A Statistical Model of Current Loops and Magnetic Monopoles
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un A STATISTICAL MODEL OF CURRENT LOOPS ANDMAGNETIC MONOPOLES
ARVIND AYYER
Abstract.
We formulate a natural model of loops and isolated vertices forarbitrary planar graphs, which we call the monopole-dimer model. We showthat the partition function of this model can be expressed as a determinant.We then extend the method of Kasteleyn and Temperley-Fisher to calculatethe partition function exactly in the case of rectangular grids. This partitionfunction turns out to be a square of a polynomial with positive integer coef-ficients when the grid lengths are even. Finally, we analyse this formula inthe infinite volume limit and show that the local monopole density, free en-ergy and entropy can be expressed in terms of well-known elliptic functions.Our technique is a novel determinantal formula for the partition function of amodel of isolated vertices and loops for arbitrary graphs. Introduction
The dimer model on a planar graph G is a statistical mechanical model whichidealises the adsorption of diatomic molecules on G . The associated combinatorialproblem is the weighted enumeration of all dimer covers of G , also known as perfectmatchings or 1-factors. This problem was solved in a beautiful and explicit way byKasteleyn [Kas61, Kas63] and by Temperley-Fisher [TF61, Fis61].The monomer-dimer model on the other hand, which idealises the adsorption ofboth monoatomic as well as diatomic molecules on G , has not had as much success.In this case, one considers the weighted enumeration of all possible matchings of G with separate fugacities for both kinds of molecules. Equivalently, this is theproblem of counting all matchings of G . There is some indirect evidence thatit is not likely to be exactly solvable [Jer87]. It has been rigorously shown thatthe monomer-dimer model does not exhibit phase transitions [GK71, HL72]. Theonly solutions so far are obtained by perturbative expansions (see the review in[HL72], for example). The asymptotics of the free energy has been studied byvarious authors, see [BW66, Ham66, HM70] for instance. There have also beenseveral numerical studies [KRS96, Kon06a, Kon06b] as well as study of monomercorrelations in a sea of dimers [FS63]. We note that there has been some success insolving restricted versions of the classical monomer-dimer model exactly, either forfinite size or in the limit of infinite size. Such is the case for a single monomer on theboundary [TW03, Wu06], arbitrary monomers on the boundary in the scaling limit[PR08] and a single monomer in the bulk in the thermodynamic limit [BBGJ07,PPR08]. More recently, after the completion of this work, there has appeared aGrassmannian approach to computing the dimer model partition function with fixedlocations of monomers exactly [AF14]. On the hexagonal lattice, a lot of work hasbeen done on monomer correlations by Ciucu, see [Ciu10] and references therein. Date : September 20, 2018.
We note in passing that signed dimer models and signed loop models have gainedattention in statistical physics recently, the former in the context of spin liquids[DDR12] and the latter as an approach towards solving the Ising model [KLM13].In this article, we will consider a signed variant of the monomer-dimer model onany planar graph, which we call the monopole-dimer model. This model will turnout to be a natural generalisation of the well-known dimer model , also definedfor any planar graph. The configurations of this model are subgraphs consistingof isolated vertices, doubled edges and oriented loops of even length on the graphsuch that each vertex is attached to exactly zero or two edges. Each configurationcan be thought of as a superposition of two monomer-dimer configurations withthe same monomer locations. The reason for the nomenclature will be explained inSection 3, when the weights associated to these configurations are specified. We willprove that the partition function of the monopole-dimer model can be written as adeterminant. This property is useful from a computational point of view becauseone can obtain a lot of information about the model using nothing more than basiclinear algebra. This approach has been extremely fruitful in studying many modelsin statistical physics, such as the Ising model in one-dimension [MW73], the sandpilemodel [Dha90] and the dimer model for planar graphs [Ken97].We will use this determinant formula to express the partition function of themonopole-dimer model on the two-dimensional grid as a product. This will turnout to give a natural generalisation of Kasteleyn’s and Temperley-Fisher’s formulafor the dimer model on the rectangular grid. It will turn out, for not obviousreasons, that the partition function will be an exact square when the sides of therectangle are even. This is in contrast to the double-dimer model [Kas61, Fis61],where the partition function is the determinant of an even anti-symmetric matrix,and hence is obviously the square of the corresponding Pfaffian.We will then derive explicit formulas for the free energy of the monopole-dimermodel in terms of known elliptic functions in the infinite size limit and compare itwith existing results for the monomer-dimer model, both rigorous and numerical.We will also calculate the entropy and the monopole density. The starting point,namely the determinant formula, is a consequence of a more general model oforiented loops, doubled edges and vertices on a general graph, which we will firstexplain.The plan of the paper is as follows. We will first define a new loop-vertex modelon arbitrary graphs in Section 2 and show that the partition function of the modelcan be written as a determinant in Theorem 2.5. We will then define the monopole-dimer model in Section 3 and use results proved in the previous section to showthat its partition function can also be written as a determinant in Theorem 3.3.We will then specialise to the two-dimensional grid graph in Section 4 and givean explicit product formula for the partition function in Theorem 4.1. We finallydiscuss the asymptotic limit of Z in Section 5.The statements of the paper can be verified using the Maple program file Monopo-le.maple available from the author’s webpage or as an ancillary file from the arXiv source. more precisely, the double-dimer model URRENT LOOPS AND MAGNETIC MONOPOLES 3
Acknowledgements
We would like to acknowledge support in part by a UGC Centre for AdvancedStudy grant. We would also like to thank C. Krattenthaler and J. Bouttier fordiscussions, T. Amdeberhan for conjecturing (4.1), K. Damle and R. Rajesh forsuggesting references, and M. Krishnapur for many helpful discussions. We alsothank the anonymous referees for several useful comments.2.
