- Email: [email protected]
Solution of Potts-3 and Potts-∞ matrix models with the equations of motion method
29 July 1999
Physics Letters B 459 Ž1999. 575–581
Solution of Potts-3 and Potts-` matrix models with the equations of motion method Gabrielle Bonnet
1
CEA r Saclay, SerÕice de Physique Theorique, F-91191 Gif-sur-YÕette Cedex, France ´ Received 8 April 1999; received in revised form 26 May 1999 Editor: L. Alvarez-Gaume´
Abstract In this letter, we show how one can solve easily the Potts-3 q branching interactions and Potts-` matrix models, by the means of the equations of motion Žloop equations.. We give an algebraic equation for the resolvents of these models, and their scaling behaviour. This shows that the equations of motion can be a useful tool for solving such models. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Random matrices are useful for a wide range of physical problems. In particular, they can be related to two-dimensional quantum gravity coupled to matter fields with a non-zero central charge C w1x. While C F 1 models are relatively well understood, the C ) 1 domain remains almost totally unknown: there is a C s 1 ‘‘barrier’’. When studying C / 0 models, we are led to consider multi-matrix models w2,3x which are often non-trivial. One class of difficult matrix models corresponds to the q-state Potts model Žin short: Potts-q . on a random surface. This model is a q-matrix model where all the matrices are coupled to each other, thus making difficult the use of usual techniques such as the saddle point or the orthogonal polynomials method. Moreover, the q ™ 4 limit corresponds to C ™ 1, thus, by solving Potts-q 1
E-mail: [email protected]
models, we shall gain a new understanding of the C s 1 barrier. In this letter, we show that, contrary to what was previously thought, one can use the loop equations to solve the Potts-3 random matrix model, and we find that the resolvent Žwhich generates many of the operators of the problem. obeys an algebraic equation that we write explicitly. We also show that this method applies when one adds branching interactions Žgluing of surfaces, also called ‘‘branched polymers’’. w4x and we derive the critical line of this extended model. The extension to the model with branching interactions and the study of its phase diagram is necessary to verify w5x’s conjecture about the C s 1 transition. Finally, we apply the method to the Potts-` matrix model, which corresponds to C s `. As this work was approaching its completion, a paper appeared on the dilute Potts model w6x, which partially overlaps our present work. In this article,
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the author also has an algebraic equation for the conventional Potts-3 model. Here, we go further as we consider the Potts-3 q branching interactions model. Moreover, his method is quite different: while he uses analytical considerations on the resolvents and large-N techniques, we solve our model by the loop equations method, which can be extended to finite N problems and is also more adapted to the use of renormalization group techniques w7,8x.
2. The Potts-3 H branching interactions model Let us define: Zs
H dF e
V ŽF . sg
yN 2 V ŽF .
trF 3 3N
F1 Fs 0 0
0 F2 0
0 d1 s 0 1
ž
qc
1 0 0
Ž 1.
ž
trF 2 trFdFd , 2N 2N
/
1 3N
²trd i Fd i F . . . d i F k : 1 2 n
Ž 4.
where i 1 , . . . , i n can be q1, y1 or 0. This trace is non-zero if and only if i 1 q . . . qi n ' 0 Žmod 3.. ² . . . : is the expectation value of Ž . . . .: ² . . . :s
1
yN 2 V ŽF .
HdF Ž . . . . e Z
Ž 5.
A trace will be said to be ‘‘of degree m’’ if there are m matrices F in it. For example, the above trace is of degree k q n y 1. Let us now use the method of the equations of motion Žor loop equations.. If we make the infinitesimal change of variables in Z:
F ™ F q e d i 1F d i 2 . . . Fd i n
Ž 6.
i 1 q i 2 q . . . qi n ' 0
Ž mod 3 .
then we obtain the expression of the general equations of motion:
0
/
t i 1 i 2 PPP i nF k s
with
Ž 2.
0 0 , F3
0 1 , 0
Let us note, for convenience:
g t i 1 . . . i nF 2 q U t i 1 . . . i nF q c Ž tŽ i 1q1.i 2 . . . Ž i ny1.F q tŽ i 1y1.i 2 . . . Ž i nq1.F .
0 dy1 s 1 0
ž
0 0 1
1 0 0
/
Ž 3.
d s d 1 q dy1. We shall also use later the notation d 0 s Id. F , d 0 , d 1 and dy1 are Ž3 N . = Ž3 N . and F 1 , F 2 , F 3 N = N hermitean matrices. c is a general two-variable function, and will mainly appear through its partial derivatives U and c with respect to trF 2r2 N and trFdFdr2 N respectively. If these are constants, then we recover the conventional Potts-3 model Žno branching interactions.. This model was given partial solution by Daul w9x, by considering the analytic structure of the resolvents. He had its critical point and its associated critical exponent. He did not know, however, if the resolvent obeyed an algebraic equation. We shall give here the expression of this equation for the conventional and extended Potts-3 model. We also derive the critical line of the extended model and check it corresponds to Daul’s result in the particular case of the conventional model.
ny1
y
Ý ti . . . i 1
j
t i jq 1 . . . i n s 0
Ž 7.
js1
The first three terms come from the transformation of V ŽF ., and the last one, from the jacobian of the transformation. Eq. Ž7. relates any expectation value of trace containing a quadratic term Ži.e. a F 2 term. to expectation values of traces of lower degrees. The problem is that we do not have any recursion relation for more general expectation values like t i 1 . . . i nF where all the i k / 0. Moreover, when one wants to compute even a very simple trace: for example tF n by using Eq. Ž7., one obtains, w 2n x steps later, a n y w 2n x degree complicated trace which does not contain quadratic terms any more. Thus, the recursion stops there. In fact, this problem can be overcome by a very simple idea: one uses the invariance of traces by cyclic permutations to get rid of the n q 1 degree term in Eq. Ž7.. Then, one obtains relations between general traces, and it is thus possible to compute the
G. Bonnetr Physics Letters B 459 (1999) 575–581
expectation values of any trace in function of the first ones. Let us see now how this idea applies to the computation of the resolvent. We denote:
v i1 i 2 . . . i n s
1 3N
²trd i Fd i F . . . d i 1 2 n
1 zyF
Ž 8.
