Nontrivial Fixed Points and Screening in theHierarchical Two-Dimensional Coulomb GasG. BenfattoDipartimento di Matematica, II Universit�a di Roma00133 Roma, ItalyJ. Renn�Department of Physics, Boston UniversityBoston, MA 02215, USAAbstractWe show the existence and asymptotic stability of two �xed pointsof the renormalization group transformation for the hierarchical two-dimensional Coulomb gas in the sine-Gordon representation and tem-peratures slightly greater than the critical one. We prove also thatthe correlations at the �xed points decay as in the hierarchical mas-sive scalar free �eld theory, that is as d�4xy . We argue that this is thenatural de�nition of screening in the hierarchical approximation.1 IntroductionIn the last ten years the renormalization group ideas have been extensivelyapplied to the two dimensional Coulomb gas of identical charges �e, in orderto rigorously understand the so called Kosterlitz-Thouless phase transition[1].�Associate of the Department of Physics, Harvard University. Supported by GruppoNazionale per la Fisica Matematica, CNR, Italy.1

For inverse temperatures � larger than the critical one, �c, and smallactivity �, it has been proved that there is no screening [2, 3]. This resultis strictly related to the fact that, in the �eld theoretical representation ofthe model (the so called sine-Gordon representation), the e�ective potentialgoes to zero as the scale goes to in�nity [4]. All these results are valid alsoin the hierarchical approximation of the model (see Sect. 2), where they areobtained much easier [5, 6].For � < �c screening is generally expected to be found, but this propertyhas been proved in the exact model only for � very small [7]. However, in thecase of the hierarchical approximation, a weak form of screening has beenproved for ��c [6].In this paper we study the hierarchical Coulomb gas in the sine-Gordonrepresentation in the region ��c, by analyzing the renormalization grouptransformation T of the e�ective potential. We prove that T has two non-trivial asymptotic stable �xed points, which have the following screeningproperty: the two-charges truncated correlations decay as d�4xy as the (hier-archical) distance dxy goes to in�nity, which is the decaying behaviour of thetruncated correlations of the massive hierarchical scalar �eld. Our analysisalso suggests that the e�ective potential for the hierarchical Coulomb gas onscale 1 is in the domain of attraction of one of the �xed points, if the activityis small enough, so that in this case as well screening should be observed.The existence of the nontrivial �xed points was proved in [6] with a dif-ferent technique, which does not use the sine-Gordon representation. Theproof of [6] extends to more general models, but allows one to study onlythe simple (not truncated) correlations. This is why the analysis of [6] wasrestricted to the screening for fractional charges; in fact, for the fractionalcharges, the truncated correlations coincide with the simple ones.Finally, we want to stress that the technique used in this paper is essen-tially based on the bound discussed in the Appendix, which is a bound forthe Ursell coe�cients of a system of arbitrary charges sitting in the samepoint and interacting with a potential cQiQj. We were unable to �nd thisestimate, which we think is interesting by itself, in the literature.2

2 The hierarchical modelLet Qj; j 2 , be a sequence of compatible pavements of 2 made of squaresof side size j, where � 2 is an integer. To each � 2 Qj we associate agaussian variable z� such thatE(z2�) = 12� log ; E(z�z�0) = 0 if � 6= �0 (2.1)Then we de�ne, 8x 2 2: 'x = 1Xk=0 z�(k)x (2.2)where �(k)x is the tessera of side size k containing x.Given x; y 2 2, let hxy be the smallest integer such that there exists a� 2 Qhxy containing both x and y. We shall call dxy � hxy , the side size of�, the hierarchical distance between x and y. By using (2.1) it is easy to seethat: E(('x � 'y)2) = E 0@hxy�1Xk=0 [z�(k)x � z�(k)y ]21A = 1� log dxy (2.3)which justi�es the claim that 'x is a reasonable approximation of the two-dimensional zero mass scalar �eld. In the following we shall denote thecorresponding Gaussian measure by P (d').If v(z); z 2 1, is a real function such thatv(0) = 0 ; v(z) = v(�z) (2.4)and � is a �nite volume belonging to QR, for some R > 0, we shall considerthe measure ��v (d') = 1Z�v P (d') Y�2Q0\� ev('�) (2.5)Z�v = Z P (d') Y�2Q0\� ev('�) (2.6)where '� is the constant value of the �eld on the tessera �.The choice v(') = �(cos(�')� 1) (2.7)3

