  Due to their immense representative power, neural network quantum states
(NQS) have gained significant interest in current research. In recent advances
in the field of NQS, it has been demonstrated that this approach can compete
with state-of-the-art numerical techniques, making NQS a compelling
alternative, in particular for the simulation of large, two-dimensional quantum
systems. In this study, we show that recurrent neural network (RNN) wave
functions can be employed to study systems relevant to current research in
quantum many-body physics. Specifically, we employ a 2D tensorized gated RNN to
explore the bosonic Hofstadter model with a variable local Hilbert space
cut-off and long-range interactions. At first, we benchmark the RNN-NQS for the
Hofstadter-Bose-Hubbard (HBH) Hamiltonian on a square lattice. We find that
this method is, despite the complexity of the wave function, capable of
efficiently identifying and representing most ground state properties.
Afterwards, we apply the method to an even more challenging model for current
methods, namely the Hofstadter model with long-range interactions. This model
describes Rydberg-dressed atoms on a lattice subject to a synthetic magnetic
field. We study systems of size up to $12 \times 12$ sites and identify three
different regimes by tuning the interaction range and the filling fraction
$\nu$. In addition to phases known from the HBH model at short-ranged
interaction, we observe bubble crystals and Wigner crystals for long-ranged
interactions. Especially interesting is the evidence of a bubble crystal phase
on a lattice, as this gives experiments a starting point for the search of
clustered liquid phases, possibly hosting non-Abelian anyon excitations. In our
work we show that NQS are an efficient and reliable simulation method for
quantum systems, which are the subject of current research.
