(ln(r)) - James Franck Institute

Nuclear Physics BI80 [FS2] (1981) 61-76
O North-Holland Publishing Company
C O R R E L A T I O N F U N C T I O N S O N T H E CRITICAL LINE O F T H E
TWO-DIMENSIONAL PLANAR MODEL: LOGARITHMS
AND CORRECTIONS TO SCALING*
Leo P. K A D A N O F F and Albert B. ZISOOK I
The James FranckInstitute, The UniverMtyof Chicago,
5640 South Ellis Avenue, Chicago,IL 60637, USA
Received 16 July 1980
Correlation functions are calculated on the critical line of the planar or XY model. A
crossover is found between simple sealing behavior for 2~rKen > 4 and more complex scaling
behavior for 2~rKeff s 4. At the Kosterlitz-Thouless critical point, 2~rKens 4, logarithmic factors
appear in the correlation functions. These corrections to simple sealing behavior axe calculated
for many correlation functions, including correlation functions not previously studied. The
calculation is carried out in the closely related generalized Villain model using a combination of
perturbation and renormali7ation techniques.
1. Introduction and summary of results
In previous discussions of the Kosterlitz-Thouless critical point, the behavior of
the order parameter correlation function has been examined in detail [5, 1]. The
basic result is that for large r the correlation function of the planar model spin,
S(r) = (cos ¢(r), sin O(e)) takes on a form involving logarithmic factors, i.e.,
(ln(r))l/S [
(cos~b(r)cos~b(0))
r-~-
l ln(ln(r))
14
16
In(r)
]
+''"
'
(1.1)
where • • • represents terms which remain much smaller than the terms shown in
the limit as r - ~ o o . In this paper, we extend the result (1.1) to the case of other
correlation functions, by using a combination of renormalization group and perturbation theoretical arguments.
We describe the calculation by using a language based upon the generalized
Villain model, which is in the same universality class as the planar or X Y model.
This model contains [2] two parameters, K and Yo.
* S u p p o r t ~ in part by the National Science Foundation.
t Robert 11. McCormick and N S F Fellow.
61
62
L.P. Kadanoff , A.B. Zisook / Two-dimensionaiplanar model
K is the nearest neighbor coupling divided by k B T a n d Y0 tells the relative
probability of a vortex excitation. When Y0 = 0, there are no vortices and the model
reduces to the exactly soluble gaussian spin wave model with coupling K. In this
vortex-free situation all correlation functions are algebraic in character and include
no logarithms. However, if Y0 ~ 0, the phase diagram changes drastically. For all K
larger than a critical value, KKx(Y0) defining the Kosterlitz-Thouless [6] transition,
the gaussian structure remains essentially unchanged. At this critical coupling, a
new structure of critical correlations emerges, including logarithms like those in eq.
(1.1). Then for K < KKx(Y0), criticality disappears.
The familiar phase diagram for this situation is shown in fig. 1. As shown in this
diagram, we parametrize this model by using the coordinates X, which is proportional to 2 ~ r K - 4 and Y which is proportional to Yo. Specifically, the hamiltonian
we consider is given by placing the variables ~ ( r ) on a square lattice and writing as
in Kadanoff [4]
eX E
H_- _ m
¢r <~/>
"]-,,)"0~ [ O0,1(/j) ''[- 00,--l(/))] •
J
(1.2)
Here the coupling constant K has been written in terms of X as 2~rK - 4e x, while
Y, the other parameter used in our analysis, is proportional to Y0- The latter
parameters are essentially fugacities which determine the ooupling strength of the
vortex operators O0, ± l(r). In general, we use the notation O.,m(r) to describe an
operator with vorticity m and spin wave quantum number n. The latter is defined
so that O.,o(r) ~ e x p [ i n ~ ( r ) ] . Thus the order parameter has n --- + 1.
u=0
u<O
, /
K,X,p
Fig. 1. Phase diagram for our model. The lines with arrows are flow lines at constant u or effective
coupling. The heavy flow line is the locus of the Kosterfitz-Thoulesstransition. The hatched line is the
stable gaussian fixed line.