corresponds to the hierarchical Coulomb gas in the volume � with activity�=2, charges �e and temperature ��1, such that�e2 = �2 (2.8)For more details on this point see [5], where a rescaled �eld was used, insteadof (2.2).Another interesting choice isv(') = �m2'2 � um(') (2.9)which should give rise to the two dimensional hierarchical scalar �eld of massm. It is important to remark, however, that this is not a good approximationof the massive scalar �eld. In fact it is easy to show thatlim�!2 Z ��um(d')'x'y / d�4xy (2.10)in disagreement with the exponential decay of the massive scalar �eld (a sim-ilar property is valid in other hierarchical models, see [8], chapter 4, exercise2). This observation will be very relevant in the following, since it impliesthat the hierarchical Coulomb gas should have power decaying correlationsalso in a screened phase, but with a power equal to 4 independently of �.Let us now de�ne the renormalization group transformation.If F (z) is a function on and < � >�v denotes the expectation w.r.t. themeasure (2.5), then< F ('0) >v� lim�!2 < F ('0) >�v=< LTk�1v � � �LvF ('0) >Tkv (2.11)where (Tv)(') = log "R P0(dz)ev('+z)R P0(dz)ev(z) # 2 (2.12)(LvF )(') = R P0(dz)ev('+z)F ('+ z)R P0(dz)ev('+z) (2.13)if P0(dz) denotes the gaussian measure on 1 of mean zero and covariance12� log . 4

The operators T and Lv appear also in the expressions similar to (2.11)valid for the expectations of any observable depending on the values of the�eld 'x in a �nite set of tesserae � 2 Q0.The operator (2.12) is the renormalization group transformation. It leavesinvariant the space C� of the continuous functions periodic of period T� =2�=� and satisfying (2.4); then, if we want to study the hierarchical Coulombgas at temperature ��1, we have to restrict T to C� with � = p�e2. In Ref.[5] it was implicitly shown that v0(') = 0 is, for �2 > 8�, a �xed point of(2.12) which is attracting for functions of the form (2.7), for � small enough.In fact one could also show that it is locally attracting in some subspace ofsu�ciently regular functions.In this paper we shall study the more di�cult case �2 � 8� and weshall prove that there are two stable �xed points v�(') 6= 0 (this result,as discussed in the Introduction, has already been obtained with a di�erenttechnique [6]) in a suitable subspace of C�.Moreover, by studying the spectrum of (2.13) for v = v� in the space ofL2 functions periodic of period T�, we shall prove that the integer chargetruncated correlations decay like d�4xy . The restriction of Lv to periodic func-tions is motivated by the fact that the truncated integer charge correlationsare given by the formula�T (x1; �1; : : : ; xn; �n) = @n@!1 � � �@!n log < expf�2 nXj=1!jei��j'xj g > j!j=0(2.14)where �i 2 f�1; 1g are the charges and xi are the positions of n particles.3 The existence of two attracting nontrivial�xed pointsThe proof of the existence of nontrivial �xed points for �2 < 8� is basedon the perturbative expansion of the renormalization group transformation(2.12). If v(') 2 C�, we can write:v(') = X06=Q2 vQ(ei�Q' � 1) (3.1)5

where vQ = v�Q. Then, if v0Q are the Fourier coe�cients of (Tv)('), (2.12)can be written in the following form:v0Q = 2��24�Q2vQ + 2 1Xn=2 1n! XQ1+���+Qn=Q vQ1 � � � vQnFn(Q1; : : : ; Qn) (3.2)where Fn(Q1; : : : ; Qn) = ET (ei�Q1z; : : : ; ei�Qnz) (3.3)if we denote ET the truncated expectation with respect to the measure P0(dz).In the appendix we prove the following nontrivial bound, which will playa crucial role in the following:jFn(Q1; : : : ; Qn)j � nn�1 " nYi=1 (4p�jQij)# e��Q2n (3.4)where � = (�2=8�) log and Q = Pni=1Qi.The linearization of (3.2) around the �xed point v = 0 has a bifurcationat �2 = 8�, where the Fourier coe�cients v�1 become unstable. This bifur-cation, as it is well known, is responsible for the Kosterlitz-Thouless phasetransition. In this paper we study the range of temperatures, immediatelyabove the critical one, given by the relation0 < " � 2� �24� � "0 (3.5)with "0 small enough. We start by proving that (3.2) has two �xed pointsdi�erent from zero.We �rst look for approximate solutions, by imposing in (3.2) the condi-tions jQj; jQij � 2 and n � 2. Taking into account the symmetry propertyvQ = v�Q, we obtain a system of two equations:v01 = "v1 + av1v2 � fv31v02 = cv2 � bv21 (3.6)where a = 2F2(�1; 2) = "(1� �4(2�"))b = �12 2F2(1; 1) = 12 �2(1�")(1� �4+2")c = �6+4"f = �12 2F3(�1; 1; 1) = 12 "(1� �4+2")2 (3.7)6

