Fermionic neural Gibbs states

TL;DR

This work introduces fermionic neural Gibbs states (fNGS), a variational scheme that combines thermofield doubling with neural-network transformations to model finite-temperature, strongly interacting fermions. Starting from a mean-field reference, fNGS uses a neural backflow-like ansatz and Jastrow factors, optimized via a Taylor-root projected imaginary-time evolution (tre-pITE) to reach target temperatures while preserving fermionic antisymmetry. Benchmarks on doped t-V and Fermi-Hubbard models show accurate thermal energies across temperatures and interaction strengths, including larger systems where exact methods are infeasible, and reveal realistic spin correlations and antiferromagnetic tendencies. The approach offers a scalable path to finite-temperature properties in higher dimensions and sets the stage for continuum extensions, real-time dynamics, and more expressive neural architectures.

Abstract

We introduce fermionic neural Gibbs states (fNGS), a variational framework for modeling finite-temperature properties of strongly interacting fermions. fNGS starts from a reference mean-field thermofield-double state and uses neural-network transformations together with imaginary-time evolution to systematically build strong correlations. Applied to the doped Fermi-Hubbard model, a minimal lattice model capturing essential features of strong electronic correlations, fNGS accurately reproduces thermal energies over a broad range of temperatures, interaction strengths, even at large dopings, for system sizes beyond the reach of exact methods. These results demonstrate a scalable route to studying finite-temperature properties of strongly correlated fermionic systems beyond one dimension with neural-network representations of quantum states.

Figures (2)

• Figure 1: Benchmark results. (A) Thermal energies of the $t$-$V$ model on a $4\times 4$ square lattice at $V=1$ and half-filling, obtained from free fermions ($V_0=0$) at various $\beta_0 = 0.25, 0.5, 1$. Exact thermal energies are shown in red, with $1\sigma$ error bars obtained from TPQ simulations with $1024$ samples. The gray data represents the expectation value of the final Hamiltonian during the work-operator evolution. (B) Thermal energies of the $4 \times 4$ strongly correlated $t$-$V$ model at doping $\delta=1/8$ and $V = 6.5$, compared to the ED and TPQ predictions. Black lines show the spectrum of $\hat{H}$. (C) Thermal energies of the $4 \times 4$ spinful correlated FH model at doping $\delta=1/8$ at various $U$, compared to the exact TPQ predictions. The ground state is indicated by the dashed horizontal lines.
• Figure 2: (A) Thermal energies at various couplings and dopings of an $8\times 8$ spinful FH model. For the $U=4$ and $\delta=18.8\%$ case, we show for various temperatures the evolution of the (B,C) connected spin-spin correlation $C_{ss}(\mathbf{d})$, and (D, E) corresponding spin structure factor $S(\mathbf{q})$.