  Numerically simulating spinful, fermionic systems is of great interest from
the perspective of condensed matter physics. However, the exponential growth of
the Hilbert space dimension with system size renders an exact parameterization
of large quantum systems prohibitively demanding. This is a perfect playground
for neural networks, owing to their immense representative power that often
allows to use only a fraction of the parameters that are needed in exact
methods. Here, we investigate the ability of neural quantum states (NQS) to
represent the bosonic and fermionic $t-J$ model - the high interaction limit of
the Fermi-Hubbard model - on different 1D and 2D lattices. Using autoregressive
recurrent neural networks (RNNs) with 2D tensorized gated recurrent units, we
study the ground state representations upon doping the half-filled system with
holes. Moreover, we present a method to calculate dispersion relations from the
neural network state representation, applicable to any neural network
architecture and any lattice geometry, that allows to infer the low-energy
physics from the NQS. To demonstrate our approach, we calculate the dispersion
of a single hole in the $t-J$ model on different 1D and 2D square and
triangular lattices. Furthermore, we analyze the strengths and weaknesses of
the RNN approach for fermionic systems, pointing the way for an accurate and
efficient parameterization of fermionic quantum systems using neural quantum
states.
