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∕r^1+σ 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∕Δ (Δ: paramagnetic gap in the disordered phase). As the power of the algebraic decay σ increases, the amplitude ratio m_H∕Δ gets enhanced gradually in agreement with the ϵ(=4−D)-expansion-renormalization-group result; here, we resort to the σ↔D relation advocated recently in order to establish a relationship between the renormalization-group result and ours.