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Low Temperature Expansion of Matrix Models

arXiv:hep-th/9303146v1 26 Mar 1993PUPT-1384Low Temperature Expansion of Matrix ModelsMark Wexler†Department of PhysicsPrinceton UniversityPrinceton, NJ 08544 USAWe show how to expand the free energy of a matrix model coupled to arbitrary matterin powers of the matter coupling constant. Concentrating on ν uncoupled Ising models—which have central charge ν/2—we work out the expansion to sixth order for ν = 1, 2, and3.

Analyzing the series by the ratio method, we exhibit the spin-ordering phase transition.We discuss the limit ν →∞, which is especially clear in the low temperature expansion;we prove that in this limit the dependence of the model on ν becomes trivial.March 1993† E-mail address: wexler@puhep1.princeton.edu

1. IntroductionOne of the unsolved questions in matrix models [1] is, how do you introduce matter?This is an important question.

If the matrix model is viewed as two-dimensional euclideanquantum gravity, the matter is the statistical system which is coupled to the randomsurface. If the matrix model is viewed as bosonic string theory, the matter represents thetarget space in which the string propagates.

And at least one phase of nonabelian gaugetheory can be reduced to a matrix model with matter [2].A general framework for coupling matter to matrix models has not been found. Theonly exactly solved cases are the open chain of matrices, which gives unitary models withcentral charge c < 1, and the one-dimensional case.

The region c > 1 remains almostcompletely unknown, although it is easy to formulate matrix models with any centralcharge. Recently there have been a number of attempts to develop alternate schemes thatone hopes could deal with matter in this region [3,4].

The method and results of [3] arepromising, but one is nervous about modeling random surfaces out of about ten squares ortriangles, which are rather unwieldy, especially if the surfaces are to be branched polymers.In the study of conventional spin models, low and high temperature expansions haveproven valuable.Maybe such expansions can prove useful for spin systems coupled tolattices as well. Before Mehta solved matrix chains [5], Itzykson and Zuber [6] suggestedexpanding the two-matrix model in low temperature series.This suggestion has beenforgotten, but the problem of coupling c > 1 matter to a fluctuating surface remainsunsolved.

Here we develop a low temperature expansion for matrix models. In each ordern in the matter coupling constant, we get arbitrarily large surfaces in which all the links joinequal spins, except for n “bridge” links.

These bridges connect the several “blobs,” opensurfaces on which all spins are frozen equal; two blobs connected by one or more bridgeshave unequal spins. This is the random surface analog of the ordinary low temperatureexpansion.

There are two advantages of this expansion: one can expand a system witharbitrary central charge to rather high (or infinite) order in the cosmological constant,obtaining surfaces which are—hopefully—close to the continuum; and one can use it tostudy the c →∞limit.1

2. The expansionThe partition function of a matrix model coupled to arbitrary matter can be writtenZQ(g, a) =Z YiDφi e−TrPi V (φi)−Pij Qijφiφj(2.1)where φi are hermitian N × N matrices.

We consider cubic models, with V (φ) = φ2/2 +gφ3/√N. The type of matter is encoded by the number of matrices in the model and thematter coupling matrix Q, which depends on a = e−β, the matter coupling constant.Here we specialize to ν uncoupled Ising models.

They are “uncoupled” only in thebare action, though; each one interacts with the fluctuating surface and they thus interactamong themselves. At the critical point of such a model, the central charge of the matteris ν/2.

For ν Ising models, we need 2ν matrices to represent every combination of spins ateach site. The coupling matrix Q can be considered as the connection matrix of the targetspace graph; the graph is a ν-dimensional hypercube, in which the connection strengthbetween any two vertices is a raised to the power of the dimension of the lowest simplexwhich contains both vertices.

