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Planar random surfaces with extrinsic action
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986
PLANAR RANDOM SURFACES WITH EXTRINSIC ACTION Bergfinnur D U R H U U S Mathematics Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark and Thordur JONSSON a Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland Received 6 April 1986
A model is considered of planar random surfaces in the lattice Z d with an action that depends on the angle that neighbouring plaquettes make with each other. It is shown that the critical exponents take their mean field value if the susceptibilities of the model and a coarse grained version of it both diverge.
1. Introduction. In this letter we describe some results about a theory of planar random surfaces in the hypercubic lattice Z d with an action that may depend on the imbedding o f the surfaces into the lattice. The surfaces considered are made up of plaquettes in Z d and are the same as those considered in the planar random surface ( P S R ) m o d e l [ 1 - 3 ] . The 'present model can be regarded as a generalization of the PRS model. In that model the action o f surfaces is proportional to the area. Continuum surface theories with extrinsic action that may be r e n t e d to the present model have recently been studied in refs. [4,5] ; see also ref. [6] for another class of random surface models. In ref. [2] it was shown that the critical exponents of the PRS model assume their mean field value if the susceptibility diverges at the critical point. In particular, the critical exponent of the susceptibility is equal to 1. We show that the critical exponents o f our model take on their mean field value if the susceptibility in the model as well as in a related coarse grained model diverge at their respective critical points. Essentially all the results about the PRS model remain valid for 1 Address after September 1, 1986: Nordita, Blegdamsvej 17, DK-2100 Copenhagen 1O, Denmark. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
the generalized version. At the end of the paper we discuss the implications of our results for the random surface model considered by Baumann and Berg (BB model) [7]. 2. Results. Let S be a connected planar random surface with boundary 7. Our model is defined by the action 3 An(S) = ~/3iXi(S), i=1
(1)
where 13 = (/31,132,/33) E R 3 are the coupling constants and the ~ki are defined as follows: XI(S ) is the number o f links (i.e. edges of pNquettes) in S such that the plaquettes that are glued along those links are overlapping in the lattice; X2(S) is the number o f links where the adjacent plaquettes are at right angles; and X3(S) is the number of links in S where the adjacent plaquettes are in the same two-plane but non-overlapping. The loop functions are given by G¢~(7) =
~ exp[-At3(S)], aS=7
(2)
where 3' is a union of loops. We adopt the convention that no action resides on the boundary links. If/31 =/32 =/33 =/3 then At~ (S) = 2/3 Area(S) up to boundary de385
Volume 180, number 4
PHYSICS LETTERS B
pendent terms and we obtain the PRS model. The limit/31 ~ ~ with 132 = ~ 3 gives the BB model where neighbouring plaquettes are forbidden to overlap. By comparison with the PRS model one can see that the loop functions are all f'mite provided min/3 i is larger than half the critical coupling of the PRS model. If we define cB C R 3 as the interior of the set of/~ for which the loop functions are finite it follows from the H61der inequality that cB is convex. We shall refer to Oq~ as the critical surface. Clearly G0(3') is analytic in q0 for any 3'. The attractive feature of the model defined by (1), (2) is its invariance under coarse graining. Applying a coarse graining operation yields a model of the same type with renormalized coupling constants, as we shall see below. This is not the case for the BB model where one coarse graining step takes us from/31 = oo to a finite value of/31. Let b be a link in 7 d and 3'(b) the loop of length 2 that consists of two copies of b. Let ci (b, b' ) be the collection of all surfaces with boundary 3"(b) U 3'(b') but no "pockets" at 3'(b) or 3'(b'), i.e., it is not possible to decompose a surface S E ~ (b, b ' ) into two nonempty surfaces S 1 and S 2 with a S 1 = 3'(b) o r ~ S 1 = 7 ( b ' ) and S 2 Ec3 (b, b ' ) by cutting along one link in S. We define the two-loop function by
Go(b, b ' ) =
~
SE~(b,b')
exp [ - A0(S)].
