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Singularrenormalization group transformations and first order phase transitions (I)
Nuclear Physics B295 [FS21] (1988) 1-20 North-Holland, Amsterdam
SINGULAR RENORMALIZATION
GROUP TRANSFORMATIONS
AND FIRST ORDER PHASE TRANSITIONS (I) Anna HASENFRATZ1 Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306, USA
Peter HASENFRATZ 2 Institute of Theoretical Physics, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland
Received 20 July 1987 (Revised 31 August 1987)
It is argued that standard position and momentum space renormalization group (RG) transformations are singular (i.e. lead to a singular hamiltonian after a finite number of RG steps) in large regions of the coupling constant space. It is shown in the d = 3, ~60(N) N ~ ~ model that the momentum space RG transformation is singular in all those points of the coupling constant space, where metastable states exist. This region includes the full first order phase transition surface and its neighbourhood. Several other examples are discussed to illustrate that this phenomenon is generic and not a specific large N effect. Some earlier and recent anomalous Monte Carlo renormalization group results are consistent with this conclusion.
1. Introduction A b a s i c a s s u m p t i o n of the R G a p p r o a c h [1] is that the t h e r m o d y n a m i c a l singularities arise b y r e p e a t i n g the e l e m e n t a r y R G steps i n f i n i t e l y m a n y times. T h e statistical p r o b l e m a s s o c i a t e d with an e l e m e n t a r y R G step (integrating out a fraction of the h i g h m o m e n t u m variables, or averaging over the short wavelength fluctuations in p o s i t i o n space) is a s s u m e d to b e non-singular. By i n v e s t i g a t i n g different b l o c k t r a n s f o r m a t i o n s in c o n f i g u r a t i o n space Griffiths a n d Pearce c o n c l u d e d [2] that this a s s u m p t i o n c a n n o t be strictly true. E x a m p l e s were f o u n d , w h e r e - starting at specific p o i n t s of the c o u p l i n g c o n s t a n t s p a c e s i n g u l a r i t i e s o c c u r r e d in the r e n o r m a l i z e d h a m i l t o n i a n after a finite n u m b e r of R G steps. A t c e r t a i n values of the original c o u p l i n g c o n s t a n t s the r e n o r m a l i z e d coup l i n g s d e v e l o p e d non-analyticities. T h e R G t r a n s f o r m a t i o n can b e singular. 1 On leave from Central Research Institute for Physics, Budapest, Hungary. 2 Supported in part by Schweizerischer Nationalfonds. 0169-6823/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
A. Hasenfratz, P. Hasenfratz / Phase transitions
The Griffiths-Pearce "peculiarities" did not raise much concern in RG practitioners as it was assumed that they do not occur in the vicinity of the fixed points of interest. Additionally, it is generally believed that no such singularities occur in momentum space renormalization [3]. In this paper we shall argue that standard RG transformations are singular in a large, continuous part of the coupling constant space if the original thermodynamical system exhibits first order phase transitions. The arguments are based on different model studies and numerical simulations. This investigation has been initiated and motivated by some unexpected M C R G results, which will be detailed in a subsequent paper [4]. Our first example (sect. 2) will be the scalar O(N) model in d = 3 dimensions:
--~=
f{ 1(0~)2"[-
) }d3x.
~ ~ "p ~(q~2)2q'- 1T/(q~2 3
(1)
The momentum space RG transformation of this model can be studied analytically in the large N limit [5, 6] and the following statement can be proved: In all those/,2, ?~, ~/> 0 points of the parameter space, where the exact effective potential of the system has several competing minima (i.e. apart from the absolute minimum (ground state) it has one or more other minima, which correspond to metastable states) the RG transformation leads to a singular hamiltonian after a finite fraction of the momenta is integrated out. The effective potential is defined here the usual way, and a priori has nothing to do with R G transformations [7]. The statement is formulated more precisely and proved in sect. 2. Metastable states certainly exist in the neighbourhood of the first order phase transition surface, therefore the statement above implies that on the whole first order surface and in its vicinity the RG transformation is singular. We emphasize that the statement above refers to the existence of metastable phases (which is an ordinary notion in first order phase transitions [8]) and not to some specific property of the large N limit. In particular the fact that a momentum space RG transformation can be singular is not specific to large N. In sect. 3 we consider a one-component scalar field theory and perform the RG transformation Acut---~ e - a ~ cut in momentum space. We discuss simple examples, where the renormalized hamiltonian is not an analytic function of t, or the original coupling constants. Unlike the large N case, this analysis is not exact. It is based on a semiclassical approximation in a coupling constant region, where the corrections are expected to be small. Sect. 4 deals with the d = 2 Ising model in an external magnetic field h. The RG transformation is defined in configuration space: 2 × 2 blocks are formed, the block spins are determined by the majority rule. The question is raised, whether the statistical problem of summing over the original spins at some given value of the
A. Hasenfratz, P. Hasenfratz / Phase transitions
3
block spins could lead to thermodynamical singularities (this would imply singular R G transformation). For certain background spin configurations and at large /3 = 1 / T it is easy to identify the configurations which dominate this constrained partition function. The resulting simplified model can be solved exactly. It predicts a first order phase transition and reproduces quantitatively our MC data on this system - including the magnetization as the function of h and the size L of the lattice, M = M(h, L). The critical field h c depends on /3, on the size of the block and on the background blockspin configuration. This indicates again that the renormalized hamiltonian is singular in a large region of the parameter space around the first order transition surface (of the original Ising model), although we are not able to make such general conclusions as in the O(N)N--* e ~ case. Sect. 5 is a summary. We give a brief account of the MCRG studies we (in collaboration with K. Decker) performed, whose results will be detailed in a subsequent paper [4]. These MCRG studies on the d = 2 Potts-10 model, on the d = 2 Ising model with magnetic field and on the d = 4 U(1) lattice gauge theory all indicate that the usually accepted RG picture of first order phase transitions with a discontinuity fixed point [9] requires modification. The results are consistent with the assumption that the RG transformation used is singular in the vicinity of first order phase transitions. We mention earlier examples also, where MCRG studies of first order phase transitions raised the possibility of singular RG transformations [101. The framework of this paper (based on renormalized hamiltonians) and that of the M C R G analysis of ref. [4] (working with blocked configurations) are somewhat different. There are points in the present analysis (like the possible connection between singular RG transformations and the existence of a metastable phase), which are not directly tested by the MCRG analysis. Similarly, there are certain regularities in the MCRG results [4], which are not explained by the present theoretical analysis. It is true, however, that the MCRG results indicate that the RG transformation used is singular on the first order transition surface. This is consistent with the conclusions of this paper and is against the standard discontinuity fixed point picture. These points are discussed further in the summary. It might be that a non-singular RG prescription exists, which works on the states rather than on the hamiltonians and which treats separately the different phases of the first order phase transitions. A RG approach of first order transitions in Ising-like models was constructed recently in terms of Peierls contours [11]. This method is free also of the singularities discussed here. 2. Relation between the existence of metastable phases and singular RG transformations in the O ( N )N-~ ~ model
We shall study the RG behaviour of the model defined by eq. (1) and prove the statement mentioned in the introduction.
4
A. Hasenfratz, P. Hasenfratz / Phase transitions
After rescaling the fields (~2 ~ y = ep2/N), the couplings (#2 ~ tt2, ~ ~ h/N, 71---> ~/N 2) and the potential (V--->V/N) in the usual way we write V(y)
= 1 2
~ t L y + ~1 y
2
+ ~1y
3.
,
y>/0,71>0,
(2)
where all the variables are dimensionless, the dimensions are carried by the cut-off A cut. We shall use the notation
u(y) = 2V'(y) and denote the inverse function of u = u(y) by F(u): y - - F ( u ) . Consider a RG transformation in momentum space, which reduces the cut-off by a factor of s( >/1):
Acut--->ACUt/s. Under this transformation the relation y =
(3)
F(u) changes into [5,6]
y=Fs(u), where, in dimension d = 3, the function
F~(u)=sF The function
(4)
Fs(u) is given as
-sf
+f(u);
s>~l.
(5)
f(u) is independent of the parameters of the potential:
f(u)=flq
d3q
1
1
1~<1(27/')3 (q2q-U)
(
(
1
)
)
(297")2 1-vrffarctan ~-u
'
u>~0
(6)
and it is defined for non-negative u-values. On the other hand, the combination -sf(u/s 2) +f(u), which enters the RG equation (5) is defined for all u > - 1 :
(u)
{1[
C(s,u)=f(u)-sf ~--
=
dq(qa+u )
2qr2
1 2~r 2 1 - s + ~ 2 u ½ 1 n
,
)]
[s+~--u 1 - ~ u s
~-u
l+~-u-
(7) '
-l
We shall assume in the following that the starting potential satisfies the u > - 1 constraint for every y >i 0 (this point will be discussed further in sect. 3).
A. Hasenfratz, P. Hasenfratz / Phase transitions
In our case the function
5
F(u) is given by
F(u)=
-o~_
+ 2~.2,
(8)
where, for later convenience, we introduced the notations X a = ~
1 + 2,/r 2 ,
w
X2
1~2
4,02
,0
(9)
The two branches of F(u) will be denoted by F -+(u). Even without going into the details of the proof it is easy to see that in large regions of the parameter space (w, a, '0) the RG transformation is singular and the relation between the existence of metastable phases and singularity is suggestive. The RG relation eq. (5) can be written as
[ ju
F/-(u)=s-,~+
w+--
-
+ C(s, u),
s2'0
(10)
where s
d(s, u) = c(s, u) + 2 T
(11)
The possible values of u are constrained by u > - 1 and by the demand that
Fs(u ) should be real and non-negative, eq. (4). For any fixed u, C(s, u) runs toward an s-independent function C*(u) as s ~ oo: C(s, u) ~ (~*(u).
