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

Abstract

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 $\sigma (\omega) =(i\omega L)^{-1}$ and capacitor $i \omega C$ 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 $\Delta^{-1}$. 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 $\Delta$ directly. Thereby, the set of critical amplitude ratios as to $C$, $L$ and $\Delta$ are estimated with the finite-size-scaling analysis for the cluster with $N \le 34$ spins.