(3)
The susceptibility X(/3) is given by
X(~) = ~b GO(b' b'),
(4)
where the summation is over all links in gd. By the same argument as in ref. [2] the one-loop function is finite at the critical surface, so the absence of pockets does not affect the critical behaviour of X. Since the action of surfaces is not additive with respect to cutting and gluing we have to pay attention to boundary conditions. Observe that for each S Ec3 (b, b ' ) there are two plaquettes in S that have links in 3'(b). Each of these plaquettes can point in 2 ( d - 1) directions, so there isa total of 2 ( d - 1) 2 possible unordered plaquette pairs that can contribute links to 3'(b) Let P denote the collection of all these pairs. At 3'(b') we have of course the same number of possible boundary plaquettes and the boundary plaquettes at b' are identical to those at b up to trans386
20 November 1986
lation and/or rotation. We ignore the dependence of P on b and label the elements of P by e,f, g, etc. Let d (b, b'; e, f ) denote the subset of c5(b, b ' ) which consists of surfaces with boundary plaquettes e at 3'(b) and f a t 3'(b'). We define
Go(b,b';e,f ) =
~
SEcJ(b,b';e,f)
exp [ - A0(S)].
(5)
Evidently,
G¢(b,b') = ~ Go(b, b';e,f). e, fEP
(6)
We now define the coarse grained surface theory. Suppose S @ ci (b, b ' ) has an internal loop 3" of length 2, i.e. 3" is made up of two links of S that are not boundary links. When S is cut along 3" two cases can occur: (1) we obtain two surfaces S 1 and S 2 with boundaries 3'(b) U 3" and 3" U ~/(b'). In this case we say that 3" is relevant; (2) we obtain two surfaces S 1 and S 2 with boundaries aS 1 = 3'(b) u 3" U 3'(b') and aS 2 = 3". In this case we say that 3" is irrelevant and $2 is a finger. Next we introduce two new classes of surfaces: cS'(b, b ' ) = {S E c3(b, b')l S has no irrelevant internal loops of length 2), cS"(b, b ' ) = {S E cS(b, b')l S has no internal loops of length 2}. The classes c5' (b, b'; e , f ) and c5 "(b,b';e,f) are defined analogously using surfaces that satisfy boundary conditions e and f a t 3'(b) and 3'(b'), respectively. We define ,, loop functions ,, , G~(b,b'),G~(b,b';e,f), Go(b, b'), Go(b , b ; e,f) for these classes of surfacesb3
G~(b, b ' ) =
~
SEcJ'(b, b')
e x p [ - Ao(S)] ,
(7)
etc. The surface theory with two-loop function
G~'(b, b') and the other loop functions analogously defined is the coarse grained theory. For this theory there is a largest open set of couplings cB" C R 3 for which {he loop functions are finite, ct~ _ q0 " andq~ " and ct~" is convex. The critical surface of the coarse grained theory is 3 ~ ". Now, summing separately over the fingers of surfaces in c3(b, b'; e,f) we find, as in ref. [2] :
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986 :
Gfl(b,b';e,f) =
~
S'E cJ ' (b,b';e,f) 3 X l-I [1 +F i(O)l hi(S')
exp [ - At~(S')] +
(14)
g,h
i=1
Let K ( t ' ) be the matrix with entries
=G~,(b,b';e,f),
(8)
where
t~ = ti -- In [1 + Fi(t)],
(9)
and
(lS)
and let the matrix L ( t ' ) be defined by the same formula with X' replaced by X". Then K(t') = L(t') +~iL!(t')K(t')
Fi(t) =
~ exp[- E(S,S') - Aft(S)]. (10) S~ cJ(7(b)) Here E(S, S') is the action that resides on the two links
that have to be cut in order to remove a finger, as well as minus the action on the link that is created when the finger is removed and the hole it leaves is closed. After summing over S in (10) the dependence on S ' is only through the character of the link in S' where the finger is glued. Next we decompose the surfaces in c5' along relevant loops of length 2 and find
G~(bl, b2; e,f) = G~'(bl, b2;e,f )
X a~(b,b2;h,f),
(11)
where the sum on b is over all links inZ d and the matrix with entries V~(g, h) carries the action on the finks that must be cut in order to obtain the above decomposition. Now we define the susceptibility matrix
Xfl(e,f) = ~ G~(b,b'; e,f). (12) b' The susceptibility matrices X'~(e,f) and Xfl(e,f) are tt
defined similarly, using the singly and doubly primed loop functions in (12). Summing over one of the boundary loops in (8) and (11) we obtain t