L.P. Kadanoff, A .B. Zisook /
Two-dimensionalplanar model
63
The main idea of the calculation is to use the Kosterhtz flow equations to resum
the perturbation series for correlation functions given a limited knowledge of the
terms in the series. The renormalization group flow equations of this system were
just derived by Kosterlitz [5] in the limit of small Y. These results m a y be written as
a flow which results when r---~ret, namely,
d_._YY= 2Y(1 - e x ) - y3 e x + . . .
dl
dX___2y
dl
2ex+...
.
(1.3)
Here • • • represents terms of order y4, y3x, y 2 x 2 ' or higher. The y3 term was
first introduced by Amit et al. [1]. An alternative derivation of terms of up to third
order in X and Y is given in the appendix. The factor of two in the second of these
equations essentially serves as a definition of the multiplicative constant relating Y0
and Y.
Before we do our detailed analysis, we state the major results we shall derive.
First consider the two-point correlation function
c.,m(R,x, r ) = (o.,,.(,.)o_. _,.(,.'))
(1.4)
for the case in which R -- [ r - r'[ is much larger than a lattice constant. At Y = 0,
the gaussian analysis gives the correlation function as
1
C,,,m(R, X, 0) = R2X.,.(x ) ,
(1.5)
the scaling index being given by
Xn,m(X) = 2m 2e x + l n 2 e - x .
(1.6)
Along the critical line which occurs for X > 0 and Y small the corrections to the
lowest-order result (1.5) are essentially trivial. If one chooses the operators O,,m(r)
so as to make their renormalization multiplicative, the correlation function (1.4)
becomes
C,,,,,( R , X , Y ) = C,,,m( R,Xeff , O)(1 + O ( W 2 ) ) .
(1.7a)
Here Xeff describes a renormalized coupling which is expandable in a power series
in y2 for sufficiently small Y. The remaining correction term is a multiplication by
1 plus a power series in W 2, where W is the small quantity
W ~ YR - 2 ( ~ - 1)
(1.7b)
64
L . P . Kadanoff , A . B . Zisook / Two.~mensional planar model
In contrast, at the Kosterhtz-Thouless critical point the correlation function is
non-trivially modified. The result is that as R---> ~ , the correlation function takes
on the form
C,,,m
WX;,.(o)
R2x ( o ) [ l + O ( W 2 ) ] .
(1.8a)
In this case, W is the small quantity
1
W -- ln(R2) + b
(
1
ln[ln(b + In(R2)) ] ]
b + ln(R 2)
,
(1.8b)
where b is a non-universal constant which sets the length scale. The X' of eq. (1.8a)
is the X derivative of the critical index
X;,m(0) =
[d
)]
•
(1.8c)
X-O
The appearance of such a derivative in defining the power of a logarithm is implicit
in the Nelson-Rudnick [9] analysis of marginal behavior.
Eq. (1.8) is one of the two major results of this paper. We shall derive it in two
steps. In sect. 2, we outline our logic by considering a simplified form of the
recursion relation which holds when X and Y are very small. We then derive eq.
(1.8a) but miss the In In R term in eq. (1.8b). After this logic is clarified, we go on in
sect. 3 to the derivation of eqs. (1.7) and (1.8).
Our methods can also be extended to the calculation of multi-point correlation
functions. It is a characteristic feature of the gaussian model that a multi-point
correlation function
Ce(rl,nl,ml;... ;re,ne,me;X,Y)=
]-[ O,j,m/(t)
j "l I
(I .9)
takes on a factorized form. In sect. 4 we show that such a factorization, i.e.,
Ce(rl,nl,ml;... ;rp,ne,mt,;X,y)=( ]-[ C2(rj,nj,mj;rk,nk,mk;X,Y))
Xj<k
X[1 + A W 2 + O ( W a ) ]
(1.10)
also applies whenever Xeff = ln(2~rKeff/4) is greater than or equal to zero. We also
evaluate the coefficients in the product, C 2, in a form very analogous to those
given in eqs. (1.7) and (1.8).