A Loop-Vertex Model on General Graphs
We begin by defining a model of isolated vertices and loops of even length onarbitrary graphs. The usefulness of the results here is that they are very general,and might be interesting in their own right. At this point, we do not know ofany relevant physical situation where this model could be applied. Part of theobjective of this section is to make the proof of the determinantal formula for thepartition function of the monopole-dimer model simpler. The reader interested inthe monopole-dimer model should feel free to skip this section.Our input data is a simple (not necessarily planar), undirected vertex- and edge-weighted labelled graph G = [ V, E ] on n vertices and an arbitrary assignment ofarrows along each edge, called the orientation O on G . We will denote vertexweights by x ( v ) for v ∈ V and edge weights as a ( v, v ′ ) ≡ a ( v ′ , v ) whenever ( v, v ′ ) ∈ E . Any labelled graph comes with a canonical orientation, the one got by directingedges from a lower vertex to a higher one. Definition 2.1. A loop-vertex configuration C consists of a subgraph of G ofedges which form directed loops of even length including doubled edges (to be thoughtof as loops of length 2), with the property that every vertex belongs to exactly zeroor two edges. Let L be the set of loop-vertex configurations. Note that the number of isolated vertices has the same parity as the size of thegraph. We first define the signed weight of a loop in C . First, the sign of an edge( v , v ), denoted sgn( v , v ) is +1 if the orientation is from v → v in O and − ℓ = ( v , . . . , v n , v ), the weight ofthe loop is(2.1) w ( ℓ ) = − n Y j =1 sgn( v j , v j +1 ) a ( v j , v j +1 ) , with the understanding that v n +1 = v . The reason for the overall minus sign willbe clear later. For now, note that the weight of a doubled edge is always + a ( v , v ) .Lastly, to each isolated vertex v , we associate the weight x ( v ). The weight w ( C ) ofa configuration C is then(2.2) w ( C ) = Y ℓ a loop w ( ℓ ) Y v anisolated vertex x ( v ) . Definition 2.2.
The loop-vertex model on a vertex- and edge-weighted graph G isthe collection L of loop-vertex configurations on G with the weight of each configu-ration given by (2.2) . Example 2.3.
For example, the weight of the configuration in Figure 1 is − x (1) · a (6 , · a (2 , a (3 , a (5 , a (7 , a (4 , a (2 , ARVIND AYYER rr rrr r rrr
123 456 789 ✲✻(cid:0)(cid:0)(cid:0)✒✻✲✲❅❅❅❘✻❅❅❅❘ ✲❅❅❅❘✻✲(cid:0)(cid:0)(cid:0)✒❅❅❅❘ ✲ rr rrr r rrr
123 456 789 ✻❅❅❅❘❅❅❅❘✻ ❤ ✲(cid:0)(cid:0)(cid:0)✠❅❅❅■ ✛ Figure 1.
A non-planar graph G with its natural orientation onthe left. A particular loop-vertex configuration is given on the rightwhere the “wrongly” oriented edges are coloured red.With a slight abuse of terminology, we say that the (signed) partition function of the loop-vertex model on the pair ( G, O ) is then(2.3) Z G, O = X C ∈L w ( C ) . Whenever the orientation is canonically defined by the labelling on the graph, wewill denote the partition function simply as Z G . Definition 2.4.
The signed adjacency matrix K associated to the pair ( G, O ) is the matrix K indexed by the vertices of G whose entries are (2.4) K ( v, v ′ ) = x ( v ) v ′ = va ( v, v ′ ) orientation is from v to v ′ in O− a ( v, v ′ ) orientation is from v ′ to v in O . Theorem 2.5.
