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

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