Machine learning to alleviate Hubbard-model sign problems

Lattice Monte Carlo calculations of interacting systems on nonbipartite lattices exhibit an oscillatory imaginary phase known as the phase or sign problem, even at zero chemical potential. One method to alleviate the sign problem is to analytically continue the integration region of the state variables into the complex plane via holomorphic flow equations. For asymptotically large flow times, the state variables approach manifolds of constant imaginary phase known as Lefschetz thimbles. However, flowing such variables and calculating the ensuing Jacobian is a computationally demanding procedure. In this paper, we demonstrate that neural networks can be trained to parametrize suitable manifolds for this class of sign problem and drastically reduce the computational cost for different severely afflicted small volume systems. In particular, we apply our method to the Hubbard model on the triangle and tetrahedron, both of which are nonbipartite. At strong interaction strengths and modest temperatures, the tetrahedron suffers from a severe sign problem that cannot be overcome with standard reweighting techniques, while it quickly yields to our method. We benchmark our results with exact calculations and comment on future directions of this work.

Figure 1

Distributions of configurations used for training data (blue-shaded region) and an ensemble produced via HMC with a neural network (orange-shaded region) trained on this data. The system is a triangle with $N_t=32, \kappa \beta =6$, and $U/\kappa =3$. The distributions were estimated from ${10}^5$ training and ${10}^6$ HMC configurations. The black crosses mark critical points of a space-time constant $\varphi$.

Figure 2

The entire spectrum of the Hubbard model on a triangle as a function of $U/\kappa$, changing from 0 to 5 in 20 equal steps, with colors corresponding to different $U/\kappa$, corresponding to the vertical direction in the color bar at right. One translucent point is plotted per state; more opaque circles correspond to more highly degenerate states, corresponding to the horizontal direction in the color bar. Energy eigenfunctions carry $Q, S, S_z$, and $L_z$ quantum numbers; each panel projects the five-dimensional space onto one quantum number and the energy. Each $U/\kappa$ is slightly offset, so that each point in the spectrum for $U/\kappa =0$ is to the left of that point's quantum numbers, while $U/\kappa =5$ is to the right. The ground-state manifold qualitatively changes at $U^{▵}/\kappa =3.61775...$, shown in a deep blue. We draw a thin horizontal line at the corresponding degenerate ground-state energy $E_0^{▵}=-1.36845...$ to make the quantum numbers of the states apparent, and thin vertical lines for each value of each quantum number at $U=U^{▵}$.

Figure 3

The absolute value and phase of (47) for $\kappa =1$ and $\mathrm{\Phi }=0$, shown as a function of two directions in field space orthogonal to the $\mathrm{\Phi }$ direction. In the left panel, yellow indicates large absolute values and purple small ones. In the right panel the color scheme is periodic; the exact zeros occur at the six points on each dark circle where the phase wraps around the point.

Figure 4

The histograms (left) of the phase for two different values of $U/\kappa$ as a function of $\beta$ and the corresponding statistical power (right) with bootstrap errors. In the right panel, different $N_t$ are shown from left to right: 16, 32, 48, and 64. The histograms (left) and statistical powers (right) are colored from red, a hot temperature, to blue, a cold temperature.

Figure 5

Left panel shows statistical power as a function of sample size on a triangle lattice with $N_t=64, U/\kappa =3, \kappa \beta =8$. Right panel shows statistical power for different parameters on a triangle lattice. Each point was estimated using ${10}^5$ configurations. The dashed line is at $\mathrm{\Sigma }=0$.

Figure 6

Single-particle correlators $\langle a{a}^{†}\left(\tau \right)\rangle$ on a triangle lattice with $N_t=64, U/\kappa =3$, and $\kappa \beta =8$. Each column shows correlators obtained from ensembles of the given number of configurations, while each row shows a different implementation of HMC. Colored points and areas show averages and error bands from HMC and black lines show exact results. The two unique correlators on a triangle are shown in blue and orange respectively—the former, low-energy correlator is the average of the doublet of two-point correlators of $O_{\pm 1}$ while the latter, high-energy correlator is a two-point correlator of the singlet $O_0$. Sample sizes list the total number of Monte Carlo configurations but correlators were measured only on every ${10}^{\mathrm{th}}$ configuration. The dotted lines are placed at $max\left(C\right)/\sqrt{\text{Sample}\text{size}}$ and indicate the scale at which even a sign-problem-free method would show sizable statistical fluctuations.

