Fermionic Neural Gibbs States (fNGS)

• Fermionic Neural Gibbs States (fNGS) is a variational framework that models finite-temperature properties in interacting fermion systems by combining mean-field TFD purifications with neural-network backflow and attention layers.
• The framework employs neural network-enhanced pair correlators and Jastrow factors to capture higher-order correlations, enabling precise computation of thermal energies and correlation functions in quantum lattice models.
• fNGS scales effectively to doped Fermi-Hubbard and t-V models, achieving quantitative accuracy beyond exact diagonalization and offering new insights into strongly correlated fermionic systems.

Fermionic neural Gibbs states (fNGS) constitute a variational framework for the modeling of finite-temperature properties in strongly interacting fermion systems. Integrating mean-field thermofield-double (TFD) purifications, expressive neural-network-based backflow architectures, and efficient projected imaginary-time evolution algorithms, fNGS enables the study of strongly correlated and doped quantum lattices at scales and parameter ranges beyond standard exact diagonalization techniques. The framework is immediately applicable to models of central interest in condensed matter physics, such as the doped Fermi-Hubbard and extended Hubbard/t-V models, in both spinless and spinful regimes, and allows for direct computation of thermal energies, correlation functions, and structure factors across a broad range of temperatures and interactions (Nys et al., 4 Dec 2025).

1. Reference Mean-Field Thermofield-Double State

fNGS builds upon the exact mean-field thermofield-double purification of a non-interacting quadratic Hamiltonian $\hat H_0$. The Hilbert space is doubled, with physical and auxiliary fermionic operators ($\hat c_\mu, \hat c_\mu^\dagger$ and $\tilde c_\mu, \tilde c_\mu^\dagger$) satisfying canonical anticommutation relations: $$\{\hat c_\mu,\hat c_\nu^\dagger\} = \{\tilde c_\mu,\tilde c_\nu^\dagger\} = \delta_{\mu\nu}, \quad \{\hat c^{(\dagger)}_\mu,\tilde c^{(\dagger)}_\nu\} = 0.$$ The mean-field TFD state at inverse temperature $\beta$ is

$$\bigl|\Psi_0(\beta)\bigr\rangle = \prod_\mu \left[u_\mu(\beta) + v_\mu(\beta)\, \hat c^\dagger_\mu \tilde c^\dagger_\mu\right] |\Omega\rangle,$$

with $w_\mu(\beta)=e^{-\beta \epsilon_\mu/2}$, $u_\mu=1/\sqrt{1+w_\mu^2}$, and $v_\mu=w_\mu u_\mu$. Canonical (fixed-particle-number) projection yields an antisymmetrized geminal-power (AGP) form,

$$\mathcal P_N|\Psi_0(\beta)\rangle = \left[\Gamma^\dagger(\beta)\right]^N |\Omega\rangle,\quad \Gamma^\dagger(\beta)=\sum_\mu w_\mu(\beta)\, \hat c_\mu^\dagger \tilde c_\mu^\dagger,$$

and the state in configuration basis is expressible as a Slater determinant of pair-orbital matrices $\varphi_{p,\tilde q}$. This exact reference anchors the variational fNGS construction (Nys et al., 4 Dec 2025).

2. Neural-Network Generalization of Correlators

The fNGS ansatz introduces neural networks to extend the pair-orbital structures beyond the mean-field reference and incorporates Jastrow factors to capture higher-order correlations. The complete variational wavefunction in the doubled occupation basis reads: $$\langle x, \tilde y\,|\,\Psi(\beta; \theta)\rangle = \det_{i,j=1}^N [\varphi_{p_i,\tilde q_j}(x, \tilde y;\theta)] \; \exp\left[-J(x, \tilde y; \theta)\right],$$ where $\varphi_{p,\tilde q}$ now depends on both the sample and neural network parameters $\theta$, and is structured using MLPs, feedforward networks (FFNs), and small Vision Transformers (ViT) for intra- and inter-species embeddings. The pair-orbital structure is enhanced via cross-species BiViT blocks employing multi-head attention to update edge features. Fermionic antisymmetry is maintained by the determinant, while the Jastrow factor incorporates both two-body and cross-species correlators.

