Zero Potts models coupled to two-dimensional quantum gravity

PHYSICS LETTERS B

Physics Letters B 277 (1992) 405--410 North-Holland

Zero Potts models coupled to two-dimensional quantum gravity D.A. J o h n s t o n Department of Mathematics, Heriot-Watt University. Riccarton, Edinburgh EH14 4AS, Scotland, UK

Received 5 January 1992 The critical exponents that have been calculated for the Ising and Potts models coupled to two-dimensional quantum gravity correspond to annealed averages in the language of solid state physics. Using the replica trick and the approach of DDK we calculate the critical exponents for the Ising and q = 3, 4 state Potts models coupled to quenched two-dimensional quantum gravity.

Although the solution o f sub-critical string theories in physical dimensions d > l has so far proved elusive the work in ref. [ 1 ] by Knizhnik, Polyakov and Zamolodchikov ( K P Z ) and in ref. [2] by Distler, David and Kawai ( D D K ) , in the light-cone and conformal gauges respectively, allowed the calculation o f conformal weights in theories with central charge c < I coupled to two-dimensional quantum gravity. The authors of ref. [ 1,2] showed that the effect of coupling c < 1 models to gravity was to "dress" an operator o f conformal weight A0 in the original theory without gravity to give a new weight A: a -A0

= - ½f~J(a

- 1),

(1)

where

¢_

v'l-c).

(2)

Eq. ( 1 ) is called the K P Z scaling relation. Various standard statistical mechanical models, such as the Ising and q < 4 Potts models in the unitary series have c ~< 1 and can thus be coupled to 2D gravity using the K P Z / D D K results. If we denote the critical temperature for the continuous spin-ordering phase transition in these models by Tc and the reduced temperature I T - Tc{/Tc by t then the critical exponents a, r , 7, v, 6, r/are defined as t ~ 0 by [3] C -~ t-~; M"

t t~,

T < TG

Z ~ t-v;

~ _,2 t-v;

Elsevier Science Publishers B.V.

(3)

M ( H , t = O) ~- H l/b,

H ~ 0;

1 ( - l g ( x ) M ( y ) ) ~ Ix - yl d-2+'7'

(3 cont'd)

t=O,

where C is the specific heat, M is the magnetization, Z is the susceptibility, ~ is the correlation length and H is an external field. It is possible to calculate ct and fl using the conformal weights of the energy density operator A, and spin operator A~ (for a review see ref. [4] ):

'~-

1 - 2A~ l-A,'

~'=

A~ 1-A----~

(4)

in both theories with and without gravity - we use either the original conformal weight or that given by the K P Z formula, respectively. Given a and fl we can now use the various scaling relations [3] c, = 2 - v d ,

y = v(2-q),

r=

½u(d-2+r/), 6_

d+2-r/ d-2+r/

(5)

to obtain the other exponents ~t . The statistical mechanical models which give the c ~< 1 conformal field theories coupled to 2D gravity as their continuum limits can be formulated as #t This, of course, assumes that the scaling relations are still valid when the models are coupled to gravity. The lsing model coupled to 2D gravity has been solved exactly and the exponents satisfy (5) providing some support for the assumption. 405

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various types of matter living on dynamical triangulations or their duals, dynamical phi-cubed graphs. Indeed, the Ising model coupled to 2D gravity was solved exactly [5,6 ] before the work of K P Z / D D K by using this approach to write the partition function as a matrix integral [7]. More recently, the matrix model approach has allowed the exact non-perturbative solution of many c < 1 models by exploiting the double scaling limit [8 ]. The Ising model partition function on random graphs G with m nodes can be written as

Z(rn) = ~ Z e x p GCm) { a }

fl Z G i s t.

x

(i j)

of [2]. The continuum equivalent o f Z (m) in eq. (6) is the fixed intrinsic area partition function

(7) where we have fixed the metric g = e ¢+ ~, with g some background metric, A is the area, 6 is the Liouville field and X is the matter. We can write the action S as S¢ + SM where the Liouville action is

a, aj+ n Z o ' i i

1 f d2z(0,6..~,6 _ ¼Qv/-~.R¢o)"

