Exact Potts model partition functions on wider arbitrary-length strips of the square lattice

Chang, Shu Chiuan; Shrock, Robert

doi:10.1016/s0378-4371(01)00142-x

Cited by 54 publications

(74 citation statements)

References 65 publications

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“…(3.70) of our earlier Ref. [8] for the determinant of the transfer matrix of the toroidal strip of the square lattice with arbitrary width L y . For the honeycomb lattice, we find…”

Section: A Determinantsmentioning

confidence: 98%

“…The linear term, denoted as λ Z,s3t,10 in [8], has multiplicity two, so the corresponding coefficient is 2b (0) = 2. The cubic equation is the same as eqs.…”

Section: Some Illustrative Calculationsmentioning

confidence: 99%

“…We recall some previous related work. The partition function Z(G, q, v) for the Potts model was calculated for arbitrary q and v on strips of the lattice Λ with toroidal or Klein bottle longitudinal boundary conditions was calculated for (i) Λ = sq, L y = 2, 3 in [8], and (ii) for L y = 3 on the square-lattice with next-nearest-neighbor spin-spin couplings in [9]. Matrix methods for calculating chromatic polynomials were developed and used in [10,11,12,13] and more recently in [14,15,16].…”

Section: Introductionmentioning

confidence: 99%

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Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

Chang

Shrock

2006

Physica A: Statistical Mechanics and its Applications

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We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G, q, v), for arbitrary q and temperature variable v, on strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width L y vertices and of arbitrarily great length L x vertices, subject to toroidal and Klein bottle boundary conditions. For the toroidal case we express the partition function Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included.

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“…(3.70) of our earlier Ref. [8] for the determinant of the transfer matrix of the toroidal strip of the square lattice with arbitrary width L y . For the honeycomb lattice, we find…”

Section: A Determinantsmentioning

confidence: 98%

“…The linear term, denoted as λ Z,s3t,10 in [8], has multiplicity two, so the corresponding coefficient is 2b (0) = 2. The cubic equation is the same as eqs.…”

Section: Some Illustrative Calculationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

Chang

Shrock

2006

Physica A: Statistical Mechanics and its Applications

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“…[28]. The free energy is given by f (sq, 3 F , q, v) = (1/3) ln λ sq,3F , where λ sq,3F is the (maximal) root of an algebraic equation of degree 4.…”

Section: F 4f 5fmentioning

confidence: 99%

“…The free energy was computed in Ref. [28] and is given by (horizontal axis) for infinite-length, finite-width strips of the square lattice. The dashed and solid curves refer to free and periodic transverse boundary conditions, respectively.…”

Section: Pmentioning

confidence: 99%

Exact results for average cluster numbers in bond percolation on lattice strips

Chang¹,

Shrock²

2004

Phys. Rev. E

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We present exact calculations of the average number of connected clusters per site, k , as a function of bond occupation probability p, for the bond percolation problem on infinite-length strips of finite width Ly, of the square, triangular, honeycomb, and kagomé lattices Λ with various boundary conditions. These are used to study the approach of k , for a given p and Λ, to its value on the two-dimensional lattice as the strip width increases. We investigate the singularities of k in the complex p plane and their influence on the radii of convergence of the Taylor series expansions of k about p = 0 and p = 1.

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Graph Polynomials and Their Applications I: The Tutte Polynomial

Ellis-Monaghan

Merino

2010

Structural Analysis of Complex Networks

117

119

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We give a number of properties and combinatorial interpretations for various evaluations of the Tutte polynomial. We recover colorings, flows, orientations, network reliability, etc., and related polynomials as specializations of the Tutte polynomial. We discuss the coefficients, zeros, and derivatives of the Tutte polynomial, and conclude with a brief discussion of computational complexity. Preliminary NotionsThe graph terminology that we use is standard and generally follows Diestel [Die00]. Graphs may have loops and multiple edges. For a graph G we denote by V (G) its set of vertices and by E(G) its set of edges. An oriented graph, G, also called a digraph, has a direction assigned to each edge. Basic ConceptsWe first recall some of the notions of graph theory most used in this chapter. Two graphsWe denote by κ(G) the number of connected components of a graph G, and by c(G) the number of non-trivial connected components, that is the number of connected components not counting isolated vertices. A graph is k-connected if at least k vertices must be removed to disconnect the graph.A cycle in a graph G is a set of edges e 1 , . . . , e k such that, if e i = (v i , w i ) for 1 ≤ i ≤ k, then w i = v i+1 for 1 ≤ i ≤ k − 1; also w k = v 1 and v i = v j for i = j. A trail is a path that may revisit a vertex, but not retrace an edge. A circuit is a closed trail, and thus a cycle is just a circuit that does not revisit any vertices. In the case of a digraph, the edges of a trail or circuit must be consistently oriented.The dual notion of a cycle is that of cut or cocycle. If {V 1 , V 2 } is a partition of the vertex set, and the set C, consisting of those edges with one end in V 1 and one end in V 2 , is not empty, then C is called a cut. A cycle with one edge is called a loop and a cocycle with one edge is called a cut-edge or bridge. We refer to an edge that is neither a loop nor a bridge as ordinary.A tree is a connected graph without cycles. A forest is a graph whose connected components are all trees. A subgraph H of a graph G is spanning if V (H) = V (G). Spanning trees in connected graphs will play a fundamental role in the theory of the Tutte polynomial. Observe that a loop in a connected graph can be characterized as an edge that is in no spanning tree, while a bridge is an edge that is in every spanning tree.If V ⊆ V (G), then the induced subgraph on V has vertex set V and edge set those edges of G with both endpoints in V . If E ⊆ E(G), then the spanning subgraph induced by E has vertex set V (G) and edge set E .

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Exact Potts model partition functions on wider arbitrary-length strips of the square lattice

Cited by 54 publications

References 65 publications

Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

Exact results for average cluster numbers in bond percolation on lattice strips

Graph Polynomials and Their Applications I: The Tutte Polynomial

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