3. Projected Imaginary-Time Evolution and Purified Gibbs States

Correlated finite-temperature states are constructed via imaginary-time evolution from the mean-field TFD state using a non-unitary "work operator,"

$$W = \hat H \otimes \tilde I - \frac{\beta_0}{\beta} I \otimes \tilde H_0,\quad |\Psi(\beta)\rangle = \exp\left(-\frac{\beta}{2} W\right) |\Psi_0(\beta_0)\rangle.$$

This operator addresses both physical and auxiliary sectors to systematically cool the state to desired temperatures. The purified Gibbs density matrix,

$$\rho(\beta;\theta) = \operatorname{Tr}_{\tilde{\mathcal H}} \left[|\Psi(\beta;\theta)\rangle\langle\Psi(\beta;\theta)|\right],$$

embodies the variational description of the finite-temperature interacting fermionic system (Nys et al., 4 Dec 2025).

4. Variational Optimization and Cost Function

The fNGS framework employs a variational time-evolution procedure known as projected imaginary-time evolution (tre-pITE). At each step $\Delta\tau$, the updated neural parameters $\theta'$ are optimized to maximize the fidelity,

$$\mathcal F(\theta';\theta) = \frac{|\langle\Psi(\beta+\Delta\tau; \theta')| e^{-\Delta\tau W/2} |\Psi(\beta; \theta)\rangle|^2}{\langle\Psi(\beta+\Delta\tau; \theta')|\Psi(\beta+\Delta\tau; \theta')\rangle\, \langle\Psi(\beta; \theta)|\Psi(\beta; \theta)\rangle},$$

using stochastic reconfiguration (SR) or Kaczmarz-inspired solvers. Gradients are estimated via Monte Carlo in the doubled Hilbert space with SR updates respecting the quantum Fisher metric. Practical implementations combine minSR with the SPRING optimizer (momentum $\mu\approx 0.95$). The formal variational free energy functional can be written as

$$\mathcal F[\rho(\theta)] = \operatorname{Tr}[\rho(\theta)\, \hat H] + \frac{1}{\beta} \operatorname{Tr}[\rho(\theta)\ln\rho(\theta)],$$

with gradients following standard variational calculus in the space of purified density matrices.

5. Application to Strongly Correlated Lattice Fermions

fNGS is benchmarked on both spinless and spinful versions of the Fermi-Hubbard and t-V models, specifically:

• Spinless t–V model on $4\times4$ lattices at half-filling ($U=0,V=1$): fNGS initialized with free-fermion TFD references at various $\beta_0$ accurately reproduces the thermal energy $E(\beta)$ throughout the temperature range, demonstrating the path independence of the imaginary-time evolution scheme.
• Spinful Hubbard model on $4\times4$ and $8\times8$ lattices with dopings $\delta=1/8$ (and up to $18.8\%$) at $U=2,4,6.5$: On $4\times4$, fNGS closely tracks exact diagonalization (ED) and thermal-pure-quantum (TPQ) benchmarks for $0\le\beta\le10$. For $8\times8$, fNGS matches expected qualitative features (onset of antiferromagnetic curvature, spin–spin correlations and structure factors) with smooth monotonic energy dependence and statistical error $\mathcal O(10^{-3})$ on single-GPU runs, outperforming the reach of ED.

These results confirm the scalability and accuracy of fNGS for two-dimensional lattice systems in regimes intractable by conventional methods (Nys et al., 4 Dec 2025).

6. Scalability and Perspectives

The fNGS approach synthesizes three essential components:

1. Exact, antisymmetric mean-field TFD references,
2. Flexible, exchange-symmetric neural network backflow and attention layers,
3. High-order projected imaginary-time evolution algorithms (tre-pITE).

Empirical benchmarks demonstrate that fNGS delivers quantitative accuracies on $8\times8$ lattice systems with significant correlation and doping, maintaining scalability with respect to both system size and network expressiveness. Prospective directions include:

• Extension to higher dimensions and continuum systems (e.g., dilute Fermi gases),
• Extraction of real-time response and transport properties from the purified fNGS states,
• Deployment of deeper neural architectures (e.g., advanced Transformers, message-passing networks) to improve low-temperature correlator fidelity,
• Utilization of alternative cooling protocols (exponential RG, XTRG),
• Adaptation to open quantum systems via neural-operator purifications.

A plausible implication is that fNGS provides a symmetry-aware, scalable variational formalism, unifying thermofield purifications, neural quantum states, and projected imaginary-time evolution for finite-temperature fermions in $>1$ dimension, facilitating progress in regimes traditionally inaccessible to both stochastic and exact diagonalization methods (Nys et al., 4 Dec 2025).