If " > 0 the system (3.7) has two solutions di�erent from zero given by:�v21 = (1� c)( " � 1)ab + f(1� c)�v2 = �b( " � 1)ab + f(1� c) (3.8)Furthermore it is possible to see that, if �v+1 and �v�1 are, respectively, thepositive and the negative solution, there is a neighbourhood S of the originin 2, containing �v+1 and �v�1 , such that S \ fv1 > 0g and S \ fv1 < 0g are inthe domain of attraction, respectively, of �v+1 and �v�1 .In the following we shall consider only the solution with �v1 > 0, but thesame considerations could be applied to the other one. We want to prove thatthere is a �xed point of (2.12) which is approximately equal to the function�v(') = 2�v1[cos'� 1] + 2�v2[cos(2')� 1] (3.9)We want to apply the contraction mapping principle; hence we need to de�nea suitable Banach space and �nd a T-invariant subset, containing (3.9), onwhich T is a contraction with respect to a suitable metric.Let us consider the functions of C�, which are analytic and bounded in asymmetric strip along the real axis of width 2~b such that� � e��~b � �a" 12 � �� < 1 (3.10)These functions form a Banach spaceB, if we de�ne the norm in the followingway: jjvjj = supQ�1 ��QjvQj (3.11)Let Bd � B be the sphere of radius d with center at the origin. We wantto choose d and �a, see (3.10), so that the smaller sphere Bd=2 contains thefunction �v, see (3.9). >From (3.8) it follows that this is possible if2s 1� cab + f(1� c) " � 1" � d�a2 bab + f(1� c) " � 1" � d�a2 (3.12)7

The bounds (3.12) and (3.10) can be satis�ed for any d, if �a is su�cientlylarge and " is su�ciently small, as we shall suppose in the following.We want now to de�ne a subset D � Bd containing the functions whichare close to �v. Let us de�ne:vi = �vi + ri ; v0i = �vi + r0i ; i = 1; 2 (3.13)and consider the linear change of coordinates which diagonalizes the lin-earization of the system (3.6) around its �xed point, that is: r1r2 ! = S u1u2 ! (3.14)where S = 1 �a��v1�2b��v1 1 ! (3.15)with � = 11� c�K( " � 1)K = 4(1� c)(1 + 2f�v21 � c) �1 +r1� 8(1�c)( "�1)(1+2f�v21�c)2 � (3.16)D is the set of functions v 2 Bd, such that:ju1j � ~d"1+�ju2j � ~d"3=2 (3.17)where 0 < � < 1=2 and ~d is any �xed positive constant.Theorem 3.1 There exist a positive constant d0 and, for any given d; ~d; �,such that 0 < d � d0 and 0 < � < 1=2, another constant "0, so that the setD is invariant under the transformation T, if " � "0.Proof - (3.2) and (3.4) imply that, if v 2 Bd and Q � 1:jv0Qj � 2�Q2(2�")d �Q + 1Xn=2(Bdp�)ne��Q2n XQ1+���+Qn=Q nYi=1 jQij�jQij� 2�Q2(2�")d �Q + 1Xn=2(B1dp�)ne��Q2n XQ1+���+Qn=QjQij�1 nYi=1 �jQij1 (3.18)8

where B;B1 are suitable positive constants and �1 is chosen so that�� < �1 < 1 (3.19)We now have to carefully bound the sum:I � 1Xn=2(B1dp�)ne��Q2n XQ1+���+Qn=QjQij�1 nYi=1 �jQij1 (3.20)We can write: I = I0 + I1 (3.21)with I0 = QXn=2(B1dp�)ne��Q2n �Q1 Nn(Q) (3.22)I1 = 1Xn=2(B1dp�)ne��Q2n n�1Xk=1 nk ! 1Xs=k �Q+2s1 Nk(s)Nn�k(Q + s) (3.23)where the combinatorial factor Nk(Q) is de�ned asNk(Q) = XQ1+���+Qk=QQi�1 1 = 8>><>>: Q� 1k � 1 ! for Q � k � 10 for 1 � Q < k (3.24)It is easy to see thatI0 � �Qe��Q(B1d)2(1 +B1dp�)Q�2�Q1 (3.25)In order to bound I1, we use the inequality sk ! � skk! � e�s�k (3.26)valid for any positive � and we choose � so that�2 � �1e� < 1 (3.27)Then we have:I1 � �Q2 (�1B1dp�)2 1Xn=2 B1dp�� !n�2 e��Q2n (1 + �22)n � 1� �2n2�22(1� �22) (3.28)9