The partition function of the loop-vertex model on ( G, O ) is givenby (2.5) Z G, O = det K. Proof.
We begin by considering the Leibniz formula for the determinant of K . Wewill consider the permutations in S n according to their cycle decomposition. Thefirst observation is that the sign of a non-trivial odd cycle c = ( c , c , . . . , c l +1 , c )is the opposite of its reverse rev( c ) = ( c , c l +1 , . . . , c , c ), but the weights are thesame. Therefore, such terms cancel out. The only odd cycles which appear arecycles of length one, also known as fixed points.It is then clear that the terms in the determinant expansion of K are in bijectionwith loop-vertex configurations of G . We now need to show that the signs are thesame. Therefore we decompose the permutation π into k fixed points and c cyclesof lengths 2 m , . . . , m c . This ensures that k has the same parity as n .A well-known combinatorial result states that if n is odd (resp. even), π is oddif and only if the number of cycles is even (resp. odd) in its cycle decomposition.In our case, the number of cycles is k + c . A short tabulation shows that the signof π is always the same as ( − c . In other words, the sign of a loop is preciselythe product of all the corresponding terms in K plus one extra sign. But this isprecisely what we have in (2.1). (cid:3) Although the loop-vertex model consists of signed weights, the following state-ment can easily be verified since the signed adjacency matrix is a sum of a diagonalmatrix and an antisymmetric matrix.
URRENT LOOPS AND MAGNETIC MONOPOLES 5
Corollary 2.6.
The partition function Z G, O is a positive polynomial in the vari-ables x ( v ) for v ∈ V and a ( v, v ′ ) for ( v, v ′ ) ∈ E . In particular, if all the weightsare positive reals, Z G, O is strictly positive. Example 2.7.
For the loop-vertex model on the complete graph with its canonicalorientation (see below (2.3) ) with vertex-weights x and edge-weights a , the signedadjacency matrix is given by K n = x a a · · · a − a x a · · · a ... . . . . . . . . . ... − a · · · − a x a − a · · · − a − a x One can compute the determinant of K n by using elementary row and column op-erations to convert it to a tridiagonal matrix. It is then easy to show that Z n satisfies the recursion Z n = 2 xZ n − − ( x − a ) Z n − . It is immediate from theinitial conditions Z = 1 and Z = x that (2.6) Z n = ⌊ n/ ⌋ X k =0 (cid:18) n k (cid:19) x n − k a k = ( x + a ) n + ( x − a ) n . The Monopole-Dimer Model on Planar Graphs
We now focus on the model of physical interest, namely the monopole-dimermodel. As we shall see, technical reasons force us to restrict our attention to planargraphs. We will first define the model for an arbitrary planar graph and state atheorem about the partition function of the model.From now on, we will use G to mean both the graph and its planar embedding.As before, G = [ V, E ] will be a labelled graph with vertex weights x ( v ) for v ∈ V and edge weights a ( v, v ′ ) whenever ( v, v ′ ) ∈ E . The configurations of the model areexactly the loop-vertex configurations L of Definition 2.1. From here on, we willuse the term monopole-dimer configurations instead of loop-vertex configurations.Let C ∈ L be a monopole-dimer configuration containing an even loop ℓ . Theweight of the loop ℓ = ( v , . . . , v n , v ) is given by(3.1) w ( ℓ ) = ( − number of vertices in V enclosed by ℓ n Y j =1 a ( v j , v j +1 ) , where, as before, v n +1 ≡ v . Notice that the planarity of the graph is used cruciallyin ensuring that w ( ℓ ) is well-defined. In the usual way, we set the weight of vertex v to be x ( v ), and the weight w ( C ) of the entire configuration C as(3.2) w ( C ) = Y ℓ a loop w ( ℓ ) Y v a vertex x ( v ) . Note that the definition of the monopole-dimer model on planar graphs is indepen-dent of any orientation, unlike the loop-vertex model.
Definition 3.1.
The monopole-dimer model on G is a model of monopole-dimer configurations L on G where the weight of each configuration is given by (3.2) . ARVIND AYYER
As before, we let the (signed) partition function of the monopole-dimer modelon G be Z G = X C a monopole-dimerconfiguration w ( C ) . Remark 3.2.
