Nuclear Physics B336 (1990) 349-362 North-Holland
THE YANG-LEE EDGE SINGULARITY ON A DYNAMICAL PLANAR RANDOM SURFACE
Matthias STAUDACHER
Loomis Laboratory of Physics, University of Illinois at Urhana-Champaign, Urbana, IL 61801, USA
Received 13 November 1989
We reconsider a recently solved Ising model on a random planar graph. The Yang-Lee edge singularity, familiar from the ordinary Ising model, is exposed. It is shown to correspond to an exactly solvable critical dimer counting problem on the random surface in the infinite temperature limit. This suggests an interesting interpretation of a recently proposed phenomenological model exhibiting multicritical behavior. The critical exponents are found to be y= -3 (string suscepti- bility) and a = '-, (edge singularity). The result is at odds with the Knizhnik-Polyakov-Zamo- lodchikov formula in conjunction with the Yang-Lee edge singularity's central charge C= - sz . Possible explanations are discussed. The result a= z coincides with the corresponding exponent of the ordinary three-dimensional spherical model, as does the set of exponents of the random
[sing critical point found previously.
1. Introduction
Over the past few years progress has been made in developing a theory of random surfaces. The idea is to generalize the Feynman path integral to a "surface integral", i.e. we want to learn to take sums over two-dimensional structures. It is hoped that such a theory could be successfully applied to string theory, gauge theory (flux tubes!), 3d critical phenomena (domain walls!), 2d quantum gravity and other fields we cannot even imagine yet -just think about the wide applicability of random walk theory. However, random surfaces are quite different from random walks and unfortunately much more difficult to handle. Oddly, the first workable proposal of how to define such a sum did not - as in the one-dimensional case - involve any limiting procedure utilizing discrete approximants, but was a continuum theory to begin with: the Polyakov string [1]. The crucial idea there is to first sum over an abstract metric and subsequently over embeddings in space. Inspired by Polyakov's formulation a promising discrete version was suggested a few years ago [2]. The sum over metrics of a given topology becomes a sum over graphs of the same topology which are also subsequently embedded. Kazakov introduced the interesting idea of placing Ising spins on the vertices of the graphs [3], thus considering an Ising model on a fluctuating, dynamical lattice. He was able to exactly solve the model and
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obtain all the critical exponents. Progress was also made in the continuum theory.
Knizhnik, Polyakov and Zamolodchikov (KPZ) gave general formulae for the critical exponents of conformal field theories in the presence of two-dimensional quantum gravity [4] . They conjectured that the string susceptibility exponent y in the case of spherical topology for a theory with central charge C be given by
7= 112 [C-1- (1-C)(25-C) ] .
(1 .1)
Subsequently, this formula and its generalization to higher topologies was also obtained by different arguments [5]. The Polyakov bosonic string in D dimensions corresponds to C =D, Kazakov's Ising model to C = ', and for the first time a direct comparison between the discrete and the continuous formulations was possi- ble. The results coincided in all cases where the discrete model could be solved exactly : C = -2 and C = 0 [2], C = '-z [3], C = 1 [6]. The other critical exponents of the Ising model also agreed.
In ref. [7], theO(N) model on the random surface was considered. The model can be critical for N E [-2, 2] and it is possible to obtain y for all N.Using a conformal field theory result relating N and C, one derives eq. (1 .1). There is thus compelling evidence that the KPZ theory and the discrete approach are equivalent in the interval CE [ - 2,1].
There remain, however, many puzzling questions. Most notably, eq. (1 .1) becomes nonsensical for C > 1 . This is exactly the regime of physical dimensions of the bosonic string. Both approaches so far fail to make solid predictions for that range of C. Also, exact solutions of the discrete model for C < -2 are so far missing, thus leaving the interval CE (-oc, - 2) unprobed.
In the present work we attempt to address the last problem. More specifically, we will be considering the Yang-Lee edge singularity which has been argued to have C= - 25 [8]. We will obtain the exact critical exponents of this model on the random surface. The result is y = - 3 (string susceptibility) and a = z (edge singularity). This value of y is at odds with the KPZ prediction because eq. (1 .1) yields y = - for C = - 52 . Two explanations are possible: Either the KPZ formula breaks down or the discrete formulation fails to describe the fluctuating surface. One might speculate that some kind of phase transition occurs at C = -2 . However, most likely this puzzling result will find its explanation through the presence of the Yang-Lee edge singularity's negative dimension operator.
