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arXiv:2002.09246 · Physical review. B./Physical review. B

Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems

Tanja Đurić, Tomislav Ševa

Published Aug 3, 2020• 9 citations
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Abstract

We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural-network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates, which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor network or artificial neural-network ansatz wave functions then allows further insight into quantum phases and phase transitions in various strongly correlated models by considering properties of these systems beyond the ground-state properties. Also, the method is in general applicable to any dimension and has no sign instability. An example that we consider is the square lattice ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg model. The model is one of the most studied models in frustrated quantum magnetism since it is closely related to the disappearance of the antiferromagnetic order in the high-Tc superconducting materials and there is still no agreement about the properties of the system in the highly frustrated regime near ${J}_{2}/{J}_{1}=0.5$. For the ${J}_{1}\ensuremath{-}{J}_{2}$ model, we write the variational ansatz in terms of the two site correlators and in the valence bond basis and calculate the lowest energy eigenstates in the highly frustrated regime near ${J}_{2}/{J}_{1}=0.5$ where the system has a paramagnetic phase. We find that our results are in good agreement with previously obtained results, which confirms applicability of the method to study frustrated spin systems.

DOI
10.1103/physrevb.102.085104

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The frustrated Heisenberg ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on a square lattice is numerically investigated by variational Monte Carlo simulations. We propose an antiferromagnetic fermion resonating-valence-bond (AF-fRVB) state that has the ability to examine the entire phase diagram in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. Two phase transition points, the second order around J2/J1=0.45{J}_{2}/{J}_{1}=0.45J2​/J1​=0.45 and the first order around J2/J1=0.6{J}_{2}/{J}_{1}=0.6J2​/J1​=0.6, can be extracted more clearly than the conventional bosonic RVB state. At the maximally frustrated point (J2/J1=0.5)({J}_{2}/{J}_{1}=0.5)(J2​/J1​=0.5), the AF-fRVB state shows the variational ground-state energy in the thermodynamic limit very close to the one estimated by the projected entangled pair state at the largest bond dimension available. On the other hand, in the frustrated regime $0.4\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, AF-fRVB states with exts2\text{ext}\phantom{\rule{0.16em}{0ex}}{s}_{2}exts2​ (using the terminology in the field of iron-based superconductors) and dxy{d}_{xy}dxy​ pairing symmetries are degenerate in the thermodynamic limit, implying the existence of gapless Dirac excitations in the spinon spectrum.
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