L.P. Kadanoff, A.B. Zisook / Two-a~mensionalplanar model
65
In eq. (1.10) A W z is an unevaluated error term in which W is given by
expressions like those in eqs. (1.7b) and (l.8b), with R being a typical value of
II) - rk l- Here A is an unknown scale invaxiant function of the 5 - rk" For Xat > 0,
this factorization is merely the well-known (see ref. [8]) statement that, aside from
corrections to scaling, the P-point correlation function is gaussian. The analysis in
this region gives us no idea of the corrections to the simple algebraic leading-order
behavior of C e.
However, for Xeff = 0, the A W 2 corrections to C e are much smaller than the
logarithmic corrections hidden in C 2. Hence in this case, eq. (1.10) represents a
non-trivial result. This result is the second major conclusion of this paper.
A preliminary report of these results is given in ref. [11].
2. The logic of the calculation
We wish to use the renormalization group to analyze the behavior of
C,,=(R, X , Y ) . To supplement the flow equations of the renormalization group we
need some additional information about the behavior of these two-point functions
for small X and Y. For X > 0, we assume that these functions are analytic in y2
near Y2 = 0. For X = 0, this analyticity must be lost since the Kosterlitz-Thouless
transition occurs for small X and Y at X 2 = y2. However, one can calculate an
asymptotic expansion of C,, m for small X and Y. We have actually carried out this
calculation for n = 2 through cubic order and we have argued out the form to all
orders. It is
1
Cn,m( R , X , Y ) = REX.,.<o-----~
×
oo oo j+2k j 2k
j+2k--I aj,k, t 1 ,
~
E X Y (ln(R))
1+ •
j~0 k=0
(2.1)
l=0
for large R. We shall make essential use of this asymptotic expansion in evaluating
the correlation function.
Notice that the leading terms here have l = 0, which make C a function of X In R
and (Yln R) 2. We cannot expect this asymptotic expansion to give good results for
large R. Therefore, we must turn to the renormalization group to give us some hint
about how the series sums up.
In order to simplify the analysis, in this section we shall hold to the case in which
X and Y are small. T o quadratic order, the Kosterlitz flow equations take the form
dX
a--f = - 2 r ~,
dY
d----[= - 2 X Y ,
(2.2a)
(2.2b)
66
L . P . Kadanoff , A . B . Zisook / Two-dimensional planar model
while to first order, the flow equation for the correlation function is
d i n C~, m = 2 X . , m ( X )
~-. 2X~,m(0) + 2XX~,,,.(0).
(2.2c)
In the remainder of this work, we shall drop the argument 0 in the An. m.
The flow equations (2.2) can be solved by making use of the flow invariants
u= ~ -
y2
(2.3)
and
19 = R 2 u ( X + u ) l / 2 ( X - u ) - 1/2
(2.4)
C~,,,(R,X, Y)= F,,,m(U,V)
(2.5)
The solution is then
R2X-.=yx~,.,~
Here F~, m is a constant of integration which must be determined.
However, it is slightly inconvenient to make direct use of the variables u and v.
The problem is that eq. (2.1) gives C~.,, as an expansion in X and y2 = u 2 _ X 2 and
v has the inconvenient feature that it is not an analytic function of u 2 at u - - 0 .
However, v - v-1 is a function of u 2 alone. If we wish to have a function which
vanishes as R ~ o0, it is convenient to choose an invariant
W ~
2u
~
12-1)
Y
[ ( X + u)/2u]R 2~- [ ( X - u)/2u]R-2~
(2.6)
This quantity is manifestly analytic in u 2, X and Y. Using it, we can rewrite eq.
(2.5) in terms of quantities with simple analyticity properties in the form
,
m(R,X, V) =
R2X,,=
(
x:'.
(2.7)
Notice that we have taken out of our constant of integration an explicit factor of
WX',-. This explicit factor is inserted so that we could make easy contact with the
Y - - 0 result, eq. (1.5). As Y--+0, W becomes simply YR -2x. Therefore, we achieve
L.P. Kadanoff, A.B. Zisook / Two-dimensionalplanar model
67
contact with eq. (5) if we assert that
Gn.m(u,O ) = 1.
