Fig. 1: Schematic diagrams near the quantum critical point.
From: Decay and renormalization of a longitudinal mode in a quasi-two-dimensional antiferromagnet
The schematic phase diagram illustrates the O(2) quantum critical point (QCP) between the antiferromagnetic (AFM) state and the quantum paramagnet (QPM) as a function of \({{{{{\rm{\alpha }}}}}}=\widetilde{J}/\widetilde{D}\) (\(\widetilde{J}\) is a Heisenberg exchange and \(\widetilde{D}\) is a easy-plane single-ion anisotropy of effective \(S\) = 1). The low-energy excitations of the QPM are two degenerate \({S}^{z}\)=\({\pm}\!1\) modes (black line) with a gap, \(\Delta\), which closes at the QCP. The spontaneous U(1) symmetry breaking leads to a gapless magnon or transverse mode (\(T\)-mode), indicated with a blue line, which is accompanied by a gapped longitudinal mode (\(L\)-mode) indicated with the orange line. Near the QCP, the energy and the lifetime of the \(L\)-mode are strongly renormalized (dashed orange line) due to the decay into the continuum of two transverse modes (shaded orange region).
