Neural-network quantum-state study of the long-range antiferromagnetic Ising chain

We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range (LR) antiferromagnetic interactions using the variational Monte Carlo method with the restricted Boltzmann machine employed as a trial wave function ansatz. First, we measure the critical exponents and the central charge through the finite-size scaling analysis, verifying the contrasting observations in the previous tensor network studies. The correlation function exponent and the central charge deviate from the short-range (SR) Ising values at a small decay exponent ${\alpha }_{\mathrm{LR}}$, while the other critical exponents examined are very close to the SR Ising exponents regardless of ${\alpha }_{\mathrm{LR}}$ examined. However, in the further test of the critical Binder ratio, we find that the universal ratio of the SR limit does not hold for ${\alpha }_{\mathrm{LR}}<2$, implying a deviation in the criticality. On the other hand, we find evidence of the conformal invariance breakdown in the conformal field theory (CFT) test of the correlation function. The deviation from the CFT description becomes more pronounced as ${\alpha }_{\mathrm{LR}}$ decreases, although a precise breakdown threshold is yet to be determined.