However, even for W 4 : 0, G,,,,, is very simple. We know that for fixed r, C,,.,,, is
expandable in an asymptotic series in X and y2. But u 2 and W 2 are analytic in X
and y2 and vanish as the square of X and Y. If C,,m is to be expandable so must be
G,, m. But since u 2 and W 2 are both small for small X 2 and y2, we can say
G,,,m(U, W ) -- 1 + O ( W 2 ) .
(2.8)
(This analysis follows, we believe, the spirit of the work of Nauenberg and
Scalapino [7].) Thence our correlation function is determined as
2
'
'
R 2x''"
× [1 +
Eq. (2.9)
equation
Thouless
First u
]
X',m
Rz"(I+¼X)+R2~(1-¼X)
O(W2)].
(2.9)
is the major result of our renormalization analysis. We now study this
for two cases u > 0 (the gaussian critical line) and u = 0 (the Kosterlitzcritical point).
> 0. Let R become very large. Then eq. (2.9) implies
Cn,m( R , X , y ) ~
R -2x",'-2ux',,
y2
× l+X.,m(u+X)2
R -a" + O(W2)]
(2.10)
The first term in the product is the expected simple scaling result. The second
factor is 1 plus a correction term. Our calculation has given us an explicit
correction term of order YZR-4U. However since W 2 is exactly of this order for
large R this explicitly calculated correction term tells us precisely nothing about
Cn, m except the fact that the correction is, indeed, of relative order W 2.
Contrast now the case u = 0. In this case from eq. (2.6)
Y
W= 1 +2Yln(R)
'
(2.11)
and from eq. (2.7)
Cn,.(R, Y, Y) --R-2X"'=(l + 2YIn(R)) -x;'*[l + O(W2)].
(2.12)
L. P. Kadanoff, A . B. Zisook / Two-dimensional planar model
68
N o w the analysis does make non-trivial statements about the correction terms
proportional to Y. To see this compare eq. (2.9) with the special case X -- Y of eq.
(2.1). The l = 0 (leading log) and 1 = 1 (first correction) terms in eq. (2.1) are fully
predicted by eq. (2.9). The W 2 term does not contribute until l = 2.
Therefore for u -- 0, the correction we have calculated (following earlier authors)
is non-trivial. It leads to a very specific prediction for the large-R form of the
correlation function, namely
El+O
However, the result (2.13) is wrong. In comparison to the result of Amit et al. [1]
given in eq. (1.1), we have a correct evaluation of the leading-order term, but we
have missed the correction, which is like
1 + 0 In(In(R))
in(R)
instead of our estimated error which is 1 + O ( l n R ) -2. To get the right result, we
must follow the previous authors and use an improved recursion equation. We shall
do this in sect. 3.
3. The actual calculation
Next we make use of recursion relations (1.3). These have been adjusted (see the
appendix) so that they are fight to first order in Y for all X and have the correct
(see ref. [1]) behavior to cubic order in X and Y. A small adjustment of quartic and
higher-order terms has been m a d e so that they can be solved.
The solution is best stated in terms of p and q defined by
p-~ 1 --e -x,
q=
(3.1)
Ye -x/2
These variables, are respectively closely analogous to our previous X and Y. The
flow equations now read
dp
- 2q 2
dl
1 -p
dq = -2pq
dl
1 -p
'
(3.2)
L . P . Kadanoff , A .B. Zisook / Two-dimensional planar model
69
F r o m these variables, one can find two flow invariants
U----C--q
2 ,
19 = R2U( p + u ) 0 + u)/2( p _ u ) ( U - I)/2.
(3.3)
If we are to follow the same trend as in our previous analysis, we must construct
from u and v a quantity which is analytic in q2 for all u > 0 and which has an
asymptotic expansion in u 2 and q2 (or p2 and q2) for small p and q. We call this
quantity W. For simplicity, we choose to normalize W so that it is linear in q for
small q. F r o m scaling, we find that for small q
q
W ~-- R 2 u / f l _ u )
'
q--~0
(3.4)
We went ~through a rather involved chain of guesswork to construct such a W.