:
v 0 s v is the usual resolvent. Using the change of variables: F™Fqe
1
Ž 9.
zyF
577
yields Ž v 1y 1 s vy1 1 for symmetry reasons. g v 10y1 q Ž U q c . v 1y1 q c z v y c s 0
Ž 16 .
and similar changes in variables lead to the equations: g vy1 0y1y1 q U vy1y1y1 q c v 1 0y1 q c z v 1y1 y c tF s 0
Ž 17 .
g v 1 0 1y1y1 q U v 1 1y1y1 q c z vy1y1y1 y c t 1y 1F q c vy1 0y1y1 y tF v s 0
Ž 18 .
z Ž U q gz . v y U y gz y g tF y v 2 q 2 c v 1y1 s 0 Ž 10 .
But, to compute vy1 y1 1y1y1 , as mentioned in the comments to Eq. Ž7., we have to substract two different changes in variables and use cyclicity of traces:
Similarly,
F ™ F q e Fdy1Fdy1 Ž z y F .
we obtain the equation:
F ™ F q ed 1Fdy1
1 zyF
Ž 11 .
z Ž U q gz . v 1y 1 y Ž U q gz . tF y g t 1y1F y v v 1y1
Ž 12 .
and, by the means of similar changes in variables, we have the equations: z Ž U q gz . vy1 y1y1 y Ž U q gz . t 1y1F y g ty1 y1y1F q c vy1 0y1y1 q c v 1 1y1y1 y v vy1 y1y1 s 0
Ž 13 .
y g t 1 1y1y1F q c v 1 0 1y1y1 q c vy1y1 1y1y1
Ž 14 .
These equations alone are not sufficient to compute v Ž z .. Indeed, if we intend to calculate v Ž z ., we generate the function v 1y 1Ž z . ŽEq. Ž10... Then, in turn, we generate the function vy1 y1y1Ž z . ŽEq. Ž12.. and so on. As for v functions containing a 0 Ži.e. a F 2 term. such as v 1 0y1 , they are easy to deal with: we know how to compute traces containing F 2 . v 1 0y1 s 31N tr d 1F 2d y 1 z y1 F , will be seen as 31N trF 2dy1 z y1 F d 1. Then the change in variables:
F ™ F q edy1
1 zyF
d1
dy1F 2
Ž 19 .
c Ž vy1 y1 1y1y1 q vy1 1y1y1y1 yvy1 y1 0y1 y vy1 0 1 1y1 . y vy1 y1y1 q tF v 1y1 s 0
Ž 20 .
This equation, as we know how to compute v 1y 1 , vy 1y 1y 1 , vy 1y 1 0y 1 and vy 1 0 1 1y 1 , relates vy1 y1 1y1y1 to vy1 1y1y1y1.
F ™ F q e Ž Fdy1Fdy1Fdy1 Ž z y F . ydy1Fdy1Fdy1 Ž z y F .
z Ž U q gz . v 1 1y1y1 y Ž U q gz . ty1y1y1F y v v 1 1y1y1 s 0
y1
dy1F
yields:
yields: q c v 1 0y1 q c vy1y1y1 s 0
ydy1Fdy1 Ž z y F .
y1
Ž 15 .
y1
y1
F
F2.
allows us to relate similarly vy1 1y1y1y1 v 1y 1y1y1y1. Then
F ™ F q e Ž Fdy1 Ž z y F . ydy1 Ž z y F .
y1
y1
to
d 1Fdy1Fd 1
d 1Fdy1Fd 1F .
Ž 21 .
allows us to relate v 1y 1y1y1y1 to v 1y1 1 1 1 , and we have v 1y 1 1 1 1 s vy1 1y1y1y1 as the roles of d 1 and dy1 are completely symmetric. Finally, as a result of these operations, we have:
vy1 1y1y1y1 q vy1 1y1y1y1 s K Ž z .
Ž 22 .
where K Ž z . only contains easy to compute v functions. We can then write an equation for vy1 1y1y1y1 and thus for vy1 y1 1y1y1 which only involves v functions that we either already know or are able to
G. Bonnetr Physics Letters B 459 (1999) 575–581
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compute similarly as was done during the two first steps of the procedure. That way, our set of equations is closed, and we obtain a degree five algebraic equation for v Ž z .. This expression applies to general expressions of U and c. For the exact expression of this equation see Appendix A. The equation only contains four unknown parameters: tF s
1 N
t 111F s
²trF 1 : , 1 N
t 1y1F s
1 N
²trF 1F 2 : ,
1 N
²trF 1F 2F 1F 3 :
Ž 23 .
These parameters are also those that would be involved if we used the renormalisation group method w10,7,8x to compute the Potts-3 model. The renormalization group flows would relate the conventional Potts-3 to the Potts-3 q branching interactions model, with arbitrary U and c; but the presence of t 111F shows us it would also be related to the dilute Potts-3 model, where one has a N1 trŽF 1 q F 2 q F 3 . 3 term. Finally, the t 1 1y1y1F term shows us it may also be related to more complicated quartic models. We are now going to derive from our equation the critical behaviour and critical line of the model when U s 1 q htrF 2r6 and c is a constant. This is the most common type of extension of a matrix model to branching interactions. The values of the unknown parameters given in Eq. Ž23. are fixed by the physical constraint that the resolvent has only one physical cut which corresponds to the support of the eigenvalues of F . Then, one can study the critical behaviour of the model. It is easy to look for the Potts critical line. Indeed, the scaling behaviour of the resolvent is then, if we denote the physical cut of v as w a,b x:
v Ž z . ; Ž z y a.
1r2
when z ; a and
v Ž z . ; Ž z yb.