(12)
The function C*(u) is shown in fig. 1. Consider now the parameter region a < 0, 0 < w < a2. In this parameter region the starting (classical or mean field) potential has a trivial and a non-trivial competing minima, the classical potential predicts a metastable state (we shall see later that this is true also beyond the classical approximation). The starting F(u) is sketched in fig. 2a. For large s we get from eq. (10)
F/-(.) -~ s ( - ~ _+~ ) + O ( . ) .
(13)
Since - a + ~ - > 0, both branches of F,+-(u) survive the large s limit (remember, Fs(u) = y should be non-negative), fig. 2b. For every Fs(u ) =y there corresponds
A. Hasenfratz, P, Hasenfratz / Phase transitions
1.0
-0.5
I
-I.
0.5
I
1.0
"~U
--0.5
--1 Fig. 1. The function C*(u) is the large s limit of the function C(s, u) defined by eqs. (11) and (7). It has a logarithmic singularity at u = - 1.
more than one u = u(y) solution, the new hamiltonian is multivalued. At some smaller s values, a u/Oy became infinite at certain y values. The R G transformation is singular. We shall prove now the following.
Statement. The R G transformation eq. (3) in singular in all those points of the parameter space (w, a, ~1), where the system has several competing minima, i.e. metastable states exist. It is singular also if the exact effective potential V~fr(~ ) has several competing branches over some region of fie- In all those points the new hamiltonian produced by the R G transformation is non-analytic, multivalued for large enough s. In any other points of the parameter space the large s hamiltonian is single valued and analytic.
First we shall identify the parameter region, where the R G transformation is singular. Then we investigate the properties of the theory in this coupling constant region using standard methods which are independent of R G transformations. Let us investigate first the possibility that the singularity arises in the region -l
Statement A. Assume, the function F(u) has a part in the - 1 < u < 0 region. Then, for - 1 < u < 0 the R G transformation is singular, if and only if F - ( u = 0) >/ 1 / 2 r r 2.
A. Hasenfratz, P. Hasenfratz / Phase transitions
I
I
-I.0
-0.5
0'.5
=U
F+(u)
F
W(u)
r" -1.0
1.o
I
:
-0.5
0.5
~
1.0
=U
Fig. 2. The starting function F(u) = F,=l(u ) corresponds to a potential having competing minima. The R G transformation breaks this function into two pieces for large s, and no single-valued inverse exists.
Proof. T h e region - 1 < u < 0 exists only if w > 0 (eq. (8)). Since Iu l is b o u n d e d here, we can write for large s: Fs+-(u)=s(-a+_v~)+(~*(u),
-l
(14)
Since F - ( u = O) = ( - a - ¢7) + 1 / 2 ~ "2, the condition F - ( u = 0) >/1/2~r 2 implies -a + ~ - a -
> 0,
vrw > / 0 .
(15) (16)
T h e r e f o r e s ( - a +_V~) is non-negative and - since C * ( u ) has also a non-negative piece for u < 0 - b o t h branches of Fs(u ) survive (fig. 2). The function u = u(y) is multivalued, the R G transformation is singular.
8
A. Hasenfratz, P. Hasenfratz / Phase transitions
Take now F-(u = 0 ) < 1/2~r 2. In this case - a - v/-w-< 0, and F]-(u) becomes negative for every - 1 < u < 0 if s is large. Only F+(u) survives, which is a monotonic function. No singularity occurs (it is easy to see that F + (u) is a monotonic increasing function of u. Both the first term and C(s, u) in eq. (10) are monotonic increasing as can be seen from the integral representation of C(s, u) in eq. (7)). Consider now the region 0 ~< u < oo:
Statement B. For 0 ~< u < oo the R G transformation is singular if and only if there exists a number A >/0 such that the equation F-(u) - f ( u ) - A = 0 has more than one (non-negative) root. The function
(17)
f(u) is defined in eq. (6).
Proof. If there is more than one root, the function F - ( u ) - f ( u ) cannot be everywhere negative, and the positive part should be a non-monotonic function of u. If there exists one root only, the positive part of F - ( u ) - f ( u ) is monotonic decreasing ( F - ( u ) - f ( u ) is negative for large u). If there is no root, F - ( u ) - f ( u ) is always negative. Let us rewrite the R G equation eq. (5) in the following form 1 s F Z ( u s 2) =
1
( F-( u) - f( u) ) - s f ( Us2) .
(18)
The function f(u) is bounded by 1/2~r 2, therefore the last term can be neglected for large s. If eq. (17) has more than one root, F-(u) - f ( u ) has a positive part, therefore F Z survives. It is, however, a non-monotonic function of u, u = u(y) is multivalued, the R G transformation is singular. If there is one root only, FT(u ) survives and is monotonic decreasing in u. Since F+(u) is monotonic increasing and F~+(u)>~FZ(u ), the full function Fs(u ) has a single valued inverse - no singularity occurs in this case. Finally, if eq. (17) has no root, F - ( u ) - f ( u ) in eq. (18) is always negative and the branch F~-(u) does not exist for large s. On the other hand, F~+(u) is monotonic - no singularity occurs. We shall show now that the condition F-(u = 0) >i 1/2~r 2, w > 0 corresponds to that part of the parameter space, where metastable state(s) exists (this region includes the first order phase transition surface and its vicinity in the phase diagram), while if eq. (17) has more than one root, then the effective potential V~ef has competing branches.
A. Hasenfratz, P. Hasenfratz / Phase transitions
9
Rather than repeating the detailed discussion of the effective potential Veff(*2) in the large N limit, we refer to refs. [5, 7] and quote the relevant result only. Let us denote the inverse function of f (eq. (6)) by g(x): g ( f ( u ) ) = u. The function g(x) is defined for 0 < x ~ 1/2qr 2. Let X = X ( , 2) the solution of the equation
u(X+*2)=g(X).
(19)
Here u(y) = 2V'(y), as before. Then the derivative of the effective potential can be written as [5, 7] Ve;f( , 2 ) =
Vt(X(*2) + ,2).
(20)
The condition of a non-trivial extremum at q]2 is Ve~f(q]2) = 0, which is equivalent to the equation
21, since the only point where g ( X ) = 0 is X = 1/2rr 2. Eqs. (19) and (21) can be expressed in terms of the inverse function F and f we defined before:
F(u) F(u =
,c2 = f ( u ) ,
(22)
1
0) = 2~r~ + q]2.
(23)
If F-(u = 0) >i 1/2~r 2 and w > 0 (these were the conditions in statement A), then F + ( u = 0 ) > F - ( u = 0 ) and eq. (23) has two nontrivial solutions. This case is sketched in fig. 3. One can see by inspection that eq. (22) always has a positive u > 0 solution at ,~ = 0, implying that V~f(, 2 = 0 ) > 0, therefore the effective potential (as the function of *c) has a minimum at the origin. We have therefore two competing minima - the ground state and a metastable state - in this case. If eq. (19), or equivalently eq. (22) has more than one solution over a region of ,2, then the effective potential has competing branches there. This is just the condition expressed by eq. (17). In eq. (17), F-(u) can be replaced by F(u), since if F+(u)-f(u)- A = 0 has a root at some Zl (it cannot have more than one) then F - ( u ) - f ( u ) - zl = 0 has zero or two roots (it cannot have one root; if F+(u)f(u)--A = 0 has a root, then F+(umi~)= F-(umin)
F(u)-f(u) This completes the proof of the statement.
-A=0.
(24)
10
A. Hasenfratz, P. Hasenfratz / Phasetransitions
( I
-1.0
I
-0.5
0.5
1.0
Fig. 3. When F-(u = 0) > 1/2~r 2 and w > 0, there always exist two non-trivial extrema and Vaf has a minimum at the origin since eq. (22) has a solution with ffc2 = 0. The function f(u) is defined in eq. (6).
3. Singular RG transformations in a one-component scalar theory T h e example discussed in the previous section runs against the often raised intuitive a r g u m e n t saying that integrating out a part of the high m o m e n t a cannot p r o d u c e singularities. Are the p h e n o m e n a of sect. 2 related to the large N limit? I n this section we shall argue and present examples to show that m o m e n t u m space R G transformations can be (and are often) singular even in a o n e - c o m p o n e n t scalar theory. There is a type of singularity whose existence is almost immediate. We have already seen in the large N model in sect. 2 that if the starting potential of the m o d e l does n o t satisfy 2 V ' ( y ) > - 1 for every y ( = ~2), the R G formulas become singular for every s > 1. These kind of singularities are present for any N. The a p p r o x i m a t e R G equation (derived in the local potential approximation, see eq. (50) in ref. [12]) exhibits singularities and makes no sense if the starting potential does not satisfy 2 V ' ( y ) > - 1 for N = ~ or 2 V ' ( y ) > - 1 and 2 V ' ( y ) + 4 y V " ( y ) > - 1 for finite N*. This problem is not related to the approximation, it is present also in the exact R G equation of Wegner and H o u g h t o n [13]. The exact equation for 9if= JC~(t) (A cut ----~e-tA cut is the R G transformation) contains a piece:
°2ae
In O q , ( q ) S q ~ ( - q )
) [q[=l'
(25)
* In eq. (50) of ref. [12] the function f corresponds to 2~- V'(y) of this paper, and f is considered there as a function of x = V%-"
A. Hasenfratz, P. Hasenfratz / Phase transitions
11
which is singular if the matrix
0 29~ Oe°(q)Oe°(-q)
Iql
(26)
=1'
has non-positive eigenvalues. The source of the problem is the following. In deriving the R G equation, one performs an infinitesimal RG transformation: Acut --, e-~tAcut. One writes ~(q) = ~0(q) + ~l(q), where q~l(q) are the field components one has to integrate out (in q~l(q) the momentum lies in the shell e -st ~< [ql ~ 1). Expanding the hamiltonian 9f'(q~0 + q~l) in q~l, Wegner and Houghton argue [13] that all the terms containing more than two q~l factors give O(& z) contributions, therefore can be neglected for infinitesimal St. The resulting gaussian integral over ~1 gives the term in eq. (25). One can write down hamiltonians, however, which give perfectly convergent functional integrals for the full theory, while the matrix eq. (26) has non-positive eigenvalues. In this case the gaussian integral does not converge, one should keep the higher order terms, which were formally O(&2). The naively derived R G equation cannot be applied and one suspects that no differential RG equation exists in this case, since there is no reason that the result of the RG transformation will be O(&). In the following we shall discuss simple explicit examples in a one-component scalar theory, which exhibit not only these singularities but also the types discussed at length in sect. 2. Consider the hamiltonian
..~= f(½(O,#)2+lt~2¢2+¼M,4)ddx,
X>O,
(27)
where q~(x) is a one-component scalar field and the cut-off constraints the momenta to lie inside a d-dimensional hypercube. In terms of dimensionless variables we have:
[qi[ ~< 1,
i = 1,2 . . . . . d.