We will also put to use the well-known [14] fact that the critical behavior of the dimer model at negative activity is in the universality class of the Yang-Lee edge singularity . The dimer model on random graphs is easily solved and we again find the above values y = - ' and a = '-, . This dimer model is identified to be the first member of the series of multicritical points found in the phenomenological ap- proach of ref. [17]. Therefore our result indicates that this series does not corre- spond to the discrete unitary series of conformal field theories [18] .
M. Staudacher / Random surface 351
2. The Yang-Lee edge singularity
The Yang-Lee edge singularity occurs in an Ising model above its critical temperature in a nonzero, purely imaginary magnetic field. Consider an Ising model on an arbitrary graph G with n vertices and incidence matrix G,(j") (i.e. G;~") = 1 if vertices i and j are nearest neighbors, G,~) = 0 otherwise). The partition function is
r
n lZ (G ) - Y_ expl'PG;'' 1a;a1+H
E
al l, (2 .1)(0) 111
where ß, H are the products of inverse temperature with the coupling constant and the magnetic field, respectively. E, ., stands for Eo The fugacity is defined to be y=exp(-2H). We can easily express Z(G ) as a polynomial in y :
Z(G )= e x ~', Pk y k _ (2 .2)
k=o
In a classic paper, Lee and Yang proved that, for any graph G , the roots of Z(G ) lie on the unit circle in the complex y-plane (Lee-Yang theorem) [9]. For a given, not too pathological family of graphs {G } -1,2, . .,, e.g. two-dimensional square lattices, we can expect the roots to become dense on smooth sections of the unit circle. Expressing the free energy F(G ) (we suppress the factor 1/kT) in terms of the roots ykof Z(G ) as
F(G ) = -logZ(G ) _ -nH -log Il (y -yk) (2 .3) k=1
one obtains for the free energy F per spin in the thermodynamic limit
F = -H-
f7
~dhp(h)log(y-e'h ), (2 .4) which defines the density of zeros p(h). It is easy to see that one has p(h) = p(-h), as Pk = P ,,. . Yang and Lee have shown that for temperatures above the critical one there is a gap h o such that p(h) = 0 for Ih I < h o. This gap vanishes exactly at the critical temperature; so the "pinching" of the y-axis characterizes the Ising phase transition. For temperatures below the critical one no gap exists. Fisher showed that h o can be regarded as a conventional critical point, albeit very different from the Ising point [10]. The characteristic critical exponent a of this edge singularity is defined byp (h) - (h- ho) Q , h --> h 1, . (2.5)
It is obvious from eqs. (2.4) and (2.5) that the magnetization dF/d H has the same singular behavior
dF
dH (H-HO)a, H-Ho . (2 .6)
In two dimensions, Cardy [8] reconsidered the Yang-Lee edge singularity from the viewpoint of conformal field theory [11,18]. He derived that under certain plausible assumptions the central charge of this nonunitary critical theory is given by C
= - s.
He was then able to predict v = - b which is in excellent agreement with older numerical results (see ref. [10]).
We will argue that the situation is quite analogous in the case of random surfaces, giving the possibility of solving a new model.
3. The Yang-Lee edge singularity on a random surface
The microcanonical partition function of an Ising model on a dynamical random lattice [3] is defined to be
Z = Y_ Z(G ),
(3 .1)
{CI }
where Z(G ) is given by eq. (2.1) and {Gn } is a sufficiently large class of planar graphs of given number of vertices n . There seems to be a great degree of freedom in choosing such a class, e.g. taking regular graphs it is immaterial whether we choose coordination number three or four. Defining the grand canonical partition function of the model to be
00
Z = E
S
nZnn=1
(3 .2)
which is seen to be the generating function of the Z , the exact solution in the case of planar, regular graphs of coordination number 4 is given by [3]
Z 'log[g(Z)
~+ 2(g(Z))2 f~dt(g(t))2- g(Z)
f~atg(t),
(3 .3a)
g(Z) = 1C2Z ,+ 3Z 1 2 -C2+ ZB 2 , (3 .3b)
(1 - Z) (1 - Z2)
where
M. Staudacher / Random surface
Z1 = _ 1c.-1V.-1/2(1 +y)
ZZ = 1 c-2 1-1[9+(16c2 +2c4 )v+912]
Z3 =- 1c-3 1-3/2[9+(18('2 +9c4 )y+(18(.2 +9c°) p2 +9`,3]
Z4= 1c-4 1-2 [189 + 432c2 +324c4 ) y + (324c.2 +600c4 +204(.6 - 6c8)12 +(432c2 + 324,.4) y3 -1189y4]
TABLE 1
The first 6 Z~.