Configurations of the model are superpositions of two configurationsof the monomer-dimer model with identical locations of monomers and thus gen-eralise the so-called double-dimer model [KW11, Ken11] . Since the weight of eachdouble-dimer loop is given a sign which is the parity of the number of monomersenclosed by it, it is reminiscent of the Dirac string representation of the monopole.Dirac had shown by integrating the flux around a curve enclosing the string that thewell-definedness of the vector potential led naturally to the quantization of charge [Dir78] . We recall the notion of a
Kasteleyn orientation for a planar graph. We willconsider the case of bipartite graphs for simplicity; the general case is similar.In this case, Kasteleyn [Kas61] showed that there exists an orientation O on G such that every basic loop enclosing a face has an odd number of clockwise orientededges. This is sometimes called the clockwise-odd property. Using this orientation O , Kasteleyn showed that the dimer partition function on G can be written as aPfaffian of an even antisymmetric matrix, now called the Kasteleyn matrix. Notethat the signed adjacency matrix K in (2.4) differs from the Kasteleyn matrix bya diagonal matrix. In what follows, we will refer to the signed adjacency matrix asa (modified) Kasteleyn matrix . Theorem 3.3.
Let O be a Kasteleyn orientation on the planar graph G and let K be the modified Kasteleyn matrix defined as (2.4) . Then the partition function ofthe monopole-dimer model on G can be written as Z G = det K. Moreover, Corollary 2.6 immediately implies that Z G is a positive polynomial inthe weights.Proof. To prove this, we have to show that the weight of a loop in a planar graphwith a Kasteleyn orientation defined by (3.1) is the same as that defined in (2.1).Suppose the loop is of length 2 ℓ and there are v internal vertices, e internal edgesand f faces. Suppose the Kasteleyn orientation is such that there are o j clockwiseedges in face j , where each o j is odd. The total number of clockwise edges on theloop is therefore P fj =1 o j − e since each internal edge contributes twice to the count,once clockwise and once counter-clockwise. The Euler characteristic v − e + f is 1on the plane since we exclude the unbounded face. Since the parity of P fj =1 o j − e is the same as that of f − e , which equals v −
1, we have shown that the totalnumber of clockwise edges on the loop is odd if and only if v is even. This showsthat the weights in (3.1) and (2.1) coincide. (cid:3) Example 3.4.
Consider the cycle graph C n where the vertices are labelled in cyclicorder with the canonical orientation with weights a to each edge and x to each vertex. URRENT LOOPS AND MAGNETIC MONOPOLES 7
The modified Kasteleyn matrix K n is then K n = x a · · · a − a x a · · · . . . . . . . . . · · · − a x a − a · · · − a x . One can then show with a little bit of work that the partition function Z n satisfies Z n = det K n = ( xa n F n (cid:0) xa (cid:1) + 2 a n +1 F n − (cid:0) xa (cid:1) , if n is odd ,xa n − F n (cid:0) xa (cid:1) + 2 a n F n − (cid:0) xa (cid:1) + 2 a n , if n is even , where F n ( x ) is the n ’th Fibonacci polynomial defined by the recurrence F n ( x ) = xF n − ( x ) + F n − ( x ) with initial conditions F ( x ) = 0 and F ( x ) = 1 . Using stan-dard properties of the Fibonacci polynomials, we can rewrite Z n = a n L n (cid:16) xa (cid:17) if n is odd ,a n (cid:16) L n/ (cid:16) xa (cid:17)(cid:17) if n ≡ ,a n − ( x + 4 a ) (cid:16) F n/ (cid:16) xa (cid:17)(cid:17) if n ≡ . where the Lucas polynomials L n ( x ) satisfy the same recurrence as the Fibonaccipolynomials but with different initial conditions, L ( x ) = 2 and L ( x ) = x . Remark 3.5.
Note that when n is divisible by 4, √ Z n is a positive polynomial andcan be considered as the partition function of a model of monomers and dimers.This phenomenon will recur in Section 4. Corollary 3.6 (Kasteleyn [Kas61]) . In the absence of vertex weights, i.e. x ( v ) =0 ∀ v ∈ V , the monopole-dimer model is exactly the double-dimer model (see Re-mark 3.2) and consequently, Z G = | Pf K | . We will now explore some consequences of the determinant formula for themonopole-dimer model. Unlike for the usual dimer model, one cannot calculateprobabilities of events for the monopole-dimer model since the measure on configu-rations here is not positive. However, one can consider expectations of observablesin this signed measure.
Definition 3.7.
