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THE DIMER PARTITION FUNCTION

arXiv:hep-lat/9303015v1 30 Mar 1993NBI-HE-93-16March 1993hep-lat/9303015THE DIMER PARTITION FUNCTIONIgor Pesando1The Niels Bohr InstituteUniversity of CopenhagenBlegdamsvej 17, DK-2100 Copenhagen ØDenmarkAbstractWe apply the Ginzburg criterion to the dimer problem and wesolve the apparent contradiction of a system with mean field α = 12,the typical value of tricritical systems, and upper critical dimensionDcr = 6. We find that the system has upper critical dimension Dcr = 6, while for D ≤4 it should undergo a first order phase transition.

Wecomment on the latter wrong result examining the approximation weused.1E-mail PESANDO@NBIVAX.NBI.DK, 22105::PESANDO, 31890::I PESANDO

In this letter we would like to show how it is possible to recover the uppercritical dimension of the dimer system without using renormalization grouparguments but with the use of the Ginzburg criterion. In this way we solvewhat could appear as a contradiction: the mean field critical exponent α = 12and the upper critical dimension Dcr = 6; in fact α = 12 is the typical value ofthe mean field critical exponent of the tricritical transitions that have uppercritical dimension 3.

Differently from previous works ([3, 4]) we do not userenormalization group arguments.We consider the following action defined on a lattice G 1 [1]Zdimer =ZYl∈Gd¯ǫl dǫl exp−µXi¯ǫiǫi −K2Xi,jAij¯ǫiǫi¯ǫjǫj(1)where Aij is the adjacency matrix of the lattice G, ¯ǫi = (ǫi)∗are complexgrassman variables defined on the vertex i.It is easy to show that thepartition function (1) is the generating function for the the dimer problemwith negative activity, more explicitly:Zdimer = µV XDN(D) (−Kµ2)D(2)where D is the number of dimers and N(D) is the number of possible con-figurations with D dimers (V is the number of lattice sites).To prove this assertion we consider the high temperature expansion (HTE)in the variable Kµ2 ∼1T . Now if a term of the HTE gives a non vanishing con-tribution to the partition function Z, it must have at most one active link(dimer) per each vertex since we cannot put two dimers on the lattice sharinga vertex because of the relation (¯ǫiǫi)2 = 0.

As an observation we want tonotice that setting µ = 0 we can get the close packing dimer problem thatit is known to be equivalent to Ising and hence, in this case, the theory hascritical dimension Dcr = 4 ([2])2.Before we can use Ginzburg criterion, we must rewrite the partition func-tion (1) in a suitable form for the application of the saddle point expansion.1 We use i, j, . .

. to indicate sites (vertices) of the lattice.2 To this purpose we have to introduce the terminal lattice of the expanded version ofthe original lattice, then we divide the vertices into two sets: the first one G1 containingthe vertices of the original lattice and the other G2 with the new vertices.

Finally we1

To this purpose we introduce the variable Ri =√K¯ǫiǫi and we use theidentityZd¯ǫ dǫ δ(R −√K¯ǫǫ) =√K dδ(R)dr(3)in order to rewrite (1) as followsZdimer = (−√K)VYk∈G∂∂Rkexp−µ√KXiRi −12Xi,jAijRiRjRk=0If we perform the substitutions Ri →−Ri andµ√K →µ, we use the factthat the derivatives act on an analytic function, we can finally rewrite theprevious equation asZdimer = (√K)VIΓYk∈Gdzk2πi expXiµzi −2 log(zi) −12Xi,jAijzizj(4)where Γ is an hypersurface surrounding the origin.We introduce the notationS=Xiµzi −2 log(zi) −12Xi,jAijzizjSi=−2zi+ µ −XjAijzjSij=2z2iδij −AijUnder the hypothesis of a translationally invariant solution, the saddle pointequations Si = 0 yield the solutionsz± = µ ± √µ2 −8q2q(5)generalize the kinetic term as follows:Aij =⇒˜Aij = Jif i and j are neighbours and both belong to lattice G11if i and j are neighboursthen there exists a critical value of J at which a second order phase transition occurs.2

where for µ > 0 z−is a minimum and z+ is a maximum and the critical valueis attained for µ2cr = 8q where the two solutions coalesce.Let us now compute the critical exponent α. It turns out that in themean field approximation this critical exponent is independent on the choiceof z+ or z−, nevertheless we must choose z−(the minimum); an explanationof that is delayed to the computation of the first loop corrections.

It is easyto obtainF (0) = log Z(0)V= −µz−−2 log(z−) −q2z2−(6)and if we identify T = µ2, we also getE(0)=12µ3z−C(0)=34µz−+ 14µ2dz−dµ(7)It is now immediate to find the behaviour at the critical point (δT = µ2 −µ2cr = µ2 −8q):E(0)=2µ2cr −2µcr√δT + O(δT)C(0)=−µcr√δT+ O(1)(8)Form these equations we can read immediately that the mean field valueα = 12 as it should be.Let us now turn to the exam of the first loop correction. In order to decidewhich of the two stationary points we must choose, we use the observationthat if we cannot find a proper path γ for the function S(z) = S|zi=z, wesurely cannot find a proper hypersurface Γ.