Z5=- 51,,-5y-5/2[729 + (1890,,2 + 1755c4 )1, + (1215c2 + 3510c4 + 2295c6 + 270c.8)1,2 +(1215c2 + 3510c4 + 2295c6 + 270cß).13 + (1890c 2 + 1755c4 )y,4 + 729y5]
353
Z6=1 c-6_V-3[8019 + (23328c2+24786c 4) r + (13608C.2 + 51030c4 + 46656c6 + 8991cß)1,2 +(11664c2 + 52488C.4 + 67032c 6 + 26352( 8 + 2808c1° + 36c12 )l,3
+(13608c2 + 51030c 4 + 46656c 6 + 8991(.8)1,4 +(23328 c2 + 24786c 4 ) ti.5 + 801916]
c=e- zß ,
(3 .4a)
B = 2(cosh H - 1) =y112 + y-1/2 -2, (3 .4b)
(3 .4c)
and z is parametrizing the solution.
Using a symbolic manipulation program we obtained Z for n = 1, . . . , 6, see table 1. To check that the orthogonal polynomial method that led to the above expressions is indeed applicable, we also computed Z1, Zz , Z3 by hand directly from the definition eq. (3.1) and compared it to the solutions obtained from eqs.
(3.3a, b) . The agreement is perfect. We then investigated numerically the location of the zeros of the polynomials ynl2Zn in the complex y plane. This was done for n = I__. . , 6 by varying the "temperature" c in steps sufficiently small to observe the flux of roots. The result is that the roots are still located on the unit circle. This is surprising for the following reason : although the Lee-Yang theorem assures us that the roots of each individual Z(G ) have modulus l, the polynomial obtained by summing the Z(G ) cannot a priori be expected to possess that property [see eq.
(3 .1)]. Unfortunately we have not found a proof for this "Lee-Yang conjecture for random planar graphs", as the existing proofs [9,12] for the original case do not seem to carry over in an obvious fashion. One might think that the Lee-Yang theorem holds for any superposition of general random graphs with a given number of vertices (not necessarily planar and/or regular), but it is possible to construct counterexamples to that conjecture. What we do have, however, is the above
"perturbative" result. Assuming the conjecture to hold to all orders, we immediately
realize that the mechanism for the phase transition on the dynamical lattice will be quite analogous to the static case. Since we possess an exact solution, we can investigate its analytical structure and see whether it is consistent with the above conjecture. As shown in ref. [3] we have in the thermodynamic limit
(1 _ c 2) 2 1
C g(Zmin)
where y defines the string susceptibility and Zn,in is the root smallest in magnitude of the equation
g'(z) =0 . The free energy F of the model is seen to be given by
F= log[-g(Zmin)]
How can phase transitions occur in this model? Call "next the root of g'(z) = 0 second smallest in magnitude. As we move in (c, B) space Zrnin and "next move on continuous curves in the z plane. Note that g(z) is analytic (except at z = 1). We
will reach a singular point when Zmin and "next merge, i.e. if g"( Zmin) = 0 .
(n - oc), (3 .5)
(3 .6)
(3 .7)
(3 .8)
Other possibilities would be g(Zmin) = 0 or g(Zmin) = oo . But from eqs. (3.1) and (3.5) we can immediately exclude g(Zmin) = 0. g(z,nin) = oo could occur only in the case Zn,in = 1, which will be seen to be included in the discussion below. So eq. (3 .8) is a necessary and sufficient condition for a phase transition.
