Chebyshev expansion of spectral functions using restricted Boltzmann machines

Calculating the spectral function of two dimensional systems is arguably one of the most pressingchallenges in modern computational condensed matter physics. While efficient techniques are avail-able in lower dimensions, two dimensional systems present insurmountable hurdles, ranging fromthe sign problem in quantum Monte Carlo, to the entanglement area law in tensor network basedmethods. We hereby present a variational approach based on a Chebyshev expansion of the spectralfunction and a neural network representation for the wave functions. The Chebyshev moments areobtained by recursively applying the Hamiltonian and projecting on the space of variational statesusing a modified natural gradient descent method. We compare this approach with a modified oneexpanding on the Kyrlov subspace using Chebyshev polynomials. We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, both with and withoutfrustration, and compare our results with those obtained by other methods in the literature.