Discovering Quantum Phase Transitions with Fermionic Neural Networks

Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.

Figure 1

The $N=27$ spin-polarized homogeneous electron gas in a body-centered cubic (bcc) simulation cell. (a) Single-determinant Slater-Jastrow backflow (SJB) ground-state total energies per electron relative to FermiNet. The “gas” and “crystal” results were obtained using SJB wave functions built using determinants of plane waves and Gaussian orbitals, respectively. Error bars are smaller than the markers. FermiNet results for $r_s\le 1$ used complex wave functions. the FermiNet VMC method yields a variational improvement over the SJB VMC and SJB DMC method results in the gas phase and over the SJB VMC method results in the crystal phase. (b) One-electron density from the FermiNet VMC method at $r_s=10$ (left) and 70 (right), projected into the (011) plane of the conventional bcc structure. Four simulation cells are shown. Length scales are normalized by $r_s$, such that the apparent length scales are equivalent and crystal sites are superimposable. (c) Order parameter averaged over crystal axes for the bcc Wigner crystal state. Error bars are smaller than the markers. At small values of $r_s$, the order parameter is $\sim 0$, corresponding to a uniform one-electron density (gaslike); the order parameter rises sharply to a finite value at $r_s=2$, corresponding to the emergence of a crystalline state.