If d is su�ciently small, we have also:B1dp�� (1 + �22) � �� < 1 (3.29)which is compatible with (3.10) and (3.12) for " small enough and �a largeenough. (3.22), (3.28) and (3.29) easily imply thatI � �A(B1d)2�Q (3.30)for a suitable positive constant �A, only depending on �� and �1.The inequalities (3.18) and (3.30) imply that, if Q � 3, jv0Qj � d�Qprovided that 2�9(2�")d+ �A(B1d)2 � d (3.31)which can be satis�ed if d is su�ciently small. Then, in order to prove thatD is invariant, we still have to check that the condition (3.17) is preserved.Let us notice that: r01r02 ! = 1� 2f�v21 a�v1�2b�v1 c ! r1r2 !+ 1 2 !+ ~v1~v2 ! (3.32)where 1 = ar1r2 � 3f�v1r21 � fr31 2 = �br21 (3.33)and ~v1 = 2 1Xn=2 1n! XQ1+���+Qn=1jQ1j+���jQnj�5 vQ1 � � � vQnFn(Q1; : : : ; Qn)~v2 = 2 1Xn=2 1n! XQ1+���+Qn=2jQ1j+���jQnj�4 vQ1 � � � vQnFn(Q1; : : : ; Qn) (3.34)Proceeding as before, it is easy to show that, if d is su�ciently small,then j~v1j � d�5 = d�a5"5=2j~v2j � d�4 = d�a4"2 (3.35)The previous considerations imply that there exists d0 > 0, such that,given d � d0, (3.10), (3.12), (3.29), (3.31) and (3.35) are satis�ed for "su�ciently small. 10

By some simple algebra and using (3.14), it is possible to show that u01u02 ! = �1u1�2u2 !+ R1R2 !+ ~v1 + a~��v1~v2~v2 + 2b~��v1~v1 ! (3.36)where �1 = 1�K( " � 1)�2 = �6+4" +K( " � 1)� 2f�v21 (3.37)and 0@ R1R2 1A = ~�0@ ar1r2 � (3f + ab�)�v1r21 � fr31�br21 + 2b~��v1(ar1r2 � 3f�v1r21 � fr31) 1A (3.38)~� = �1� 2ab�2�v21 (3.39)By using (3.8), (3.14), (3.35), and (3.36), it is easy to prove that, if0 < � < 1=2, and " is su�ciently small, say " � "0, thenju01j � �1 ~d"1+� + c1 ~d3"5=2+� + c2d�a5"5=2ju02j � �2 ~d"3=2 + c3 ~d3"2+2� + c4d�a5"2 (3.40)for suitable constants ci; i = 1; : : : ; 4. Then the conditions (3.17) are pre-served if �1 + c1 ~d2"3=2 + c2d~d�a5"3=2�� � 1�2 + c3 ~d2"1=2+2� + c4d~d�a5"1=2 � 1 (3.41)By looking at (3.37), it is immediate to see that (3.41) can be satis�ed, givenany 0 < � < 1=2, if " is small enough. In order to prove that Tv 2 D, westill have to check that jv01j � d� ; jv02j � d�2 (3.42)which is again true for any � < 1=2, if " is small enough, by (3.13), (3.17)and the fact that �v 2 Bd=2.We now want to show that D contains a �xed point v� of the transfor-mation T and that, given any v 2 D, jjTnv � v�jj ! 0 as n!1.11

Given two elements of D, v(1) and v(2), we de�ne r(j)i = v(j)i � �v(j)i , j; i =1; 2, and u(j)i as in (3.13) and (3.14); then we de�ne:m(v(1); v(2)) = maxf��1ju(1)1 �u(2)1 j; ��2ju(1)2 �u(2)2 j; supQ�3 ��Qjv(1)Q �v(2)Q jg (3.43)It is easy to see thatjv(1)1 � v(2)1 j = jr(1)1 � r(2)1 j � c5 �m(v(1); v(2))jv(1)2 � v(2)2 j = jr(1)2 � r(2)2 j � c5 �2m(v(1); v(2)) (3.44)and ju(1)1 � u(2)1 j � c5 � jjv(1) � v(2)jjju(1)2 � u(2)2 j � c5 �2 jjv(1) � v(2)jj (3.45)for some constant c5.The inequalities (3.44) and (3.45) imply that m and the norm (3.11)generate the same topology. Therefore, in order to prove the existence ofa �xed point v� in D and its asymptotic stability, the following theorem issu�cient.Theorem 3.2 There exist a positive constant d1 � d0 and, for any givend � d1 and 0 < � < 1=2, another constant "1 � "0, such that, for any " � "1:m(Tv(1);Tv(2)) � �"m(v(1); v(2)) (3.46)with �" < 1 and �" ! 1 as "! 0.Proof - By proceeding as in the proof of Theorem 3.1 and using the identityv(1)Q1 � � �v(1)Qn � v(2)Q1 � � � v(2)Qn = nXk=1 v(1)Q1 � � �v(1)Qk�1 [v(1)Qk � v(2)Qk ]v(2)Qk+1 � � � v(2)Qn (3.47)and (3.44), it is easy to show that, if Q � 3 and d is su�ciently small, then��Qj(Tv(1))Q � (Tv(2))Qj � [ 2�9(2�") +B2d]m(v(1); v(2)) (3.48)for some constant B2, depending only on �� and �1.12