6r5
when z ; b
105 c 3 q 4 g 2 s 0, 2480625 c 2 Ž y1 y 4 c q 43c 2 . q 296100 c Ž 15 q 113c . h y692968 h 2 s 0
Ž 25 . Let us note here that when h s 0 Žno branching interactions. we recover the Potts-3 bicritical point which agrees with Daul’s result w9x:
²trF 1F 2F 3 : ,
t 1 1y1y1F s
presence of the 65 exponent leads to simple conditions on the derivatives of the algebraic equation. We obtain:
cs
43
,
gs
'105 2
ž
y3 q '47 41 y '47
3r2
/
Thus, we have shown that the resolvent for the model of Potts-3 plus branching interactions obeys a degree five algebraic equation. We have found the critical line and exponent of this extended model. This extends the results of Daul w9x who had only derived the position of the critical point and exponent of the conventional model. Finally, let us recall that, in a recent paper w6x, Zinn-Justin obtains independently algebraic equations for similar problems. His method, though, does not involve loop equations, and is rather in the spirit of w9x. Moreover, it does not address the problem of branching interactions and thus overlaps our results only in the case of the conventional Potts-3 model.
3. The Potts-` model We are now going to briefly derive the solution for the Potts-` model, from the equations of motion point of view. The purpose of this part is mainly to show the efficiency of our method on this c s ` model. This model was previously studied by Wexler in w11x. Let us denote
Ž 24 .
The corresponding exponent gs is y 15 , which corresponds to the C s 54 central charge of the model. Rather than looking for the resolvent for any values of the coupling constants, it is easier to search for the resolvent only on this critical line where the
2 y '47
Fs
Xs
F1
0 ..
.
Fq
0
Ž F 1 q . . . qFq . N
0
, and
m 1 q= q
G. Bonnetr Physics Letters B 459 (1999) 575–581
We shall define the Potts-q partition function as Z s dF eyN V ŽF .
ZLC s
V ŽF . sg
trF
3
3N
qU
trF
2
2N
qc
tr X
2
Ž 26 .
2N
V ŽF . is of order q when q ™ `. First, let us use the equations of motion to relate aŽ x . s bŽ y. s
have n independent matrices F˜ 1 . . . F˜n . Each of them has the partition function
where
H
1 qN 1 qN
²tr
²tr
1 xyF
: to
Ž 27 .
:
Let us also denote: dŽ x , y. s
1 qN
F™Fqe
²tr 1
H dF˜
g k
eyNŽ 3
trF˜ 3k q
U
trF˜ 2k q ctrF˜ k L C .
2
1
xyF yyX
Ž 28 .
:
xyF yyX 1
Ž 32 .
This must give us LC , provided we calculate ²F˜ 1 :LC in function of LC . This is a solvable problem, but it is much faster to note that
LC s tF 1 q N=q N
1
yields
Ž 31 .
As trF 1 . . . Fn s trF˜ 1 . . . F˜n , we have ²tr X n : s tr²F˜ 1 :LC . . . ²F˜n :L C where ² . . . :L C is the expectation value obtained with the partition function ZLC Žcf. Eq. Ž31... The matrices F˜ k , k s 1, . . . n all play the same part, and ²tr X n : s tr LCn , thus tr LCn s tr²F˜ 1 :n
1 yyX
579
Ž 33 . 1 qN
n:
.n
²tr X s Ž tF , and bŽ y . is is solution. Thus, simply Ž y y tF .y1 . This gives us immediately the solution for aŽ x .: it obeys a second order equation and reads: aŽ x .
ž
x Ž gx q U . q c y y a Ž x . y
bŽ y. q
/
s 12 x Ž U q gx . q c tF
dŽ x , y.
ž y(Ž x Ž U q gx . q ct
q g y gy b Ž y . y c a Ž x . y b Ž y . Ž gx q U . s 0 Ž 29 . We can get rid of dŽ x, y . since, when x Ž gx q U . q y c y y aŽ x . y b Žq . s 0, dŽ x, y . remains finite, thus g y gy bŽ y . y c aŽ x . y bŽ y . Ž gx q U . s 0. This is sufficient to relate aŽ x . to bŽ y .. Moreover, the value of bŽ y . is easy to compute when q s `. Let us briefly summarize this computation: we calculate the value of ²tr X n : in the q ™ ` limit. First: ²tr X n : s ²trF 1 . . . Fn : q O
F
ž/
Ž 30 .
q
Žrecall that all the F i play the same role.. If we now separate the first n matrices from the remaining q y n Žwith q 4 n., and suppose there is a saddle point for the eigenvalues of F nq 1 q . . . q F q , q
then this saddle point is Žin the q ™ ` limit. independent from the matrices F 1 , . . . , Fn . Then, in this limit, up to a change in variables: F˜ k s UF k Uy1 , we
2
y 4 Ž U q gx q gtF .
/
Ž 34 . The Potts-` plus branched polymers model is thus very similar to an ordinary pure gravity model. As previously, we compute the parameter tF by imposing that the resolvent aŽ x . has only one physical cut. The model is critical when aŽ x . behaves as Ž x y x 0 . 3r2 , x 0 being a constant, and the critical point verifies Žas in w11x.: gc s
1
.
1 4'2
and
cc s y
1 2
Ž 35 .
Let us note finally that the loop equations method used here is appropriate for the renormalization group method of w10,7,8x.
4. Conclusion In this letter, we have shown that it is possible to solve the Potts-3 and Potts-` models on two-dimen-
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sional random lattices through the method of the equations of motion. We have obtained a closed set of loop equations for the Potts-3 model, which was thought to be impossible. We have shown that the Potts-3 resolvent obeys an order five equation, and this new knowledge opens the door to the calculation of expectation values of the operators of the model. We have extended the Potts-3 conventional model to Potts-3 plus branching interactions, and given the general algebraic equation and the Potts critical line of this model. Finally, we have shown our method also applies successfully to another Potts model: the Potts-` model. We hope to generalize soon our method to more general Potts-q models, in particular for large-q Potts q branching interactions models.
