Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states

First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow-inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared with exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge-ordered phases as a function of the interaction strength and the particle density.

Figure 1

Diagram of the wave-function ansatz as defined in Eq. (20). From left to right: square lattice of side length $L=10$ with 20 particles, occupation map in the lattice that the sign and amplitude networks take as an input. Top and bottom convolutional networks are amplitude and sign factors, respectively. The number of convolutional filters is indicated above each layer.

Figure 2

Benchmark of the proposed variational ansatz (VMC) and Hartree-Fock approximation (HF) using the exact ED states. (a), (b) Color-maps show the relative error of the energy in the square lattice of size (a) $L=4$ and in the square lattice of size (b) $L=6$. Relative error is shown at different fillings of the lattice and values of the coupling constant. Lines separate regions of the phase diagram with different orders of magnitude of the relative error as indicated. (c) Shows the error of the proposed ansatz in panels (a) and (b), relative to the error of the Hartree-Fock approximation, in the square lattice of size $L=4$ and $L=6$.

Figure 3

Benchmark of the proposed variational ansatz (VMC) using ED eigenpairs in the square lattice of side length $L=6$. Two-point density correlation functions [Eq. (5)] are shown as a function of graph distance $r$ at different values of $V/t$: $V/t=0.01, V/t=0.599, V/t=2.15$, and $V/t=5.99$, as shown by the color scale in the top of the panel. Solid lines are the correlations from ED and the dots correspond to the correlations computed with our approach. Different panels correspond to different fillings, as indicated in each panel. The results obtained using ED and neural networks are visually indistinguishable.

Figure 4

Two-point density correlation function defined in Eq. (5) and the moduli of their renormalized Fourier modes defined in Eq. (6). Different colors indicate different values of $V/t$, as indicated in the legend in the top of the plot. (a) Two-point density correlation functions as a function of the graph distance $r$. System size is $L=10$. Dots represent the obtained values for the correlations and black solid lines are for visual guidance. Each panel corresponds to a different filling as indicated. (b) Modulus of the Fourier transform of the correlation functions displayed in the corresponding (a) panels, as a function of $k$. Note that $|{\tilde{C}}_{\mathrm{ren}}\left(k\right)|$ is only shown for $k\in [0,\pi ]$ as $|{\tilde{C}}_{\mathrm{ren}}\left(k\right)|$ is symmetrical with respect to $k=\pi$. (c) Amplitude peak in $|{\tilde{C}}_{\mathrm{ren}}\left(k\right)|$ shown in panel (a) as a function of the lattice size. For different system sizes the peak is identified as the maximum of $|{\tilde{C}}_{\mathrm{ren}}\left(k\right)|$ at the largest value of $V/t$ considered. Different panels correspond to different lattice fillings, as indicated.

Figure 5

Order parameter as defined in Eq. (3) as a function of the coupling constant. Each panel shows the order parameter in system sizes $L=6, L=8$, and $L=10$, as indicated in the legend with different colors. Lines connecting dots are for visual guidance. Different panels correspond to different fillings as indicated.

Figure 6

Phase diagram of the two-dimensional interacting spinless fermion model under consideration. The Hartree-Fock and IPEPS transition lines are from Ref. [30]. The orange dots correspond to the transition points from the finite-size scaling of $o\left[\rho \right]$, using the proposed variational ansatz. Lines connecting the dots are for visual guidance. The color-map represents the CDW order parameter $o\left[\rho \right]$ in the largest system size studied ($L=10$) at different values of the particle density and interaction strength.