Moreover, by using (3.36), it is easy to show that��1ju0(1)1 � u0(2)1 j � [�1 + c6"3=2 + c6"2]m(v(1); v(2))��2ju0(1)2 � u0(2)2 j � [�2 + c6"1=2+� + c6"]m(v(1); v(2)) (3.49)for d small enough, say d � d1 � d0, and some constant c6 depending on ~d; �aand d1.All the claims of the theorem easily follow.Theorem 3.2 is not completely satisfactory, since we are interested in theproperties of the measure (2.5) with potential v(') = �(cos'� 1) � v(�)(').The properties of the approximate transformation (3.6) (see discussion after(3.8)) suggest that v(�) is in the domain of attraction of v�, for � positive andsu�ciently small, and that a similar result holds for � < 0; moreover, thecomputer simulation is in complete agreement with this conjecture. In orderto rigorously prove this claim, however, we should investigate more accuratelythe properties of the equation (3.36), trying to show that Tnv(�) 2 D for any� su�ciently small, if n is large enough. We think that this is possible, butwe did not try to �ll in the details.4 The correlation functionsLet us suppose that ", � and d are chosen so that there is in D a �xed pointv� of the transformation T. We want to study the linear operator Lv, see(2.13), for v = v�; let us simply call it L.We shall consider the action of L on the Hilbert space H of the functionsperiodic of period T� = 2�=� with inner product(G;F ) = 1T� Z T�0 d' q(')G�(')F (') (4.1)where q(') = ev(') Z P0(dz)ev('+z) � ev(')N(') (4.2)Proposition 4.1 L is a trace class positive self-adjoint operator of norm 1.13

Proof - It is very easy to verify that L is self-adjoint, by using the fact thatthe measure P0(dz) is even in z.Let us now observe that the functions Q(') = q(')� 12 ei�Q' ; Q 2 (4.3)are a base of H and that we can write:ev(')q(')� 12 =XQ gQei�Q' ; XQ jgQj2 <1 (4.4)Then we have:Tr(L) = PQ( Q;L Q) ==XQ Z P0(dz) 1T� Z T�0 d'[ev(') �Q(')][ev('+z) Q('+ z)] == XQ;Q0 jgQ0�Qj2 Z P0(dz)ei�Q0z = XQ;Q0 jgQ0�Qj2 ��24�Q02 == (XQ jgQj2) (XQ0 ��24�Q02) <1 (4.5)which proves that L is trace class.If F 2 H, evF 2 L2 and therefore we can writeev(')F (') =XQ ~fQei�Q' ; XQ j ~fQj2 <1 (4.6)Then, by proceeding as before, we can check that(F;LF ) =XQ j ~fQj2 ��24�Q2 (4.7)which proves that L is a positive operator. Moreover(F;LF ) = j(F;LF )j �� Z P0(dz) 12T� Z T�0 d'ev(')+v('+z)[jF (')j2 + jF ('+ z)j2] == 1T� Z T�0 d'ev(')jF (')j2Nv(') = (F; F ) (4.8)

14

Since LF = F , if F is a constant function, jjLjj = 1.By Prop. 4.1 L is a positive compact operator, then it has a pure discretepoint spectrum with positive eigenvalues, at most �nitely degenerate. Fur-thermore the subspaces H+ and H� of H, which contain the functions evenand odd in ' respectively, are invariant under the action of L. Let L� bethe restriction of L to H�.For " = 0 (and hence v = 0) the eigenvalues of L+ and L� are the same,that is: ��n (0) = �2n2 ; n = 0; 1; : : : (4.9)and they are all simple. By using the properties of v proven in Sect. 3and known results about the perturbation theory of compact operators, itis possible to show that the eigenvalues of L� can be written as suitablefunctions ��n ("), which are continuous in " = 0.Since the constants are eigenfunctions of L for any " � "1:�+0 = ��0 = 1 (4.10)and 1 is a simple eigenvalue of L. All the other eigenvalues are strictly lessthan 1.Let f�ngn�0 be the set of all eigenvalues, ordered so that �n+1 � �n, andlet Gn the corresponding eigenfunctions, normalized so that Gn is real and(Gn; Gm) = �nm (4.11)In particular �0 = 1 and G0 = " 1T� Z T�0 d'q(')#� 12 (4.12)Moreover, it is possible to show, by the technique used below in the proof ofTheorem 4.1, that the Gn are smooth functions.Let us consider a function F 2 H such that its expansion in terms of theGn: F (') = 1Xn=0 fnGn(') (4.13)15