y 36 c 5 g 2 tf x y 2 c 2 g 4 tf x y 52 c 5 g U x y 4 c 2 g 3 U x q 36 t 1y 1 f c 3 g 3 U x y 18 c 4 g 2 tf U x q 2 c 4 g U 2 x q 54 c 3 g 2 tf U 2 x q 22 c 3 g U 3 x y 24 c 5 g 2 x 2 y c 2 g 4 x 2 q 8 t 1y1 f c 3 g 4 x 2 q 2 c 4 g 3 tf x 2 q 24 c7 U x 2 q 20 c 4 g 2 U x 2 q 26 c 3 g 3 tf U x 2 y 48 c 6 U 2 x 2 q 12 c 3 g 2 U 2 x 2 q 24 c 5 U 3 x 2 q 24 c7 g x 3 q 6 c 4 g 3 x 3 q 4 c 3 g 4 tf x 3 y 64 c 6 g U x 3 q 2 c 3 g 3 U x 3 q 44 c 5 g U 2 x 3 y 16 c 6 g 2 x 4 q 24 c 5 g 2 U x 4 q 4 c 5 g 3 x 5
Acknowledgements
q Ž 24 c 5 g y 12 t 1y 1 f c 3 g 3 q 8 t 111 f c 2 g 4 We thank P. Zinn-Justin for discussing his ideas with us, and we are grateful to F. David and J.-B. Zuber for useful discussions and careful reading of the manuscript.
y36 c 4 g 2 tf y 2 c g 4 tf y 40 c 4 g U y 2 c g 3 U q36 t 1y1 f c 2 g 3 U y 18 c 3 g 2 tf U y 10 c 3 g U 2 q54 c 2 g 2 tf U 2 q 26 c 2 g U 3 q 24 c7 x
Appendix A. The equation for the Potts-3 resolvent Here is the degree five equation for the resolvent of this model, where W Ž x . is related to v Ž x . by W Ž x . s v Ž x . y gx 2 y Ux. y24 c7 q 4 c 4 g 2 y 16 tqy f c 5 g 2 y 12 t 111 f c 4 g 3
y24 c 4 g 2 x y 2 c g 4 x q 16 t 1y 1 f c 2 g 4 x q4 c 3 g 3 tf x y 36 c 6 U x q 4 c 3 g 2 U x q52 c 2 g 3 t y f U x q 12 c 5 U 2 x q36 c 2 g 2 U 2 x y 12 c 4 U 3 x q 12 c 3 U 4 x y4 c 6 g x 2 q 14 c 3 g 3 x 2 q 12 c 2 g 4 tf x 2 y36 c 5 g U x 2 q 10 c 2 g 3 U x 2 q 30 c 4 g U 2 x 2
y 4 t 1y 1 f c 2 g 4 q 8 t 11y1y1 f c 3 g 4 q 68 c 6 g tf
q22 c 3 g U 3 x 2 y 40 c 5 g 2 x 3 q 60 c 4 g 2 U x 3
q 2 c 3 g 3 tf q 3 c 2 g 4 tf2 q 60 c 6 U q 2 c 3 g 2 U
q12 c 3 g 2 U 2 x 3 q 18 c 4 g 3 x 4
y 52 t 1y 1 f c 4 g 2 U q 20 t 111 f c 3 g 3 U
q2 c 3 g 3 U x 4 . W Ž x . q Ž 12 c 6 y 6 c 3 g 2 q 8 t 1y 1 f c g 4 q 2 c 2 g 3 tf
y 20 c 5 g tf U y 4 c 2 g 3 tf U y 36 c 5 U 2 y 3 c 2 g 2 U 2 q 36 t 1y1 f c 3 g 2 U 2 y 36 c 4 g tf U 2 4
3
3
3
3
4
y12 c 5 U y 4 c 2 g 2 U q 26 c g 3 tf U y 12 c 4 U 2 q18 c g 2 U 2 q 12 c 3 U 3 y 44 c 5 g x y 2 c 2 g 3 x
6
y 12 c U q 36 c g tf U q 12 c U q 28 c g x q 2 c 3 g 3 x y 12 t 1y 1 f c 4 g 3 x q 8 t 111 f c 3 g 4 x
q12 c g 4 tf x q 30 c 4 g U x q 20 c g 3 U x q26 c 2 g U 3 x q 6 c 4 g 2 x 2 q 2 c g 4 x 2
G. Bonnetr Physics Letters B 459 (1999) 575–581
q12 c 3 g 2 U x 2 q 39 c 2 g 2 U 2 x 2 q 20 c 3 g 3 x 3 q14 c 2 g 3 U x 3 q c 2 g 4 x 4 . W Ž x .
q4 g 3 U q 26 c 2 g U 2 q 20 c 3 g 2 x q 4 g 4 x q12 c 2 g 2 U x q 18 c g 2 U 2 x q 22 c g 3 U x 2 3
q Ž y Ž c 2 g 2 . q 18 c g 2 U q 4 c g 3 x q 4 g 3 U x 4
References
2
q Ž y12 c 4 g y 2 c g 3 q 4 g 4 tf y 10 c 3 g U
q4 c g 4 x 3 . W Ž x .
581
5
q4 g 4 x 2 . W Ž x . q 4 g 3 W Ž x . s 0 Note that U and c may depend, in the most general case, on t 1y 1F and tF 2 , the latter being related to tF through the equation of motion: g tF 2 q ŽU q 2 c . tF s 0. In this article, we have computed explicitly the critical line for the particular case of c constant and U s 1 q 2h tF 2 .
w1x For general reviews and more references see for instance: The large N Expansion in Quantum Field Theory and Statistical Physics, from Spin Systems to 2-Dimensional Gravity, E. Brezin, S.R. Wadia Eds. ŽWorld Scientific, Singapore, ´ 1993.. w2x M. Staudacher, Phys. Lett. B 305 Ž1993. 332. w3x B. Eynard, cond-matr9801075. w4x S.R. Das, A. Dhar, A.M. Sengupta, S.R. Wadia, Mod. Phys. Lett. A 5 Ž1990. 1041. w5x F. David, Nucl. Phys. B 487 wFSx Ž1997. 633. w6x P. Zinn-Justin, cond-matr9903385. w7x S. Higuchi, C. Itoi, S. Nishigaki, N. Sakai, Phys. Lett. B 318 Ž1993. 63, Nucl. Phys. B 434 Ž1995. 283; Phys. Lett. B 398 Ž1997. 123. w8x G. Bonnet, F. David, Nucl. Phys. B 6166 Ž1999. 1. w9x J.M. Daul hep-thr9502014. w10x E. Brezin, J. Zinn-Justin, Phys. Lett. B 288 Ž1992. 54. ´ w11x M. Wexler, Nucl. Phys. B 410 Ž1993. 377; J. Ambjorn, G. Thorleifsson, M. Wexler, Nucl. Phys. B 439 Ž1995. 187.
