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

TL;DR

This paper numerically investigates the critical behavior of low-frequency conductivity in the bilayer quantum Heisenberg model, revealing scaling relations and critical amplitude ratios using exact diagonalization.

Contribution

It provides the first direct calculation of critical amplitude ratios among conductivity constants and the paramagnetic gap in the bilayer quantum Heisenberg model.

Findings

Conductivity exhibits inductor and capacitor behaviors in ordered and disordered phases.

Critical amplitude ratios among C, L, and Δ are estimated via finite-size scaling.

The study employs exact diagonalization for N ≤ 34 spins to analyze criticality.

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.