(6) where fl here is the inverse temperature (not to be confused with the critical exponent), G (m) are the adjacency matrices of the graphs, H is an external field and the as are the usual +1 Ising spins. The sum over the graphs, ~ a , that appears is the discretization of the integral over the two dimensional metrics in the continuum approach. Various simulations of the Ising model on dynamical triangulations and phi-cubed graphs have been carried out [9], finding reasonable agreement with the exponents predicted from the exact solution and the work of K P Z / D D K . Recently, simulations of the q = 3, 4 Ports models (where no exact solution is available) have also given good agreement with the K P Z / D D K values for exponents [ 10]. If we look at these results we see that they correspond to annealed dynamics in the standard solid state terminology because the graphs or triangulations are changing on a comparable time scale to the spin models. It is also interesting from the statistical mechanics point of view to ask whether we obtain a new set of critical exponents when the graphs are changing on a much slower timescale than the spin-models and the simulations are quenched. In this case we would carry out simulations on different fixed random graphs generated by 2D gravity and then average over these to obtain our measurements. In this paper we will calculate the conformal weights in such a theory, and hence the critical exponents, by making use of the conformat gauge Liouville theory approach of D D K and the replica trick [ 11 ]. We first review the calculation of the conformal weights in the annealed case, following the methods 406

12 March 1992

(8)

and the matter action in a Feigin-Fuchs [ 12 ] representation is

SM = ~-~ If

d2z(OX OX + ½i% V/-~RX).

(9)

The central charge of the matter is then given by c = 1 12ag. If we demand that the total central charge of the Liouville plus matter theory, vanish we find Q = V/½(25 - c). Similarly, demanding that e ¢~ be a ( 1, 1 ) operator to make g invariant gives eq. (2) for and demanding that dressed conformal fields q~ e v* have weight ( 1, 1 ) gives

P± --

1 (V/~5-c:Fv/1-c+24Ao) ,

2v~

(lO)

where A0 is the "bare" conformal weight of ~ . Using a scaling argument on the one-point function F,(A)

=

[Z(A)]-~[fD,6DXexp(-S) (11)

then shows that the new con formal weight or "gravitational scaling dimension" defined by F~ (A) "~ A l-~ is given by A= = 1 - (p~/¢) or A± =

V/I-c+24Ao,/~-

e - ,/-1-

v/l-c c

which satisfies the KPZ relation.

,

(12)

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PHYSICS LETTERS B

In the quenched case we must calculate the free energies before summing over the disorder, which in this case is the random graphs, so we have

12 March 1992

one point function of an operator • in the quenched theory by considering

F~(A)"~ ° F = E

log Z,

[z.(n)]-' If D ~ D X e x p ( - S

n)

G(m)

Z = ~exp|fl~)_ t% lot L - (ij)

+H

ai

, (13)

which is hard to calculate directly because of the unfamiliar logarithm. The replica trick consists of replacing l o g Z = ( Z ~ - l )/n, n ---, 0 to give something that looks like a more familiar statistical mechanical partition function. We thus calculate quantities of interest with n identical copies o f the original theory in this approach and take the n ~ 0 limit to obtained the desired quenched averages. An Ising spin, for instance, acquires an extra replica label a, ~ a,~, a = 1. . . . . n. These methods have led to great advances in the theory of spin-glasses and disordered systems [13]. The usual statistical mechanical quantities such as the magnetization can also be calculated in this formalism [ 14]. For example, if we denote the thermal (spin) averages by ( ) and the averages over the disorder (graphs) by [ ]a then the magnetizalion per spin, M, is

(17) where we have inserted one copy ~ a of tb. S" now contains n copies of the matter action S:

a=l

(18) and we have divided by Z, (A), the fixed area partition function with the action S", because we shall be interested in obtaining scaling dimensions. The background charge Q in the Liouville theory is modified to Q' = x/½ (25 - nc) for n copies ofconformal matter with central charge c. Applying the same scaling arguments as before gives us the gravitational scaling dimensions

A'~ = dz V/l - nc + 243o - x/1 - nc x/~-

M=

[(a,)]a = [ T r a ~ e x p [ - S ( G ) ] ] Z(G)

