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arXiv:1904.11406 · Physical review. B./Physical review. B

Long-range entangled-plaquette states for critical and frustrated quantum systems on a lattice

Jérôme Thibaut, Tommaso Roscilde, Fabio Mezzacapo

Published Oct 30, 2019• 21 citations
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Abstract

We explore a variational Ansatz for lattice quantum systems---named long-range entangled-plaquette state---in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of correlations built in this wave function makes it fit to reproduce critical states with long-range entanglement. The use of complex weights in the Ansatz allows for an efficient optimization of nonpositive definite states in a fully variational fashion, namely without any additional bias (arising, e.g., from pre-imposed sign structures) beyond that imposed by the parametrization of the state coefficients. These two features render the Ansatz particularly appropriate for the study of quantum phase transitions in frustrated systems. Moreover, the Ansatz can be systematically improved by increasing the long-range plaquette size, as well as by the inclusion of even larger adjacent-site plaquettes. We validate our Ansatz and its optimization procedure in the case of the XX and Heisenberg chain, and further apply it to the case of a simple, yet paradigmatic model of frustration, namely the ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg chain. For this model we provide clear evidence that our trial wave function faithfully describes both the short-range physics (particularly in terms of ground-state energy) and the long-range one expressed by the Luttinger exponent, and the central charge of the related conformal field theory, which govern the decay of correlations and the scaling of the entanglement entropy, respectively. Finally we successfully reproduce the incommensurate correlations developing in the system at strong frustration, as a result of the flexible representation of sign (phase) structures via complex weights.

DOI
10.1103/physrevb.100.155148

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We explore a variational Ansatz for lattice quantum systems---named long-range entangled-plaquette state---in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of correlations built in this wave function makes it fit to reproduce critical states with long-range entanglement. The use of complex weights in the Ansatz allows for an efficient optimization of nonpositive definite states in a fully variational fashion, namely without any additional bias (arising, e.g., from pre-imposed sign structures) beyond that imposed by the parametrization of the state coefficients. These two features render the Ansatz particularly appropriate for the study of quantum phase transitions in frustrated systems. Moreover, the Ansatz can be systematically improved by increasing the long-range plaquette size, as well as by the inclusion of even larger adjacent-site plaquettes. We validate our Ansatz and its optimization procedure in the case of the XX and Heisenberg chain, and further apply it to the case of a simple, yet paradigmatic model of frustration, namely the J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ antiferromagnetic Heisenberg chain. For this model we provide clear evidence that our trial wave function faithfully describes both the short-range physics (particularly in terms of ground-state energy) and the long-range one expressed by the Luttinger exponent, and the central charge of the related conformal field theory, which govern the decay of correlations and the scaling of the entanglement entropy, respectively. Finally we successfully reproduce the incommensurate correlations developing in the system at strong frustration, as a result of the flexible representation of sign (phase) structures via complex weights.
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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 J2/J1=0.5{J}_{2}/{J}_{1}=0.5J2​/J1​=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 J2/J1=0.5{J}_{2}/{J}_{1}=0.5J2​/J1​=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.
We make extensive simulations over a spin chain model that combines the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ spin chain and the long-range nonfrustrated ${(\ensuremath{-}1)}^{(r\ensuremath{-}1)}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ decay interactions through the variational matrix product state method for both finite and infinite lengths. We study both the ground-state entanglement and phase diagram. We find that it is most entangled in the rotation invariant long-range ordered antiferromagnetic phase, where the entanglement scales approximately logarithmically. We determine the development of the Majudar-Ghosh point to a disorder line from entanglement. And we approximately determine the transition from the dimerized and incommensurate phase of the ${J}_{1}\ensuremath{-}{J}_{2}$ model to a decoupled phase by studying spin correlation and the dimerization order parameter. Some implications for entanglement in systems with long-range interactions are stated.
We study the suitability of correlator product states for describing ground-state properties of two-dimensional spin models. Our ansatz for the many-body wave function takes the form of either plaquette or bond correlator product states, and the energy is optimized by varying the correlators using Monte Carlo minimization. For the Ising model, we find that plaquette correlators are best for estimating the energy, while bond correlators capture the expected long range correlations and critical behavior of the system more faithfully. For the antiferromagnetic Heisenberg model, however, plaquettes outperform bond correlators at describing both local and long range correlations because of the substantially larger number of local parameters they contain. These observations have quantitative implications for the application of correlator product states to other more complex systems, and they give important heuristic insights, in particular, the necessity of carefully tailoring the choice of correlators to the system considered, and its interactions and symmetries.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.