The joint correlation of a subconfiguration of monopoles v , . . . , v j and loops ℓ , . . . , ℓ k in the graph G is h v , . . . , v j ; ℓ , . . . , ℓ k i = j Y r =1 x ( v r ) k Y s =1 w ( ℓ s ) b Z G ′ Z G where G ′ is the subgraph of G with the vertices v , . . . , v j and those in loops ℓ , . . . , ℓ k removed; and b Z G ′ is the partition function of the monopole-dimer model in G ′ withthe caveat that the sign of loops in G ′ given by (3.1) be taken by considering verticesin all of G ARVIND AYYER
To give a formula for joint correlations, we recall the complementary minoridentity of Jacobi. For a k × k nonsingular matrix A , 1 ≤ i ≤ k , sequences [ p ] =( p , . . . , p i ) , [ q ] = ( q , . . . , q i ) where 1 ≤ p < · · · < p i ≤ k and 1 ≤ q < · · · < q i ≤ k , let A [ p ][ q ] be the i × i submatrix of A consisting of rows p j and columns q j . Also, let[¯ p ] (resp. [¯ q ]) be the complementary sets { , . . . , k }\ [ p ] (resp. { , . . . , k }\ [ q ]). Recallthat the determinant of such a submatrix is called a minor , and when [ p ] = [ q ],both the submatrix and its minor are qualified by the adjective principal . Theorem 3.8 (Jacobi, see § .
16 of [GR00]) . det (cid:16) ( A − ) [¯ q ][¯ p ] (cid:17) = ( − p + q + ··· + p i + q i det A det A [ p ][ q ] . Remark 3.9.
In the case of an unsigned combinatorial model on a graph (i.e.with nonnegative weights) whose partition function can be written as a determinant,Theorem 3.8 implies that probabilities of local events can be computed in terms ofprincipal minors of the inverse. This fact has been used with great success for thedimer model [Ken97] . Lemma 3.10.
Let [¯ p ] be the set of vertices of monopoles v , . . . , v j and of loops ℓ , . . . , ℓ k . Then the joint correlation is given by h v , . . . , v j ; ℓ , . . . , ℓ k i = j Y r =1 x ( v r ) k Y s =1 w ( ℓ s ) det (cid:16) ( K − ) [¯ p ][¯ p ] (cid:17) . Furthermore, it is positive if the subconfiguration has positive weight.Proof.
The sum over all configurations with these prescribed monopoles and loopsis given by the appropriate principal minor of the modified Kasteleyn matrix, K .By Theorem 3.8, this is exactly the complementary minor of the inverse, whichexists because of Corollary 2.6. The minor is the determinant of a matrix which isthe sum of an antisymmetric matrix and a diagonal matrix and is positive, againusing Corollary 2.6. Thus, the only way for the joint correlation to be negative isif the subconfiguration itself has negative weight. (cid:3) We will use Lemma 3.10 in the following to compute monopole correlations. Thejoint correlation of the monopole-loop configuration on the right side of Figure 2 ina large m × n square is the simplest example of one which is negative.4. The Monopole-Dimer model on the Rectangular Grid
We will now calculate the partition function for the monopole-dimer model inSection 3 on the rectangular grid graph, thereby generalising the famous productformula of Temperley-Fisher [Fis61] and Kasteleyn [Kas61].Consider the m × n grid Q m,n = { ( i, j ) | ≤ i ≤ m, ≤ j ≤ n } with hori-zontal edge-weights a , vertical edge-weights b and vertex-weights z . For the sakeof completeness, we recall the Kasteleyn orientation O prescribed independentlyby Fisher [Fis61] and Kasteleyn [Kas61]. The arrow always points in the direction( i, j ) → ( i, j + 1), i.e., towards the positive x -axis. In the y -direction, the arrowpoints from ( i, j ) → ( i, j + 1) (i.e. towards the positive y -axis) whenever i is oddand in the reverse direction when i is even. This orientation can be easily seen tobe induced by a “snake-like” labelling, as seen in Figure 2. URRENT LOOPS AND MAGNETIC MONOPOLES 9
Define the function Y m ( b ; z ) = ⌊ m/ ⌋ Y j =1 (cid:18) z + 4 b cos jπm + 1 (cid:19) . Theorem 4.1.
The partition function of the monopole-dimer model on Q m,n isgiven by Z m,n = ⌊ m/ ⌋ Y j =1 ⌊ n/ ⌋ Y k =1 (cid:18) z + 4 b cos jπm + 1 + 4 a cos kπn + 1 (cid:19) × if m and n are even ,Y m ( b ; z ) if m is even and n is odd ,Y n ( a ; z ) if m is odd and n is even ,zY m ( b ; z ) Y n ( a ; z ) if m and n are odd . (4.1) Proof.