(14)

a'

where we have explicitly written the dependence on the random graph G. The average over the disorder looks awkward to perform because of the dependence on the random graph G in both the numerator and denominator but the problem can be sidestepped by noting (15)

The denominator in this expression tends to one as n --~ 0 which allows us to write M ,,~0 ((a a)),

(16)

where the a ~ are the spins in one replica and the (()) denotes an average with an S" composed of n copies of the original S. If we now translate this approach into the continuum language of D D K we see that we can calculate the

-

,

(19)

nc

3q~e,ched = ¼+V/1 + 2 4 3 0 - 1,

(20)

where A0 is again the bare conformal weight of q5. These new quenched dimensions satisfy --

,~4 = [ ( Z ( G ) ) n - t T r a i e x p [ - S ( G ) ] ] G

41

where, because we intend taking n ~ 0, we do not worry about nc exceeding 1. If we now take this limit we find

,dquenched

[Z(G)]"

nc -

330 =

2 - 2~quenched

,

(21 )

which is just the KPZ relation for pure gravity which has c = 0, or ~ = - 2 / v ~ . We can now use eq. (20) to calculate thc dimensions of the energy and spin operators in the Ising model coupi'ed to quenched 2D quantum gravity, giving 3, = ~ ( x / ] - 3 - 1 ) =0.6513878, A~

_-

,((5g -

~

1) ~- 0.1452847.

(22)

We do not expect to find spin-glass like behaviour in the model because the bonds on the graph, although 407

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PHYSICS LETTERS B

12 March 1992

Table 1 Critical exponents for lsing models. Type

a

fl

y

~

v

~/

fixed annealed quenched

0 - I -0.8685169

1/8 1/2 0.4167516

7/4 2 2.0350137

15 5 5.8830375

I

1/4

3/d

2 - 2d/3

2.8685169/d

2 - 0.7094306d

8

u

Table 2 Critical exponents for q = 3 Potts models. Type

~

fl

?

fixed annealed quenched

1/3 - 1/2 -0.2932676

1/9 1/2 0.3511286

13/9 3/2 1.5910104

14 4 5.531133

5/6

4/15

5/2d 2.2932676/d

2 - 0.6937744d

r/

2 - 3d/5

Table 3 Critical exponents for q = 4 Ports models.

Type

a

fl

~'

~

v

fixed annealed quenched

2/3 0 0.3009442

1/12 1/2 0.2468468

7/6 1 1.2053662

15 3 5.8830375

2/3

1/4

2/d 1.6990558/d

2 - d/2

randomly connected, do not alternate in sign to give rise to frustration. We would however expect to see a standard spin ordering transition and it therefore makes sense to calculate the exponents ~ and fl using the above values for d~, do. This gives a =

6 - 2v53 5-vq3

p_ -i 5- ~

"~-0.8685170,

-~ 0.4167516.

(23)

It is interesting to note that that the values for a and fl in the quenched model lie between those for the lsing model on a fixed lattice a = 0, fl = 1/8 and on an annealed dynamical lattice c~ = - 1 , fl = 1/2. The negative value of c~ shows that the transition is higher than second order on the quenched lattices (the exact solution in the annealed case displays a third order transition). Assuming the validity of the scaling relations in eq. (5) once more, we can calculate the other exponents from these. We summarize the exponents for the standard lsing model, the Ising model 408

2 - 0.7094306d

coupled to 2D gravity and the lsing model coupled to quenched 2D gravity in table 1. It is, of course, also possible to calculate the quenched exponents for other models, such as q = 3, 4 Potts models coupled to quenched 2D gravity and we have done this in tables 2 and 3 #2 In both the annealed and quenched cases the internal fractal dimension d is not known a priori so we have expressed ;7 and v as functions o f d . An interesting side issue arising from numerical simulations is that there does not appear to be an internal fractal dimension at all in pure 2D gravity if one uses a naive geometrical definition of distance (simply counting lattice links) to measure it. It was found in ref. [15] that the relation between the area A and the radius of a circle in pure two-dimensional gravity (i.e. graphs alone, with no spins) was of the form logA = a + b l o g r + clog2 r,

(24)

,r2 Note that ~/ for q = 4 is misprinted in rcfi [10] - the value of 0.25 here is correct.