Physics Letters B 459 Ž1999. 575–581
Solution of Potts-3 and Potts-` matrix models with the equations of motion method Gabrielle Bonnet
1
CEA r Saclay, SerÕice de Physique Theorique, F-91191 Gif-sur-YÕette Cedex, France ´ Received 8 April 1999; received in revised form 26 May 1999 Editor: L. Alvarez-Gaume´
Abstract In this letter, we show how one can solve easily the Potts-3 q branching interactions and Potts-` matrix models, by the means of the equations of motion Žloop equations.. We give an algebraic equation for the resolvents of these models, and their scaling behaviour. This shows that the equations of motion can be a useful tool for solving such models. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Random matrices are useful for a wide range of physical problems. In particular, they can be related to two-dimensional quantum gravity coupled to matter fields with a non-zero central charge C w1x. While C F 1 models are relatively well understood, the C ) 1 domain remains almost totally unknown: there is a C s 1 ‘‘barrier’’. When studying C / 0 models, we are led to consider multi-matrix models w2,3x which are often non-trivial. One class of difficult matrix models corresponds to the q-state Potts model Žin short: Potts-q . on a random surface. This model is a q-matrix model where all the matrices are coupled to each other, thus making difficult the use of usual techniques such as the saddle point or the orthogonal polynomials method. Moreover, the q ™ 4 limit corresponds to C ™ 1, thus, by solving Potts-q 1
E-mail: [email protected]
models, we shall gain a new understanding of the C s 1 barrier. In this letter, we show that, contrary to what was previously thought, one can use the loop equations to solve the Potts-3 random matrix model, and we find that the resolvent Žwhich generates many of the operators of the problem. obeys an algebraic equation that we write explicitly. We also show that this method applies when one adds branching interactions Žgluing of surfaces, also called ‘‘branched polymers’’. w4x and we derive the critical line of this extended model. The extension to the model with branching interactions and the study of its phase diagram is necessary to verify w5x’s conjecture about the C s 1 transition. Finally, we apply the method to the Potts-` matrix model, which corresponds to C s `. As this work was approaching its completion, a paper appeared on the dilute Potts model w6x, which partially overlaps our present work. In this article,
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 7 0 1 - 7
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576
the author also has an algebraic equation for the conventional Potts-3 model. Here, we go further as we consider the Potts-3 q branching interactions model. Moreover, his method is quite different: while he uses analytical considerations on the resolvents and large-N techniques, we solve our model by the loop equations method, which can be extended to finite N problems and is also more adapted to the use of renormalization group techniques w7,8x.
2. The Potts-3 H branching interactions model Let us define: Zs
H dF e
V ŽF . sg
yN 2 V ŽF .
trF 3 3N
F1 Fs 0 0
0 F2 0
0 d1 s 0 1
ž
qc
1 0 0
Ž 1.
ž
trF 2 trFdFd , 2N 2N
/
1 3N
²trd i Fd i F . . . d i F k : 1 2 n
Ž 4.
where i 1 , . . . , i n can be q1, y1 or 0. This trace is non-zero if and only if i 1 q . . . qi n ' 0 Žmod 3.. ² . . . : is the expectation value of Ž . . . .: ² . . . :s
1
yN 2 V ŽF .
HdF Ž . . . . e Z
Ž 5.
A trace will be said to be ‘‘of degree m’’ if there are m matrices F in it. For example, the above trace is of degree k q n y 1. Let us now use the method of the equations of motion Žor loop equations.. If we make the infinitesimal change of variables in Z:
F ™ F q e d i 1F d i 2 . . . Fd i n
Ž 6.
i 1 q i 2 q . . . qi n ' 0
Ž mod 3 .
then we obtain the expression of the general equations of motion:
0
/
t i 1 i 2 PPP i nF k s
with
Ž 2.
0 0 , F3
0 1 , 0
Let us note, for convenience:
g t i 1 . . . i nF 2 q U t i 1 . . . i nF q c Ž tŽ i 1q1.i 2 . . . Ž i ny1.F q tŽ i 1y1.i 2 . . . Ž i nq1.F .
0 dy1 s 1 0
ž
0 0 1
1 0 0
/
Ž 3.
d s d 1 q dy1. We shall also use later the notation d 0 s Id. F , d 0 , d 1 and dy1 are Ž3 N . = Ž3 N . and F 1 , F 2 , F 3 N = N hermitean matrices. c is a general two-variable function, and will mainly appear through its partial derivatives U and c with respect to trF 2r2 N and trFdFdr2 N respectively. If these are constants, then we recover the conventional Potts-3 model Žno branching interactions.. This model was given partial solution by Daul w9x, by considering the analytic structure of the resolvents. He had its critical point and its associated critical exponent. He did not know, however, if the resolvent obeyed an algebraic equation. We shall give here the expression of this equation for the conventional and extended Potts-3 model. We also derive the critical line of the extended model and check it corresponds to Daul’s result in the particular case of the conventional model.
ny1
y
Ý ti . . . i 1
j
t i jq 1 . . . i n s 0
Ž 7.
js1
The first three terms come from the transformation of V ŽF ., and the last one, from the jacobian of the transformation. Eq. Ž7. relates any expectation value of trace containing a quadratic term Ži.e. a F 2 term. to expectation values of traces of lower degrees. The problem is that we do not have any recursion relation for more general expectation values like t i 1 . . . i nF where all the i k / 0. Moreover, when one wants to compute even a very simple trace: for example tF n by using Eq. Ž7., one obtains, w 2n x steps later, a n y w 2n x degree complicated trace which does not contain quadratic terms any more. Thus, the recursion stops there. In fact, this problem can be overcome by a very simple idea: one uses the invariance of traces by cyclic permutations to get rid of the n q 1 degree term in Eq. Ž7.. Then, one obtains relations between general traces, and it is thus possible to compute the
G. Bonnetr Physics Letters B 459 (1999) 575–581
expectation values of any trace in function of the first ones. Let us see now how this idea applies to the computation of the resolvent. We denote:
v i1 i 2 . . . i n s
1 3N
²trd i Fd i F . . . d i 1 2 n
1 zyF
Ž 8.