Xfl(e,f) =Xfl,(e,f),
(13)
(16)
and K(t') = L(t') [1-- L (t')] -1,
(17)
where 1 is the identity matrix, provided 1 - L(fl') is invertible, which is the case for all fl' = fl'(t) with t C q6, because for those t ' the expansion to
K ( t ' ) = ~ L (t') / /=1
(18)
is convergent. Taking the trace o f e q . (17) we find
Tr [K(fl')] =
+ ~ ~ G~(bl,b;e,g ) V¢(g,h) b g,h
and
K(t')eS :
i~=~l1 -~i(t') vi(t' ) '
(19)
where the ui are the eigenvalues of the matrix L ( t ' ) . The eigenvalues are all smaller than 1 in absolute value for t E q3 (because (18) is convergent) and come in complex conjugate pairs, since L is a real matrix. The elements of the matrix V~,(e,f) are analytic functions c t ' so we see that if the susceptibility x(t) diverges as t approaches the critical surface, the divergence is determined by how fast the largest eigenvalue of L ( t ' ) approaches 1. Note that X(t) = ~,e,f Xfl(e,f) and x~(e,f) Xfl(g, h) for any boundary conditions e,f, g, h, and t in a neighbourhood of 0q3. It is not hard to check that the eigenvalues of L ( t ' ) are analytic functions of t for t ~ q 3 . Let us now assume that the susceptibility x(t) diverges at some tc E 0q0 and the coarse grained theory is noncritical at t (tic), 1.e., t ' (tic) E q3 . Choose a smooth path t(X), X>~ O, inside q5 such that t ( 0 ) = tc. We can assume that t(X) is parametrized by arclength and is transverse to 0q~ at tc. If ~o(O) is any thermodynamic function in the random surface theo•
•
t
.
It
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Volume 180, number 4
ry, and ff diverges at/3e, we define its critical exponent co at/3c (if there is any such) by @(/3(X)) ~ X- t ° + o(X- w )
as X-+ 0.
(20)
One can check that the exponent 6o depends only on the endpoint of/3(X). We claim that with our assumptions the critical exponent of the susceptibility is ~ at /3c" To prove the claim suppose there exists 3' > 0 such that X(/3(X)) ~ X- ' r
(21)
asX+0.
Then as X-+0,
n
x_~+l) ~ _ d
1)i(/3t(/3 (~k)))
i = 11 -- ~ ) ) "
(23)
By assumption the vi are analytic functions of/3' at /3' =/3'(/3(0)). Furthermore, by the same argument as in ref. [2], lemma B6,
3 a/3~(/3(x)) X(/3(X))
(24)
i}/3/.
for/" = 1,2, 3. Retaining only the most singular term on the right hand side of (23) it follows that X- (3'+1) ~< const.
X(/3fa)) I1 - Vmax (/3' (/3(X)))l 2
~ X- 3~,
(25)
which implies 7 ~ 7- By a straightforward adaptation of the arguments in ref• [3] one can prove a "tree inequality" for the present model. As in ref. [3] the i n 1 equality implies 3, ~<~, and we conclude that (26)
Armed with the assumption that the coarse grained theory is noncritical at/3' =/3'(/3c) and the result (26) we can proceed as in ref. [2] and show that the other critical exponents take their mean field values. The only result which does not carry over immediately is the positivity of the string tension r(/3)which is defined by
388
lim
R, T-***
(RT) -1 log
GO(TR,T),
(27)
where 7R, T is a rectangular loop with sides of length R and T, lying in a coordinate plane. The existence of the limit (27) is proven by a sub-additivity argument as in ref. [2]• In order to prove the positivity of the string tension, let us assume that the rectangular loop ')'R,T contains a fixed link £. Let Q~R,T be the collection of all surfaces with boundary 7R, T and with the property that the boundary plaquettes all lie in the plane of the loop '~R,T and in its interior. Define
SEQ~R, T
(22)
where Vmax(/3' ) is the eigenvalue of I_ (/3') that is closest to 1. Differentiating (19) we obtain
T = ~1.
r(/3) = -
F/3(R,T)= ~
I1 - Vmax(/3'(/3CA)))I ~ X~
i=1
20 November 1986
PHYSICS LETTERS B
exp[-A~(S)].
(28)
If we take two surfaces S 1 and S 2 fromC~?R, T and glue them together along the boundary links but let £ remain unglued, then we obtain a surface S that contributes to the one-loop function G/3(7(£)). From any S constructed in this fashion we can uniquely recover S 1 and S 2. Hence, there is a constant C(/3) such that Ge(3'R_2 ' r _ 2 ) 2 <
C(/3)R+rFa(R,
T) 2
C(/3)R +T G (7(t)).
