A neural-network quantum state study of the long-range Ising chain

Employing the restricted Boltzmann machine as a ground-state wave function ansatz, we explore the critical line of the antiferromagnetic transverse-field Ising chain with a power-law decaying long-range (LR) coupling. Performing extensive variational Monte Carlo calculations with the aid of GPU, we examine how the LR coupling alters the characterization of the criticality. We first verify that the critical exponents of the standard observables are very close to the values of the 2D Ising universality class, regardless of the exponent of the LR coupling, which corroborates some of the previous numerical estimates. However, we argue that below a certain threshold of the LR exponent, the conformal invariance becomes broken at the critical point by providing numerical evidence from the test of the CFT description of the spin-spin correlation function. We also briefly discuss how it differs from the ferromagnetic counterpart.