Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis

TL;DR

This paper introduces an efficient loop update method for quantum Monte Carlo simulations in the valence-bond basis, significantly reducing computational effort and enabling the study of larger systems.

Contribution

It combines loop updates with ground-state projector and variational schemes in the valence bond basis, improving efficiency from O(N^2) to O(N) and from O(m^2) to O(m).

Findings

Successfully calculated sublattice magnetization for large lattices.

Achieved high-precision results extrapolated to the thermodynamic limit.

Demonstrated improved computational efficiency enabling larger system studies.

Abstract

We show how efficient loop updates, originally developed for Monte Carlo simulations of quantum spin systems at finite temperature, can be combined with a ground-state projector scheme and variational calculations in the valence bond basis. The methods are formulated in a combined space of spin z-components and valence bonds. Compared to schemes formulated purely in the valence bond basis, the computational effort is reduced from up to O(N^2) to O(N) for variational calculations, where N is the system size, and from O(m^2) to O(m) for projector simulations, where m>> N is the projection power. These improvements enable access to ground states of significantly larger lattices than previously. We demonstrate the efficiency of the approach by calculating the sublattice magnetization M_s of the two-dimensional Heisenberg model to high precision, using systems with up to 256*256 spins. Extrapolating the results to the thermodynamic limit gives M_s=0.30743(1). We also discuss optimized variational amplitude-product states, which were used as trial states in the projector simulations, and compare results of projecting different types of trial states.