A Short Introduction to Numerical Linked-Cluster Expansions

TL;DR

This paper introduces numerical linked-cluster expansions (NLCEs), explaining their algorithm, comparing results with exact diagonalization, and discussing methods to improve computational efficiency and convergence for studying quantum many-body systems.

Contribution

It provides a comprehensive pedagogical overview of NLCEs, including algorithmic details, symmetry considerations, and techniques for extending to the thermodynamic limit.

Findings

NLCEs accurately reproduce thermodynamic properties of the Heisenberg model.

Symmetry and topology considerations reduce computational cost.

Resummation techniques improve convergence of NLCE series.

Abstract

We provide a pedagogical introduction to numerical linked-cluster expansions (NLCEs). We sketch the algorithm for generic Hamiltonians that only connect nearest-neighbor sites in a finite cluster with open boundary conditions. We then compare results for a specific model, the Heisenberg model, in each order of the NLCE with the ones for the finite cluster calculated directly by means of full exact diagonalization. We discuss how to reduce the computational cost of the NLCE calculations by taking into account symmetries and topologies of the linked clusters. Finally, we generalize the algorithm to the thermodynamic limit, and discuss several numerical resummation techniques that can be used to accelerate the convergence of the series.