  The spin-1/2 Heisenberg model on the pyrochlore lattice is an iconic
frustrated three-dimensional spin system with a rich phase diagram. Besides
hosting several ordered phases, the model is debated to possess a spin-liquid
ground state when only nearest-neighbor antiferromagnetic interactions are
present. Here, we contest this hypothesis with an extensive numerical
investigation using both exact diagonalization and complementary variational
techniques. Specifically, we employ a RVB-like many-variable Monte Carlo ansatz
and convolutional neural network quantum states for (variational) calculations
with up to $4\times 4^3$ and $4 \times 3^3$ spins, respectively. We demonstrate
that these techniques yield consistent results, allowing for reliable
extrapolations to the thermodynamic limit. Our main results are (1) the
determination of the phase transition between the putative spin-liquid phase
and the neighboring magnetically ordered phase and (2) a careful
characterization of the ground state in terms of symmetry-breaking tendencies.
We find clear indications of spontaneously broken inversion and rotational
symmetry, calling the scenario of a featureless quantum spin-liquid into
question. Our work showcases how many-variable variational techniques can be
used to make progress in answering challenging questions about
three-dimensional frustrated quantum magnets.

Strongly Correlated Electrons;Disordered Systems and Neural Networks