For one or two Ising models, for example, the matrices canbe written†:Q(1) = 120aa0Q(2) = 120aaa2a0a2aaa20aa2aa0(2.2)In the spherical limit, the free energy of these models isF (ν)(g, a) = limN→∞1N 2 log Z(ν)(g, a)Z(ν)(g, 0)(2.3)(which we normalize for later convenience). The problem is to expand F (ν) in powers ofa.

We begin by expanding Z(ν) = z(0) + z(ν)1 a + z(ν)2 a2 + · · ·; of course, we will generateboth disconnected and connected terms, but the former will be canceled by the logarithm.Consider z(1)1 :z(1)1= ⟨Tr φ1φ2⟩= ⟨φαβ⟩⟨φβα⟩,(2.4)where in the last expression the averages are with respect to a single matrix model. Byexpressing the traces in terms of components, we have reduced the calculation to twosurfaces (blobs), each of which has uniform spins, with one link (bridge) joining the two.†A superscript in parentheses will refer to ν.2

Similarly, z2 will contain contributions from two blobs joined by two bridges, or three blobsjoined in an open chain. Obviously, we can repeat this procedure for all multi-matrix modelaverages, yielding contractions of one-matrix model Green’s functions.The object of interest, then, is the one-matrix model average tensor ⟨φα1β1 · · · φαnβn ⟩.Because of the φ 7→U †φU symmetry, it can only depend on δαiβj , and we must keepseparate the upper and lower indices In general,⟨φα1β1 · · · φαnβn ⟩=Xπ∈Πnλn,π T α1...αnβ1...βn (π)T α1...αnβ1...βn (π) = δα1βπ1· · · δαnβπn(2.5)where Πn is the set of permutations of n objects.

Because the matrix components commute,many of the coefficients in (2.5) are equal. We can characterize this by a mapping frompermutations to the partitions of the integer n, f : Πn →Pn, where f(π) is the set of thelengths of the cycles of permutation π; if f(π) = f(π′), then λπ = λπ′.

Therefore we define⟨φα1β1 · · · φαnβn ⟩=Xp∈Pnκn,ph(T α1...αnβ1...βn (π1) + T α1...αnβ1...βn (π2) + · · ·if(π1) = f(π2) = · · · = p(2.6)For n = 3, for example, we have⟨φα1β1 φα2β2 φα3β3 ⟩=κ3,1δα1β1 δα2β2 δα3β3 + κ3,2δα1β1 δα2β3 δα3β2 + δα1β3 δα2β2 δα3β1 + δα1β2 δα2β1 δα3β3+ κ3,3δα1β3 δα2β1 δα3β2 + δα1β2 δα2β3 δα3β1(2.7)(for a given n, the κ’s will be numbered in lexicographic order of the partitions).The task now is to calculate the coefficients κn,p(g). (The reader may wonder whywe do not simply use one-matrix model connected Green’s functions [7] to represent theblobs.

We wish that this were possible, but—despite one’s intuition—blobs that make upa planar surface may themselves be counted by nonplanar Green’s functions.) The moststraightforward method is to contract (2.6) with the various tensors T α1...αnβ1...βn (π), obtaininga closed set of linear equations for κn,p, where the inhomogeneous terms are one-matrixaverages of products of traces.

This becomes quite cumbersome, though, when n gets large.Another method is to contract (2.6) with Λβ1α1 · · · Λβnαn, where Λ is some N × N hermitiantensor; this gives⟨(Tr Λφ)n⟩=Xp∈Pnµn,pκn,pTr Λp1Tr Λp2 · · ·(2.8)3

where p1, p2, . .