has good convergence properties. Then, by (2.11) we have:< F ('0) >v=< (LkF )('0) >v==Xn fn�kn < Gn('0) >v ���!k!1 f0G0 = (G0; F )G0 (4.14)Let us now suppose that we want to calculate the correlation betweenF ('0) and F �('x). This problem arises, for example, if we are interestedin the two charges correlation; in this case F (') = ei�', whose expansionhas the needed convergence properties, as it is possible to show with somestandard calculation, using the smoothness of the functions Gn and the factthat they are small perturbations of the functions ei�Q.If h is the smallest integer so that there exists a � 2 Qh containing both0 and x, we can write:< F ('0)F �('x) >v=< j(ThF )('0)j2 >v==Xnm fnfm�hn�hm < Gn('0)Gm('0) >v (4.15)But, by (4.11) and (4.14):< Gn('0)Gm('0) >v= (G0; GnGm)G0 = G20(Gn; Gm) = G20�nm (4.16)Then< F ('0)F �('x) >Tv = < F ('0)F �('x) >v �j < F ('0) >v j2 == G20 1Xn=1�2hn jfnj2 (4.17)and, as a consequence< F ('0)F �('x) >Tv 'x!1 c�2h1 = cd��0x (4.18)with c a suitable constant and � = �2 log �1 (4.19)We now want to show that �1 = �2 (4.20)16

at least for " small enough. We start by observing that, since Tv = v, by(2.12): e 1 2 v(') = R P0(dz)ev('+z)R P0(dz)ev(z) (4.21)By calculating the '-derivative of both sides, it is easy to check that:L dvd' = �2 dvd' (4.22)Since dvd' 2 H�, this implies that��1 (") = �2 ; 8" � "1 (4.23)But �1, for " small enough, is the minimum between �+1 (") and ��1 ("); hence,in order to show (4.20) it is su�cient to prove that �+1 (") is smaller than �2.We notice that (2.13) can be written also in the following way:(LvF )(') = dd� log Z P0(dz)ev('+z)+�F ('+z)�����=0 (4.24)If F 2 H, F 2 L2, then we can write:F (') =XQ fQei�Q' (4.25)Moreover, since (LvF )(') does not change if we add a constant to v(') , wecan replace in (4.24) v(') by the expansion PQ6=0 vQei�Q', whose coe�cientsare the same as in (3.1). Then we obtain:(LvF )(') =XQ fQ ��24�Q2ei�Q'++ 1Xn=2 1(n� 1)! XQ1;:::;Qn vQ1 � � �vQn�1fQnFn(Q1; : : : ; Qn)ei�(Pni=1 Qi)' (4.26)If � < 1, the eigenvalue equation LvF = �F is satis�ed if f0 = 0 and�fQ = fQ ��24�Q2++ P1n=2 1(n�1)! PQ1+���+Qn=Q vQ1 � � � vQn�1fQnFn(Q1; : : : ; Qn) (4.27)17

for jQj 6= 0.We are interested in the dependence on " of the eigenvalue �+1 (") and ofthe Fourier coe�cients fQ(") = f�Q(") of the corresponding eigenfunction,which we shall normalize so thatf1(") = f�1(") = 1 (4.28)For " = 0, we havef1(0) = f�1(0) = 1 ; fQ = 0 ifjQj 6= 1 (4.29)We shall now rewrite (4.27), for � = �+1 ("), as a �xed point equation ina suitable Banach space, where the existence of a unique solution will followfrom the contraction mapping principle.Let us de�ne:GQ(F ) = 1Xn=2 1(n� 1)! �� XjQ0j�2 fQ0 XQ1+���+Qn�1=Q�Q0 vQ1 � � �vQn�1Fn(Q1; : : : ; Qn�1; Q0) (4.30)Eq. (4.27) gives, for Q = 1:�+1 (") = �2 + r(") +G1(F ) (4.31)where r(") = �2( " � 1) + v2F2(2;�1) + 1Xn=3 1(n� 1)! �� XQ=�1 XQ1+���+Qn�1=1�Q vQ1 � � � vQn�1Fn(Q1; : : : ; Qn�1; Q) (4.32)If jQj � 2, we have: [�+1 (")� ��24�Q2]fQ = hQ +GQ(F ) (4.33)wherehQ = 1Xn=2 1(n� 1)! XjQ0j=1 XQ1+���+Qn�1=Q�Q0 vQ1 � � � vQn�1Fn(Q1; : : : ; Qn�1; Q0)(4.34)18

By proceeding as in the proof of Theorem 3.1, it is easy to show thatthere exist constants a1 and A1, only depending on �� and �1, such thatjr(")j � a1" (4.35)jhQj � A1d�jQj�1 ; if jQj � 2 (4.36)Let us now consider the Banach space E of the even functions F ('),periodic of period T�, such that f0 = 0, f1 = f�1 = 1 andjjF jj � supQ�2 ��Q+1jfQj <1 (4.37)We denote ED the sphere of radiusD and center at the origin and we considerthe operator K from ED to H+, de�ned so that, if (KF )(') = PQ6=0 f 0Qei�Q',then f 0Q = f 0�Q = hQ +GQ(F ) �2 + r(") +G1(F )� ��24�Q2 ; if Q � 2f 01 = f 0�1 = 1 ; f 00 = 0 (4.38)By (4.31) and (4.33) F is a solution of (4.27) for � = �+1 ("), belonging toED, if and only if F is a �xed point of the operator K.Theorem 4.1 There exist a positive constant d2 � d1 and, for any givend � d2 and 0 < � < 1=2, other constants "2, D0 and D1, such that ED isinvariant under the transformationK, if " � "2 and D0 � D � D1; moreoverK is a contraction as an operator from ED to ED.Proof - By proceeding as in the proof of Theorem 3.1, it is possible toshow that, if Q � 2 and Dd is small enough:jf 0Qj � A1d(1 +D)�Q�1 �2 � a1"�DdA1� � ��2� (4.39)Therefore ED is invariant under the transformation K ifD � A1d(1 +D) �2 � a1"�DdA1� � ��2� (4.40)19

