Multiplicative logarithmic corrections to quantum criticality in three-dimensional dimerized antiferromagnets

We investigate the quantum phase transition in an $S = 1/2$ dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks O($N$) symmetry with $N = 3$ in $D = 3 + 1$ dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization, $m_s$, and the N\'eel temperature, $T_N$. We present a scaling Ansatz for $T_N$, including logarithmic corrections, which agrees with our data and indicates exact linearity with $m_s$, implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted $N$-independent leading and subleading logarithmic corrections in the size-dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as TlCuCl$_3$.