Criticality of the magnon-bound-state hierarchy for the quantum Ising chain with the long-range interactions

The quantum Ising chain with the interaction decaying as a power law $1/r^{1+\sigma}$ of the distance between spins $r$ was investigated numerically. A particular attention was paid to the low-energy spectrum, namely, the single-magnon and two-magnon-bound-state masses, $m_{1,2}$, respectively, in the ordered phase. It is anticipated that for each $\sigma$, the scaled bound-state mass $m_2/m_1$ should take a universal constant (critical amplitude ratio) in the vicinity of the critical point. In this paper, we calculated the amplitude ratio $m_2/m_1$ with the exact diagonalization method, which yields the spectral information such as $m_{1,2}$ directly. As a result, we found that the scaled mass $m_2/m_1$ exhibits a non-monotonic dependence on $\sigma$; that is, the bound state is stabilized by an intermediate value of $\sigma$. Such a feature is accordant with a recent observation based on the non-perturbative-renormalization-group method.