:
v 0 s v is the usual resolvent. Using the change of variables: F™Fqe
1
Ž 9.
zyF
577
yields Ž v 1y 1 s vy1 1 for symmetry reasons. g v 10y1 q Ž U q c . v 1y1 q c z v y c s 0
Ž 16 .
and similar changes in variables lead to the equations: g vy1 0y1y1 q U vy1y1y1 q c v 1 0y1 q c z v 1y1 y c tF s 0
Ž 17 .
g v 1 0 1y1y1 q U v 1 1y1y1 q c z vy1y1y1 y c t 1y 1F q c vy1 0y1y1 y tF v s 0
Ž 18 .
z Ž U q gz . v y U y gz y g tF y v 2 q 2 c v 1y1 s 0 Ž 10 .
But, to compute vy1 y1 1y1y1 , as mentioned in the comments to Eq. Ž7., we have to substract two different changes in variables and use cyclicity of traces:
Similarly,
F ™ F q e Fdy1Fdy1 Ž z y F .
we obtain the equation:
F ™ F q ed 1Fdy1
1 zyF
Ž 11 .
z Ž U q gz . v 1y 1 y Ž U q gz . tF y g t 1y1F y v v 1y1
Ž 12 .
and, by the means of similar changes in variables, we have the equations: z Ž U q gz . vy1 y1y1 y Ž U q gz . t 1y1F y g ty1 y1y1F q c vy1 0y1y1 q c v 1 1y1y1 y v vy1 y1y1 s 0
Ž 13 .
y g t 1 1y1y1F q c v 1 0 1y1y1 q c vy1y1 1y1y1
Ž 14 .
These equations alone are not sufficient to compute v Ž z .. Indeed, if we intend to calculate v Ž z ., we generate the function v 1y 1Ž z . ŽEq. Ž10... Then, in turn, we generate the function vy1 y1y1Ž z . ŽEq. Ž12.. and so on. As for v functions containing a 0 Ži.e. a F 2 term. such as v 1 0y1 , they are easy to deal with: we know how to compute traces containing F 2 . v 1 0y1 s 31N tr d 1F 2d y 1 z y1 F , will be seen as 31N trF 2dy1 z y1 F d 1. Then the change in variables:
F ™ F q edy1
1 zyF
d1
dy1F 2
Ž 19 .
c Ž vy1 y1 1y1y1 q vy1 1y1y1y1 yvy1 y1 0y1 y vy1 0 1 1y1 . y vy1 y1y1 q tF v 1y1 s 0
Ž 20 .
This equation, as we know how to compute v 1y 1 , vy 1y 1y 1 , vy 1y 1 0y 1 and vy 1 0 1 1y 1 , relates vy1 y1 1y1y1 to vy1 1y1y1y1.
F ™ F q e Ž Fdy1Fdy1Fdy1 Ž z y F . ydy1Fdy1Fdy1 Ž z y F .
z Ž U q gz . v 1 1y1y1 y Ž U q gz . ty1y1y1F y v v 1 1y1y1 s 0
y1
dy1F
yields:
yields: q c v 1 0y1 q c vy1y1y1 s 0
ydy1Fdy1 Ž z y F .
y1
Ž 15 .
y1
y1
F
F2.
allows us to relate similarly vy1 1y1y1y1 v 1y 1y1y1y1. Then
F ™ F q e Ž Fdy1 Ž z y F . ydy1 Ž z y F .
y1
y1
to
d 1Fdy1Fd 1
d 1Fdy1Fd 1F .
Ž 21 .
allows us to relate v 1y 1y1y1y1 to v 1y1 1 1 1 , and we have v 1y 1 1 1 1 s vy1 1y1y1y1 as the roles of d 1 and dy1 are completely symmetric. Finally, as a result of these operations, we have:
vy1 1y1y1y1 q vy1 1y1y1y1 s K Ž z .
Ž 22 .
where K Ž z . only contains easy to compute v functions. We can then write an equation for vy1 1y1y1y1 and thus for vy1 y1 1y1y1 which only involves v functions that we either already know or are able to
G. Bonnetr Physics Letters B 459 (1999) 575–581
578
compute similarly as was done during the two first steps of the procedure. That way, our set of equations is closed, and we obtain a degree five algebraic equation for v Ž z .. This expression applies to general expressions of U and c. For the exact expression of this equation see Appendix A. The equation only contains four unknown parameters: tF s
1 N
t 111F s
²trF 1 : , 1 N
t 1y1F s
1 N
²trF 1F 2 : ,
1 N
²trF 1F 2F 1F 3 :
Ž 23 .
These parameters are also those that would be involved if we used the renormalisation group method w10,7,8x to compute the Potts-3 model. The renormalization group flows would relate the conventional Potts-3 to the Potts-3 q branching interactions model, with arbitrary U and c; but the presence of t 111F shows us it would also be related to the dilute Potts-3 model, where one has a N1 trŽF 1 q F 2 q F 3 . 3 term. Finally, the t 1 1y1y1F term shows us it may also be related to more complicated quartic models. We are now going to derive from our equation the critical behaviour and critical line of the model when U s 1 q htrF 2r6 and c is a constant. This is the most common type of extension of a matrix model to branching interactions. The values of the unknown parameters given in Eq. Ž23. are fixed by the physical constraint that the resolvent has only one physical cut which corresponds to the support of the eigenvalues of F . Then, one can study the critical behaviour of the model. It is easy to look for the Potts critical line. Indeed, the scaling behaviour of the resolvent is then, if we denote the physical cut of v as w a,b x:
v Ž z . ; Ž z y a.
1r2
when z ; a and
v Ž z . ; Ž z yb.
6r5
when z ; b
105 c 3 q 4 g 2 s 0, 2480625 c 2 Ž y1 y 4 c q 43c 2 . q 296100 c Ž 15 q 113c . h y692968 h 2 s 0
Ž 25 . Let us note here that when h s 0 Žno branching interactions. we recover the Potts-3 bicritical point which agrees with Daul’s result w9x:
²trF 1F 2F 3 : ,
t 1 1y1y1F s
presence of the 65 exponent leads to simple conditions on the derivatives of the algebraic equation. We obtain:
cs
43
,
gs
'105 2
ž
y3 q '47 41 y '47
3r2
/
Thus, we have shown that the resolvent for the model of Potts-3 plus branching interactions obeys a degree five algebraic equation. We have found the critical line and exponent of this extended model. This extends the results of Daul w9x who had only derived the position of the critical point and exponent of the conventional model. Finally, let us recall that, in a recent paper w6x, Zinn-Justin obtains independently algebraic equations for similar problems. His method, though, does not involve loop equations, and is rather in the spirit of w9x. Moreover, it does not address the problem of branching interactions and thus overlaps our results only in the case of the conventional Potts-3 model.