Now since the path γ has tosurround the origin and z+ is a maximum, we cannot find a proper path γfor S(z) that crosses its saddle but only paths that raise and descend thesaddle on the same side, it follows that there is no proper path γ and henceno proper Γ. It follows that we must choose z−; it is not difficult to check thatthe path γ(t) = z−eit with 0 ≤t < 2π has all the characteristics necessaryfor the application of the saddle point method.If we compute the first loop correction on an hypercubic lattice, we findF (1) = 12logπ −12Z π−πdDp(2π)D log 2z2−−2Xνcos(pν)!

(9)3

from which we can compute both the correction to the energy and to thespecific heat due to the first loop correction, explicitly we haveE(1)=− µz−!3 dz−dµdF (1)d(1/z2−) ==− µz−!3 1q1 −2µqµ2 −µ2crZ π−πdDp(2π)D12z2−−2Pν cos(pν)(10)andC(1)=−3µ2z3−dz−dµdF (1)d 1z2−+ 2µ2z4− dz−dµ!2 dF (1)d 1z2−−µ22z3−d2z−dµ2dF (1)d 1z2−−µ22z3−dz−dµd2F (1)d1z2−2 ==−µ4qz3−1qµ2 −µ2cr3Z π−πdDp(2π)D12z2−−2 Pν cos(pν)−2µ2qz3−1qµ2 −µ2crZ π−πdDp(2π)D1( 2z2−−2 Pν cos(pν))2 + . .

. (11)Using the fact that1z2−−1z2−cr ∼2qµ2cr√δT we easily deduce the critical behaviourof these quantities:F (1)∼(√δT)D2 log(δT)E(1)∼(√δT)D2 −2C(1)∼(√δT)D2 −4(12)From the first equation we deduce immediately that we should have a firstorder transition for D ≤4, while from the comparison between the secondone and the second of (8) we get the condition D2 −4 > −1, i.e.

D > 6.Now we would comment about the rigour of these results.The firstobservation is that we applied the saddle point expansion without any ”large”parameter and hence we should keep in account also the higher order, as wewill show the higher order give milder divergences on α provided D ≥6. Toshow this we notice that the coefficient of a generic n-vertex is ∼1zn−and thatthe generic contribution to the L loop correction to F (L) for µ ∼µcr isF (L) ∼Yn≥3 1zn−!Vn ZdDp1 .

. .

dDpL1∆( 1z2−) + q21. .

.1∆( 1z2−) + q2I(13)4

where Vn is the number of n vertices and I is the number of internal lines.Using ∆( 1z2−) ∼√δT, we get immediatelyF (L)∼√δTDL2 −IzI−E(L)∼√δTDL2 −I−1C(L)∼√δTDL2 −I−2(14)With the help of 2I =Pn nVn (the graphs have no external legs) and ofL = I −Pn Vn + 1, we find DL −2I = D + 12Pn≥3(D(n −2) −2n)Vn andhence for D ≥6 it follows that DL −2I ≥D. Finally we get −α(L) ≥D2 −2 ≥1 and similarly we find that the energy cannot diverge.

All thismeans that the method is selfconsistent and that for D < 6 the fluctuation areimportant and the renormalization group is needed. As far as the dimensionwhere the first order phase transition takes place, we cannot be sure thatthe found value 4 is the correct one because from the previous discussionwe know that this value is deep inside the region where the analysis is notmore reliable.The proof of the fact that in D = 4 no first order phasetransition takes place was done long time ago in ([4]) in connection to theYang-Lee edge singularity.

In this work it was performed an analysis of thescaling dimension of the irrelevant operators in D = 6, like φ4, near the fixedpoint in lower dimension: the result is that the φ4 term is still irrelevantnear four dimension. Obviously the dimer problem and the Yang-Lee edgesingularity problem are equivalent, i.e.

are in the same universality class,until the dimers have a second order phase transition but for D < 3 this isno longer true and they can be different as they actually are: for instance inD = 1 we have σY L = −12 and αdimer −1 = ”σdimer” = 1.There is also another possible approach in order to justify this result: wecould imagine of replacing S by NS in (4) and then let N →∞, this isthe point of view advocated in ([5]), for instance. Performing backward thesteps that lead from (1) to (4), we discover that this substitution is equivalentto perform the substitutions3 Ql∈G d¯ǫldǫl →Ql∈GQA=2N−1A=1d¯ǫAl dǫAl and ¯ǫǫ →3 This is easily understandable looking to the number of derivatives implied by−2N log(z) in (4) and then looking for the number of grassman variables needed to havesuch a number of derivatives form a formula similar to (3).5

PA=2N−1A=1¯ǫAǫA in (1) ,where A labels different replica of the original grassmanvariables ¯ǫ and ǫ. This in turn would imply a different HTE expansion: wewould get a gas of branched polymers with loops made of N different typesof monomers, each type being selfavoiding and having a negative activity.Consequently we would describe very different objects when N >> 1 andhence this point of view is not completely satisfactory in this case.In conclusion we show that there is no contradiction between α =12and Dcr = 6 and that the apparent paradox is due to a the fact that thedependence on√δT does not cancel in the free energy differently from whathappens in the usual Landau-Ginzburg.References[1] S. Samuel, J.

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Wallace, J. Phys. A12 (1979) L47[5] E. Br´ezin, J.C. Guillou and J. Zinn-Justin, Phase transitions and criticalphenomena, editors C. Domb and M.S.

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