and it is easy to see that there exist d2, "2, D0 and D1 such that (4.40) issatis�ed if d � d2, " � "2 and D0 � D � D1.Let us now consider two elements F1; F2 2 ED such that, for Q � 2,jf1Q � f2Qj � ��Q�1 (4.41)It is easy to check that, for any Q � 1:jGQ(F1)�GQ(F2)j � �dA1�Q�1 (4.42)Moreover, if Q � 2, by (4.38):f 01Q � f 02Q = fhQ[G1(F2)�G1(F1)] + bQ[GQ(F1)�GQ(F2)]++GQ(F1)[G1(F2)�G1(F1)] +G1(F1)[GQ(F1)�GQ(F2)]g==f[bQ +G1(F1)][bQ +G1(F2)]g (4.43)where bQ = r(") + �2 � ��24�Q2 (4.44)Then it is very easy to show that, for d small enough,jf 01Q � f 02Qj � ��� ; �� < 1 (4.45)which immediately implies, together with Theorem 3.1, all the claims of thistheorem.>From (4.30), (4.31), (4.32), Theorem 4.1 and some simple algebra followsthat: �+1 (") = �2 � " log (1 + abab + f(1� c)) +O("3=2) (4.46)Then, if " is small enough �+1 (") < ��1 (") (4.47)so that (4.20) is satis�ed.This means that, if v = v�, the integer charge truncated correlationsdecay as d�4xy . With some more computational e�ort one could show that thisresult is true also if v is in the domain of attraction of v�.We conclude by two remarks. The �rst remark, anticipated in the dis-cussion preceding (4.13), is that the technique used in this section can be20

applied to any eigenvalue of Lv with similar results. In particular, one canshow that, for any given n and " small enough (how small depends on n),��n (") < ��n (0).The second remark is that we could study the fractional charges correla-tions, by using similar arguments, in spite of the fact that the function ei�'must be substituted by ei��', 0 < � < 1, which is not periodic of period T�.It is only su�cient to observe that, if �F (') = ei��'F ('), with F 2 H, then(Lv �F )(') = ei��'(L(�)v F )(') (4.48)(L(�)v F )(') = R P0(dz)ei��'ev('+z)F ('+ z)R P0(dz)ev('+z) (4.49)and L(�)v is again a self-adjoint operator from H to H, whose spectrum can bestudied in the same way as the spectrum of Lv, obtaining the same resultsreported in Ref. [6].Acknowledgements.We are indebted to G. Gallavotti and F. Nicol�o for many discussions andsuggestions.A Proof of the bound (3.4)Let I = f1; : : : ; ng be the set of the �rst n positive integers and, for eachi 2 I, let Qi be a �xed integer (Qi 2 Z). If zt is a random gaussian variablewith mean 0 and covariance E(z2t ) = t � 1 (A1)and c is a �xed positive constant, we de�ne:F (I; t) = ET (eipcQ1zt; : : : ; eipcQnzt) (A2)where ET denotes the truncated expectation with respect to zt. It is a wellknown fact that: F (I; t) = e� c2Pni=1Q2i tf(I; t) (A3)21

with f(I; t) =XG Yij2G(e�cQiQjt � 1) (A4)where G is the family of all connected graphs on n vertices labeled by theelements of I, with bonds denoted by ij(i; j 2 I).Using the results of Ref. [9], in particular Lemma 3.3 and the recurrencerelation for f(I; t) which follows from it (see pag. 40 of [9]), it is very easyto show that:F (I; t) = 8>>>>>><>>>>>>: � c2 Z t0 dse� c2Q2I(t�s) X;6=J$IQJQInJF (J; s)F (InJ; s)if jIj = n � 2e� c2Q2I t if jIj = 1 (A5)where we used the de�nition, for J a subset of I:QJ =Xi2J Qi (A6)We want now to describe the solution of the recurrence relation (A5). Letus consider, for n � 2, the family �n of all planar binary trees with root rand n endpoints labeled by the elements of I, oriented from the root to theendpoints, see Fig. 1.We call vertices the root, the endpoints (e:p: in the following) and thebranch points of the tree; the branch points will be called also nontrivial (n:t:in the following) vertices. If v is a vertex di�erent from r, we shall denoteby v0 the vertex immediately preceding it in the tree and we shall say thati 2 v if the e:p: with label i follows v; moreover v0 will denote the verteximmediately following the root. We de�ne:Qv =Xi2v Qi (A7)Finally we label each vertex v with a real number sv such that:t � sv0 � sv � 0sr = tsv = 0 if v is an e:p: (A8)22