The matrix K m,n is exactly the regular Kasteleyn matrix added to z timesthe identity matrix of size mn . Therefore, the inversion technique described ineither of these papers works identically when m or n are even. The case when both m and n are odd is a special case, which has to be worked out separately. Bothcases can be analysed simultaneously.We use Fisher’s labelling [Fis61]. The modified Kasteleyn matrix can be writtenin m × m tridiagonal block form K = X Y · · · − Y X Y · · · − Y X Y · · · · · · − Y X Y · · · − Y X , where each of the blocks is an n × n matrix with X = z a · · · − a z a · · · − a z a · · · · · · − a z a · · · − a z , and Y = · · · b · · · b b · · · b · · · . Fisher showed [Fis61] that the matrix K can be simplified considerably by theunitary transformation U = u m ⊗ u n where u s is an s × s matrix with entries(4.2) ( u s ) p,q = r s + 1 i p sin πpqs + 1 , where i = √− u − s also has an equally simple formula(4.3) ( u s ) − p,q = r s + 1 ( − i ) q sin πpqs + 1 . One can show that U − K U can be written as a block diagonal matrix of χ s ’s for s = 1 , . . . , m , where(4.4) χ s = z +2 ia cos πn +1 . . . − n − i n b cos πsm +1 z +2 ia cos πn +1 ( − n − i n b cos πsm +1 ... . . . . . . ...... . . . . . . ... − i n b cos πsm +1 z +2 ia cos ( n − πn +1 − i n b cos πsm +1 . . . z +2 ia cos nπn +1 Let us now look at each of the cases. When n is even, the determinant of χ s iseasily expressed as a product of 2 × χ s = n Y p =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + 2 ia cos pπn + 1 ( − n − p i n b cos sπm +1 ( − p − i n b cos sπm + 1 z + 2 ia cos ( n +1 − p ) πn +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n Y p =1 (cid:18) z + 4 b cos sπm + 1 + 4 a cos pπn + 1 (cid:19) = det χ m +1 − s . When m is also even, we get after multiplying over all s , precisely the formula inthe first case (4.1). When m is odd, we get an additional factor | χ ( m +1) / | , whichis easy to compute because it is a diagonal matrix. The factor we get isdet χ ( m +1) / = n Y p =1 (cid:18) z + 2 ia cos pπn + 1 (cid:19) = n Y p =1 (cid:18) z + 4 a cos pπn + 1 (cid:19) = Y n ( a ; z ) . This also matches with (4.1). When n is odd and m is even, we have the additionalfactor contributing to each | χ s | from the central term, z + 2 ib cos πsm + 1 . Multiplying this factor for all s gives us Y m ( b ; z ) as needed. The last case whenboth m and n are odd gives us both the factors above and an additional termcorresponding to the central entry of the central block matrix, which can be seento be z . (cid:3) Example 4.2.
Consider the first nontrivial case: m = n = 3 . Figure 2 shows Q , with a Kasteleyn orientation and one of two configurations which contribute with a URRENT LOOPS AND MAGNETIC MONOPOLES 11 rr rrr r rrr
123 6 54 789 ✲✻ ❄ ✲✻✲✻ ✲❄ ✻✲ ✲ rr rrr r rrr
123 6 54 789 ✲ ✲✻✻✛✛❄❄ ❤ Figure 2.
The graph Q , of Example 4.2 with its natural orien-tation on the left and a particular monopole-dimer configurationon the right. weight − za b . The modified Kasteleyn matrix is given by K , = z a b − a z a b − a z b − b z a b − b − a z a b − b − a z b − b z a
00 0 0 0 − b − a z a − b − a z The partition function is given by Z , = z (cid:0) a + z (cid:1) (cid:0) b + z (cid:1) (cid:0) a + 2 b + z (cid:1) , in agreement with (4.1) . Remark 4.3.
The fact that the partition function Z m,n is an exact square of apositive polynomial when m and n are even is nontrivial since K m,n is not anti-symmetric. We will now use Theorem 3.8 to calculate joint correlations in the monopole-dimer model on the grid graph. We focus on the case when m, n are even forsimplicity. To do so, we will first need to calculate the matrix entries for the inverseof the modified Kasteleyn matrix. We will now calculate this in full generality. Sincewe have used the snake-like Kasteleyn orientation explained at the beginning of thisSection, the relationship between the entries of the matrix and coordinates on thegrid depends on the parity of the abscissa. To simplify notation, we define thefunctions φ g,h ( c, d ; e, f ) = 4 i c + d ( − i ) e + f ( m + 1)( n + 1) sin πcgm + 1 sin πegm + 1 sin πdhn + 1 sin πf hn + 1 × z − ia cos πhn +1 + ( − f + h i n b cos πgm +1 z + 4 a cos πhn +1 + 4 b cos πgm +1 ! ,ψ g,h ( c, d ; e, f ) = 4 i c + d ( − i ) e + f ( m + 1)( n + 1) sin πcgm + 1 sin πegm + 1 sin πdhn + 1 sin πf hn + 1 × z − ia