(28)
Consider the R G transformation, where the field components with e a t < Iqi[ ~ 1 are integrated out. Here At is assumed to be small. We write for the Fourier field ¢(q): q~(q) = •0(q) + q~l(q),
(29)
where q~o(q) is defined only for Iqi] < e-at, while ffl(q) are the variables we have to integrate over. In this functional integral the components q~0(q) are external parameters. According to the basic assumption of the RG procedure, for any set of q~o(q) the result should be an analytic function of the parameters ~2 and ~ and, for
12
A. Hasenfratz, P. Hasenfratz / Phase transitions
infinitesimal At, should be proportional to At (otherwise)~"(t) is not analytic in the R G parameter t). Take, as an example ~0(q)--0. In this case the hamiltonian in the functional integral over qh(q) fe
Dq~l(q) exp ( -at~lqil~< 1
f [ 1 (0g,l~l)2 q_ ~1.21~.)2[ ' 1I,x ,q_ ~1 M h4 (. x .)], ddx ) ,
(30)
has the original form, the field qh(x) contains high momentum components only
~l(X)
fe -----
eiqxdpl(q)" _at~lqil<~1
(31)
Take small positive )t. Then a systematic perturbative expansion can be set up for the functional integral in eq. (30), where the propagator is (q2+ ~t2)-1 and the momentum integrals run over the e-ate< IqA ~< 1 shell only. If, however, /~2 is negative and large, q2+ #2 becomes negative and the gaussian integrals of the perturbative expansion do not converge. We cannot expand around ~a = 0, we have to find the new saddle-point. The variation of ~ in eq. (30) gives
a~= f(-o,O~Ol(X) + ~12~l(X) -[- Xeo](x))3q,,(x)ddx=O.
(32)
In an unconstrained theory eq. (32) would imply the vanishing of the expression in the bracket. This is not true in our case since the variation 6qh(x) is not arbitrary, it contains momentum components in the shell only. The usual solution qffx) = constant (or - 3(q) in Fourier space) is not available for qh(x). The closest we can get to this is to take the superposition of oppositely running plane waves having "q = 0 in average". Consider first d --- 1, it will be easy to generalize to arbitrary dimensions. Take the ansatz
eOl,¢,=c(eipx+e-ip~),
p ~ [e-at, l ] ,
c real.
(33)
The term q~3(x) in eq. (32) gives
3Xc3(e ipx + e-ip x) + 2tc3(e3ip x + e- 3ipx).
(34)
For any allowed variation 3cOl(x), the second term of eq. (34) gives zero in eq. (32): no Fourier component exists in 3qh(x ) with Iql- 3. The remaining terms are proportional to (e ipx + e -ipx) and 3a'~°= 0 gives (/72 -t- ~£2)C -t- 3)kC 3 = 0,
(35)
A. Hasenfratz, P. Hasenfratz / Phase transitions
13
which (for large negative/,2) has the solution p 2 + t,2
c2 =
(36)
3)`
At this solution, the hamiltonian density takes the value
,)W~1 T
(/x2 +p2)2 =
([/,21
6),
_p2) 2 6)`
(37)
and the lowest saddle requires p =Pmin = e-at- By writing
(38)
~ l ( X ) = ~I,cI(X) "}- ~ I ( X )
one expands the hamiltonian in qgl(x). There is no linear term (eq. (32) is satisfied if ~x --+ qh,cl, 8'#1 ~ ~1 is written), the quadratic part is positive*. One can calculate the (small) corrections to the semiclassical result in the usual way. These considerations can be easily generalized to higher dimensions d. The dominant saddle is written as d
qq,c, = c 1-I ( eip'x'+ e-ip'xj),
pj = e - a ' ,
(39)
j=l
where the constant c satisfies the equation (p2 +/.t2)c + 3d)`c3 = 0.
(40)
The classical energy is given as ")~cl
Volume
(~2 +P2) 2 2a-2
)`
3a
(41)
As we discussed before, the result of the functional integral eq. (30) can be written as
e A"~(ga°(q) ~ 0),
(42)
* More precisely, there is a zero-mode due to translation invariance: ~ l , c l ( X -- ~X) is also a saddle. This problem is similar to the one in soliton's quantization. Here, however, (unlike in the case of solitons) the corresponding eigenfunction is not a normalizable state and, presumably, one can forget about it. For the purpose of this paper it is an irrelevant technicality anyhow.
14
A. Hasenfratz, P. Hasenfratz / Phase transitions
where AJ~' is the change of the hamiltonian under the RG transformation*. Since the leading contribution to AOff is just the semiclassical piece ~ 1 in eq. (41), A ~ will not be proportional to At even if At is infinitesimal. Therefore, an analytic function Jg'(t) does not exist, when #2 in eq. (27) is negative and large. An infinitesimal shell in momentum space contributes a finite amount - similar happens in the full, unconstrained model when the field gets a constant ( - 6 ( q ) in Fourier space) expectation value - a sign of condensation. 1 6 One can introduce an g~/q~ term (7/> 0, small) in eq. (27) (and the corresponding term in eq. (30)), and repeat the analysis with /*2> 0 and small negative ~. The saddle point equation eq. (40) is replaced by ( p 2 + / x 2 ) c + 3a~c 3 + lOarlc 5 = 0 ,
(43)
while the value of the classical energy density at c is
- - = 2 a [ ½ ( ~ 2 + p 2 ) c 2 + a ^1TM c . ~4"~d + l~/c610a]. Volume
(44)
Apart from trivial factors these equations are the same as the usual mean-field equations in a standard formulation. As far as the semiclassical approximation is a good guide (which is expected for small ~ and 2,) we get a first order phase transition as ), (or/,2) is changed: c jumps from a zero to a non-zero value (~l.cl(X) oscillates around zero with an amplitude c, eq. (39)). There will be a corresponding non-analyticity in the renormalized hamiltonian. Due to the extra factors in eqs. (43) and (44), the position of the singularities in the (/,2, 2~, 7) space will be different from those occurring in the full theory (which can be studied independently of any RG transformations). However, we considered until now the case ~ 0 ( q ) - 0 only. There is no reason to expect that the case of other 00(q) configurations will be free of non-analyticities. In general, the position of singularities will depend on the ~0(q) configuration and the RG transformation will be singular over a domain of the parameter space as it happened in the case of the large N limit. 4. S i n g u l a r R G transformations in the d = 2 Ising m o d e l
We shall discuss on this model more explicitly how the singularity arises and how its position in the coupling constant space depends on the background block-spin configuration and the block-transformation. The hamiltonian reads (1/kbT included)
- ~ = /3 E siss + h ESi, * There are other trivial contributions coming from rescaling the fields and momenta.
(45)
A. Hasenfratz, P. Hasenfratz / Phase transitions
15
where S i = +_1, nearest-neighbour spins are coupled, h is the magnetic field, while 13 is proportional to the inverse temperature. The R G transformation forms 2 x 2 blocks and uses a simple majority rule: if, in a block the sum of the four spins is positive, the block spin S' will have the value + 1; if the sum is zero, S' = + 1 ( - 1) with probability 1 / 2 (1/2), finally S' = - 1 if the sum is negative. The new hamiltonian is defined as
e -~e'({s'))=
~
(constraints) e -je(s},
(46)
{S,= +_1}
where only those { S } configurations are summed over, which are consistent with the { S'} block-spin ("background spin") configuration. Consider the block-spin configuration, where all the block-spins have the value - 1. Take fi well above the Curie point (/3c = 0.4406... ). When h < 0, almost all the spins point downwards. This is consistent with the constraints, and a flip costs large energy, 8/3 + 21h I. When h becomes positive, but small, there begins a competition between the volume and the surface energy when a blob of + 1 spins is formed. In an unconstrained system thermal fluctuations would produce large enough droplets [8], where the gain in volume energy dominates, and the whole configuration would flip into a predominantly + 1 configuration. Here, however, the constraints (which demand that in every 2 × 2 block at least two spins should be - 1 ) prevent that. The system tries to form droplets of +1 spins maximizing the gain on the volume energy relative to the loss in the surface energy under this constrained condition. It is easy to see that long strips of width 2 of + 1 spins running vertically or horizontally on the lattice (fig. 4) have a suppression factor per unit length (relative to the sea of - 1 spins) e-(4~-4h+ln 2),
-
--i "= _ _ d ' P I+._ _-.j
- P--"i
P----]
' _'t.i' L+__d t-_
_
-
-
-
4-
÷
+
+
+
+
+
-I-
+
-~-
Fig. 4. A long strip of width two of + 1 spins running through the lattice.
