. Instead of trying to calculate every single possibility at once, which is impossible for large chains, the network uses a sampling method to explore the most important configurations and build an accurate picture of the system's ground state, or its lowest energy condition.

The researchers first tested their method on small chains where they could compare the results against exact, traditional calculations. They found that their neural network approach was remarkably accurate, reproducing the energy levels and magnetic properties with high precision. This validation gave them the confidence to scale up their simulations to much larger chains, containing up to one hundred spins, a size that is far beyond the reach of traditional exact calculation methods. As they increased the size of the system and adjusted the strength of the forward-pushing magnetic field, they observed a distinct change in the system's behavior. Around a specific critical value of the field, the system underwent a phase transition. Before this point, the system maintained a certain type of symmetry, but as the field crossed the threshold, this symmetry broke spontaneously. This breaking of symmetry was accompanied by the emergence of a magnetic order, where the spins aligned in a specific direction across the entire chain.

A key discovery in this work was the identification of "exceptional points." These are special conditions where the system's energy levels and their corresponding states merge together, acting as a boundary between two different regimes of behavior. The researchers mapped out how these points appear and disappear as they changed the strength of the magnetic fields. They found that the forward-pushing field had a complex relationship with these points: in some cases, increasing the field made the region where these points exist shrink, while in other cases, it made them grow. This revealed that the interplay between the different magnetic forces and the system's ability to exchange energy creates a rich landscape of behaviors that cannot be predicted by simpler models. By using their neural network framework, the team was able to track these transitions and the resulting magnetic order with a level of detail that was previously unattainable for such large, interacting systems.

The study demonstrates that neural networks are not just tools for recognizing images or playing games, but are becoming powerful engines for exploring the frontiers of quantum physics. By successfully modeling a system that is both interacting and non-Hermitian, the researchers have provided a scalable way to investigate critical phenomena that were previously out of reach. Their work confirms that even in systems where traditional symmetry is broken and energy levels become complex, it is possible to find clear signatures of phase transitions and magnetic ordering. This approach opens the door to studying a wide range of exotic quantum materials and devices, from optical lattices to superconducting circuits, where similar non-Hermitian effects play a crucial role. The findings suggest that the combination of machine learning and quantum simulation offers a robust path forward for understanding the complex, many-body world that lies at the heart of modern condensed matter physics.

Technical Summary: Longitudinal-Field-Driven Transition in a Non-Integrable Non-Hermitian Transverse-Field Ising Chain via RBMs

Problem Statement
The paper addresses the challenge of investigating ground-state properties and quantum critical behavior in interacting non-Hermitian many-body systems. Specifically, it focuses on a one-dimensional non-Hermitian transverse-field Ising chain (TFIM) subjected to both transverse and longitudinal magnetic fields. The introduction of a longitudinal field ( $h$ ) fundamentally alters the physics of the standard TFIM by destroying its free-fermion structure (rendering the system non-integrable) and explicitly breaking the underlying $Z_2$ symmetry. Furthermore, the system incorporates a complex transverse field ( $g_j = \eta + i\xi$ ), introducing non-Hermiticity associated with effective gain and loss processes.

Theoretical investigation of such systems is hindered by the exponential growth of the Hilbert space dimension, which restricts exact diagonalization (ED) to small system sizes. While tensor-network methods (e.g., Matrix Product States) are effective for 1D systems, their efficiency is often limited by the entanglement structure of the target state, particularly in strongly correlated or highly entangled regimes. The paper seeks a scalable alternative to characterize the interplay between transverse quantum fluctuations, longitudinal interactions, and non-Hermiticity, specifically focusing on parity-time (PT) symmetry breaking and the emergence of exceptional points (EPs).

Methodology
To overcome the limitations of ED and tensor networks, the authors employ a neural-network quantum state (NQS) approach based on Restricted Boltzmann Machines (RBM), optimized via Variational Monte Carlo (VMC) sampling.

