Biorthogonal Neural Network Approach to Two-Dimensional Non-Hermitian Systems

1. Problem Statement

Non-Hermitian (NH) quantum many-body systems exhibit unique physical phenomena such as the non-Hermitian skin effect, exceptional points (EPs), and unconventional phase transitions. However, studying these systems numerically is challenging due to several factors:

• Breakdown of Variational Principles: The standard Rayleigh-Ritz variational principle, which relies on real energy spectra and orthogonal eigenstates, fails in NH settings where eigenvalues are complex and eigenstates are non-orthogonal.
• Limitations of Existing Methods: Traditional techniques like Density Matrix Renormalization Group (DMRG) and Matrix Product States (MPS) struggle with two-dimensional (2D) systems and strong correlations. Quantum Monte Carlo (QMC) methods often suffer from the "sign problem" in NH systems.
• Optimization Landscapes: Direct application of Variational Monte Carlo (VMC) to NH systems often leads to poor convergence due to complex energy landscapes, saddle points, and the difficulty of distinguishing the ground state (defined as the eigenstate with the smallest real energy part) from excited states.

2. Methodology

The authors propose a Biorthogonal Neural Network Quantum State (NQS) framework combined with a Self-Consistent Variance Minimization strategy.

A. Biorthogonal Formalism

Instead of treating the system with a single wavefunction, the method utilizes the biorthogonal structure of NH systems:

• Right Eigenstates ($|R_n\rangle$): $\hat{H}|R_n\rangle = \lambda_n|R_n\rangle$
• Left Eigenstates ($|L_n\rangle$): $\hat{H}^\dagger|L_n\rangle = \lambda_n^*|L_n\rangle$
• Dual State: A quantum state $|\psi\rangle$ is defined alongside its dual $|\tilde{\psi}\rangle$. Expectation values are calculated using the biorthogonal inner product: $\langle \hat{O} \rangle = \frac{\langle \tilde{\psi} | \hat{O} | \psi \rangle}{\langle \tilde{\psi} | \psi \rangle}$.

B. Variance Minimization Loss Function

Since energy minimization is ill-defined for complex spectra, the authors minimize the variance of the energy. They define two Hermitian, positive semi-definite operators:

• Right operator: $\hat{V}_R(\varepsilon) = (\hat{H}^\dagger - \varepsilon^*)(\hat{H} - \varepsilon)$
• Left operator: $\hat{V}_L(\varepsilon) = (\hat{H} - \varepsilon)(\hat{H}^\dagger - \varepsilon^*)$

The loss function is the sum of the variances for both states:
$$L[\psi, \tilde{\psi}] = \frac{\langle \psi | \hat{V}_R(\varepsilon) | \psi \rangle}{\langle \psi | \psi \rangle} + \frac{\langle \tilde{\psi} | \hat{V}_L(\varepsilon) | \tilde{\psi} \rangle}{\langle \tilde{\psi} | \tilde{\psi} \rangle}$$
Crucially, $\varepsilon$ is defined as the biorthogonal expectation value of the energy: $\varepsilon = \frac{\langle \tilde{\psi} | \hat{H} | \psi \rangle}{\langle \tilde{\psi} | \psi \rangle}$.

C. Optimization Strategies

To address the challenges of optimizing this loss function, the authors introduce three key components:

1. Self-Consistent Optimization: Unlike previous methods that treat $\varepsilon$ as a free variational parameter (which introduces saddle points), this method updates $\varepsilon$ self-consistently. In each step, the parameters of $|\psi\rangle$ and $|\tilde{\psi}\rangle$ are updated while keeping $\varepsilon$ fixed, then $\varepsilon$ is recalculated. This ensures the wavefunctions converge to true biorthogonal eigenstates with complex-conjugate eigenvalues.
2. Initialization Schemes:
   • Warm-Start: Inspired by the adiabatic theorem, the system is evolved from a Hermitian limit ($k=0$) to the full NH Hamiltonian by gradually increasing the non-Hermitian parameter.
   • Fixed-Start: Uses a prior estimate (e.g., mean-field or lower bound) to fix $\varepsilon$ initially, then smoothly transitions to the self-consistent regime.
   • Combined Approach: A hybrid strategy is used to navigate near Exceptional Points (EPs), where the energy gap closes and states coalesce.
3. Neural Network Architecture: The wavefunctions are represented by Restricted Boltzmann Machines (RBM) in the $\sigma^z$ basis. Stochastic Reconfiguration (SR) is used to precondition gradients for faster convergence.

3. Key Contributions

• Novel Framework: Development of a self-consistent variance minimization framework specifically tailored for NH systems using NQS, overcoming the breakdown of the Rayleigh-Ritz principle.
• Handling Exceptional Points: Introduction of combined initialization strategies (Warm-Start + Fixed-Start) that allow the algorithm to converge to the ground state even in the vicinity of EPs, where standard variational methods fail.
• Scalability: Demonstration that this method scales effectively to 2D systems, outperforming traditional tensor network methods (like DMRG) in terms of variance scaling with system size.
• Biorthogonal Expectation Values: Rigorous derivation and application of biorthogonal estimators for observables, showing distinct physical behaviors (e.g., magnetization jumps) compared to standard "right-only" estimators.

4. Results

The method was benchmarked on the Non-Hermitian Transverse-Field Ising Model (NH-TFIM) with a complex longitudinal field.

• Accuracy: The method achieved high accuracy across both Parity-Time (PT) symmetric and PT-broken phases. It showed an order-of-magnitude improvement in energy accuracy compared to treating $\varepsilon$ as a free parameter.
• Exceptional Points: The combined initialization strategy successfully navigated the phase transition at EPs, accurately capturing the ground state where the energy gap vanishes and eigenstates coalesce.
• 2D Scaling:
  • On a $6 \times 6$ lattice, the NQS results matched DMRG (with high bond dimension $\chi=1000$) for both energy and magnetization.
  • On an $8 \times 8$ lattice, the variance per site for the NQS remained nearly constant as system size increased, whereas DMRG variance increased significantly. This indicates the NQS approach is more scalable for 2D NH systems.
• Physical Insights:
  • The method correctly identified the PT-symmetric phase (paramagnetic, real energy) and the broken phase (ferromagnetic, complex energy).
  • Analysis of the connected correlation function $C_z(r)$ showed exponential decay in both phases, but the magnetization behavior revealed a qualitative difference: the real part of the magnetization vanishes in the PT phase but becomes finite in the broken phase, signaling spontaneous symmetry breaking.

5. Significance

This work establishes Neural Network Quantum States (NQS) trained via self-consistent variance minimization as a powerful, scalable tool for investigating interacting, higher-dimensional non-Hermitian systems.

• Overcoming Limitations: It bypasses the sign problem of QMC and the bond-dimension limitations of MPS/DMRG in 2D.
• Theoretical Advancement: It provides a robust numerical framework to study exotic phenomena like the NH skin effect and exceptional points in strongly correlated regimes, which were previously inaccessible.
• Future Directions: The authors suggest this approach can be extended to real-time dynamics, non-Hermitian fermionic models, and more expressive ansätze (e.g., autoregressive models or transformers) to capture long-range correlations.

In summary, the paper bridges the gap between machine learning-based quantum simulation and non-Hermitian physics, offering a reliable computational pathway to explore the rich phase diagrams of open and driven quantum systems.