3. The Potts-` model We are now going to briefly derive the solution for the Potts-` model, from the equations of motion point of view. The purpose of this part is mainly to show the efficiency of our method on this c s ` model. This model was previously studied by Wexler in w11x. Let us denote
Ž 24 .
The corresponding exponent gs is y 15 , which corresponds to the C s 54 central charge of the model. Rather than looking for the resolvent for any values of the coupling constants, it is easier to search for the resolvent only on this critical line where the
2 y '47
Fs
Xs
F1
0 ..
.
Fq
0
Ž F 1 q . . . qFq . N
0
, and
m 1 q= q
G. Bonnetr Physics Letters B 459 (1999) 575–581
We shall define the Potts-q partition function as Z s dF eyN V ŽF .
ZLC s
V ŽF . sg
trF
3
3N
qU
trF
2
2N
qc
tr X
2
Ž 26 .
2N
V ŽF . is of order q when q ™ `. First, let us use the equations of motion to relate aŽ x . s bŽ y. s
have n independent matrices F˜ 1 . . . F˜n . Each of them has the partition function
where
H
1 qN 1 qN
²tr
²tr
1 xyF
: to
Ž 27 .
:
Let us also denote: dŽ x , y. s
1 qN
F™Fqe
²tr 1
H dF˜
g k
eyNŽ 3
trF˜ 3k q
U
trF˜ 2k q ctrF˜ k L C .
2
1
xyF yyX
Ž 28 .
:
xyF yyX 1
Ž 32 .
This must give us LC , provided we calculate ²F˜ 1 :LC in function of LC . This is a solvable problem, but it is much faster to note that
LC s tF 1 q N=q N
1
yields
Ž 31 .
As trF 1 . . . Fn s trF˜ 1 . . . F˜n , we have ²tr X n : s tr²F˜ 1 :LC . . . ²F˜n :L C where ² . . . :L C is the expectation value obtained with the partition function ZLC Žcf. Eq. Ž31... The matrices F˜ k , k s 1, . . . n all play the same part, and ²tr X n : s tr LCn , thus tr LCn s tr²F˜ 1 :n
1 yyX
579
Ž 33 . 1 qN
n:
.n
²tr X s Ž tF , and bŽ y . is is solution. Thus, simply Ž y y tF .y1 . This gives us immediately the solution for aŽ x .: it obeys a second order equation and reads: aŽ x .
ž
x Ž gx q U . q c y y a Ž x . y
bŽ y. q
/
s 12 x Ž U q gx . q c tF
dŽ x , y.
ž y(Ž x Ž U q gx . q ct
q g y gy b Ž y . y c a Ž x . y b Ž y . Ž gx q U . s 0 Ž 29 . We can get rid of dŽ x, y . since, when x Ž gx q U . q y c y y aŽ x . y b Žq . s 0, dŽ x, y . remains finite, thus g y gy bŽ y . y c aŽ x . y bŽ y . Ž gx q U . s 0. This is sufficient to relate aŽ x . to bŽ y .. Moreover, the value of bŽ y . is easy to compute when q s `. Let us briefly summarize this computation: we calculate the value of ²tr X n : in the q ™ ` limit. First: ²tr X n : s ²trF 1 . . . Fn : q O
F
ž/
Ž 30 .
q
Žrecall that all the F i play the same role.. If we now separate the first n matrices from the remaining q y n Žwith q 4 n., and suppose there is a saddle point for the eigenvalues of F nq 1 q . . . q F q , q
then this saddle point is Žin the q ™ ` limit. independent from the matrices F 1 , . . . , Fn . Then, in this limit, up to a change in variables: F˜ k s UF k Uy1 , we
2
y 4 Ž U q gx q gtF .
/
Ž 34 . The Potts-` plus branched polymers model is thus very similar to an ordinary pure gravity model. As previously, we compute the parameter tF by imposing that the resolvent aŽ x . has only one physical cut. The model is critical when aŽ x . behaves as Ž x y x 0 . 3r2 , x 0 being a constant, and the critical point verifies Žas in w11x.: gc s
1
.
1 4'2
and
cc s y
1 2
Ž 35 .
Let us note finally that the loop equations method used here is appropriate for the renormalization group method of w10,7,8x.
4. Conclusion In this letter, we have shown that it is possible to solve the Potts-3 and Potts-` models on two-dimen-
G. Bonnetr Physics Letters B 459 (1999) 575–581
580
sional random lattices through the method of the equations of motion. We have obtained a closed set of loop equations for the Potts-3 model, which was thought to be impossible. We have shown that the Potts-3 resolvent obeys an order five equation, and this new knowledge opens the door to the calculation of expectation values of the operators of the model. We have extended the Potts-3 conventional model to Potts-3 plus branching interactions, and given the general algebraic equation and the Potts critical line of this model. Finally, we have shown our method also applies successfully to another Potts model: the Potts-` model. We hope to generalize soon our method to more general Potts-q models, in particular for large-q Potts q branching interactions models.
y 36 c 5 g 2 tf x y 2 c 2 g 4 tf x y 52 c 5 g U x y 4 c 2 g 3 U x q 36 t 1y 1 f c 3 g 3 U x y 18 c 4 g 2 tf U x q 2 c 4 g U 2 x q 54 c 3 g 2 tf U 2 x q 22 c 3 g U 3 x y 24 c 5 g 2 x 2 y c 2 g 4 x 2 q 8 t 1y1 f c 3 g 4 x 2 q 2 c 4 g 3 tf x 2 q 24 c7 U x 2 q 20 c 4 g 2 U x 2 q 26 c 3 g 3 tf U x 2 y 48 c 6 U 2 x 2 q 12 c 3 g 2 U 2 x 2 q 24 c 5 U 3 x 2 q 24 c7 g x 3 q 6 c 4 g 3 x 3 q 4 c 3 g 4 tf x 3 y 64 c 6 g U x 3 q 2 c 3 g 3 U x 3 q 44 c 5 g U 2 x 3 y 16 c 6 g 2 x 4 q 24 c 5 g 2 U x 4 q 4 c 5 g 3 x 5
Acknowledgements
q Ž 24 c 5 g y 12 t 1y 1 f c 3 g 3 q 8 t 111 f c 2 g 4 We thank P. Zinn-Justin for discussing his ideas with us, and we are grateful to F. David and J.-B. Zuber for useful discussions and careful reading of the manuscript.
