Two-dimensional frustrated $J_1\text{−}{J}_2$ model studied with neural network quantum states

The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ansätz, we study the frustrated spin-1/$2J_1\text{−}{J}_2$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ansätz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.

Figure 1

Network architecture. (a) Shape of the filter which we apply across the square lattice. (b) Full architecture of the convolutional neural network used in this work. There are six convolutional layers followed by an output layer which simply sums the values of the prenultimate layer. In the first layer, we use the logarithm of the hyperbolic cosine as the activation function $g_{\mathrm{lncosh}}\left(z\right)=log[cosh\left(z\right)]$, while in all other layers the activation function is given by the complex generalization of the ReLU [51, 52]. The total number of complex-valued parameters in the network is 3838.

Figure 2

Simulation results on the $6\times 6$ square lattice with periodic boundary conditions. (a) Energy comparison with ED results from Ref. [47]. The CNN energies are indicated by the blue line while the black stars gives the exact values. The relative errors are plotted with respect to the right axis. The blue dashed line shows the relative error for our CNN ansätz and the green dashed line corresponds the the Gutzwiller-projected mean-field fermionic variational ansätz from Ref. [40]. (b) Spin-spin structure factor as defined in Eq. (14). (c) The total spin $\langle {\widehat{\mathit{S}}}^2\rangle$, an extensive quantity. In the exact case, this value should be zero since the ground state is in the singlet sector.

Figure 3

Simulation results on the $10\times 10$ square lattice with periodic boundary conditions. (a) Energy comparison with other state-of-the-art methods. We plot the energy difference per site with respect to the energy obtained from our CNN variational ansätz. The red open circles show the values for the DMRG results of Ref. [41] and the green triangles correspond to the the Gutzwiller-projected mean-field fermionic variational ansätz. The black star shows the exact result from Green's function quantum Monte Carlo for the sign-problem free point $\left(J_2=0\right)$. The red stars show the results of a restricted Boltzmann machine with 3600 hidden units (3636 parameters). (b) Spin-spin structure factor as defined in Eq. (14). (c) The total spin $\langle {\widehat{\mathit{S}}}^2\rangle$. In the exact case, this value should be zero since the ground state is in the singlet sector.