Let us now look for the singularities generated by eq. (3.8). From eqs. (3.3b), (3.6) and (3 .8) we need to satisfy gô(zmin) = 0 and g0r'(Zmin) = 0 where
go(z ) =(1+Z)4[1-c2(1-z)4] +2Bz(l+z 2 ), (3.9a) g~(Z) 3 (1 -1 z) 3 (1 +")3gô(Z) (3 .9b)
gô(z) will have a double root precisely when its discriminant (that is, the resultant of
go(z)
and gô'(z)) is zero . This yields an equation algebraic in c and B,which is fourth order in Band can therefore be solved. The calculations are very tedious andM. Staudacher / Random surface 35 5 0.2 0.4 0.6c 0.8 1 .0
Fig. 1. The critical line B , , , (c) of the Yang-Lee edge singularity .O :Ising model, o : dimer model.
were done utilizing symbolic computation; let us therefore just state the result :
(3 .10)
with x = 16c2.
We now need to check whether the obtained double roots are the smallest modulus solutions of gô(z) = 0 ; this was done numerically by varying c and observing the flow of roots. The result is that the double roots are smallest if and only if (see fig. 1)
c > 4+ B=B + . +(c) . (3 .11)
When c =4 and B=B, , +(â) = 0 we recover the well-known Ising critical point; at c = 4, B=B- ~(4) _ -4 its "mirror image", where each Z is weighted by a factor (-1)" due to the magnetic field H =i7r . The latter values of c and B correspond to the above `special' case z,,,;,, = 1 ; it is easily seen that g(1) < oe despite the poles.
For '-, < c < 1 we obtain singularities at B ~, +(c)which are nicely identified as the endpoints of the Yang-Lee cut along [BB+,+ ] in the complex B plane. It is obvious that B (-- ( -4, 0) corresponds to purely imaginary values of the magnetic field : H = iH' with H' real gives B = 2(cos H'- 1) [eq. (3.4b)]. Note how the cut closes in onto the c-axis at the critical "temperature" c = 4 .
Let us also point out the different nature of the singularities at c = 4 (Ising) and along the "critical line"B +,. +(c), c > '(Yang-Lee edge singularity) : For B =0 and c = 4, varying c causes the double root to split into two real roots z,.;. and z,ext.
For c > 4 and B = B+.+ (c) increasing B on the real axis (see fig. 1) causes the double root to branch into two real roots while decreasing it causes splitting into a pair of complex conjugates (with analogous behavior at B = B-, + (c)) . This clearly indicates the cut in the free energy.
Let us now address the problem of extracting the critical exponents of the Yang-Lee edge singularity. From ref. [3] it is clear that
due to g"(zm;n )= 0. Actually, it could be more generally - ' - 1, -
s - 2. .. . .
However, it is easy to see from eq. (3 .3a) that d 3Zldg s diverges [13], so, as Z _(g - gc)z-Y [see eq. (3.5)], those values are excluded.
To compute the critical exponent Q it is necessary to eliminate z between eqs.
(3.3b) and (3.6). This is done by calculating a resultant. The ensuing equation is algebraic in g, B and c and is solved by g(Zmin). The singularities of g(zmin) as a function of B and c are then found by again calculating a discriminant. It can thus be shown that we have for all c > â the branch-point behavior
which gives, using eqs. (3.7) and (2.6)
g(Zmin) -(B-B +, +(c))31z
We will omit the calculations as they are extremely cumbersome. Note the tempera- ture independence of the edge singularity exponent Q .
4. The Yang-Lee edge singularity and the dimer model
The point c = 1, B = - 2 on the critical line found in sect. 3 is special (see fig. 1) and one can expect to obtain explicit expressions for the Yang-Lee edge singularity in the limit c - i (gâ(z) factorizes in this limit). Here we prefer to proceed in a different way, exploiting a well-known result [14] : the infinite-temperature limit of the high-temperature expansion of the Ising model on a given graph G reduces to the hard dimer model on the same graph. This model is defined through the partition function [15]
(c,,)
>~--(G,,) = 1 + r_ 0 (i)~',
(3 .13)
where ~ is the dimer activity and 0ji) is the number of ways of placing i hard dimers on the graph G with n vertices and e(G ) links. A dimer is a "rod" placed on an edge linking two vertices. It is hard if at most one dimer is allowed to be
attached to each vertex.