(29)
From (29) we conclude that r(/3) ~> 0 for any/3 Ect~. It is not hard to see that r(/3) is strictly monotone on the intersection of q~ with any line through the origin in R 3. Thus, r(/3) > 0 for/3 E c B . As in ref. [2] we now deduce that the string tension does not vanish at 13e and the surfaces collapse to branched polymers provided the coarse grained theory is noncritical at/3'(/3e)" By the methods of ref. [8] one can prove OrnsteinZernike decay of the two-loop function and establish the existence of the scaling limit, which is a free field theory of a single species of particles. Let us finally consider the BB model in the light of the above results. We take/32 =/33 and let/31 -~oo in (19). The existence of the limit is clear because the susceptibilities decrease as ~1 increases. Taking the limit in (9) we find that lim /3'1(/3)
/31 --+oo
=ln(#]ilno. a ~S = ,
exp[-/3i-/3/-A(l(S)]) '
(30)
where/3i and/31 are the couplings on the boundary of
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PHYSICS LETTERS B
the fmger, cf. the discussion after eq. (10). The limit under the logarithm in (27) exists o f course and gives essentially the one-loop function of the BB model. T h u s / ~ has a finite limit when/~1 tends to infinity. Let now f12 and/~3 approach the critical coupling o f the BB model. The renormalized couplings approach some limit ~'c. We can go through the same argument as before and conclude that the critical exponent o f the susceptibility in the BB model ls-~ " 1 iffl'c i s n o t in the critical surface o f the coarse grained theory. Thus, taking for granted the numerical results of ref. [7] that 7 ~ 1, we find that ~'c must be a critical point for the coarse grained theory, and at this critical point the susceptibility is finite.
3. Discussion. If the susceptibility o f our model diverges for all fl E ~ c~, it is reasonable to expect from universality that the susceptibility o f the coarse g a i n e d model diverges for any/~ E a ~ '. In this case the models have mean field behaviour at any critical/3 and the result of BB certainly implies a violation o f universality at/31 = oo. At present it is known from numerical studies that the susceptibility diverges in low dimensions for/3E ~ q~ and ~]1 = ~2 =/J3 [1] or fll = oo and/~2 =/33 [7]. In high dimensions mean field theory analysis also indicates a divergent susceptibility [2]. It is clear that more analytical and/or numerical work is needed in order to ob-
20 November 1986
tain a clear picture o f the behaviour of the susceptibility along the boundary of ct~ or ~ '. We would like to thank K. Gawedski for his warm hospitality at IHES.
References [1] T. Eguchi andH. Kawai, Phys. Lett. B 110 (1982) 143; B 114 (1982) 247; H. Kawai and Y. Okamoto, Phys. Lett. B 130 (1983) 415; B. Duthuus, J. FriShlich and T. Jonsson, Nucl. Phys. B 225 [FS91 (1983) 185. [2] B. Durhuus, J. Fr6hlich and T. Jonsson, Nucl. Phys. B 240 [FS12] (1984) 453. [3] B. Durhuus, J. Fr~Shlichand T. Jonsson, Nucl. Phys. B 247 [FS14] (1985) 779;Phys. Lett. B 137 (1984) 93. [4] F. David, Rigid surfaces in space with large dimensionality Saclay preprint SPhT/86/052. [5] A. Polyakov, Nucl. Phys. B 268 (1986) 406. [6] J. Ambj~6rn,B. Durhuus and J. Fr~Shlich,Nucl. Phys. B 25~ [FS14] (1985) 433; F. David, Nucl. Phys. B 257 [FS14] (1985) 543; V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. B 157 (1985) 295; J. Ambj~brn,B. Durhuus, J. Fr~Shlichand P. Orland, Nucl. Phys. B 270 [FS16] (1986) 457; D.V. Bulatov, V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. B 174 (1986) 87. [7] B. Baumann and B. Berg, Phys. Lett. B 164 (1985) 131. [8] T. Jonsson, Ornstein-Zernike theory for the planar random surface mode, University of Iceland preprint RH-01-86, Commun, Math. Phys., to be published.