(47)
16
A. Hasenfratz, P. Hasenfratz / Phase transitions
and energetically one cannot do better. The term In 2 in eq. (47) comes as follows (fig. 4). A length two piece of the strip creates two blocks, where the sum of the spins is zero. This gives a - 1 block-spin with probability ½ only, which results in a reducing factor (½)2 for the two blocks. It is a factor ¼ for a piece of length two - a factor ½= e -In2 for a piece of strip of unit length. Breaking a strip costs e-8a, a very small number when/3 is large - it will occur rarely. Bending a strip, or producing dislocations are also very costly energetically, while the gain in entropy is small in this constrained geometry. This suggests to consider the following strip model to describe the situation: On a L × L periodic lattice the partition function is saturated by vertically* running strips of width 2, which wrap around the lattice and have energy/unit length = 4fl - 4h + In 2.
(48)
The strips are always separated by an even number of lattice sites in transversal direction, the minimum distance between them is 2. We compared the predictions of this simple, exactly solvable model with the results of a MC simulation of this constrained model at fl = 1.2 on 20 × 20 and 32 × 32 lattices. The MC configurations contained almost exclusively strips of the type described above. The predicted magnetization as the function of h and the size of the lattice agreed quantitatively with the MC data. 1 Let the size of the system L be a multiple of 4, and n = 5L. The number of ways one can put k (k = 1,2 . . . . . ½n) vertical strips on the lattice is given by
k(nkkl
1).
(49)
The energy of a strip is A = L (413 - 4h + In 2).
(50)
The partition function is written as
Z=I+
n/2 ~2 k ( n k k=l
-k1- l )
e ~.
(51)
This sum can be given in closed form Z = 2e -na/2cosh[ n A r s h ( l e ~ / 2 ) ] .
* The horizontally running strips give an overall, irrelevant factor of 2 in the partition function.
(52)
A. Hasenfratz, P. Hasenfratz / Phasetransitions
17
T h e m a g n e t i z a t i o n of a configuration with k strips is
2k-2(n-k)
2k =
2n
--
n
-
1,
(53)
which gives
112n,2n'n k ,)e kA]x
M= ~
n k'~-'_lkk~
-1
or, in closed form 1 M
e 1/2a
2 ~1 + ¼e z
tanh[ n Arsh(½e a / 2 ) ] .
(55)
Eqs. (50) a n d (55) predict a first order phase transition for L---, ~ at the critical m a g n e t i c field
h c = f l + ¼1n2.
(56)
Eqs. (50) a n d (55) give the magnetization as the function of r , h and L. In fig. 5 we c o m p a r e the M C data on 20 x 20 and 32 x 32 lattices at fl = 1.2 and h in the n e i g h b o u r h o o d of the predicted critical magnetic field h c = 1.373. There is an excellent agreement.
-(M), 1.0 0.8
0.6
0.4 0.2
1.30
1.34
1.38
1.62
Fig. 5. The strip model prediction for the magnetization M = M(fl, h, L) is compared with the MC data at fl = 1.2 and L = 20,32. Dots and triangles give the data points on 20 x 20 and 32 x 32 lattices respectively. The dotted and continuous fines give the corresponding predictions.
18
A. Hasenfratz, P. Hasenfratz / Phase transitions
First order phase transition in eq. (46) means, of course, singular R G transformation. YF is a non-analytic function of h and/3. For the block-spin configuration we considered this singularity occurs at h c =/3 + 4~ In 2. However, this value depends on the fixed block-spin configuration and on the size of the block in the R G transformation. Consider, for example, the same block-spin configuration, but take the block size to be 2k × 2k, k = 1, 2 . . . . Then the relevant configurations will be strips of width 2k. The energy per unit length is given as energy/unit length = 4/3 - 4kh + ~ In 2,
k = 1,2 . . . . .
(57)
while the predicted critical magnetic field is 1
1
h e = ~-/3+ ~ - ~ ln2,
k=l,2 .....
(58)
Finally, consider 2 × 2 blocks and a block-spin configuration, where in a row two + 1 block-spins are followed by two - 1 spins and this is repeated (fig. 6). In this case already the block-spins form a strip structure. Take/3 very large. Then all the dislocations will be suppressed and there will be straight Bloch walls between the positive and negative regions. The only question is, where are these Bloch walls. For h = 0 a strip of four positive spins will be separated by a Bloch wall from a strip of four negative spins: . . . J + + + + I I "'" • Moving the wall one step to the left or right like . . . I + + + I I "'" costs e z/=]n2, and it is completely suppressed when L is large. Increase now the magnetic field. When h exceeds h c = ¼1n2, the gain in volume energy over-compensates the loss in ln2's, the Bloch wall moves one step and the magnetization jumps from zero to + ½. This picture should become exact as /3 ~ oo when the density of dislocations (isolated spin flips or distorted Bloch walls) drops to zero.
['-,~,=1 ~:.-'q ~3.-; ~TZ'-I
fi--fi
,i ~~_l_ u ," e,
"
7--'fi !I .~_ _l ~ ' ! ~ I' "
L~__tl - - - i . *-'I ~ - - ~
"'
,~'_-_'~l ,,.y_j
7,-,'-I I ~'
,-__-_1
fiefi ,"--" ,
~__~
. . . . . i ~j i i~
~__.J
i
L~_ L-_I
Fig. 6. A specific block-spin configuration, where block-spin strips of width two run vertically.
A. Husenfratz, P. Hasenfratz
/
Phase transitions
19
5. Summary We argued in this paper that standard RG transformations are in general singular in a large part of the coupling constant space. Singular RG transformations occur typically in the neighbourhood of first order phase transitions. The relation between first order phase transitions (more precisely the existence of metastable phases(s)) and singular RG transformations is shown explicitly in the O(N), N + cc case. We presented several examples in different other models, which support the existence of this relation in general. This work was initiated by Monte Carlo RG studies of first order phase transitions in four- and two-dimensional models, whose results will be detailed in a separate paper [4]. A striking common feature of the results is that the RG transformation drives the system rapidly away from the first order transition surface even if the starting hamiltonian was very close to this surface. This means that two points which are very (infinitesimally) close to each other but lie on opposite sides of the transition surface, will be at a finite distance from each other after a few (one) RG steps. This implies that the RG transformation is singular. We are aware of two previously published MCRG works, where the results are consistent with this scenario [lo]. In their paper on the d = 3, Potts-3 model Blote and Swendsen [lo] actually raised the possibility that the transformation they studied is singular. Another common feature of our MCRG results is that the speed of diverging away from the first order transition surface is consistent with an index v = l/d. This is like in the case of a discontinuity fixed point, which result, however, requires a new interpretation now, since the RG transformation, when played in the space of renormalized hamiltonians, is singular. There exists, perhaps, a non-singular RG prescription, which works on the states rather than on the hamiltonians - a challenging, unexplored possibility. We have received a preprint by Gonzales-Arroyo et al. [14] on a MCRG study in the d = 4 Z, gauge theory. The results show similar anomalies as found in ref. [lo] and in ref. [4] and are consistent with the conclusions of this paper.
We are indebted for useful discussions to K. Binder, V. Gribov, C. Lang, T. Neuhaus, F. Niedermayer and R. Swendsen. One of us (A.H.) was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the US Department of Energy through Contract No. DE-FCOS85ER250000.
References [l] K. Wilson and J. Kogut, Phys. Reports 12C (1974) 75; K. Wilson, Rev. Mod. Phys. 47 (1985) 589, 55 (1983) 583; L.P. Kadanoff, Rev. Mod. Phys. 49 (1977) 267
20
A. Hasenfratz, P. Hasenfratz / Phase transitions
[2] R.B. Griffiths and P.A. Pearce, Phys. Rev. Lett. 41 (1978) 917; J. Stat. Phys. 20 (1979) 499 [3] T.W. Burkhardt and J.M.J. van Leeuwen, in Real space renormalization, (Springer 1982) p. 1 [4] K. Decker, A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B295 [FS21] 21; K. Decker, Lattice Gauge Theory Meeting, Brookhaven, 1986; A. Hasenfratz, Nuc. Phys. B [Proc. Supp.] 1A (1987) 127 [5] S. Ma, Rev. Mod. Phys. 45 (1973) 589; Phys. Lett. 43A (1973) 475 [6] F. David, D.A. Kessler and H. Neuberger, Nucl. Phys. B257 [FS14] (1985) 695 [7] T. Applequist and U. Heinz, Phys. Rev. D25 (1982) 2620; W.A. Bardeen and M. Moshe, Phys. Rev. D25 (1983) 1372; W. A. Bardeen, M. Moshe and M. Bander, Phys. Rev. Lett. 52 (1983) 1188 [8] J.D. Gunton, M. San Miguel and P.S. Sahni, in Phase transitions and critical phenomena, vol. 8, eds C. Domb and J.L. Lebowitz (Academic Press, 1983) p. 269 [9] B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35 (1975) 477; M.E. Fisher and A.N. Berker, Phys. Rev. B26 (1982) 2507 [10] H.W.J. Bl~Steand R.H. Swendsen, Phys. Rev. Lett. 43 (1979) 799; C.B. Lang, Nucl. Phys. B280 [FS18] (1987) 255 [11] K. Gawedzki, A. Kupiainen and R. Koteck~, Bures-sur Yvette preprint, IHES/P/86/57; K. Gawedzki, Physica 140A (1986) 78 [12] A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B270 [FS16] (1986) 687 [13] F.J. Wegner and A. Houghton, Phys. Rev. A8 (1972) 401 [14] A. Gonzales-Arroyo, M. Okawa and Y. Shimizu, KEK-TH-160 (1987)
SINGULAR RENORMALIZATION
GROUP TRANSFORMATIONS
AND FIRST ORDER PHASE TRANSITIONS (I) Anna HASENFRATZ1 Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306, USA
Peter HASENFRATZ 2 Institute of Theoretical Physics, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland
Received 20 July 1987 (Revised 31 August 1987)
It is argued that standard position and momentum space renormalization group (RG) transformations are singular (i.e. lead to a singular hamiltonian after a finite number of RG steps) in large regions of the coupling constant space. It is shown in the d = 3, ~60(N) N ~ ~ model that the momentum space RG transformation is singular in all those points of the coupling constant space, where metastable states exist. This region includes the full first order phase transition surface and its neighbourhood. Several other examples are discussed to illustrate that this phenomenon is generic and not a specific large N effect. Some earlier and recent anomalous Monte Carlo renormalization group results are consistent with this conclusion.