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PHYSICS LETTERS B

which means that there is no internal fractal dimension. Quenched simulations would be done on fixed graphs generated by pure gravity, so the internal geometry would be just as singular here. Indeed, it is difficult to see how any sort of thermodynamic behaviour could emerge in annealed simulations as well if there is no internal fractal dimension. It is interesting to note, however, that the geometrical definition of the fractal dimension gives _~ 2.8 in the Potts models coupled to (annealed) gravity simulated in ref. [ 10], but these were on very small meshes in comparison with the pure gravity simulations. If these results are not confirmed on larger meshes the resolution of the difficulty may well be that the spin models are sensitive to the "physical geometry" where one defines distances by using the asymptotics o f a massive propagator. This appears to be much less singular - indeed a finite fractal dimension o f around 2.7 n3 has been found using this definition for pure 2D gravity [16]. To obtain the exponents calculated here one would have to simulate the Ising model on a collection of graphs generated by 2D gravity and take the average over the different simulations at the end. We sec from the table that the numerical values of the quenched exponents are closer to those of the annealed exponents than the fixed exponents. However, it appears to be a very challenging task to distinguish numerically whether we are seeing new behaviour in a quenched simulation instead of a collection of fixed simulations. C.F. Baillie has found that a set often lsing model simulations carried out on graphs generated by 2D gravity is insufficient to distinguish between the fixed and quenched exponents [17 ]. At least one hundred individual simulations may be needed to get reasonable statistics. Various simulations have been carried out previously on single fixed random lattices that have a "wellbehaved" distribution of points [ 18,19 ] finding resuits consistent with those on a regular lattices. In these it appears that we are dealing with "weak disorder" and it is possible to consider the continuum limit on a single lattice. The Harris criterion for whether critical properties are unchanged by weak disorder that is discussed in ref. [ 18] suggests that disorder will have no effect if c~ < 0 (or c~ < ve if there are ,~3 Remarkably similar to the --~ 2.8 of ref. [ 10]!

12 March 1992

correlations with tr 2 ~ ( d - , ) . l f a > 0 it suggests that a sharp transition with the critical behaviour of the original system is inconsistent. This might apparently rule out critical behaviour in the quenched q = 3, 4 state Potts models, with the Ising model as a marginal case. However, the disorder displayed by graphs generated in 2D gravity can in no sense be regarded as weak, as the measurements in ref. [ 15 ] make clear, so it is probable that this criterion does not apply. The differing quenched exponents indicate that we are, in any case, seeing a different sort of critical behaviour that is closer to that on the dynamical graphs. It is not clear whether it is possible to take the continuum limit on a single graph arising from a simulation of 2D gravity because of its rather singular nature. A simulation of the lsing model on such a graph has, however, found values consistent with the fixed mesh exponents [20]. In summary, we have calculated the critical exponents for the Ising and q = 3, 4 Ports models coupled to quenched 2D gravity and found values differing from both the fixed and annealed values. We might expect to find these exponents when averaging over a large number of simulations on fixed graphs generated by 2D gravity. We have suggested that the irregular nature of these graphs allows us to sidestep the Harris criterion for the (ir)relevance of weak disorder. I would like to thank C.F. Baillie for useful discussions.

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D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127; E. Br6zin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [9] Z. Burda and J. Jurkiewicz, Acta Phys. Polon. B 20 (1989) 949; V.A. Kazakov, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 93; J. Jurkiewicz, A. Krzywicki, B. Petersson and B. S6derberg, Phys. Lett. B 213 (1988) 511; R. Ben-Av, J. Kinar and S. Solomon, Nucl. Phys. B (Proc. Suppl.)20 (1991) 711; S.M. Catterall, J.B. Kogut and R.L. Renken, Scaling behaviour of the Ising model coupled to twodimensional gravity, Illinois preprint ILL- (TH)-91-19 (1991). [10]C. Baillie and D. Johnston, A numerical test of KPZ scaling: Ports models coupled to two-dimensional quantum gravity, Colorado preprint COLO-HEP-269.

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