cos πhn +1 + ( − f + h − i n b cos πgm +1 z + 4 a cos πhn +1 + 4 b cos πgm +1 ! , for fixed m, n and weights a, b, z . The only difference between the two functions isthe power of − Lemma 4.4. If n is even, the entries of the inverse matrix are given by (cid:0) K − m,n (cid:1) ( c,d ) , ( e,f ) = X ( g,h ) ∈ Q m,n φ g,h ( c, d ; e, f ) if both c and e are odd, φ g,h ( c, n + 1 − d ; e, f ) if c is even and e is odd, φ g,h ( c, d ; e, n + 1 − f ) if c is odd and e is even, φ g,h ( c, n + 1 − d ; e, n + 1 − f ) if both c and e are even,and if n is odd, the entries are given by (cid:0) K − m,n (cid:1) ( c,d ) , ( e,f ) = X ( g,h ) ∈ Q m,n ψ g,h ( c, d ; e, f ) if both c and e are odd, ψ g,h ( c, n + 1 − d ; e, f ) if c is even and e is odd, ψ g,h ( c, d ; e, n + 1 − f ) if c is odd and e is even, ψ g,h ( c, n + 1 − d ; e, n + 1 − f ) if both c and e are even.Proof. Since U − K m,n U = Diag( χ , . . . , χ m ), where χ s is given in (4.4) and U = u m ⊗ u n , U − are given explicitly in (4.2),(4.3), one starts by inverting χ s andobtains K − m,n as U − Diag( χ − , . . . , χ − m ) U by a somewhat lengthy but straightfor-ward calculation.The parities of c and e enter in the calculation simply because in the Kasteleynorientation, the coordinates d, f increase from left to right when c, e are odd andfrom right to left, when c, e are even; see Figure 2 for example. (cid:3) Corollary 4.5.
The one-point monopole correlation (informally the density) at ( c, d ) in the m × n grid is given by z ( m + 1)( n + 1) X ( g,h ) ∈ Q m,n sin πcgm +1 sin πdhn +1 z + 4 a cos πhn +1 + 4 b cos πgm +1 . Proof.
As per the definition of the correlation and Theorem 3.8, the one-pointmonopole correlation at ( c, d ) is given by z K − c,d ) , ( c,d ) . We use Lemma 4.4 anduse the symmetry of φ g,h ( c, d ; c, d ) and ψ g,h ( c, d ; c, d ) under the transformations g m + 1 − g and h n + 1 − h to obtain the result. (cid:3) As expected, the prefactor of z ensures that there is one additional monopolewhen either m or n is even. See, for example, Figure 3 for the density plot when m = n = 20. One can also compute joint correlations of monopoles. For instance,the two-point correlation of monopoles at positions ( c, d ) and ( e, f ) far apart isgiven by det K − c,d ) , ( c,d ) K − c,d ) , ( e,f ) K − e,f ) , ( c,d ) K − e,f ) , ( e,f ) ! . Discussion on Asymptotic Behaviour
We will now focus on asymptotic results for the monopole-dimer model on thegrid graph. The results in this section will be less formal and will focus more onobtaining rough estimates for the equivalent of quantities in standard thermody-namics, such as the free energy, density and the entropy. Part of the reason for theinformality of this section is that we are manipulating Z m,n as if it were the stan-dard partition function in statistical physics. This is not strictly allowed because URRENT LOOPS AND MAGNETIC MONOPOLES 13
Figure 3.
A contour plot of exact monopole densities as a func-tion of location in a 20 ×
20 grid for a = b = z = 1. Note theuniformity of the density in the interior of the system.our partition functions are signed sums. However, as we shall see, we can justifythis a posteriori by showing that the results are sensible.Just as at the end of the previous section, m, n are assumed to be even forsimplicity. The free energy is then given by F ( a, b, z ) = lim m,n →∞ mn ln Z m,n . Using (4.1), one can treat the right hand side as a Riemann sum, which tends tothe limit F ( a, b, z ) = 2 π Z π/ d θ Z π/ d φ ln (cid:0) z + 4 a cos θ + 4 b cos φ (cid:1) . Following standard thermodynamic relations, the density of a -type of dimers (andsimilarly, the b -type) and that of monopoles is given, after differentiating under theintegral sign, by ρ a = a ∂∂a F ( a, b, z ) = 2 π Z π/ d θ Z π/ d φ a cos θz + 4 a cos θ + 4 b cos φ , (5.1) ρ z = z ∂∂z F ( a, b, z ) = 2 π Z π/ d θ Z π/ d φ z z + 4 a cos θ + 4 b cos φ . (5.2)It is easy to see that ρ a + ρ b + ρ z = 1 . This is to be expected since each vertexeither contains a monopole or is part of a loop adjacent to either an a or a b dimer.One of the integrals in each case is easily done. Surprisingly, ρ a is easier toevaluate than ρ z even though the final formula will turn out to be simpler for thelatter. ρ a = 2 π Z π/ d θ a cos θ p ( z + 4 a cos θ )( z + 4 b + 4 a cos θ ) After the change of variables