1. Introduction A b a s i c a s s u m p t i o n of the R G a p p r o a c h [1] is that the t h e r m o d y n a m i c a l singularities arise b y r e p e a t i n g the e l e m e n t a r y R G steps i n f i n i t e l y m a n y times. T h e statistical p r o b l e m a s s o c i a t e d with an e l e m e n t a r y R G step (integrating out a fraction of the h i g h m o m e n t u m variables, or averaging over the short wavelength fluctuations in p o s i t i o n space) is a s s u m e d to b e non-singular. By i n v e s t i g a t i n g different b l o c k t r a n s f o r m a t i o n s in c o n f i g u r a t i o n space Griffiths a n d Pearce c o n c l u d e d [2] that this a s s u m p t i o n c a n n o t be strictly true. E x a m p l e s were f o u n d , w h e r e - starting at specific p o i n t s of the c o u p l i n g c o n s t a n t s p a c e s i n g u l a r i t i e s o c c u r r e d in the r e n o r m a l i z e d h a m i l t o n i a n after a finite n u m b e r of R G steps. A t c e r t a i n values of the original c o u p l i n g c o n s t a n t s the r e n o r m a l i z e d coup l i n g s d e v e l o p e d non-analyticities. T h e R G t r a n s f o r m a t i o n can b e singular. 1 On leave from Central Research Institute for Physics, Budapest, Hungary. 2 Supported in part by Schweizerischer Nationalfonds. 0169-6823/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
A. Hasenfratz, P. Hasenfratz / Phase transitions
The Griffiths-Pearce "peculiarities" did not raise much concern in RG practitioners as it was assumed that they do not occur in the vicinity of the fixed points of interest. Additionally, it is generally believed that no such singularities occur in momentum space renormalization [3]. In this paper we shall argue that standard RG transformations are singular in a large, continuous part of the coupling constant space if the original thermodynamical system exhibits first order phase transitions. The arguments are based on different model studies and numerical simulations. This investigation has been initiated and motivated by some unexpected M C R G results, which will be detailed in a subsequent paper [4]. Our first example (sect. 2) will be the scalar O(N) model in d = 3 dimensions:
--~=
f{ 1(0~)2"[-
) }d3x.
~ ~ "p ~(q~2)2q'- 1T/(q~2 3
(1)
The momentum space RG transformation of this model can be studied analytically in the large N limit [5, 6] and the following statement can be proved: In all those/,2, ?~, ~/> 0 points of the parameter space, where the exact effective potential of the system has several competing minima (i.e. apart from the absolute minimum (ground state) it has one or more other minima, which correspond to metastable states) the RG transformation leads to a singular hamiltonian after a finite fraction of the momenta is integrated out. The effective potential is defined here the usual way, and a priori has nothing to do with R G transformations [7]. The statement is formulated more precisely and proved in sect. 2. Metastable states certainly exist in the neighbourhood of the first order phase transition surface, therefore the statement above implies that on the whole first order surface and in its vicinity the RG transformation is singular. We emphasize that the statement above refers to the existence of metastable phases (which is an ordinary notion in first order phase transitions [8]) and not to some specific property of the large N limit. In particular the fact that a momentum space RG transformation can be singular is not specific to large N. In sect. 3 we consider a one-component scalar field theory and perform the RG transformation Acut---~ e - a ~ cut in momentum space. We discuss simple examples, where the renormalized hamiltonian is not an analytic function of t, or the original coupling constants. Unlike the large N case, this analysis is not exact. It is based on a semiclassical approximation in a coupling constant region, where the corrections are expected to be small. Sect. 4 deals with the d = 2 Ising model in an external magnetic field h. The RG transformation is defined in configuration space: 2 × 2 blocks are formed, the block spins are determined by the majority rule. The question is raised, whether the statistical problem of summing over the original spins at some given value of the
A. Hasenfratz, P. Hasenfratz / Phase transitions
3
block spins could lead to thermodynamical singularities (this would imply singular R G transformation). For certain background spin configurations and at large /3 = 1 / T it is easy to identify the configurations which dominate this constrained partition function. The resulting simplified model can be solved exactly. It predicts a first order phase transition and reproduces quantitatively our MC data on this system - including the magnetization as the function of h and the size L of the lattice, M = M(h, L). The critical field h c depends on /3, on the size of the block and on the background blockspin configuration. This indicates again that the renormalized hamiltonian is singular in a large region of the parameter space around the first order transition surface (of the original Ising model), although we are not able to make such general conclusions as in the O(N)N--* e ~ case. Sect. 5 is a summary. We give a brief account of the MCRG studies we (in collaboration with K. Decker) performed, whose results will be detailed in a subsequent paper [4]. These MCRG studies on the d = 2 Potts-10 model, on the d = 2 Ising model with magnetic field and on the d = 4 U(1) lattice gauge theory all indicate that the usually accepted RG picture of first order phase transitions with a discontinuity fixed point [9] requires modification. The results are consistent with the assumption that the RG transformation used is singular in the vicinity of first order phase transitions. We mention earlier examples also, where MCRG studies of first order phase transitions raised the possibility of singular RG transformations [101. The framework of this paper (based on renormalized hamiltonians) and that of the M C R G analysis of ref. [4] (working with blocked configurations) are somewhat different. There are points in the present analysis (like the possible connection between singular RG transformations and the existence of a metastable phase), which are not directly tested by the MCRG analysis. Similarly, there are certain regularities in the MCRG results [4], which are not explained by the present theoretical analysis. It is true, however, that the MCRG results indicate that the RG transformation used is singular on the first order transition surface. This is consistent with the conclusions of this paper and is against the standard discontinuity fixed point picture. These points are discussed further in the summary. It might be that a non-singular RG prescription exists, which works on the states rather than on the hamiltonians and which treats separately the different phases of the first order phase transitions. A RG approach of first order transitions in Ising-like models was constructed recently in terms of Peierls contours [11]. This method is free also of the singularities discussed here. 2. Relation between the existence of metastable phases and singular RG transformations in the O ( N )N-~ ~ model
We shall study the RG behaviour of the model defined by eq. (1) and prove the statement mentioned in the introduction.
4
A. Hasenfratz, P. Hasenfratz / Phase transitions
After rescaling the fields (~2 ~ y = ep2/N), the couplings (#2 ~ tt2, ~ ~ h/N, 71---> ~/N 2) and the potential (V--->V/N) in the usual way we write V(y)
= 1 2
~ t L y + ~1 y
2
+ ~1y
3.
,
y>/0,71>0,
(2)
where all the variables are dimensionless, the dimensions are carried by the cut-off A cut. We shall use the notation
u(y) = 2V'(y) and denote the inverse function of u = u(y) by F(u): y - - F ( u ) . Consider a RG transformation in momentum space, which reduces the cut-off by a factor of s( >/1):
Acut--->ACUt/s. Under this transformation the relation y =
(3)
F(u) changes into [5,6]
y=Fs(u), where, in dimension d = 3, the function
F~(u)=sF The function
(4)
Fs(u) is given as
-sf
+f(u);
s>~l.
(5)
f(u) is independent of the parameters of the potential:
f(u)=flq
d3q
1
1
1~<1(27/')3 (q2q-U)
(
(
1
)
)
(297")2 1-vrffarctan ~-u
'
u>~0
(6)
and it is defined for non-negative u-values. On the other hand, the combination -sf(u/s 2) +f(u), which enters the RG equation (5) is defined for all u > - 1 :
(u)
{1[
C(s,u)=f(u)-sf ~--
=
dq(qa+u )
2qr2
1 2~r 2 1 - s + ~ 2 u ½ 1 n
,
)]
[s+~--u 1 - ~ u s
~-u
l+~-u-
(7) '
-l
We shall assume in the following that the starting potential satisfies the u > - 1 constraint for every y >i 0 (this point will be discussed further in sect. 3).
A. Hasenfratz, P. Hasenfratz / Phase transitions
In our case the function
5
F(u) is given by
F(u)=
-o~_
+ 2~.2,
(8)
where, for later convenience, we introduced the notations X a = ~
1 + 2,/r 2 ,
w
X2
1~2
4,02
,0
(9)
The two branches of F(u) will be denoted by F -+(u). Even without going into the details of the proof it is easy to see that in large regions of the parameter space (w, a, '0) the RG transformation is singular and the relation between the existence of metastable phases and singularity is suggestive. The RG relation eq. (5) can be written as
[ ju
F/-(u)=s-,~+
w+--
-
+ C(s, u),
s2'0
(10)
where s
d(s, u) = c(s, u) + 2 T
(11)
The possible values of u are constrained by u > - 1 and by the demand that
Fs(u ) should be real and non-negative, eq. (4). For any fixed u, C(s, u) runs toward an s-independent function C*(u) as s ~ oo: C(s, u) ~ (~*(u).
