Criticality of the Higgs mass for the long-range quantum $XY$ chain: Amplitude ratio between the Higgs and paramagnetic gaps

The quantum $XY$ spin chain with the interactions decaying as a power law $1\mathrm{/}{r}^{1+\sigma }$ of the distance between spins $r$ was investigated with the exact diagonalization method. Here, the constituent spin is set to $S=1$, which enables us to incorporate the biquadratic interactions so as to realize the order-disorder transition with the O$(2)$ symmetry maintained. Thereby, in the ordered phase, we resolved the Higgs mass $m_H$ out of the Goldstone-excitation continuum by specifying Higgs-particle’s quantum numbers to adequate indices. We then turn to the analysis of the critical amplitude ratio $m_H\mathrm{/}\mathrm{\Delta }$ ($\mathrm{\Delta }$: paramagnetic gap in the disordered phase). As the power of the algebraic decay $\sigma$ increases, the amplitude ratio $m_H\mathrm{/}\mathrm{\Delta }$ gets enhanced gradually in agreement with the $ϵ(=4-D)$-expansion-renormalization-group result; here, we resort to the $\sigma \leftrightarrow D$ relation advocated recently in order to establish a relationship between the renormalization-group result and ours.