Nonlinear Network Description for Many-Body Quantum Systems in Continuous Space

We show that the recently introduced iterative backflow wave function can be interpreted as a general neural network in continuum space with nonlinear functions in the hidden units. Using this wave function in variational Monte Carlo simulations of liquid ${}^4\mathrm{He}$ in two and three dimensions, we typically find a tenfold increase in accuracy over currently used wave functions. Furthermore, subsequent stages of the iteration procedure define a set of increasingly good wave functions, each with its own variational energy and variance of the local energy: extrapolation to zero variance gives energies in close agreement with the exact values. For two dimensional ${}^4\mathrm{He}$, we also show that the iterative backflow wave function can describe both the liquid and the solid phase with the same functional form—a feature shared with the shadow wave function, but now joined by much higher accuracy. We also achieve significant progress for liquid ${}^3\mathrm{He}$ in three dimensions, improving previous variational and fixed-node energies.

Figure 1

(a) Schematic representation of a SWF as a nonlinear network. The input layer is formed by the coordinates of the wave function, $\mathbf{R}$, and we have to integrate over the coordinates in the hidden layer, ${\mathbf{R}}^{\prime }$. Input and hidden layer coordinates are connected via a Gaussian, whereas the coordinates inside each layer are connected by the many-body correlation potentials, $V(\mathbf{R})$ and $U({\mathbf{R}}^{\prime })$. Including several hidden layers correspond to the application over several projection steps. (b) Structure of the iterated backflow wave function obtained after approximated integration over the hidden layers of SWF and projector Monte Carlo wave functions. Each layer introduces a new set of nonlinear functions $U^{(n)}$ (here, two- and three-body Jastrow forms, $U_2^{(n)}$, $U_3^{(n)}$) and backflow coordinates ${\mathbf{Q}}^{(n)}$ which depend only on the coordinates of the previous layers ${\mathbf{Q}}^{(m<n)}$.

Figure 2

Difference between the variational energy and the exact value in units of the kinetic energy $T$ for increasing number of hidden layers $M$ of our nonlinear network function for liquid ${}^4\mathrm{He}$ in three dimensions starting from a Jastrow wave function (J) with two- and three-particle correlations, $M=0$.

Figure 3

Pair correlation functions $g(r)$ for liquid ${}^4\mathrm{He}$ in three dimensions at equilibrium density. Dashed lines (left scale) show variational results without backflow terms (blue) and with three backflow iterations (red), as well as DMC results (black, barely visible behind the red dashes; extrapolated estimate [16] using the BF3 trial wave function). Solid lines (right scale) show a tenfold magnification of the deviation between the VMC and DMC results.

Figure 4

Error of the ground state energy of liquid and solid ${}^4\mathrm{He}$ in two dimension in units of the kinetic energy $T$ across the liquid-solid coexistence region (shaded) [36], using various wave functions: Shadow [37], Jastrow, Nosanow, and iterative backflow network (5 iteration layers at coexistence, 4 layers otherwise).