r v0���HHHHHHHH������v0hhhhhhhhhhhhhhhhPPPPPPPPPP����v ``````(((((((((((hhhhhh(((((hhhhhFig. 1It is easy to see that, if jIj � 2:F (I; t) = X�2�n Z Yv n:t: �dsv2 !��W� (A9)where �� is the characteristic function of the set (A8) andW� = nYi=1(pcQi) Yv e:p: e� c2Q2i sv0! �� 264 Yv n:t:v 6=v0 (pcQv)e� c2Q2v(sv0�sv)375 e� c2Q2I(t�sv0 ) (A10)We want to show thatjF (I; t)j � nn�1 nYi=1(2pcjQij)e� c4nQ2It (A11)The �rst step in the proof is to get rid of the \bad" factors Qv in (A10),using the boundpcjQvje� c2Q2v(sv0�sv) � 1psv0 � sv e� c4Q2v(sv0�sv) (A12)23

Then, if jIj � 2, we can write, using also that 0 � t� sv0 � 1:jF (I; t)j � nYi=1(pcjQij)E(I; t) (A13)whereE(I; t) = 8>>>>>><>>>>>>: 12 Z t0 dspt� se� c4Q2I(t�s) X;6=J$IE(J; s)E(InJ; s)if jIj = n � 2e� c2Q2I t if jIj = 1 (A14)We now prove, by induction on n = jIj, that:E(I; t) � (2n)n�1e� c4nQ2I t (A15)In fact (A15) is true for n = 1; moreover, if we suppose that it is true for1 � k < n, we have, using (A14):E(I; t) � 2n�3 X;6=J$I kk�1(n� k)n�k�1G(I; J; t) (A16)where k = jJ j andG(I; J; t) = e� c4Q2I t Z t0 dspt� se c4 s[Q2I�Q2J=k�Q2InJ=(n�k)] (A17)If [Q2I �Q2J=k �Q2InJ=(n� k)] � 0, thenG(I; J; t) � e� c4 t[Q2J=k+Q2InJ=(n�k)] Z t0 dspt� s � 2e� c4n tQ2I (A18)where we used the fact that t � 1 and the inequality, valid for two arbitraryreal numbers a and b: a2k + b2n� k � 1n(a+ b)2 (A19)If [Q2I �Q2J=k �Q2InJ=(n� k)] � 0, thenG(I; J; t) � e� c4 tQ2I Z t0 dspt� s � 2e� c4n tQ2I (A20)24

(A16), (A18) and (A20) imply that:E(I; t) � 2n�2e� c4nQ2I t n�1Xk=1 nk!kk�1(n� k)n�k�1 � (2n)n�1e� c4nQ2It (A21)where we used the identity (see [9]: Lemma 4.2):n�1Xk=1 nk!kk�1(n� k)n�k�1 = 2(n� 1)nn�2 (A22)Then (A15) is proved; if we insert it into (A13), we obtain the bound(A11).References[1] J.M. Kosterlitz and D.J. Thouless. Ordering, metastability and phasetransitions in two-dimensional systems. J. Phys., C6:1181{1203, 1973.[2] J. Fr�ohlich and T. Spencer. The Kosterlitz-Thouless transition in twodimension Abelian spin systems and the Coulomb gas. Commun. Math.Phys., 81:527{602, 1981.[3] D.H.U. Marchetti, A. Klein, and J. Fernando Perez. Power law fall o� inthe Kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas.J. Stat. Phys., 60:137, 1990.[4] J. Dimock and T. R. Hurd. A renormalization group analysis of theKosterlitz-Thouless phase. Commun. Math. Phys., 137:263{287, 1991.[5] G. Benfatto, G. Gallavotti, and F. Nicol�o. The dipole phase in twodimensional hierarchical Coulomb gas: analyticity and correlations decay.Commun. Math. Phys., 106:227{288, 1986.[6] D.H.U. Marchetti and J. Fernando Perez. The Kosterlitz-Thouless phasetransition in two-dimensional hierarchical Coulomb gases. J. Stat. Phys.,55:141{156, 1989.[7] D. Brydges and P. Federbush. Debye screening. Commun. Math. Phys.,73:197{246, 1980. 25

[8] K. Gawedski and A. Kupiainen. Asymptotic freedom beyond perturba-tion theory. In Les Houches Summer School, 1984.[9] D. Brydges and T. Kennedy. Mayer expansions and the Hamilton-Jacobiequation. J. Stat. Phys., 48:19{49, 1987.

26