. are elements of the partition p, and µn,p is the number of different per-mutations π ∈Πn such that f(π) = p.The averages on the left hand side of (2.8) can be calculated by expanding the externalfield integralZ(g, Λ) =ZDφ e−Tr[V (φ)−Λφ](2.9)in powers of Λ. Fortunately, this integral has been computed in the spherical limit byKazakov and Kostov [8] and by Gross and Newman [9] using loop equations.† Using Grossand Newman’s notation,F(g, Λ) = limN→∞1N 2 log Z(g, Λ) = −12N 2Xa,blog(µa + µb) −16g(σ−2 −x) +2√27g σ−3+ σ1σ−1 +r g48σ31 −1108g2 −14 log 3g(2.10)where µa = √λa + x, σk = 1NPa 1/(λa+x)k/2, λa are the eigenvalues of Λ, and x satisfiesthe equationx = 1/(12g) −p3gσ1(x).

(2.11)We first expand x = x0 + x1Tr Λ + x21(Tr Λ)2 + x22Tr Λ2 + · · ·; the correct root of(2.11) has x0 = 1/(12g) −6g + · · ·. Plugging x into (2.10) and comparing with (2.8), wecan calculate the required coefficients.

There is only one subtlety: we do not want Green’sfunctions where two external legs are connected directly to each other, as this would “shortcircuit” that surface; i.e., the number of dissident links would be one less than required.Since the Green’s functions generated by F are connected, the problem only occurs insecond order; therefore we must subtract the constant term from that Green’s functionbefore we exponentiate F to obtain Z. Proceeding confidently, we can now read offthecoefficients.

The first few are:κ1,1 =−16g +14x0+rx03g √Nκ2,1 =116x20N +−16g +14x0+rx03g2Nκ2,2 =12√3gx0−116x20−1(2.12)† This integral seems to be much harder for a quartic than for a cubic potential. That is thereason why we use a cubic potential in this work.4

and x0 satisfies x0 = 1/(12g) −p3g/x.Equipped with the coefficients κ, we can directly evaluate the free energies (2.3) forany value of ν. The a0 term is of course just 2ν times the free energy of the one-matrixmodel [7].In the Appendix, we give F (ν) through order a3 where the coefficients aregiven for arbitrary ν as exact functions of ν and g; in other words, we have includedarbitrarily large surfaces.

Beyond that, we have calculated F (1), F (2), and F (3) throughorder a6, where the series coefficients have been expanded to order g32, meaning that wehave included surfaces made of up to 32 triangles. We will present a brief analysis of theseries in the next section, but first we will give a summary of the difficulties encounteredin the calculation, and how we can check it.When one expands the interaction term in Z(ν) in powers of a, in each order one getslarge numbers of graphs in target space, which is a ν-dimensional hypercube for ν Isingmodels.

The number of graphs increases rapidly with both ν and the order in a. Many ofthe graphs are isomorphic, and so give the same results.

The choice is between generatingmany labeled graphs, or many fewer unlabeled graphs. In the latter case (which is whatwe did up to order 3), however, one has to solve the difficult combinatorical problem ofhow many ways there are to embed each unlabeled graph in the hypercubic target space.The other major computational difficulty is in expanding the external field integral (2.9);one could, perhaps, speed things up by dropping the quadratic term from V (φ) and thenshifting φ to induce it [10].We can check F (1), of course, against the exact result [11].

For more than one Isingmodel, we can regroup the series into powers of g, and compare with small-surface expan-sions similar to the ones developed by Br´ezin and Hikami [3]. This is a series to muchhigher order in a but much lower order in g than what we have.

One may, of course, alsouse the low temperature expansion to check the small-surface expansion.3. Series AnalysisTo exhibit critical behavior, we perform a series analysis modeled after those in [3].Having expanded to sixth order, we do not expect to calculate the critical exponents, butwe will learn something about the critical behavior.

We first regroup the series into powersof g. The coefficient An of gn is then the sum of the matter partition functions on all the5

n-th order graphs, i.e., surfaces of n triangles; we know it only to order a6, though. If wehad calculated it to all orders in a, its asymptotic behavior at large n would beAn ≈g−ncnζ(3.1)where the exponent ζ = γstr −3.