y36 c 4 g 2 tf y 2 c g 4 tf y 40 c 4 g U y 2 c g 3 U q36 t 1y1 f c 2 g 3 U y 18 c 3 g 2 tf U y 10 c 3 g U 2 q54 c 2 g 2 tf U 2 q 26 c 2 g U 3 q 24 c7 x
Appendix A. The equation for the Potts-3 resolvent Here is the degree five equation for the resolvent of this model, where W Ž x . is related to v Ž x . by W Ž x . s v Ž x . y gx 2 y Ux. y24 c7 q 4 c 4 g 2 y 16 tqy f c 5 g 2 y 12 t 111 f c 4 g 3
y24 c 4 g 2 x y 2 c g 4 x q 16 t 1y 1 f c 2 g 4 x q4 c 3 g 3 tf x y 36 c 6 U x q 4 c 3 g 2 U x q52 c 2 g 3 t y f U x q 12 c 5 U 2 x q36 c 2 g 2 U 2 x y 12 c 4 U 3 x q 12 c 3 U 4 x y4 c 6 g x 2 q 14 c 3 g 3 x 2 q 12 c 2 g 4 tf x 2 y36 c 5 g U x 2 q 10 c 2 g 3 U x 2 q 30 c 4 g U 2 x 2
y 4 t 1y 1 f c 2 g 4 q 8 t 11y1y1 f c 3 g 4 q 68 c 6 g tf
q22 c 3 g U 3 x 2 y 40 c 5 g 2 x 3 q 60 c 4 g 2 U x 3
q 2 c 3 g 3 tf q 3 c 2 g 4 tf2 q 60 c 6 U q 2 c 3 g 2 U
q12 c 3 g 2 U 2 x 3 q 18 c 4 g 3 x 4
y 52 t 1y 1 f c 4 g 2 U q 20 t 111 f c 3 g 3 U
q2 c 3 g 3 U x 4 . W Ž x . q Ž 12 c 6 y 6 c 3 g 2 q 8 t 1y 1 f c g 4 q 2 c 2 g 3 tf
y 20 c 5 g tf U y 4 c 2 g 3 tf U y 36 c 5 U 2 y 3 c 2 g 2 U 2 q 36 t 1y1 f c 3 g 2 U 2 y 36 c 4 g tf U 2 4
3
3
3
3
4
y12 c 5 U y 4 c 2 g 2 U q 26 c g 3 tf U y 12 c 4 U 2 q18 c g 2 U 2 q 12 c 3 U 3 y 44 c 5 g x y 2 c 2 g 3 x
6
y 12 c U q 36 c g tf U q 12 c U q 28 c g x q 2 c 3 g 3 x y 12 t 1y 1 f c 4 g 3 x q 8 t 111 f c 3 g 4 x
q12 c g 4 tf x q 30 c 4 g U x q 20 c g 3 U x q26 c 2 g U 3 x q 6 c 4 g 2 x 2 q 2 c g 4 x 2
G. Bonnetr Physics Letters B 459 (1999) 575–581
q12 c 3 g 2 U x 2 q 39 c 2 g 2 U 2 x 2 q 20 c 3 g 3 x 3 q14 c 2 g 3 U x 3 q c 2 g 4 x 4 . W Ž x .
q4 g 3 U q 26 c 2 g U 2 q 20 c 3 g 2 x q 4 g 4 x q12 c 2 g 2 U x q 18 c g 2 U 2 x q 22 c g 3 U x 2 3
q Ž y Ž c 2 g 2 . q 18 c g 2 U q 4 c g 3 x q 4 g 3 U x 4
References
2
q Ž y12 c 4 g y 2 c g 3 q 4 g 4 tf y 10 c 3 g U
q4 c g 4 x 3 . W Ž x .
581
5
q4 g 4 x 2 . W Ž x . q 4 g 3 W Ž x . s 0 Note that U and c may depend, in the most general case, on t 1y 1F and tF 2 , the latter being related to tF through the equation of motion: g tF 2 q ŽU q 2 c . tF s 0. In this article, we have computed explicitly the critical line for the particular case of c constant and U s 1 q 2h tF 2 .
w1x For general reviews and more references see for instance: The large N Expansion in Quantum Field Theory and Statistical Physics, from Spin Systems to 2-Dimensional Gravity, E. Brezin, S.R. Wadia Eds. ŽWorld Scientific, Singapore, ´ 1993.. w2x M. Staudacher, Phys. Lett. B 305 Ž1993. 332. w3x B. Eynard, cond-matr9801075. w4x S.R. Das, A. Dhar, A.M. Sengupta, S.R. Wadia, Mod. Phys. Lett. A 5 Ž1990. 1041. w5x F. David, Nucl. Phys. B 487 wFSx Ž1997. 633. w6x P. Zinn-Justin, cond-matr9903385. w7x S. Higuchi, C. Itoi, S. Nishigaki, N. Sakai, Phys. Lett. B 318 Ž1993. 63, Nucl. Phys. B 434 Ž1995. 283; Phys. Lett. B 398 Ž1997. 123. w8x G. Bonnet, F. David, Nucl. Phys. B 6166 Ž1999. 1. w9x J.M. Daul hep-thr9502014. w10x E. Brezin, J. Zinn-Justin, Phys. Lett. B 288 Ž1992. 54. ´ w11x M. Wexler, Nucl. Phys. B 410 Ž1993. 377; J. Ambjorn, G. Thorleifsson, M. Wexler, Nucl. Phys. B 439 Ž1995. 187.