# Long-Range Entangled-Plaquette States for Critical and Frustrated Quantum Systems on a Lattice

## TL;DR

This paper introduces a novel variational wave function called long-range entangled-plaquette state, capable of accurately modeling critical and frustrated quantum lattice systems by capturing long-range correlations and complex sign structures.

## Contribution

The authors develop and validate a flexible, scale-free variational Ansatz that efficiently describes long-range entanglement and frustration effects in quantum lattice models.

## Key findings

- Accurately reproduces ground state energies of XX and Heisenberg chains.
- Captures long-range properties like Luttinger exponent and central charge.
- Successfully models incommensurate correlations in frustrated systems.

## 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 non positive definite states in a fully variational fashion, namely without any additional bias (arising \emph{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 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 $J_1-J_2$ 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.