From the exact solution for ν = 1 one finds [11] thatζ(a) = −7/2, the pure gravity value, for all 0 ≤a ≤1 except the critical point a∗(1) =(2√7 −1)/27 .= 0.1589, which is where the spin ordering phase transition takes place andwhere ζ = −10/3. This is how we will look for the phase transition.To calculate ζ(c), we will use a ratio method [12].

We define r[1]n = An/An−2, q[1]n =n(r[1]n −r[2]n )/2, and for u > 1,r[u]n = nr[u−1]n−(n −2u + 2)r[u−1]n−22p −2q[u]n= nq[u−1]n−(n −2u + 2)q[u−1]n−22p −2s[u]n = r[u]n /q[u]n(3.2)Naively, the asymptotic behavior as n →∞should ber[u]n ≈g−2c1 + O(n−u)s[u]n ≈ζ1 + O(n−u)(3.3)This is true, however, only if there are no confluent singularities. Another difficulty withthe method is that as u gets large, the coefficient of the n−u term can get large as well.We start with one Ising model.

In fig. 1 we plot s[u]32 ≈ζ(a) from the sixth-orderseries for various iterations of the ratio method, u = 3, .

. ., 7.

The known value of a∗(1)is shown as a vertical line. At a = 0, ζ is very close to its exact value for pure gravity,−7/2; for u = 4, for example, we have ζ .= −3.4991.

These graphs are not exactly whatwe would have expected; the peak, for one, is much too high. Nonetheless, fig.

1 does givequalitative evidence for the spin-ordering phase transition. Moreover, as we increase u,we obtain increasingly accurate values for a∗(the closest one being 0.1547 at u = 6),† aswell as lower peaks.

The ratio approximants seem to deteriorate, however, after u = 4,developing a second peak. We will therefore use a compromise value, u = 4.† One can also get very good estimates for a∗from the peak in the specific heat, C ≈d2da2 log g∗.6

In figs. 2–4, we plot ζ(ν)(a) for ν = 1, 2, and 3, keeping terms up to orders 3–6 in a.The right-hand side of the graphs is given only for completeness, as there is no reason totrust the expansion when a is not small; if we had expanded to all orders in a, the curvewould come back to −7/2 at a = 1, because at infinite temperature each spin fluctuatesindependently.

Two things can be claimed with certainty: the plots are showing evidencefor the spin-ordering phase transition which becomes stronger and more realistic as weexpand to higher orders in a; and the critical point gets closer to zero as the central chargeincreases.4. The ν →∞limitAs can be seen for the first three orders from the expression for the free energy in theAppendix, apart from an overall normalization, the coefficient of an in the free energy is ann-th degree polynomial in ν.

This property, although not a priori obvious, can be shownto hold to all orders. The easiest way to see it is through the small-surface expansion [3].There, the coefficient of gm in the free energy is a sum of the (one Ising model) partitionfunctions of all surfaces of area m, with each partition function raised to the power ν. Eachof those partition functions is a 3m/2-degree polynomial in a, so its n-th power cannothave any higher power of ν than νn.

This proves the assertion.We can see immediately that the asymptotic dependence of F(ν) on ν becomes trivialas ν (and therefore the central charge) approaches ∞, since ν will then simply be a multi-plicative renormalization for a. The critical exponents of the system will be independentof ν, and the critical temperature will have the asymptotic behaviora∗(ν) →a∞νasν →∞.

(4.1)This is already approximately true in our results for up to ν = 3. In fig.

5 we plot a∗(ν)for ν = 1, 2, and 3, showing the plausibility of the asymptotics (4.1). It is also intriguingto note that in the third-order expression for the free energy in the Appendix, the graphswhich appear as coefficients of νn in order n are all trees.

If this behavior can be shown tohold in all orders, this will prove that surfaces in the c →∞limit are branched polymers.7

5. DiscussionIn this letter, we have shown how to expand a matrix model coupled to arbitrarymatter in powers of the matter coupling constant.