t = (2 a cos θ ) − , we obtain ρ a = 1 πayz Z ∞ / a d tt q ( t − a )( t + z )( t + y ) , where y = z + 4 b . Using a known formula for elliptic integrals [GR00][(3.137),Formula 8] and an amazing transformation [AS64][Formula 17.7.14], it turns outthat ρ a can be concisely expressed in terms of a single known special function, theHeuman Lambda function Λ ( θ, k ), defined in [AS64, Formula 17.4.39], as(5.3) ρ a = 1 − Λ ( θ a , k ) , where all the complexity has been absorbed in the parameters(5.4) θ a = tan − r b + z a ! , and k = 4 ab p (4 a + z )(4 b + z ) , where k is the standard notation for the elliptic modulus. We remark that theHeuman Lambda function is an elliptic function related to the Jacobi Zeta functionand has come up in various physical problems. It turns out that the monopoledensity can be written, using a miraculous addition formula for Λ [BF71, Formula153.01] as(5.5) ρ z = K ( k ) k z π a b , where K ( k ) is the complete elliptic integral of the first kind. Now that we haveexpressions for ρ a and ρ z , we would like to obtain a simple expression for the freeenergy F ( a, b, z ) by integrating ρ z z .Starting with the series expansion for K ( k ) in [AS64, Formula 17.3.11], we get(5.6) ρ z = z p (4 a + z )(4 b + z ) ∞ X j =0 (cid:18) (2 j − j j ! (cid:19) (cid:18) a b (4 a + z )(4 b + z ) (cid:19) j . We now integrate ρ z z term by term assuming a > b and obtain, using a standardcomputer algebra package, an infinite sum involving 2 F − hypergeometric func-tions, F ( a, b, z ) = r b + z a − b ∞ X j =0 j − (cid:18) jj (cid:19) × (cid:18) a b ( a − b )(4 b + z ) (cid:19) j F " − j + j − j ; 4 b + z b − a (5.7)Since the integrands are symmetric in a and b , we can obtain the free energy when b > a by interchanging a and b in (5.7). We handle the a = b case separately. Inparticular, we set them equal to 1 without loss of generality. In that case, eachintegral in (5.6) is easier because of the absence of square roots and it turns outthat we can write F (1 , , z ) again using a computer algebra package as(5.8) F (1 , , z ) = 12 ln(4 + z ) − z ) F (cid:20)
32 32 z ) (cid:21) As expected from general statistical physical considerations, F (1 , , z ) growsmonotonically in z and is concave [Ham66] as seen in Figure 4. In accordance with URRENT LOOPS AND MAGNETIC MONOPOLES 15
Figure 4.
A plot of F (1 , , z ) for z varying between 0 and 1000.the intuition developed for the monomer-dimer model [GK71, HL72], F (1 , , z ) issmooth and there are no phase transitions. One can verify that F (1 , ,
0) = 2
G/π ,where G is Catalan’s constant. This is expected since this model reduces, when z =0, to the double-dimer model, which is the square of the dimer model [Kas61, Fis61].Since configurations of the monopole-dimer model are superpositions of twodimer model configurations with fixed monopole locations, and since Z m,n turnsout to be a perfect square, we can compare p Z m,n with existing literature on themonomer-dimer model. F (1 , , z ) / ρ = ρ a + ρ b , not z . The transformation is a classicexercise in demonstrating equivalence of ensembles. See [Kon06b, Appendix A] forexample. Figure 5.
Comparison of F (1 , , z ) / One can also calculate the entropy using standard thermodynamic relations, S ( a, b, z ) = F ( a, b, z ) − z ln z ∂∂z F ( a, b, z )= F ( a, b, z ) − ρ z ln z. In the special case of equal dimer weights, this leads to S (1 , , z ) = F (1 , , z ) − z ln zπ (4 + z ) K (cid:18)
44 + z (cid:19) . Using (5.8), one can show that the entropy is maximum when z = 1, at which pointthe monopole density using (5.5) is ρ z = 25 π K (cid:18) (cid:19) ≈ . . This compares very well with the exact result for the 20 ×
20 grid in Figure 3.Many qualitative properties of the monopole-dimer model on grids are similar tothose of the classical monomer-dimer model, which is of much interest to scientistsin various fields. The exact formulas for grids presented here might be used togain further insight about the monomer-dimer model. The determinantal charac-ter of the partition function for the loop-vertex model on general graphs and themonopole-dimer model on planar graphs might also prove useful in other contexts.
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Department of Mathematics, Indian Institute of Science, Bangalore 560012, India.
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