  Recently significant progress has been made in $(2+1)$-dimensional conformal
field theories without supersymmetry. In particular, it was realized that
different Lagrangians may be related by hidden dualities, i.e., seemingly
different field theories may actually be identical in the infrared limit. Among
all the proposed dualities, one has attracted particular interest in the field
of strongly-correlated quantum-matter systems: the one relating the easy-plane
noncompact CP$^1$ model (NCCP$^1$) and noncompact quantum electrodynamics (QED)
with two flavors ($N = 2$) of massless two-component Dirac fermions. The
easy-plane NCCP$^1$ model is the field theory of the putative deconfined
quantum-critical point separating a planar (XY) antiferromagnet and a dimerized
(valence-bond solid) ground state, while $N=2$ noncompact QED is the theory for
the transition between a bosonic symmetry-protected topological phase and a
trivial Mott insulator. In this work we present strong numerical support for
the proposed duality. We realize the $N=2$ noncompact QED at a critical point
of an interacting fermion model on the bilayer honeycomb lattice and study it
using determinant quantum Monte Carlo (QMC) simulations. Using stochastic
series expansion QMC, we study a planar version of the $S=1/2$ $J$-$Q$ spin
Hamiltonian (a quantum XY-model with additional multi-spin couplings) and show
that it hosts a continuous transition between the XY magnet and the
valence-bond solid. The duality between the two systems, following from a
mapping of their phase diagrams extending from their respective critical
points, is supported by the good agreement between the critical exponents
according to the proposed duality relationships.