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PHYSICS LETTERS B
20 November 1986
PLANAR RANDOM SURFACES WITH EXTRINSIC ACTION Bergfinnur D U R H U U S Mathematics Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark and Thordur JONSSON a Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland Received 6 April 1986
A model is considered of planar random surfaces in the lattice Z d with an action that depends on the angle that neighbouring plaquettes make with each other. It is shown that the critical exponents take their mean field value if the susceptibilities of the model and a coarse grained version of it both diverge.
1. Introduction. In this letter we describe some results about a theory of planar random surfaces in the hypercubic lattice Z d with an action that may depend on the imbedding o f the surfaces into the lattice. The surfaces considered are made up of plaquettes in Z d and are the same as those considered in the planar random surface ( P S R ) m o d e l [ 1 - 3 ] . The 'present model can be regarded as a generalization of the PRS model. In that model the action o f surfaces is proportional to the area. Continuum surface theories with extrinsic action that may be r e n t e d to the present model have recently been studied in refs. [4,5] ; see also ref. [6] for another class of random surface models. In ref. [2] it was shown that the critical exponents of the PRS model assume their mean field value if the susceptibility diverges at the critical point. In particular, the critical exponent of the susceptibility is equal to 1. We show that the critical exponents o f our model take on their mean field value if the susceptibility in the model as well as in a related coarse grained model diverge at their respective critical points. Essentially all the results about the PRS model remain valid for 1 Address after September 1, 1986: Nordita, Blegdamsvej 17, DK-2100 Copenhagen 1O, Denmark. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
the generalized version. At the end of the paper we discuss the implications of our results for the random surface model considered by Baumann and Berg (BB model) [7]. 2. Results. Let S be a connected planar random surface with boundary 7. Our model is defined by the action 3 An(S) = ~/3iXi(S), i=1
(1)
where 13 = (/31,132,/33) E R 3 are the coupling constants and the ~ki are defined as follows: XI(S ) is the number o f links (i.e. edges of pNquettes) in S such that the plaquettes that are glued along those links are overlapping in the lattice; X2(S) is the number o f links where the adjacent plaquettes are at right angles; and X3(S) is the number of links in S where the adjacent plaquettes are in the same two-plane but non-overlapping. The loop functions are given by G¢~(7) =
~ exp[-At3(S)], aS=7
(2)
where 3' is a union of loops. We adopt the convention that no action resides on the boundary links. If/31 =/32 =/33 =/3 then At~ (S) = 2/3 Area(S) up to boundary de385
Volume 180, number 4
PHYSICS LETTERS B
pendent terms and we obtain the PRS model. The limit/31 ~ ~ with 132 = ~ 3 gives the BB model where neighbouring plaquettes are forbidden to overlap. By comparison with the PRS model one can see that the loop functions are all f'mite provided min/3 i is larger than half the critical coupling of the PRS model. If we define cB C R 3 as the interior of the set of/~ for which the loop functions are finite it follows from the H61der inequality that cB is convex. We shall refer to Oq~ as the critical surface. Clearly G0(3') is analytic in q0 for any 3'. The attractive feature of the model defined by (1), (2) is its invariance under coarse graining. Applying a coarse graining operation yields a model of the same type with renormalized coupling constants, as we shall see below. This is not the case for the BB model where one coarse graining step takes us from/31 = oo to a finite value of/31. Let b be a link in 7 d and 3'(b) the loop of length 2 that consists of two copies of b. Let ci (b, b' ) be the collection of all surfaces with boundary 3"(b) U 3'(b') but no "pockets" at 3'(b) or 3'(b'), i.e., it is not possible to decompose a surface S E ~ (b, b ' ) into two nonempty surfaces S 1 and S 2 with a S 1 = 3'(b) o r ~ S 1 = 7 ( b ' ) and S 2 Ec3 (b, b ' ) by cutting along one link in S. We define the two-loop function by
Go(b, b ' ) =
~
SE~(b,b')
exp [ - A0(S)].
