  We use the functional renormalization group (FRG) to derive analytical
expressions for thermodynamic observables (density, pressure, entropy, and
compressibility) as well as for single-particle properties (wavefunction
renormalization and effective mass) of interacting bosons in two dimensions as
a function of temperature $T$ and chemical potential $\mu$. We focus on the
quantum disordered and the quantum critical regime close to the dilute Bose gas
quantum critical point. Our approach is based on a truncated vertex expansion
of the hierarchy of FRG flow equations and the decoupling of the two-body
contact interaction in the particle-particle channel using a suitable
Hubbard-Stratonovich transformation. Our analytic FRG results extend previous
analytical renormalization group calculations for thermodynamic observables at
$\mu =0$ to finite values of $\mu$. To confirm the validity of our FRG
approach, we have also performed quantum Monte Carlo simulations to obtain the
magnetization, the susceptibility, and the correlation length of the
two-dimensional spin-$1/2$ quantum $XY$ model with coupling $J$ in a regime
where its quantum critical behavior is controlled by the dilute Bose gas
quantum critical point. We find that our analytical results describe the Monte
Carlo data for $\mu \leq 0$ rather accurately up to relatively high
temperatures $T \lesssim 0.1 J$.