Figure 7

Single-particle correlators $\langle a{a}^{†}\left(\tau \right)\rangle$ on a triangle lattice for a sample size of ${10}^5$. The low energy correlators are averages of two degenerate results. The top panels show the correlators for different $N_t$ from HMC with a neural network and the results from exact diagonalization of the Hamiltonian. The lower panels show the relative error (48). Errors for the low and high energy correlators are labeled $E^{-}$ and $E^{+}$, respectively.

Figure 8

Correlation functions between two charge operators $\rho$ or two $S^3$ spin operators separated by euclidean time $\kappa \tau$ on a triangle for different discretization scales given by $N_t$ and the exact result (in black). The bottom panels show the relative error (48) for the constant correlator (0) and the average of two heavy correlators $(+)$ which have $k=0$ and $\pm 1$, respectively.

Figure 9

The order parameters $Q^2$ (left) and $S^2$ (right) calculated with the triangle system with various couplings $U$ and inverse temperatures $\beta$. Shown is a comparison between results calculated with the neural network and on the standard real plane. The dashed lines show results from exact diagonalization.

Figure 10

Left panel shows the statistical power as a function of sample size on a tetrahedron lattice with $N_t=64, U/\kappa =3$, and $\kappa \beta =6$. Right panel shows statistical power for different parameters on a triangle lattice. Each point was estimated using ${10}^6$ configurations. The dashed line is at $\mathrm{\Sigma }=0$.

Figure 11

Single-particle correlators $\langle a{a}^{†}\left(\tau \right)\rangle$ on a tetrahedron lattice with $N_t=64, U/\kappa =3, \kappa \beta =6$. Each column shows correlators obtained from ensembles of the given number of configurations, while each row shows a different implementation of HMC. Colored points and areas show averages and error bands from HMC and black lines show exact results. The two unique correlators on a tetrahedron are shown in blue and orange respectively—the former, low-energy correlator is the average of the three degenerate two-point correlators of the triplet of $O_1$ while the latter, high-energy correlator is a two-point correlator of the $O_0$ singlet. Correlators are computed only on every tenth configuration. The dotted lines are placed at $max\left(C\right)/\sqrt{\text{Sample}\text{size}}$ and indicate the scale at which even a sign-problem-free method would show sizable statistical fluctuations.

Figure 12

Single-particle correlators $\langle a{a}^{†}\rangle$ on a tetrahedron lattice. The low energy correlators are averages of three degenerate results. The top panels show the correlators for different $N_t$ from HMC with neural network and the results from exact diagonalization of the Hamiltonian. The lower panels show the relative error (48). Errors for the low and high energy correlators are labeled $E^{-}$ and $E^{+}$, respectively.

Figure 13

Correlation functions between two charge operators $\rho$ or two $S^3$ spin operators on a tetrahedron. The bottom panels show the relative error (48) for the constant correlator $E^0$ and the average of three heavy correlators $E^{+}$, respectively.

Figure 14

The order parameters $Q^2$ (left) and $S^2$ (right) calculated with the tetrahedron system with different couplings $U$ and inverse temperature $\beta$. Shown is a comparison between results calculated with the neural network and on the standard real plane. The dashed lines show results from exact diagonalization.

Figure 15

The statistical power $\mathrm{\Sigma }$ for the triangle, diamond, and tetrahedron, with and without a neural-network-enhanced HMC, with $U/\kappa =3$ and $N_t=64$, as a function of inverse temperature $\kappa \beta$. The histograms (left) and statistical powers (right) are colored from red, a hot temperature, to blue, a cold temperature.

Figure 16

Numerical results for some correlation functions on a triangle lattice with $U/\kappa =3, \kappa \beta =8$, and ${10}^5$ configurations.

Figure 17

Numerical results for some correlation functions on a tetrahedron lattice with $U/\kappa =3, \kappa \beta =8,$ and ${10}^5$ configurations.