(3)
The susceptibility X(/3) is given by
X(~) = ~b GO(b' b'),
(4)
where the summation is over all links in gd. By the same argument as in ref. [2] the one-loop function is finite at the critical surface, so the absence of pockets does not affect the critical behaviour of X. Since the action of surfaces is not additive with respect to cutting and gluing we have to pay attention to boundary conditions. Observe that for each S Ec3 (b, b ' ) there are two plaquettes in S that have links in 3'(b). Each of these plaquettes can point in 2 ( d - 1) directions, so there isa total of 2 ( d - 1) 2 possible unordered plaquette pairs that can contribute links to 3'(b) Let P denote the collection of all these pairs. At 3'(b') we have of course the same number of possible boundary plaquettes and the boundary plaquettes at b' are identical to those at b up to trans386
20 November 1986
lation and/or rotation. We ignore the dependence of P on b and label the elements of P by e,f, g, etc. Let d (b, b'; e, f ) denote the subset of c5(b, b ' ) which consists of surfaces with boundary plaquettes e at 3'(b) and f a t 3'(b'). We define
Go(b,b';e,f ) =
~
SEcJ(b,b';e,f)
exp [ - A0(S)].
(5)
Evidently,
G¢(b,b') = ~ Go(b, b';e,f). e, fEP
(6)
We now define the coarse grained surface theory. Suppose S @ ci (b, b ' ) has an internal loop 3" of length 2, i.e. 3" is made up of two links of S that are not boundary links. When S is cut along 3" two cases can occur: (1) we obtain two surfaces S 1 and S 2 with boundaries 3'(b) U 3" and 3" U ~/(b'). In this case we say that 3" is relevant; (2) we obtain two surfaces S 1 and S 2 with boundaries aS 1 = 3'(b) u 3" U 3'(b') and aS 2 = 3". In this case we say that 3" is irrelevant and $2 is a finger. Next we introduce two new classes of surfaces: cS'(b, b ' ) = {S E c3(b, b')l S has no irrelevant internal loops of length 2), cS"(b, b ' ) = {S E cS(b, b')l S has no internal loops of length 2}. The classes c5' (b, b'; e , f ) and c5 "(b,b';e,f) are defined analogously using surfaces that satisfy boundary conditions e and f a t 3'(b) and 3'(b'), respectively. We define ,, loop functions ,, , G~(b,b'),G~(b,b';e,f), Go(b, b'), Go(b , b ; e,f) for these classes of surfacesb3
G~(b, b ' ) =
~
SEcJ'(b, b')
e x p [ - Ao(S)] ,
(7)
etc. The surface theory with two-loop function
G~'(b, b') and the other loop functions analogously defined is the coarse grained theory. For this theory there is a largest open set of couplings cB" C R 3 for which {he loop functions are finite, ct~ _ q0 " andq~ " and ct~" is convex. The critical surface of the coarse grained theory is 3 ~ ". Now, summing separately over the fingers of surfaces in c3(b, b'; e,f) we find, as in ref. [2] :
Volume 180, number 4
PHYSICS LETTERS B
20 November 1986 :
Gfl(b,b';e,f) =
~
S'E cJ ' (b,b';e,f) 3 X l-I [1 +F i(O)l hi(S')
exp [ - At~(S')] +
(14)
g,h
i=1
Let K ( t ' ) be the matrix with entries
=G~,(b,b';e,f),
(8)
where
t~ = ti -- In [1 + Fi(t)],
(9)
and
(lS)
and let the matrix L ( t ' ) be defined by the same formula with X' replaced by X". Then K(t') = L(t') +~iL!(t')K(t')
Fi(t) =
~ exp[- E(S,S') - Aft(S)]. (10) S~ cJ(7(b)) Here E(S, S') is the action that resides on the two links
that have to be cut in order to remove a finger, as well as minus the action on the link that is created when the finger is removed and the hole it leaves is closed. After summing over S in (10) the dependence on S ' is only through the character of the link in S' where the finger is glued. Next we decompose the surfaces in c5' along relevant loops of length 2 and find
G~(bl, b2; e,f) = G~'(bl, b2;e,f )
X a~(b,b2;h,f),
(11)
where the sum on b is over all links inZ d and the matrix with entries V~(g, h) carries the action on the finks that must be cut in order to obtain the above decomposition. Now we define the susceptibility matrix
Xfl(e,f) = ~ G~(b,b'; e,f). (12) b' The susceptibility matrices X'~(e,f) and Xfl(e,f) are tt
defined similarly, using the singly and doubly primed loop functions in (12). Summing over one of the boundary loops in (8) and (11) we obtain t
