Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-lat/9604021 · International Journal of Modern Physics A

Linked Cluster Expansions Beyond Nearest Neighbor Interactions: Convergence and Graph Classes

Andreas Pordt, T. Reisz

Published Aug 20, 1997• 9 citations
arXivHTMLPDF

Abstract

We generalize the technique of linked cluster expansions on hypercubic lattices to actions that couple fields at lattice sites which are not nearest neighbors. We show that in this case the graphical expansion can be arranged in such a way that the classes of graphs to be considered are identical to those of the pure nearest neighbor interaction. The only change then concerns the computation of lattice imbedding numbers. All the complications that arise can be reduced to a generalization of the notion of free random walks, including hopping beyond nearest neighbor. Explicit expressions for combinatorical numbers of the latter are given. We show that under some general conditions the linked cluster expansion series have a nonvanishing radius of convergence.

DOI
10.1142/s0217751x97001924

Related papers

Linked cluster expansion on trees

Deepak Iyer, Yuyi Wan

The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.

arXiv:2303.07754•Physical review. E•Feb 9, 2023•1 citation

Linked cluster expansion and 1/d expansions for classical spin systems

Einar Halvorsen, M. Bartkowiak

We consider $1/d$ expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate $1/d$ expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in $1/d.$ The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in $1/d.$

Physical review. B, Condensed matter•Dec 11, 2000•8 citations

Computer Aided Series Expansions for Critical Phenomena

Hildegard Meyer‐Ortmanns, T. Reisz

Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics.

arXiv:hep-lat/9612022•Dec 20, 1996•0 citations

Renormalized finite-cluster method for lattice models. I. Site renormalization and star cluster expansions

William J. Shugard, John D. Weeks

We give a simple derivation of a finite cluster expansion for the free energy of a general inhomogeneous lattice system which uses only connected clusters. Two possible ways of defining these clusters are discussed in detail, one most useful for obtaining high-temperature series expansions and another low-temperature expansions. A renormalized finite cluster expansion which uses only multiply connected or star clusters follows from a Legendre transformation on the free energy in which single-site potentials are replaced as independent variables by single-site probability distribution functions. Certain special cases of these results have been known before and successfully used to derive high-temperature series expansions. The relationship of the present approach to the renormalized linked-cluster perturbation expansion used by Wortis and co-workers is discussed, and comparison is also made with the work of Rushbrook and Scoins. Truncations of the renormalized cluster expansion yield closed-form approximations which, except in the critical region, offer a systematic way of improving mean-field theory.

Physical review. B, Condensed matter•Dec 1, 1980•4 citations

Linked-cluster expansion around the atomic limit of the Hubbard model

Walter Metzner

We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.

Physical review. B, Condensed matter•Apr 1, 1991•239 citations

Variational localized-site cluster expansions. II. Trees and near trees

Douglas J. Klein, M. A. García‐Bach

Cluster expanded wavefunctions involving localized-site states and only pair excitations between nearest neighbor sites are considered. In the case that the bonds connecting these nearest neighbor sites form a tree graph (without rings), exact matrix element formulas are obtained in terms of the inverse of a modified type of topological matrix associated with this tree graph. The treatment of some near-tree cases, without any touching rings, is also discussed.

The Journal of Chemical Physics•Jun 15, 1976•8 citations
We generalize the technique of linked cluster expansions on hypercubic lattices to actions that couple fields at lattice sites which are not nearest neighbors. We show that in this case the graphical expansion can be arranged in such a way that the classes of graphs to be considered are identical to those of the pure nearest neighbor interaction. The only change then concerns the computation of lattice imbedding numbers. All the complications that arise can be reduced to a generalization of the notion of free random walks, including hopping beyond nearest neighbor. Explicit expressions for combinatorical numbers of the latter are given. We show that under some general conditions the linked cluster expansion series have a nonvanishing radius of convergence.
The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.
We consider 1/d1/d1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate 1/d1/d1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in 1/d.1/d.1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in 1/d.1/d.1/d.
Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics.
We give a simple derivation of a finite cluster expansion for the free energy of a general inhomogeneous lattice system which uses only connected clusters. Two possible ways of defining these clusters are discussed in detail, one most useful for obtaining high-temperature series expansions and another low-temperature expansions. A renormalized finite cluster expansion which uses only multiply connected or star clusters follows from a Legendre transformation on the free energy in which single-site potentials are replaced as independent variables by single-site probability distribution functions. Certain special cases of these results have been known before and successfully used to derive high-temperature series expansions. The relationship of the present approach to the renormalized linked-cluster perturbation expansion used by Wortis and co-workers is discussed, and comparison is also made with the work of Rushbrook and Scoins. Truncations of the renormalized cluster expansion yield closed-form approximations which, except in the critical region, offer a systematic way of improving mean-field theory.
We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.