(12)
The function C*(u) is shown in fig. 1. Consider now the parameter region a < 0, 0 < w < a2. In this parameter region the starting (classical or mean field) potential has a trivial and a non-trivial competing minima, the classical potential predicts a metastable state (we shall see later that this is true also beyond the classical approximation). The starting F(u) is sketched in fig. 2a. For large s we get from eq. (10)
F/-(.) -~ s ( - ~ _+~ ) + O ( . ) .
(13)
Since - a + ~ - > 0, both branches of F,+-(u) survive the large s limit (remember, Fs(u) = y should be non-negative), fig. 2b. For every Fs(u ) =y there corresponds
A. Hasenfratz, P, Hasenfratz / Phase transitions
1.0
-0.5
I
-I.
0.5
I
1.0
"~U
--0.5
--1 Fig. 1. The function C*(u) is the large s limit of the function C(s, u) defined by eqs. (11) and (7). It has a logarithmic singularity at u = - 1.
more than one u = u(y) solution, the new hamiltonian is multivalued. At some smaller s values, a u/Oy became infinite at certain y values. The R G transformation is singular. We shall prove now the following.
Statement. The R G transformation eq. (3) in singular in all those points of the parameter space (w, a, ~1), where the system has several competing minima, i.e. metastable states exist. It is singular also if the exact effective potential V~fr(~ ) has several competing branches over some region of fie- In all those points the new hamiltonian produced by the R G transformation is non-analytic, multivalued for large enough s. In any other points of the parameter space the large s hamiltonian is single valued and analytic.
First we shall identify the parameter region, where the R G transformation is singular. Then we investigate the properties of the theory in this coupling constant region using standard methods which are independent of R G transformations. Let us investigate first the possibility that the singularity arises in the region -l
Statement A. Assume, the function F(u) has a part in the - 1 < u < 0 region. Then, for - 1 < u < 0 the R G transformation is singular, if and only if F - ( u = 0) >/ 1 / 2 r r 2.
A. Hasenfratz, P. Hasenfratz / Phase transitions
I
I
-I.0
-0.5
0'.5
=U
F+(u)
F
W(u)
r" -1.0
1.o
I
:
-0.5
0.5
~
1.0
=U
Fig. 2. The starting function F(u) = F,=l(u ) corresponds to a potential having competing minima. The R G transformation breaks this function into two pieces for large s, and no single-valued inverse exists.
Proof. T h e region - 1 < u < 0 exists only if w > 0 (eq. (8)). Since Iu l is b o u n d e d here, we can write for large s: Fs+-(u)=s(-a+_v~)+(~*(u),
-l
(14)
Since F - ( u = O) = ( - a - ¢7) + 1 / 2 ~ "2, the condition F - ( u = 0) >/1/2~r 2 implies -a + ~ - a -
> 0,
vrw > / 0 .
(15) (16)
T h e r e f o r e s ( - a +_V~) is non-negative and - since C * ( u ) has also a non-negative piece for u < 0 - b o t h branches of Fs(u ) survive (fig. 2). The function u = u(y) is multivalued, the R G transformation is singular.
8
A. Hasenfratz, P. Hasenfratz / Phase transitions
Take now F-(u = 0 ) < 1/2~r 2. In this case - a - v/-w-< 0, and F]-(u) becomes negative for every - 1 < u < 0 if s is large. Only F+(u) survives, which is a monotonic function. No singularity occurs (it is easy to see that F + (u) is a monotonic increasing function of u. Both the first term and C(s, u) in eq. (10) are monotonic increasing as can be seen from the integral representation of C(s, u) in eq. (7)). Consider now the region 0 ~< u < oo:
Statement B. For 0 ~< u < oo the R G transformation is singular if and only if there exists a number A >/0 such that the equation F-(u) - f ( u ) - A = 0 has more than one (non-negative) root. The function
(17)
f(u) is defined in eq. (6).
Proof. If there is more than one root, the function F - ( u ) - f ( u ) cannot be everywhere negative, and the positive part should be a non-monotonic function of u. If there exists one root only, the positive part of F - ( u ) - f ( u ) is monotonic decreasing ( F - ( u ) - f ( u ) is negative for large u). If there is no root, F - ( u ) - f ( u ) is always negative. Let us rewrite the R G equation eq. (5) in the following form 1 s F Z ( u s 2) =
1
( F-( u) - f( u) ) - s f ( Us2) .
(18)
The function f(u) is bounded by 1/2~r 2, therefore the last term can be neglected for large s. If eq. (17) has more than one root, F-(u) - f ( u ) has a positive part, therefore F Z survives. It is, however, a non-monotonic function of u, u = u(y) is multivalued, the R G transformation is singular. If there is one root only, FT(u ) survives and is monotonic decreasing in u. Since F+(u) is monotonic increasing and F~+(u)>~FZ(u ), the full function Fs(u ) has a single valued inverse - no singularity occurs in this case. Finally, if eq. (17) has no root, F - ( u ) - f ( u ) in eq. (18) is always negative and the branch F~-(u) does not exist for large s. On the other hand, F~+(u) is monotonic - no singularity occurs. We shall show now that the condition F-(u = 0) >i 1/2~r 2, w > 0 corresponds to that part of the parameter space, where metastable state(s) exists (this region includes the first order phase transition surface and its vicinity in the phase diagram), while if eq. (17) has more than one root, then the effective potential V~ef has competing branches.
A. Hasenfratz, P. Hasenfratz / Phase transitions
9
Rather than repeating the detailed discussion of the effective potential Veff(*2) in the large N limit, we refer to refs. [5, 7] and quote the relevant result only. Let us denote the inverse function of f (eq. (6)) by g(x): g ( f ( u ) ) = u. The function g(x) is defined for 0 < x ~ 1/2qr 2. Let X = X ( , 2) the solution of the equation
u(X+*2)=g(X).
(19)
Here u(y) = 2V'(y), as before. Then the derivative of the effective potential can be written as [5, 7] Ve;f( , 2 ) =
Vt(X(*2) + ,2).
(20)
The condition of a non-trivial extremum at q]2 is Ve~f(q]2) = 0, which is equivalent to the equation
21, since the only point where g ( X ) = 0 is X = 1/2rr 2. Eqs. (19) and (21) can be expressed in terms of the inverse function F and f we defined before:
F(u) F(u =
,c2 = f ( u ) ,
(22)
1
0) = 2~r~ + q]2.
(23)
If F-(u = 0) >i 1/2~r 2 and w > 0 (these were the conditions in statement A), then F + ( u = 0 ) > F - ( u = 0 ) and eq. (23) has two nontrivial solutions. This case is sketched in fig. 3. One can see by inspection that eq. (22) always has a positive u > 0 solution at ,~ = 0, implying that V~f(, 2 = 0 ) > 0, therefore the effective potential (as the function of *c) has a minimum at the origin. We have therefore two competing minima - the ground state and a metastable state - in this case. If eq. (19), or equivalently eq. (22) has more than one solution over a region of ,2, then the effective potential has competing branches there. This is just the condition expressed by eq. (17). In eq. (17), F-(u) can be replaced by F(u), since if F+(u)-f(u)- A = 0 has a root at some Zl (it cannot have more than one) then F - ( u ) - f ( u ) - zl = 0 has zero or two roots (it cannot have one root; if F+(u)f(u)--A = 0 has a root, then F+(umi~)= F-(umin)
F(u)-f(u) This completes the proof of the statement.
-A=0.
(24)
10
A. Hasenfratz, P. Hasenfratz / Phasetransitions
( I
-1.0
I
-0.5
0.5
1.0
Fig. 3. When F-(u = 0) > 1/2~r 2 and w > 0, there always exist two non-trivial extrema and Vaf has a minimum at the origin since eq. (22) has a solution with ffc2 = 0. The function f(u) is defined in eq. (6).
3. Singular RG transformations in a one-component scalar theory T h e example discussed in the previous section runs against the often raised intuitive a r g u m e n t saying that integrating out a part of the high m o m e n t a cannot p r o d u c e singularities. Are the p h e n o m e n a of sect. 2 related to the large N limit? I n this section we shall argue and present examples to show that m o m e n t u m space R G transformations can be (and are often) singular even in a o n e - c o m p o n e n t scalar theory. There is a type of singularity whose existence is almost immediate. We have already seen in the large N model in sect. 2 that if the starting potential of the m o d e l does n o t satisfy 2 V ' ( y ) > - 1 for every y ( = ~2), the R G formulas become singular for every s > 1. These kind of singularities are present for any N. The a p p r o x i m a t e R G equation (derived in the local potential approximation, see eq. (50) in ref. [12]) exhibits singularities and makes no sense if the starting potential does not satisfy 2 V ' ( y ) > - 1 for N = ~ or 2 V ' ( y ) > - 1 and 2 V ' ( y ) + 4 y V " ( y ) > - 1 for finite N*. This problem is not related to the approximation, it is present also in the exact R G equation of Wegner and H o u g h t o n [13]. The exact equation for 9if= JC~(t) (A cut ----~e-tA cut is the R G transformation) contains a piece:
°2ae
In O q , ( q ) S q ~ ( - q )
) [q[=l'
(25)
* In eq. (50) of ref. [12] the function f corresponds to 2~- V'(y) of this paper, and f is considered there as a function of x = V%-"
A. Hasenfratz, P. Hasenfratz / Phase transitions
11
which is singular if the matrix
0 29~ Oe°(q)Oe°(-q)
Iql
(26)
=1'
has non-positive eigenvalues. The source of the problem is the following. In deriving the R G equation, one performs an infinitesimal RG transformation: Acut --, e-~tAcut. One writes ~(q) = ~0(q) + ~l(q), where q~l(q) are the field components one has to integrate out (in q~l(q) the momentum lies in the shell e -st ~< [ql ~ 1). Expanding the hamiltonian 9f'(q~0 + q~l) in q~l, Wegner and Houghton argue [13] that all the terms containing more than two q~l factors give O(& z) contributions, therefore can be neglected for infinitesimal St. The resulting gaussian integral over ~1 gives the term in eq. (25). One can write down hamiltonians, however, which give perfectly convergent functional integrals for the full theory, while the matrix eq. (26) has non-positive eigenvalues. In this case the gaussian integral does not converge, one should keep the higher order terms, which were formally O(&2). The naively derived R G equation cannot be applied and one suspects that no differential RG equation exists in this case, since there is no reason that the result of the RG transformation will be O(&). In the following we shall discuss simple explicit examples in a one-component scalar theory, which exhibit not only these singularities but also the types discussed at length in sect. 2. Consider the hamiltonian
..~= f(½(O,#)2+lt~2¢2+¼M,4)ddx,
X>O,
(27)
where q~(x) is a one-component scalar field and the cut-off constraints the momenta to lie inside a d-dimensional hypercube. In terms of dimensionless variables we have:
[qi[ ~< 1,
i = 1,2 . . . . . d.
