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

TL;DR

This work uses variational Monte Carlo with a restricted Boltzmann machine neural-network quantum-state ansatz to study the long-range antiferromagnetic transverse-field Ising chain with algebraically decaying interactions $J_{ij}\propto |i-j|^{-\alpha_{\mathrm{LR}}}$. By combining finite-size scaling, Binder ratios, second Rényi entropy, and a conformal-field-theory (CFT) test of the correlation function, it shows that many critical exponents remain near the SR Ising values, but the correlation-function exponent $\eta$ and the central charge $c$ deviate as $\alpha_{\mathrm{LR}}$ decreases, and the universal Binder ratio also drifts away from SR expectations for $\alpha_{\mathrm{LR}}<2$. A CFT-based test reveals breakdown of conformal invariance more pronounced for smaller $\alpha_{\mathrm{LR}}$, suggesting LR interactions modify criticality beyond standard Ising universality and that the precise breakdown threshold may differ from related LR models. Overall, the study demonstrates the viability of neural-network quantum states to probe LR criticality and highlights the nuanced interplay between exponents, entanglement, and conformal symmetry in LR quantum systems.

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 $α_\mathrm{LR}$, while the other critical exponents examined are very close to the SR Ising exponents regardless of $α_\mathrm{LR}$ 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 $α_\mathrm{LR} < 2$, 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 $α_\mathrm{LR}$ decreases, although a precise breakdown threshold is yet to be determined.

Figures (7)

• Figure 1: Convergence test of the RBM wave function in the VMC search for the ground state. The case with the system of the size $L = 64$ for $\alpha_\mathrm{LR} = 0.5$ is shown for example. The estimates of (a) energy density $E_0 / L$ and (b) relative variance $(\langle \hat{H}^2 \rangle - \langle \hat{H} \rangle^2) / \langle \hat{H} \rangle^2$ measured after $2 \times 10^5$ iterations are plotted as a function of $n_h$. The insets show the same quantities for a fixed number of filters $n_h = 16$ monitored during the iterations of the parameter updates. The data points in the insets represent the averages measured in the logarithmic bins of iteration numbers. The error bars are measured with ten independent RBM wave function samples.
• Figure 2: FSS analysis of RBM observables. (a) The estimates of the critical exponents $\nu$, $\beta$, $\gamma$ are plotted in the range of $\alpha_\mathrm{LR}$ between $0.5$ and $3$. The dotted lines are given for comparison with the SR Ising values. The FSS collapse tests with the critical exponents are demonstrated at $\alpha_\mathrm{LR} = 0.5$ for the data of (b) the Binder's cumulant $U_4$, (c) the AF order parameter $m_s$, and (d) the susceptibility $\chi_s$. The inset of (b) shows the crossing point of $U_4$ locating the critical point $\theta_c$.
• Figure 3: Critical exponent of the spin-spin correlation function. The correlation function $C_{xx}(r)$ at $r=L/4$ is plotted as a function of the system size $L$. The inset shows the exponent $\eta$ extracted from the data fitting to $C_{xx}(L/4) \propto L^{-\eta}$. The dotted lines given for comparison indicate the SR Ising exponent $\eta = 1/4$.
• Figure 4: Estimate of the central charge. (a) The second Rényi entropy $S_2$ of a half chain is plotted at the critical point $\theta_c$ as a function of system size $L$. (b) The effective central charge $c_\mathrm{eff}$ (empty symbols) is plotted as a function of $1/L$. Solid symbols and their error bars indicate our estimate of the central charge $c_\infty$ and its standard error obtained from the linear fit (dotted lines) of the $c_\mathrm{eff}$ data. In (c), $c_\infty$ is plotted as a function of $\alpha_\mathrm{LR}$.
• Figure 5: Test of the critical Binder ratio. The self-combined ratio $S_\mathrm{SR}^*$ and the Binder ratio $Q^*$ at the critical point are plotted as a function of $\alpha_\mathrm{LR}$. The data points are extrapolated to infinite size. The horizontal solid lines indicate the SR limit.