A Short Introduction to Numerical Linked-Cluster Expansions

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.

💡 Research Summary

This paper offers a pedagogical introduction to Numerical Linked‑Cluster Expansions (NLCEs), a powerful numerical technique for obtaining thermodynamic properties of quantum many‑body systems directly in the thermodynamic limit. The authors begin by outlining the conceptual foundation of NLCE: the observable of an infinite lattice can be expressed as a sum over contributions (weights) of all connected clusters that can be embedded in the lattice. Each cluster’s weight is defined by the inclusion‑exclusion principle – the observable computed on the cluster minus the contributions of all its subclusters. Because each cluster is finite, its properties can be obtained exactly by full exact diagonalization (ED).

The algorithmic section focuses on Hamiltonians that involve only nearest‑neighbor interactions and on clusters with open boundary conditions. The procedure for generating clusters proceeds in three steps: (i) enumerate all possible site subsets up to a given size, (ii) test each subset for connectivity, and (iii) eliminate duplicate clusters by identifying topologically equivalent graphs. The authors emphasize the use of graph‑isomorphism checks to recognize topological equivalence, which dramatically reduces the number of distinct clusters that must be diagonalized.

Once the cluster list is built, each cluster is diagonalized to obtain the spectrum and thermodynamic quantities such as internal energy, specific heat, and magnetization. The weight of a cluster is then computed by subtracting the weights of all its subclusters. Summing the weights of all clusters up to order N yields the NLCE estimate of the infinite‑system observable at that order.

To illustrate the method, the paper applies NLCE to the spin‑½ Heisenberg model on one‑ and two‑dimensional lattices. Results for orders 1 through 6 are compared with direct ED on finite clusters of the same size. The comparison shows rapid convergence of the NLCE series: even at relatively low order the NLCE reproduces the exact thermodynamic curves across a wide temperature range, outperforming conventional high‑temperature expansions especially at low temperatures.

A substantial portion of the manuscript is devoted to practical cost‑reduction strategies. By exploiting point‑group symmetries and spin‑rotation invariance, clusters that are symmetry‑related can be grouped, allowing a single diagonalization to serve multiple embeddings. Moreover, the topological reduction described earlier typically cuts the total number of clusters by 30 % or more, leading to sizable savings in memory and CPU time.

The authors then discuss how to extend the formalism to the true thermodynamic limit. Instead of treating each finite cluster as a separate system, they introduce the concept of “infinite clusters,” which automatically accounts for the translational embedding multiplicities and eliminates double‑counting. This formulation enables a clean mapping from the finite‑order NLCE coefficients to the infinite‑lattice observable without the need for extrapolation from finite‑size data.

Because NLCE series can converge slowly near critical points or at very low temperatures, the paper reviews several resummation techniques designed to accelerate convergence. Padé approximants, Dlog‑Padé analysis, Borel transforms, and a beta‑smoothing scheme are implemented and benchmarked on the Heisenberg data. Padé and Dlog‑Padé are found to be particularly effective in extending the usable temperature range and in providing reliable estimates of critical behavior.

In the concluding section, the authors summarize the advantages of NLCE—exact treatment of quantum correlations within each cluster, systematic inclusion‑exclusion bookkeeping, and the ability to reach the thermodynamic limit without finite‑size scaling. They argue that the combination of symmetry reduction, topological classification, infinite‑cluster formulation, and advanced resummation makes NLCE a versatile tool for a broad class of problems, including systems with longer‑range interactions, disordered lattices, and dynamical response functions. The paper thus serves as both a tutorial and a reference for researchers seeking to implement NLCE in their own studies of strongly correlated quantum matter.