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+σ 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 σ, the scaled bound-state mass m₂∕m₁ should take a universal constant (critical amplitude ratio) in the vicinity of the critical point. In this paper, we calculated the amplitude ratio m₂∕m₁ 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 m₂∕m₁ exhibits a non-monotonic dependence on σ; that is, the bound state is stabilized by an intermediate value of σ. Such a feature is accordant with a recent observation based on the non-perturbative-renormalization-group method.