Xfl(e,f) =Xfl,(e,f),
(13)
(16)
and K(t') = L(t') [1-- L (t')] -1,
(17)
where 1 is the identity matrix, provided 1 - L(fl') is invertible, which is the case for all fl' = fl'(t) with t C q6, because for those t ' the expansion to
K ( t ' ) = ~ L (t') / /=1
(18)
is convergent. Taking the trace o f e q . (17) we find
Tr [K(fl')] =
+ ~ ~ G~(bl,b;e,g ) V¢(g,h) b g,h
and
K(t')eS :
i~=~l1 -~i(t') vi(t' ) '
(19)
where the ui are the eigenvalues of the matrix L ( t ' ) . The eigenvalues are all smaller than 1 in absolute value for t E q3 (because (18) is convergent) and come in complex conjugate pairs, since L is a real matrix. The elements of the matrix V~,(e,f) are analytic functions c t ' so we see that if the susceptibility x(t) diverges as t approaches the critical surface, the divergence is determined by how fast the largest eigenvalue of L ( t ' ) approaches 1. Note that X(t) = ~,e,f Xfl(e,f) and x~(e,f) Xfl(g, h) for any boundary conditions e,f, g, h, and t in a neighbourhood of 0q3. It is not hard to check that the eigenvalues of L ( t ' ) are analytic functions of t for t ~ q 3 . Let us now assume that the susceptibility x(t) diverges at some tc E 0q0 and the coarse grained theory is noncritical at t (tic), 1.e., t ' (tic) E q3 . Choose a smooth path t(X), X>~ O, inside q5 such that t ( 0 ) = tc. We can assume that t(X) is parametrized by arclength and is transverse to 0q~ at tc. If ~o(O) is any thermodynamic function in the random surface theo•
•
t
.
It
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Volume 180, number 4
ry, and ff diverges at/3e, we define its critical exponent co at/3c (if there is any such) by @(/3(X)) ~ X- t ° + o(X- w )
as X-+ 0.
(20)
One can check that the exponent 6o depends only on the endpoint of/3(X). We claim that with our assumptions the critical exponent of the susceptibility is ~ at /3c" To prove the claim suppose there exists 3' > 0 such that X(/3(X)) ~ X- ' r
(21)
asX+0.
Then as X-+0,
n
x_~+l) ~ _ d
1)i(/3t(/3 (~k)))
i = 11 -- ~ ) ) "
(23)
By assumption the vi are analytic functions of/3' at /3' =/3'(/3(0)). Furthermore, by the same argument as in ref. [2], lemma B6,
3 a/3~(/3(x)) X(/3(X))
(24)
i}/3/.
for/" = 1,2, 3. Retaining only the most singular term on the right hand side of (23) it follows that X- (3'+1) ~< const.
X(/3fa)) I1 - Vmax (/3' (/3(X)))l 2
~ X- 3~,
(25)
which implies 7 ~ 7- By a straightforward adaptation of the arguments in ref• [3] one can prove a "tree inequality" for the present model. As in ref. [3] the i n 1 equality implies 3, ~<~, and we conclude that (26)
Armed with the assumption that the coarse grained theory is noncritical at/3' =/3'(/3c) and the result (26) we can proceed as in ref. [2] and show that the other critical exponents take their mean field values. The only result which does not carry over immediately is the positivity of the string tension r(/3)which is defined by
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lim
R, T-***
(RT) -1 log
GO(TR,T),
(27)
where 7R, T is a rectangular loop with sides of length R and T, lying in a coordinate plane. The existence of the limit (27) is proven by a sub-additivity argument as in ref. [2]• In order to prove the positivity of the string tension, let us assume that the rectangular loop ')'R,T contains a fixed link £. Let Q~R,T be the collection of all surfaces with boundary 7R, T and with the property that the boundary plaquettes all lie in the plane of the loop '~R,T and in its interior. Define
SEQ~R, T
(22)
where Vmax(/3' ) is the eigenvalue of I_ (/3') that is closest to 1. Differentiating (19) we obtain
T = ~1.
r(/3) = -
F/3(R,T)= ~
I1 - Vmax(/3'(/3CA)))I ~ X~
i=1
20 November 1986
PHYSICS LETTERS B
exp[-A~(S)].
(28)
If we take two surfaces S 1 and S 2 fromC~?R, T and glue them together along the boundary links but let £ remain unglued, then we obtain a surface S that contributes to the one-loop function G/3(7(£)). From any S constructed in this fashion we can uniquely recover S 1 and S 2. Hence, there is a constant C(/3) such that Ge(3'R_2 ' r _ 2 ) 2 <
C(/3)R+rFa(R,
T) 2
C(/3)R +T G (7(t)).