Since it is hard to describe guesswork, we merely report the result. This is expressed
as an implicit equation for W, of the form
W ffi (~k dr u ) I + u / ( i - ~ ) _ (~k - / , / ) 1 - u / ( l - 2 k )
DI/(I-~)_U-I/(1--X)
(3.5)
with the implicit nature of the equation being expressed by the statement
2, = ~
+ W2
(3.6)
This expression has several very attractive features. First, for fixed p > 0, it
defines W as an analytic function of u 2 for all u 2 > 0. [Notice that from eq. (3.3),
when u ~ - u , v ~ , - 1 . ] Also, for fixed u > 0, as q--)0, W becomes proportional to
q times an analytic function of q2 for all q2 > 0. I n fact, the two limiting forms are
reasonably simple. For u > 0, as q--)0, W ~ 0 and h = u so that eq. (3.4) is satisfied.
As u--, 0, for q > 0, h---> W and then
1 - W+lnW
Wffi
q-l + lnq+ lnR2
'
which can be rearranged to read
W-I(1 +lnW)-F o r small
1+q-l+lnq+lnR
2,
(u---O).
(3.7)
q, eq. (3.7) gives a W of the form of an analytic function of q, specifically
W--
q(1 - q)
1 + q h i R - ½q3(tnfR 2) + 1) 2
which holds for q In R2<< 1.
(1 + O ( q 4 ) ) ,
70
L . P . Kadanoff, A .B. Zisook / Two-dimensional planar model
In order to get large R correlation functions, we need the form of W as R---> oo.
For u = 0, eq. (3.7) gives for q In R >> 1
1
W= ln(RZ)+q-l+lnq + 1
1 -
ln(R2)+q-~+lnq+ 1
.
(3.8)
Notice that this result is exactly the form of eq. (1.8b), with the constant, b, having
the value b = 1 + q - i + I n q.
N o w that we have the right flow-invariant variables, it is an easy matter to solve
the flow equation for Cn,,~. If we neglect the small effects which result from terms
of order y2 we find
d In Cn . = 2 X . , m ( X )
dl
----¼n2(1 - p ) + 4 m 2 / ( 1 - p ) ,
(3.9)
so that we may integrate the flow equations (3.2) and obtain a result directly
analogous to that in eq. (2.7), namely
c.,.(R,x,
r) =
e-p"2/SG(u, W)
R 4 . 2 +n2(l +u2)/4
x
(3.10)
As before, we took out an explicit power of I4" to neutralize the singularity which
appears as a power of q. The factor e-~,n~/s is not singular and m a y be neglected.
We know that G(u, W) equals one at Y = 0 since at that point, we must have the
gaussian result
Cn,m( R , X , O) = R -n20--P)/4+4m2(l --P)
(3.11)
According to eqs. (3.4) and (3.10), eq. (3.11) can only be true if
G(u, O) = 1.
(3.12)
By the same argument as before we m a y set G(u, W ) = 1 + O ( W 2) since any
corrections must be analytic in W 2. Since W is always small for large enough R,
these corrections are also small.
L.P. Kadanoff, A.B. Zisook / Two-dimensionalplanar model
71
We need only now collect our terms. For u =P 0 we find that in the limit of large
R
C.,m(R,X, Y) = C.,,,,(R,Xa,,O)[1 + O ( W 2 ) ] ,
(3.13)
.Yen = ln(2 ~rKaf/4) = - ln(1 - u ) .
(3.14)
where
This essentially trivial result could have been derived from simple scaling arguments. However, for u = 0, we obtain a non-trivial conclusion, namely
C.,m(R,X, Y) = (W/q)X""R2X.,.
[1 + O ( W 2 ) ] .
(3.15)
Since W is now given by eq. (3.8), we have now derived our desired result, namely
eq. (1.8). Thus, the correlation function has been evaluated at the KosterlitzThouless critical point.