Variational Ansatz: The authors utilize a real-valued RBM framework. Although non-Hermitian systems generally require independent optimization of left ( $\langle \Psi_L |$ ) and right ( $| \Psi_R \rangle$ ) eigenstates, the authors leverage the PT-symmetry of the Hamiltonian in the unbroken phase. This allows them to reformulate the problem using a single real-valued neural network ansatz, significantly reducing computational cost while maintaining physical accuracy.
Optimization: The variational energy is minimized using the Adam optimizer with Gibbs sampling. The local energy is evaluated as a statistical average, where the imaginary components of complex conjugate pairs cancel out, yielding a real-valued ground-state energy estimate consistent with the PT-symmetric phase.
Ground State Definition: The ground state is defined as the eigenstate with the smallest real part of the eigenvalue that continuously connects to the Hermitian ground state as the non-Hermitian parameter is adiabatically introduced.
Observables: The study computes ground-state energy, longitudinal magnetization ( $M_z$ ), and spin-spin correlation functions ( $C_{zz}(r)$ ) to map out phase transitions and critical behavior.

Key Contributions

Framework Extension: The work extends the RBM neural quantum state framework to a non-integrable, interacting non-Hermitian system by explicitly including a longitudinal magnetic field, a scenario not previously covered in existing NQS studies of non-Hermitian Ising models.
Benchmarking: The authors provide a rigorous benchmark of the RBM-VMC method against exact diagonalization for finite lattice sizes ( $N=4$ to $N=10$ ). The results demonstrate that the RBM approach accurately reproduces ground-state energies, magnetization, and spin-spin correlations.
Scalability: The methodology is successfully scaled to larger system sizes ( $N=100$ ), surpassing the reach of exact diagonalization, thereby enabling the study of macroscopic quantum criticality in non-Hermitian systems.
Phase Transition Characterization: The study identifies and characterizes a non-Hermitian quantum phase transition driven by the longitudinal field, marked by PT-symmetry breaking and the emergence of magnetic order.

Results

Spectral Analysis: Spectral analysis of finite chains reveals the emergence of exceptional points (EPs), where eigenvalues and eigenvectors coalesce. The study maps the EP structure in the $(h, \xi)$ parameter space, finding that the longitudinal field $h$ has a dual effect: it suppresses the EP region in the low- $\xi$ regime but enhances it in the high- $\xi$ regime.
Accuracy: For small systems ( $N=10$ ), the RBM-calculated ground-state energy and its variance converge precisely to the ED results. The imaginary part of the local energy fluctuates around zero, confirming the validity of the real-valued ansatz in the PT-symmetric regime.
Phase Transition: For larger systems ( $N=100$ $N = 100$ ), the authors identify a critical longitudinal field strength $h_c \approx 0.3$ $h_{c} \approx 0.3$ .
- Energy Scaling: The real part of the ground-state energy exhibits a distinct change in behavior around $h_c$ , signaling a phase transition.
- Magnetization: The longitudinal magnetization $M_z$ shows a spontaneous symmetry breaking around $h_c$ , transitioning from a disordered to an ordered magnetic phase.
- Correlations: The spin-spin correlation function $C_{zz}(r)$ indicates long-range ferromagnetic order in the low-field regime ( $h < h_c$ ), with correlations persisting across the entire chain. The correlation length is comparable to the system size, consistent with the finite magnetization observed.
PT-Symmetry Breaking: The transition at $h_c$ is accompanied by the spontaneous breaking of PT symmetry, evidenced by the emergence of complex-conjugate energy pairs and changes in the correlation profiles.

Significance and Claims
The paper claims to establish real-valued neural quantum states as an efficient and scalable framework for investigating critical phenomena in interacting non-Hermitian quantum systems. By successfully modeling a non-integrable system with a longitudinal field, the authors demonstrate that RBMs can accurately characterize PT-symmetry-breaking transitions and the emergence of magnetic order without the computational bottlenecks of exact diagonalization. The work provides a powerful computational tool for exploring quantum criticality and emergent magnetic order in regimes where traditional analytical and numerical methods are insufficient. The authors position their findings as a step toward understanding the rich interplay between coherent spin interactions, quantum fluctuations, and non-Hermitian dynamics in experimental platforms such as Rydberg atom arrays and superconducting qubits.

Technical Summary: Longitudinal-Field-Driven Transition in a Non-Integrable Non-Hermitian Transverse-Field Ising Chain via RBMs

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