M. Staudacher / Random surface 35 7
It is readily seen by expanding exp[ZG~")a;aj ] in the Ising partition function Z(Gn) [eq. (2.1)] in powers of ß that
Z(G ) = (e ll +e-H)n{1 +tanh2 H[8
n
(I)ß+O(ß 2 ) ] +(tanh2 H) 2[en
(2)# 2 +0(ß3)] +[cf. eq. (3 .1)] which will be expected to have the asymptotic behavior
d log [kMI - (~ - W Q
(~ 0
(4.2) so defining = ß tanh2H and taking ß -~ 0, tanh2H - oc such that - oo < ~ < 0 we get
Z(G,,)(e"'+e-H)-n=I+B (1)~+On(2)~2+ . . . = z(G1,) . (4 .3) Clearly the above limit requires H --j i7/2, which corresponds to c - 1, B - 2, as announced .
On the dynamical lattice we obviously have
yn= f =(Gn) (4 .4)
(G )
n
..n-nY-3 [j ~ (n oc) (4 .5)
9W
[cf. eq. (3 .5)] with some function g(~) determining the string susceptibility. The critical exponent a, defined in eq. (2.6), will be obtained by noting that in this infinite temperature limit the role of magnetic field H and free energy F are now played by ~ and -log[1/g(~)], respectively ; thus
(4 .6)
Fortunately, this random dimer problem can be easily solved, utilizing well-estab- lished methods [7,16]. This is done by considering the infinite number of compo- nents limit ("large N " limit) of a hermitian matrix model
Zmatrix
=
f-94p _!2Mexp {NTr [ - ZCp2+ ;9T4- zM2+g~MrP3~ ~ (4.7)Here cp and M are NXN hermitian matrices with invariant measures 9T, _!2M.
The perturbative expansion in the "coupling constant" g(not to be confused with the "g " of sect. 3) yields Feynman diagrams whose components are shown in fig. 2 .
I I
with [cf. eq. (4.4)]
suppressed)
9,/T
Fig. 2. (a) (p-propagator (empty bond). (b) M-propagator (bond with dimer) . (c) Unoccupied vertex . (d) Vertex occupied by dimer.
In the N -> oc limit only planar diagrams survive, and it is easily seen that the class of connected diagrams in order g" corresponds precisely to the set of all planar 4-regular random graphs with n vertices and attached dimers (fig. 3). Each diagram is properly weighted with a factor ~`, if the corresponding graph possesses i dimers.
Therefore we have
Zmatrix - eN2-(~), (N-oc) (4.8)
M - yrr(0 911 . (4.9)
~r=t
The gaussian integral over M can be performed, giving (irrelevant constants are
Zmatrix -
f
.9q) exp {NTr[ - 2(P2+ 4 4 + zg2~9~6~ (4 .10) Following ref. [16], in the N - oc limit we obtain a saddle-point equation for theFig. 3. Section of a cp° random planar graph with random dimer configuration.
M. Staudacher / Random surface 359
spectral density u(p), with support [-2a,2a] :
discriminant D is given by
/'2adft u(JA) =- 1 X+'1 gÂ2?a 2 2 + 32g2~X5 . Using the normalization condition J2
Z
ady u(,u) = 1 we derivea 2 - 3g(a 2 )2 - 30g20 a2)3 - 1 =0 . (4.12) This third-order equation in a 2 is readily solved by Cardano's method; the
D= (3 ) 3+ () 2,2
P 30g2~(1 + 1q) '
1 1 1
q 30g2~ (450g~2 + 30g~ +
a=
gq) - 1 [(1 + 1q) 3/2 - 1~ - 1
450 30~
(4 .13a)
(4.13b)
(4.13c)
The radius of convergence g(~) of a 2( g) and therefore
-M
can be found from the condition D = 0, which yields(4 .14)
which displays, via eqs . (4.5) and (4.9) the exact solution of the dimer problem in the thermodynamic limit. Let us note that we have a branch point singularity at
~c = - tô . The negative value corresponding to an "unphysical" activity is expected from the limiting procedure eq. (4.3). It is interesting to observe the "Yang-Lee"
cut in the free energy. In view of eq. (4.6), the exponent a can now be extracted : (4.15) It is also easy to check that y [see eq. (4.5)] "survived" the ß - 0 limit: a 2 is a linear combination of (- zq +tß)113 and (- 11 q-- rD)113, so we have for g --1 g
a2(g) _ (g-g)1/2, a2(g) _(g-g)113,
~ =t ~~C (4.16a)
~=~~ (4 .16b)
where we have used the result that p = 0 at ~ = ~,. This leads to
This completes the task of computing the critical exponents.