4. Multi-point correlation functions
The very same mode of analysis can be applied to a multi-operator correlation
function, containing P different operators:
Ce(r,,n,,m,;.
. .
;re,neme;X,r)= (j.,fi O.pmj(rj)
(4.1)
This is non-vanishing in the case in which the total n and m quantum numbers add
up to zero
P
P
E nj = E mj =
j=l
O.
(4.2)
j--1
In that case, the correlation function (20) has a very simple form in the case in
which Y = 0, namely
C e-- II C2(~,nj,mj;rk,nk,mk;X,O),
j<k
(4.3)
L.P. Kadanoff,A.B. Zisook / Two-dimensionaiplanarmodel
72
where C 2 is given by [4]
eiOi2(nlm2--n2ml)
C2(rl,nl,ml; r2,n2,m2; X, O) --
(4.4)
It, -
r2l -<"'"'° -x/" +4mlmzeX)
and
tan 012 = XI _ X2 .
(4.5)
Consider once again the case in which X and Y are close to zero. In the P - - 2
case, we calculated several expansions of C in a power series in X and Y and
convinced ourselves that for fixed coordinates r,. there was a perfectly good
expansion of C in X and Y, in which the leading contribution i n j t h order in X and
2 k t h order in Y had terms in ( l n R ) J+zk and all lower powers. We assume without
proof that a similar expansion holds for p > 2 where now R refers to some typical
separation ~k = [~ rk[ in the correlation function.
The flow equation for C e has the form
-
d lnCe=~.[4mje2
dl
J
__
X
1 2
+~nje_X]
(4.6)
The basic flow invariants are u = (p2 _ q2)1/2 and
vjk = I~ - rk 12~(p + u)<l + u>/2(p _ u)<~- 1>/2
(4.7)
Starting from any l)jk , o n e can form a Wjk which has nice analyticity properties by
performing the construction (3.5). The ratio of any two distances is also a flow
invariant, which can be formed from the v's via
- -
=
r12
.
(4.8)
DI2 ]
Using these invariants, one can write the solution to the flow equation (4.6) in a
form which closely mirrors eq. (4.3) and also has the same basic structure as eq.
(3.10), namely
Ce(rl,nl,ml; ' ' ' ;re,ne,me;X, Y)
= ( H C2(t~,nj,mj; r k , n k , m k ; X ,
x j< k
Y))Ge( u, W~2, (rj_~g}),
/12
(4.9)
L.P. Kadanoff, A.B. Ziaook / Two-dimensiontdplanar model
73
with the factor C 2 having the suggestive form
eiOjk(njmk--n~mj)÷pnjnk/g
C2(/), nj , m£; rk, nk, m k , X ,
Y ) -~
19-
rkl
-,,,,,~(i +u'~)/4-4mjm k
This particular form has been chosen precisely so that when Y = 0 and u v~ 0, eXlS.
(4.4) and (4.10) will be identical, except for the irrelevant analytic factor
e x p ( p j % n k / 8 ). Given this identity, eq. (4.3) implies that
GP( u ' O ' I ~r12
k l l ]f f]i l ' (
(4.11)
Once again we use the anaiyticity of C e in y2 for u=/=0 and its asymptotic
expandability in Y for u -- 0 to then argue that G e is always dose to unity, i.e.,
Here, A is an unknown (non-universal) function of u 2 and ratios of coordinates
which becomes a constant, independent of u 2 as u 2--.0,
Once again, our analysis gives no more than sealing results for u ~ 0 but gives
substantial information for u--->0. For u ~ 0, eq. (4.10) implies
C2(rj, nj, mj; rk, nk, mk; X, Y ) =
e",.",".--,-,,÷.-,"./s
f
X -- n j n l t ( l -- u ) / 4 -- 4 m y r a k / ( 1
~9k)
[1 +
-- u)
~,
(4.13)
The first factor in eq. (4.13) is simple scaling, the next a known correction to this
simple scaling. However, the unknown correction A W 2 in eq. (4.12) is of exactly
the same order as the known correction in eq. (4.13). Hence for u ~ 0 our results
have no more content than the simple statement that the first correction to the
known Gaussian behavior--with a renormalized coupling--is of order
y2/R4U/O-u).