(28)
Consider the R G transformation, where the field components with e a t < Iqi[ ~ 1 are integrated out. Here At is assumed to be small. We write for the Fourier field ¢(q): q~(q) = •0(q) + q~l(q),
(29)
where q~o(q) is defined only for Iqi] < e-at, while ffl(q) are the variables we have to integrate over. In this functional integral the components q~0(q) are external parameters. According to the basic assumption of the RG procedure, for any set of q~o(q) the result should be an analytic function of the parameters ~2 and ~ and, for
12
A. Hasenfratz, P. Hasenfratz / Phase transitions
infinitesimal At, should be proportional to At (otherwise)~"(t) is not analytic in the R G parameter t). Take, as an example ~0(q)--0. In this case the hamiltonian in the functional integral over qh(q) fe
Dq~l(q) exp ( -at~lqil~< 1
f [ 1 (0g,l~l)2 q_ ~1.21~.)2[ ' 1I,x ,q_ ~1 M h4 (. x .)], ddx ) ,
(30)
has the original form, the field qh(x) contains high momentum components only
~l(X)
fe -----
eiqxdpl(q)" _at~lqil<~1
(31)
Take small positive )t. Then a systematic perturbative expansion can be set up for the functional integral in eq. (30), where the propagator is (q2+ ~t2)-1 and the momentum integrals run over the e-ate< IqA ~< 1 shell only. If, however, /~2 is negative and large, q2+ #2 becomes negative and the gaussian integrals of the perturbative expansion do not converge. We cannot expand around ~a = 0, we have to find the new saddle-point. The variation of ~ in eq. (30) gives
a~= f(-o,O~Ol(X) + ~12~l(X) -[- Xeo](x))3q,,(x)ddx=O.
(32)
In an unconstrained theory eq. (32) would imply the vanishing of the expression in the bracket. This is not true in our case since the variation 6qh(x) is not arbitrary, it contains momentum components in the shell only. The usual solution qffx) = constant (or - 3(q) in Fourier space) is not available for qh(x). The closest we can get to this is to take the superposition of oppositely running plane waves having "q = 0 in average". Consider first d --- 1, it will be easy to generalize to arbitrary dimensions. Take the ansatz
eOl,¢,=c(eipx+e-ip~),
p ~ [e-at, l ] ,
c real.
(33)
The term q~3(x) in eq. (32) gives
3Xc3(e ipx + e-ip x) + 2tc3(e3ip x + e- 3ipx).
(34)
For any allowed variation 3cOl(x), the second term of eq. (34) gives zero in eq. (32): no Fourier component exists in 3qh(x ) with Iql- 3. The remaining terms are proportional to (e ipx + e -ipx) and 3a'~°= 0 gives (/72 -t- ~£2)C -t- 3)kC 3 = 0,
(35)
A. Hasenfratz, P. Hasenfratz / Phase transitions
13
which (for large negative/,2) has the solution p 2 + t,2
c2 =
(36)
3)`
At this solution, the hamiltonian density takes the value
,)W~1 T
(/x2 +p2)2 =
([/,21
6),
_p2) 2 6)`
(37)
and the lowest saddle requires p =Pmin = e-at- By writing
(38)
~ l ( X ) = ~I,cI(X) "}- ~ I ( X )
one expands the hamiltonian in qgl(x). There is no linear term (eq. (32) is satisfied if ~x --+ qh,cl, 8'#1 ~ ~1 is written), the quadratic part is positive*. One can calculate the (small) corrections to the semiclassical result in the usual way. These considerations can be easily generalized to higher dimensions d. The dominant saddle is written as d
qq,c, = c 1-I ( eip'x'+ e-ip'xj),
pj = e - a ' ,
(39)
j=l
where the constant c satisfies the equation (p2 +/.t2)c + 3d)`c3 = 0.
(40)
The classical energy is given as ")~cl
Volume
(~2 +P2) 2 2a-2
)`
3a
(41)
As we discussed before, the result of the functional integral eq. (30) can be written as
e A"~(ga°(q) ~ 0),
(42)
* More precisely, there is a zero-mode due to translation invariance: ~ l , c l ( X -- ~X) is also a saddle. This problem is similar to the one in soliton's quantization. Here, however, (unlike in the case of solitons) the corresponding eigenfunction is not a normalizable state and, presumably, one can forget about it. For the purpose of this paper it is an irrelevant technicality anyhow.
14
A. Hasenfratz, P. Hasenfratz / Phase transitions
where AJ~' is the change of the hamiltonian under the RG transformation*. Since the leading contribution to AOff is just the semiclassical piece ~ 1 in eq. (41), A ~ will not be proportional to At even if At is infinitesimal. Therefore, an analytic function Jg'(t) does not exist, when #2 in eq. (27) is negative and large. An infinitesimal shell in momentum space contributes a finite amount - similar happens in the full, unconstrained model when the field gets a constant ( - 6 ( q ) in Fourier space) expectation value - a sign of condensation. 1 6 One can introduce an g~/q~ term (7/> 0, small) in eq. (27) (and the corresponding term in eq. (30)), and repeat the analysis with /*2> 0 and small negative ~. The saddle point equation eq. (40) is replaced by ( p 2 + / x 2 ) c + 3a~c 3 + lOarlc 5 = 0 ,
(43)
while the value of the classical energy density at c is
- - = 2 a [ ½ ( ~ 2 + p 2 ) c 2 + a ^1TM c . ~4"~d + l~/c610a]. Volume
(44)
Apart from trivial factors these equations are the same as the usual mean-field equations in a standard formulation. As far as the semiclassical approximation is a good guide (which is expected for small ~ and 2,) we get a first order phase transition as ), (or/,2) is changed: c jumps from a zero to a non-zero value (~l.cl(X) oscillates around zero with an amplitude c, eq. (39)). There will be a corresponding non-analyticity in the renormalized hamiltonian. Due to the extra factors in eqs. (43) and (44), the position of the singularities in the (/,2, 2~, 7) space will be different from those occurring in the full theory (which can be studied independently of any RG transformations). However, we considered until now the case ~ 0 ( q ) - 0 only. There is no reason to expect that the case of other 00(q) configurations will be free of non-analyticities. In general, the position of singularities will depend on the ~0(q) configuration and the RG transformation will be singular over a domain of the parameter space as it happened in the case of the large N limit. 4. S i n g u l a r R G transformations in the d = 2 Ising m o d e l
We shall discuss on this model more explicitly how the singularity arises and how its position in the coupling constant space depends on the background block-spin configuration and the block-transformation. The hamiltonian reads (1/kbT included)
- ~ = /3 E siss + h ESi, * There are other trivial contributions coming from rescaling the fields and momenta.
(45)
A. Hasenfratz, P. Hasenfratz / Phase transitions
15
where S i = +_1, nearest-neighbour spins are coupled, h is the magnetic field, while 13 is proportional to the inverse temperature. The R G transformation forms 2 x 2 blocks and uses a simple majority rule: if, in a block the sum of the four spins is positive, the block spin S' will have the value + 1; if the sum is zero, S' = + 1 ( - 1) with probability 1 / 2 (1/2), finally S' = - 1 if the sum is negative. The new hamiltonian is defined as
e -~e'({s'))=
~
(constraints) e -je(s},
(46)
{S,= +_1}
where only those { S } configurations are summed over, which are consistent with the { S'} block-spin ("background spin") configuration. Consider the block-spin configuration, where all the block-spins have the value - 1. Take fi well above the Curie point (/3c = 0.4406... ). When h < 0, almost all the spins point downwards. This is consistent with the constraints, and a flip costs large energy, 8/3 + 21h I. When h becomes positive, but small, there begins a competition between the volume and the surface energy when a blob of + 1 spins is formed. In an unconstrained system thermal fluctuations would produce large enough droplets [8], where the gain in volume energy dominates, and the whole configuration would flip into a predominantly + 1 configuration. Here, however, the constraints (which demand that in every 2 × 2 block at least two spins should be - 1 ) prevent that. The system tries to form droplets of +1 spins maximizing the gain on the volume energy relative to the loss in the surface energy under this constrained condition. It is easy to see that long strips of width 2 of + 1 spins running vertically or horizontally on the lattice (fig. 4) have a suppression factor per unit length (relative to the sea of - 1 spins) e-(4~-4h+ln 2),
-
--i "= _ _ d ' P I+._ _-.j
- P--"i
P----]
' _'t.i' L+__d t-_
_
-
-
-
4-
÷
+
+
+
+
+
-I-
+
-~-
Fig. 4. A long strip of width two of + 1 spins running through the lattice.
