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arXiv:2308.09709 · Physical review. E

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

Jicheol Kim, D. Kim, DH Kim

Published Jun 10, 2024• 3 citations
arXivHTMLPDF

Abstract

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 α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{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 α_{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 α_{LR} decreases, although a precise breakdown threshold is yet to be determined.

DOI
10.1103/physreve.109.064123

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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 α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{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 α_{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 α_{LR} decreases, although a precise breakdown threshold is yet to be determined.
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We study the phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent α \\alpha , in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents α > 1 \\alpha > 1 up to small deviations in some critical exponents. We also address the elusive regime α < 1 \\alpha < 1 , where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at α = 0 \\alpha = 0 . Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
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