As an universality check it is also interesting to consider the dimer model on a planar random lattice with coordination number three. Instead of eq. (4.7) we have
Zmatrix- f_ -gpgMexp{NTr [- 3y~2 + 3gw3
2M
2 +gV, M~2~ ~ (4.18) and eq. (4.10) becomesZmatrix= fQqqexp{NTr [ -39~2+ 3gq93+ 3g 3JIP4 1 (4.l9) Using the methods of sect. 3 it is established that the model undergoes a phase transition at
3, 1055
V15535 + 15360 - 3 -113 = -0 .1347 .
15535 - 15360
The critical exponents are seen to be the same as the ones found above.
5. Discussion
(4 .17b)
We have shown that the YLES exists on a random dynamical lattice in a manner quite analogous to the static case. We succeeded in calculating the exact critical exponents, y = - ; and a = z . Curiously the result is at odds with Cardy's [8]
central charge assignment in conjunction with the KPZ [4] formula: Evaluating eq.
(1 .1) with C = - 5 yields Y = - z .
The first explanation that comes to mind is that the Yang-Lee edge singularity's central charge might be actually C = 3 . Indeed ref. [8] makes certain ad hoc assumptions in order to derive C = - 25 . However, numerical investigations em- ploying the method of finite size scaling have independently confirmed Cardy's result [19]. We have also performed numerical finite size scaling computations on the dimer system, again confirming C = - 5 and reinforcing that the dimer model is in the universality class of the Yang-Lee edge singularity [21].
So we are left with two possibilities : The lattice formulation fails or the KPZ approach breaks down or has to be modified. One might speculate that some kind
M. Staudacher / Random surface 361
of phase transition occurs at C= -2. It is certainly curious that so far all exact solutions have been for the interval CE [ -2, lJ (e .g., [2, 3, 6, 7]). However, the most conspicuous difference between the models solved so far and the Yang-Lee edge singularity is that model's negative dimension operator. It seems probable that this
fact is responsible for our unexpected results.
In ref. [17] it was pointed out that large N single matrix models with general potentials can exhibit multicritical behavior: By fine-tuning the potential it is possible to obtain y = -1/m, m = 2,3. . . It was conjectured that these multicriti- cal points may correspond to the discrete unitary series [18] ; however our identifica- tion of the YLES with a single matrix model (with potential = V(q)) = a(P4 + b996, see eq. (4.10)) seems to disprove this conjecture.
Let us conclude with two further remarks :
- It is gratifying to observe that the hard dimer problem does not exhibit a phase transition at a positive activity ~, just like its static relative [14,15]. This strengthens the general notion that the phase diagrams of models on rigid and dynamic lattices have the same structure; albeit their critical exponents differ.
- The YLES exponent a = z calculated in the present work has an interesting interpretation . It was observed in ref. [3] that the system of critical exponents of the random Ising model coincides for hitherto unknown reasons with that of the ordinary three-dimensional spherical model. However, the exponent a of that model is well known [20]. It turns out to be a = z ! So a neatly completes the set of previously calculated critical exponents.
It is a pleasure to thank J. Cardy, V. Kazakov, T. Klassen, J. Kogut, R. Renken, S. Shenker and especially M. Douglas for interesting discussions. Special thanks to T. Klassen for pointing out ref. [19] . Some of this work was completed during my stay at the University of Chicago, which was made possible through the support of my advisor, J. Kogut. This work was supported in part by the National Science Foundation under grant number NSF PHY87-01775.
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