(47)
16
A. Hasenfratz, P. Hasenfratz / Phase transitions
and energetically one cannot do better. The term In 2 in eq. (47) comes as follows (fig. 4). A length two piece of the strip creates two blocks, where the sum of the spins is zero. This gives a - 1 block-spin with probability ½ only, which results in a reducing factor (½)2 for the two blocks. It is a factor ¼ for a piece of length two - a factor ½= e -In2 for a piece of strip of unit length. Breaking a strip costs e-8a, a very small number when/3 is large - it will occur rarely. Bending a strip, or producing dislocations are also very costly energetically, while the gain in entropy is small in this constrained geometry. This suggests to consider the following strip model to describe the situation: On a L × L periodic lattice the partition function is saturated by vertically* running strips of width 2, which wrap around the lattice and have energy/unit length = 4fl - 4h + In 2.
(48)
The strips are always separated by an even number of lattice sites in transversal direction, the minimum distance between them is 2. We compared the predictions of this simple, exactly solvable model with the results of a MC simulation of this constrained model at fl = 1.2 on 20 × 20 and 32 × 32 lattices. The MC configurations contained almost exclusively strips of the type described above. The predicted magnetization as the function of h and the size of the lattice agreed quantitatively with the MC data. 1 Let the size of the system L be a multiple of 4, and n = 5L. The number of ways one can put k (k = 1,2 . . . . . ½n) vertical strips on the lattice is given by
k(nkkl
1).
(49)
The energy of a strip is A = L (413 - 4h + In 2).
(50)
The partition function is written as
Z=I+
n/2 ~2 k ( n k k=l
-k1- l )
e ~.
(51)
This sum can be given in closed form Z = 2e -na/2cosh[ n A r s h ( l e ~ / 2 ) ] .
* The horizontally running strips give an overall, irrelevant factor of 2 in the partition function.
(52)
A. Hasenfratz, P. Hasenfratz / Phasetransitions
17
T h e m a g n e t i z a t i o n of a configuration with k strips is
2k-2(n-k)
2k =
2n
--
n
-
1,
(53)
which gives
112n,2n'n k ,)e kA]x
M= ~
n k'~-'_lkk~
-1
or, in closed form 1 M
e 1/2a
2 ~1 + ¼e z
tanh[ n Arsh(½e a / 2 ) ] .
(55)
Eqs. (50) a n d (55) predict a first order phase transition for L---, ~ at the critical m a g n e t i c field
h c = f l + ¼1n2.
(56)
Eqs. (50) a n d (55) give the magnetization as the function of r , h and L. In fig. 5 we c o m p a r e the M C data on 20 x 20 and 32 x 32 lattices at fl = 1.2 and h in the n e i g h b o u r h o o d of the predicted critical magnetic field h c = 1.373. There is an excellent agreement.
-(M), 1.0 0.8
0.6
0.4 0.2
1.30
1.34
1.38
1.62
Fig. 5. The strip model prediction for the magnetization M = M(fl, h, L) is compared with the MC data at fl = 1.2 and L = 20,32. Dots and triangles give the data points on 20 x 20 and 32 x 32 lattices respectively. The dotted and continuous fines give the corresponding predictions.
18
A. Hasenfratz, P. Hasenfratz / Phase transitions
First order phase transition in eq. (46) means, of course, singular R G transformation. YF is a non-analytic function of h and/3. For the block-spin configuration we considered this singularity occurs at h c =/3 + 4~ In 2. However, this value depends on the fixed block-spin configuration and on the size of the block in the R G transformation. Consider, for example, the same block-spin configuration, but take the block size to be 2k × 2k, k = 1, 2 . . . . Then the relevant configurations will be strips of width 2k. The energy per unit length is given as energy/unit length = 4/3 - 4kh + ~ In 2,
k = 1,2 . . . . .
(57)
while the predicted critical magnetic field is 1
1
h e = ~-/3+ ~ - ~ ln2,
k=l,2 .....
(58)
Finally, consider 2 × 2 blocks and a block-spin configuration, where in a row two + 1 block-spins are followed by two - 1 spins and this is repeated (fig. 6). In this case already the block-spins form a strip structure. Take/3 very large. Then all the dislocations will be suppressed and there will be straight Bloch walls between the positive and negative regions. The only question is, where are these Bloch walls. For h = 0 a strip of four positive spins will be separated by a Bloch wall from a strip of four negative spins: . . . J + + + + I I "'" • Moving the wall one step to the left or right like . . . I + + + I I "'" costs e z/=]n2, and it is completely suppressed when L is large. Increase now the magnetic field. When h exceeds h c = ¼1n2, the gain in volume energy over-compensates the loss in ln2's, the Bloch wall moves one step and the magnetization jumps from zero to + ½. This picture should become exact as /3 ~ oo when the density of dislocations (isolated spin flips or distorted Bloch walls) drops to zero.
['-,~,=1 ~:.-'q ~3.-; ~TZ'-I
fi--fi
,i ~~_l_ u ," e,
"
7--'fi !I .~_ _l ~ ' ! ~ I' "
L~__tl - - - i . *-'I ~ - - ~
"'
,~'_-_'~l ,,.y_j
7,-,'-I I ~'
,-__-_1
fiefi ,"--" ,
~__~
. . . . . i ~j i i~
~__.J
i
L~_ L-_I
Fig. 6. A specific block-spin configuration, where block-spin strips of width two run vertically.
A. Husenfratz, P. Hasenfratz
/
Phase transitions
19
5. Summary We argued in this paper that standard RG transformations are in general singular in a large part of the coupling constant space. Singular RG transformations occur typically in the neighbourhood of first order phase transitions. The relation between first order phase transitions (more precisely the existence of metastable phases(s)) and singular RG transformations is shown explicitly in the O(N), N + cc case. We presented several examples in different other models, which support the existence of this relation in general. This work was initiated by Monte Carlo RG studies of first order phase transitions in four- and two-dimensional models, whose results will be detailed in a separate paper [4]. A striking common feature of the results is that the RG transformation drives the system rapidly away from the first order transition surface even if the starting hamiltonian was very close to this surface. This means that two points which are very (infinitesimally) close to each other but lie on opposite sides of the transition surface, will be at a finite distance from each other after a few (one) RG steps. This implies that the RG transformation is singular. We are aware of two previously published MCRG works, where the results are consistent with this scenario [lo]. In their paper on the d = 3, Potts-3 model Blote and Swendsen [lo] actually raised the possibility that the transformation they studied is singular. Another common feature of our MCRG results is that the speed of diverging away from the first order transition surface is consistent with an index v = l/d. This is like in the case of a discontinuity fixed point, which result, however, requires a new interpretation now, since the RG transformation, when played in the space of renormalized hamiltonians, is singular. There exists, perhaps, a non-singular RG prescription, which works on the states rather than on the hamiltonians - a challenging, unexplored possibility. We have received a preprint by Gonzales-Arroyo et al. [14] on a MCRG study in the d = 4 Z, gauge theory. The results show similar anomalies as found in ref. [lo] and in ref. [4] and are consistent with the conclusions of this paper.
We are indebted for useful discussions to K. Binder, V. Gribov, C. Lang, T. Neuhaus, F. Niedermayer and R. Swendsen. One of us (A.H.) was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the US Department of Energy through Contract No. DE-FCOS85ER250000.
References [l] K. Wilson and J. Kogut, Phys. Reports 12C (1974) 75; K. Wilson, Rev. Mod. Phys. 47 (1985) 589, 55 (1983) 583; L.P. Kadanoff, Rev. Mod. Phys. 49 (1977) 267
20
A. Hasenfratz, P. Hasenfratz / Phase transitions
[2] R.B. Griffiths and P.A. Pearce, Phys. Rev. Lett. 41 (1978) 917; J. Stat. Phys. 20 (1979) 499 [3] T.W. Burkhardt and J.M.J. van Leeuwen, in Real space renormalization, (Springer 1982) p. 1 [4] K. Decker, A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B295 [FS21] 21; K. Decker, Lattice Gauge Theory Meeting, Brookhaven, 1986; A. Hasenfratz, Nuc. Phys. B [Proc. Supp.] 1A (1987) 127 [5] S. Ma, Rev. Mod. Phys. 45 (1973) 589; Phys. Lett. 43A (1973) 475 [6] F. David, D.A. Kessler and H. Neuberger, Nucl. Phys. B257 [FS14] (1985) 695 [7] T. Applequist and U. Heinz, Phys. Rev. D25 (1982) 2620; W.A. Bardeen and M. Moshe, Phys. Rev. D25 (1983) 1372; W. A. Bardeen, M. Moshe and M. Bander, Phys. Rev. Lett. 52 (1983) 1188 [8] J.D. Gunton, M. San Miguel and P.S. Sahni, in Phase transitions and critical phenomena, vol. 8, eds C. Domb and J.L. Lebowitz (Academic Press, 1983) p. 269 [9] B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35 (1975) 477; M.E. Fisher and A.N. Berker, Phys. Rev. B26 (1982) 2507 [10] H.W.J. Bl~Steand R.H. Swendsen, Phys. Rev. Lett. 43 (1979) 799; C.B. Lang, Nucl. Phys. B280 [FS18] (1987) 255 [11] K. Gawedzki, A. Kupiainen and R. Koteck~, Bures-sur Yvette preprint, IHES/P/86/57; K. Gawedzki, Physica 140A (1986) 78 [12] A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B270 [FS16] (1986) 687 [13] F.J. Wegner and A. Houghton, Phys. Rev. A8 (1972) 401 [14] A. Gonzales-Arroyo, M. Okawa and Y. Shimizu, KEK-TH-160 (1987)