Criticality of the low-frequency conductivity for the bilayer quantum Heisenberg model

The criticality of the low-frequency conductivity for the bilayer quantum Heisenberg model was investigated numerically. The dynamical conductivity (associated with the O$(3)$ symmetry) displays the inductor $\sigma (\omega) =(i\omega L)^{-1}$andcapacitor$i \omega C$ behaviors for the ordered and disordered phases, respectively. Both constants, $C$ and $L$, have the same scaling dimension as that of the reciprocal paramagnetic gap ${\mathrm{\Delta }}^{-1}$. Then, there arose a question to fix the set of critical amplitude ratios among them. So far, the O$(2)$ case has been investigated in the context of the boson-vortex duality. In this paper, we employ the exact diagonalization method, which enables us to calculate the paramagnetic gap $\mathrm{\Delta }$ directly. Thereby, the set of critical amplitude ratios as to $C$, $L$ and $\mathrm{\Delta }$ are estimated with the finite-size-scaling analysis for the cluster with $N\le 34$ spins.