Artificial Intelligence for Quantum Matter: Finding a Needle in a Haystack

TL;DR

A general and efficient method for learning the NN representation of an arbitrary many-body complex wave function from its N-particle probability density and probability current density and successfully test on (non-Abelian) fractional quantum Hall states and chiral BCS wavefunction.

Abstract

Neural networks (NNs) have great potential in solving the ground state of various many-body problems. However, several key challenges remain to be overcome before NNs can tackle problems and system sizes inaccessible with more established tools. Here, we present a general and efficient method for learning the NN representation of an arbitrary many-body complex wave function from its N-particle probability density and probability current density and successfully test on (non-Abelian) fractional quantum Hall states and chiral BCS wavefunction. Having reached overlaps as large as 99.9%, we employ our neural wave function for pre-training to effortlessly solve the fractional quantum Hall problem with Coulomb interactions and realistic Landau-level mixing for as many as 25 particles and uncover distinctive features of the edge. Our work demonstrates efficient, scalable and accurate simulation of highly-entangled quantum matter using general-purpose deep NNs enhanced with physics-informed initialization.

Figures (9)

• Figure 1: Fermionic neural network and VMC: Illustration of our fermionic attention-based architecture (left), and its role inside the NN variational Monte Carlo (right).
• Figure 2: Laughlin and MR wave-functions: Comparison of the wave functions for the Laughlin $(a)$ and MR $(b)$ state, Eqs. \ref{['eq:psiL']}-\ref{['eq:psiMR']}, with the output of the neural network ($2$ self-attention layers and $4$ determinants). These plots are obtained by keeping $N - 1$ particles at fixed positions (black/white dots), obtained from Monte Carlo sampling, and moving the remaining particle away from its "original" position (red dot) across the $2$D plane (positions in units of the droplet radius $R_{ L} = \sqrt{6N\ell_M^2}$$(a)$ and $R_{ M\!R} = \sqrt{4N\ell_M^2}$$(b)$, with $\ell_M^2 = \phi_0/2\pi H$ the magnetic length associated to the out of plane field $H$).
• Figure 3: Chiral BCS wave-function: Comparison of the reference wave function \ref{['eq:chiralBCS_odd']} (odd particle number) with the output of the neural network ($5$ self-attention layers and $4$ determinants). For evaluating $\psi_{ B\!C\!S}$, we set the chemical potential $\mu = 0.5 \times (k_F^2/2m)$ (Fermi momentum $k_F = \sqrt{4\pi N/ L^2}$) and the gap function to $\Delta_k = \mu (k/k_F)\, \exp\!\left[-(k/2 k_F)^2\right]$, using a soft UV cutoff to regularize the Fourier summation.
• Figure 4: FQH ground state: Spatial density profiles of the Laughlin droplet $(a)$ and FQH ground state for mixing parameter $\lambda = 1$$(b)$ ($N=25$ and positions expressed in units of $R_{ L}$). $(c)$ In the FQH ground state, Coulomb forces induce long-ranged oscillations of the density $\rho(r)$ that are slowly decaying on the scale of the system size, contrary to the exponential decay for the Laughlin state. $(d)$ Evolution of the variational energy (green) and "distance" from the Laughlin state (blue), as measured by $1 - |\braket{\psi_{ L}|\psi_{\theta}}|$. As the energy gradually decreases, the wave function $\psi_{\theta}$ diverges away from $\psi_{ L}$.
• Figure 5: Scaling analysis: Overlap with the Laughlin wave function as a function of the particle number, for three different architectures (dark blue, green and yellow). The light blue curve is for the Moore-Read state, and the purple for the chiral BCS state. The two different type of lines represent two training protocols: minimizing only loss function \ref{['eq:totalLoss']} (dashed); further improving the result by optimizing fidelity loss \ref{['eq:lossF']} (full).