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

Physical review. B, Condensed matter

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

William J. Shugard, John D. Weeks

Published Dec 1, 1980• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.22.5245

Related papers

Renormalized finite-cluster expansions

Sherwin J. Singer, John D. Weeks

There are many statistical systems in which degrees of freedom can be grouped according to their degree of correlation. We develop a cluster-expansion method for these systems based on selecting finite clusters of the most strongly interacting degrees of freedom. The starting point is Kubo's generalized cumulant formalism. We show how this expansion can be renormalized; that is, how the contributions of infinite classes of articulated finite clusters can be resummed. The elimination of articulated clusters from the renormalized expansion generates modified interactions in the finite clusters remaining in the expansion. When truncated at different orders, the final result is a hierarchy of mean-field approximations, which generally reproduce an increasing number of terms in exact series expansions for the system. High-order series-expansion coefficients can be obtained by perturbative renormalization of very large clusters, a convenient alternative to the complicated combinatorial analysis associated with standard expansion techniques for lattice models. We compare our work to previous finite-cluster techniques, such as the cluster-variation method of Kikuchi.

Physical review. B, Condensed matter•Aug 1, 1987•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

Some Algebraic Techniques for obtaining Low-temperature Series Expansions

Ian Graham Enting

It is shown that low-temperature series expansions for lattice models in statistical mechanics can be obtained from a consideration of only connected strong subgraphs of the lattice. This general result is used as the basis of a linked-cluster form of the method of partial generating functions and also as the basis for extending the finite lattice method of series expansion to low-temperature series.

Australian Journal of Physics•Dec 1, 1978•38 citations

LINKED CLUSTER SERIES EXPANSION TECHNIQUE FOR QUANTUM SPIN SYSTEMS

The linked-cluster series expansion is developed fur use on models of quantum spin systems with single-ion potentials. The linked-cluster series is defined as a statistical mechanical perturbation series in which the mean-field Hamiltonian and the single-ion potential are treated exactly with the perturbation representing the effects of fluctuation correlations. All terms of the free energy up to a given finite order are calculated, and estimates of thermodynamic functions are made based on extrapolation methods including the ratio method and Pade approximants.

•Jan 1, 1986•0 citations

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

Linked-cluster Expansions for Lattice Spin Models

Yuyi Wan

Similar to various series expansions that are used to approximate mathematical func- tions, the linked-cluster expansion is an approximation method that allows us to approach the actual values of a very large physical system’s different physical quan- tities by systematically studying smaller systems embedded in this larger system. The main concept in linked-cluster expansion, weight, represents the additional con- tribution to a certain physical quantity by increasing the system size by one unit. These weights are used to eventually build up the result on a larger system. In our case, we focus on the partition function, a quantity that can be used to calculate several essential thermodynamic aspects of the system such as average energy, spe- cific heat, magnetic susceptibility, etc. We study these weights for the Ising and Potts models on a one dimensional lattice and Bethe lattice, with and without an external magnetic field. Previous studies have shown that in the one dimensional Ising model, adding more than two lattice sites to the system does not create any ad- ditional contribution to the partition function (i.e. their weights are all zero), giving us the result for an infinitely large system with just two lattice sites, a remarkable simplification. In our study, we prove that this property holds not only for the Ising model, but also its generalization, the Potts model, and show that it is a result of a spin flip symmetry inherent in the system. In order to test this, we break this symmetry using an externally applied magnetic field and show that in this case, for magnetic dipole energies comparable with the exchange energy between neighboring spins, this special property vanishes.

•Jan 1, 2020•0 citations
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.
There are many statistical systems in which degrees of freedom can be grouped according to their degree of correlation. We develop a cluster-expansion method for these systems based on selecting finite clusters of the most strongly interacting degrees of freedom. The starting point is Kubo's generalized cumulant formalism. We show how this expansion can be renormalized; that is, how the contributions of infinite classes of articulated finite clusters can be resummed. The elimination of articulated clusters from the renormalized expansion generates modified interactions in the finite clusters remaining in the expansion. When truncated at different orders, the final result is a hierarchy of mean-field approximations, which generally reproduce an increasing number of terms in exact series expansions for the system. High-order series-expansion coefficients can be obtained by perturbative renormalization of very large clusters, a convenient alternative to the complicated combinatorial analysis associated with standard expansion techniques for lattice models. We compare our work to previous finite-cluster techniques, such as the cluster-variation method of Kikuchi.
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.
The linked-cluster series expansion is developed fur use on models of quantum spin systems with single-ion potentials. The linked-cluster series is defined as a statistical mechanical perturbation series in which the mean-field Hamiltonian and the single-ion potential are treated exactly with the perturbation representing the effects of fluctuation correlations. All terms of the free energy up to a given finite order are calculated, and estimates of thermodynamic functions are made based on extrapolation methods including the ratio method and Pade approximants.
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.
Similar to various series expansions that are used to approximate mathematical func- tions, the linked-cluster expansion is an approximation method that allows us to approach the actual values of a very large physical system’s different physical quan- tities by systematically studying smaller systems embedded in this larger system. The main concept in linked-cluster expansion, weight, represents the additional con- tribution to a certain physical quantity by increasing the system size by one unit. These weights are used to eventually build up the result on a larger system. In our case, we focus on the partition function, a quantity that can be used to calculate several essential thermodynamic aspects of the system such as average energy, spe- cific heat, magnetic susceptibility, etc. We study these weights for the Ising and Potts models on a one dimensional lattice and Bethe lattice, with and without an external magnetic field. Previous studies have shown that in the one dimensional Ising model, adding more than two lattice sites to the system does not create any ad- ditional contribution to the partition function (i.e. their weights are all zero), giving us the result for an infinitely large system with just two lattice sites, a remarkable simplification. In our study, we prove that this property holds not only for the Ising model, but also its generalization, the Potts model, and show that it is a result of a spin flip symmetry inherent in the system. In order to test this, we break this symmetry using an externally applied magnetic field and show that in this case, for magnetic dipole energies comparable with the exchange energy between neighboring spins, this special property vanishes.