(29)
From (29) we conclude that r(/3) ~> 0 for any/3 Ect~. It is not hard to see that r(/3) is strictly monotone on the intersection of q~ with any line through the origin in R 3. Thus, r(/3) > 0 for/3 E c B . As in ref. [2] we now deduce that the string tension does not vanish at 13e and the surfaces collapse to branched polymers provided the coarse grained theory is noncritical at/3'(/3e)" By the methods of ref. [8] one can prove OrnsteinZernike decay of the two-loop function and establish the existence of the scaling limit, which is a free field theory of a single species of particles. Let us finally consider the BB model in the light of the above results. We take/32 =/33 and let/31 -~oo in (19). The existence of the limit is clear because the susceptibilities decrease as ~1 increases. Taking the limit in (9) we find that lim /3'1(/3)
/31 --+oo
=ln(#]ilno. a ~S = ,
exp[-/3i-/3/-A(l(S)]) '
(30)
where/3i and/31 are the couplings on the boundary of
Volume 180, number 4
PHYSICS LETTERS B
the fmger, cf. the discussion after eq. (10). The limit under the logarithm in (27) exists o f course and gives essentially the one-loop function of the BB model. T h u s / ~ has a finite limit when/~1 tends to infinity. Let now f12 and/~3 approach the critical coupling o f the BB model. The renormalized couplings approach some limit ~'c. We can go through the same argument as before and conclude that the critical exponent o f the susceptibility in the BB model ls-~ " 1 iffl'c i s n o t in the critical surface o f the coarse grained theory. Thus, taking for granted the numerical results of ref. [7] that 7 ~ 1, we find that ~'c must be a critical point for the coarse grained theory, and at this critical point the susceptibility is finite.
3. Discussion. If the susceptibility o f our model diverges for all fl E ~ c~, it is reasonable to expect from universality that the susceptibility o f the coarse g a i n e d model diverges for any/~ E a ~ '. In this case the models have mean field behaviour at any critical/3 and the result of BB certainly implies a violation o f universality at/31 = oo. At present it is known from numerical studies that the susceptibility diverges in low dimensions for/3E ~ q~ and ~]1 = ~2 =/J3 [1] or fll = oo and/~2 =/33 [7]. In high dimensions mean field theory analysis also indicates a divergent susceptibility [2]. It is clear that more analytical and/or numerical work is needed in order to ob-
20 November 1986
tain a clear picture o f the behaviour of the susceptibility along the boundary of ct~ or ~ '. We would like to thank K. Gawedski for his warm hospitality at IHES.
References [1] T. Eguchi andH. Kawai, Phys. Lett. B 110 (1982) 143; B 114 (1982) 247; H. Kawai and Y. Okamoto, Phys. Lett. B 130 (1983) 415; B. Duthuus, J. FriShlich and T. Jonsson, Nucl. Phys. B 225 [FS91 (1983) 185. [2] B. Durhuus, J. Fr6hlich and T. Jonsson, Nucl. Phys. B 240 [FS12] (1984) 453. [3] B. Durhuus, J. Fr~Shlichand T. Jonsson, Nucl. Phys. B 247 [FS14] (1985) 779;Phys. Lett. B 137 (1984) 93. [4] F. David, Rigid surfaces in space with large dimensionality Saclay preprint SPhT/86/052. [5] A. Polyakov, Nucl. Phys. B 268 (1986) 406. [6] J. Ambj~6rn,B. Durhuus and J. Fr~Shlich,Nucl. Phys. B 25~ [FS14] (1985) 433; F. David, Nucl. Phys. B 257 [FS14] (1985) 543; V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. B 157 (1985) 295; J. Ambj~brn,B. Durhuus, J. Fr~Shlichand P. Orland, Nucl. Phys. B 270 [FS16] (1986) 457; D.V. Bulatov, V.A. Kazakov, I.K. Kostov and A.A. Migdal, Phys. Lett. B 174 (1986) 87. [7] B. Baumann and B. Berg, Phys. Lett. B 164 (1985) 131. [8] T. Jonsson, Ornstein-Zernike theory for the planar random surface mode, University of Iceland preprint RH-01-86, Commun, Math. Phys., to be published.
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