We have shown that already at ordera6 one can qualitatively observe the spin-ordering phase transition, with good quantitativeresults for the critical coupling. The immediate extension to this work would be to expandto higher orders (in a and g), and to refine the series analysis.

The reader should know thatthe present work was carried out with rather simple algorithms programmed in the usefulbut very slow Mathematica language, running on an overloaded Iris 4D/480S; altogether,the calculations took about a day of real time to execute. For conventional spin models,the low temperature series are typically known to order 20 or 30, which gives accurateresults for exponents; perhaps one could push the present expansion as far?Another possibility is to expand other thermodynamic quantities than the free energy.The magnetic susceptibility series, for instance, is known to converge rapidly.

It is not hardto add a magnetic field to a matrix model [11]. One could, of course, also experiment withdifferent types of matter such as Potts models.Finally, the limit ν →∞is worth studying.

This limit becomes quite clear in thecontext of the low temperature expansion: one must calculate the νn coefficient of the anterm. Many of the calculations leading to the low temperature series drastically simplifyin this limit.The combinatorics on a hypercube, for example, is much simpler whenits dimension goes to infinity.

There exist predictions for behavior of a random surfaceembedded in D →∞dimensions [13] which can be tested. In this way, perhaps we couldlearn whether the central charge c →∞limit is universal.AppendixHere we give the free energy for arbitrary ν to first three orders.† κn,p,ℓis the coefficient† As in eq.

(2.3), the expression here does not include the homogeneous (pure gravity) term.8

of the N ℓterm in κn,p.F2ν =12 κ21,1, 12 ν a +14h κ21,1, 12 + 2κ21,1, 12 κ2,1,−1 + 2κ21,1, 12 κ2,2,0ν2κ22,2,0 −κ21,1, 12νia2 +112hκ21,1, 121 + 6 (κ2,1,−1 + κ2,2,0) + 6 (κ2,1,−1 + κ2,2,0)2+ 2κ1,1, 12 (κ3,1,−52 + 3κ3,2,−32 + 2κ3,3,−12 )ν3 ++ 3κ1,1, 12 (2κ2,2,0κ3,2,−32 + 4κ2,2,0κ3,3,−12 −κ1,1, 12−2κ1,1, 12 κ2,1,−1 −2κ1,1, 12 κ2,2,0) ν2 +2(κ21,1, 12 + κ23,3,−12 ) νia3 + Oc4(1)The κn,p for n ≤0 were given in the text; the third-order coefficients are:κ3,1 =3g + 2x0√3gx08x30(3g −4x30)N −5/2+ 3κ1,1, 12 κ2,1,−1N −1/2 + κ31,1, 12 N 3/2κ3,2 = −N −3/216x30+ κ1,1, 12 κ2,2,0N 1/2κ3,3 =132x30−18x0√3gx0N −1/2(2)9

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Figure CaptionsFig. 1.ζ(a) for ν = 1 (one Ising model), different iterations of the ratio method: u =3, .

. ., 7.

The vertical line marks the known critical point.Fig. 2.ζ(a) for ν = 1, to orders a3, a4, a5, and a6Fig.

3.Same as fig. 2, ν = 2Fig.

4.Same as fig. 2, ν = 3Fig.

5.The critical point a∗(ν) for ν = 1, 2, and 311

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00.20.40.60.81a-3.5-3-2.5-2-1.5-1-0.50zetaFigure 134567

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This figure "fig3-1.png" is available in "png" format from:http://arxiv.org/ps/hep-th/9303146v1

00.20.40.60.81a-3.5-3-2.5-2-1.5-1-0.50zetaFigure 2654300.20.40.60.81a-3.5-3-2.5-2-1.5-1-0.50zetaFigure 3654300.20.40.60.81a-3.5-3-2.5-2-1.5-1-0.50zetaFigure 46543

0.20.40.60.811/nu00.0250.050.0750.10.1250.150.175critical aFigure 5

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