Neural-network quantum state study of the long-range antiferromagnetic Ising chain
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We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range antiferromagnetic interactions by using the variational Monte Carlo method with the restricted Boltzmann machine being employed as a trial wave function ansatz. In the finite-size scaling analysis with the order parameter and the second Rényi entropy, we find that the central charge deviates from 1/2 at a small decay exponent α_LR in contrast to the critical exponents staying very close to the short-range (SR) Ising values regardless of α_LR examined, supporting the previously proposed scenario of conformal invariance breakdown. To identify the threshold of the Ising universality and the conformal symmetry, we perform two additional tests for the universal Binder ratio and the conformal field theory (CFT) description of the correlation function. It turns out that both indicate a noticeable deviation from the SR Ising class at α_LR < 2. However, a closer look at the scaled correlation function for α_LR≥ 2 shows a gradual change from the asymptotic line of the CFT verified at α_LR = 3, providing a rough estimate of the threshold being in the range of 2 ≲α_LR < 3.
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