However, at u ffiO, eq. (4.10) m a y be combined with eq. (3.8) to give the
non-trivial statement
eiOjk(njmk --nkmj)
C2(rj,nj,mj;rk,nk,mk ;X' Y)
X [ lln(92k) + b [1
: ,--4mj,,k,njnk/4
[gk)
(ln(ln(~2))
l +nb ] (] - 2mjm~+
~ 2 nJnka[/ )
1 + O ln(9kl)2 + b )2]
(4.14)
74
L.P. Kadanoff , A.B. Zisook / Two.~menaional planar model
This result holds for large §'k and u = 0. To see how much non-trivial data is
actually contained in our results write
and consider ~ k / R to be of order unity. Then, we can express our conclusion in the
statement that at u = 0 one can expand the correlation function in the small
parameter
(In(In(R2))
1
W = ln(R2) + b
l
)
ln(R 2) + b
(4.15)
in the form
C P ( r l , hi, ml; ... ; X, Y ) ~ Ce(rt, n i, m l;. • • ; 0, 0)W -X)L, x~j.,j
In ~ - ( 2 m j m k - i n j n k ) + O ( W
X I+W~
2) ,
]<k
(4.16)
which holds for u = 0. The term of order W 2 is of course an unknown function of
ratios of coordinates §'k/r!2. The known term, of order W, represents the detailed
effect of the factorizability of the correlation function.
Appendix
DERIVATION OF FLOW EQUATIONS
We rederive the flow equations of Amit et al. [1] by making use of a calculational
device like the one used in the appendix of ref. [3]. We consider flows in X, Y and
Y4, the last being the conjugate to the operator 04, o + 0_4, o. The flow equations
read
dY
d--[ = - Fr( X ' Y' r, ) ,
dr,
dl = -F,.(X,
dX
r,
dl = - F x ( X' Y' Y4).
(A.1)
L.P. Kadanoff, A .B. Zisook / Two-dimen~ionalplanar model
75
The necessary symmetry of these equations under Y--->- Y or Y4--->- Y 4 imply that
YFr, Y4 Fv, and Fx are all even under the change of sign of Y or Y4. The additional,
dual symmetry under X---> - X Y,~->Ya then gives
F y ( X , Y, r , ) - - F r , ( - X , Y4, Y ) ,
F x ( X , Y, }'4) -- - F x ( - X ,
Y4, V).
(A.2)
Since we know scaling indices on the gaussian line Y4 = Y - - 0 , we know that to
first order in Y and I14,
F r ( X , Y, Y4) = - 2 ( 1 - e X ) V .
(A.3)
Furthermore, it is possible to adjust the normalization of Y and Y4 so that to
quadratic order
Fx = 2 ( y 2 _ y 2 ) .
(A.4)
Assume that the F ' s can be expanded in a power series in Y, Y4 and X. These
data imply an expansion of the form
Fr = Y(2X + X 2 + aY 2 + by2),
F v , = Y4(2X + X 2 + a y 2 + b y 2 ) ,
Fx = 2 ( y 2 _ y 2 ) + c ( y 2 + y 2 ) x '
(A.5)
where a, b and c are unknown constants.
According to Zisook [10], there is a fixed line at X = 0, Y = ___Y4. This can only
be true if a = - b . T o see the properties of this fixed line, rewrite eqs. (A. 1) in terms
of
H 1 = Y + I14,
H2= Y - Y,,
H 3 = X.
(A.6)
Then the equations read
dH~ = -F~,
dl
(A.7)
76
L.P. Kadanoff, A .B. Zisook / Two-dimensionalplanar model
with
Fl = 2H2H 3 + HI(H2 + all?),
F 2 -~ 2 H i H 3 + H 2 ( H f + a n t i ) ,
(A.S)
F3 = 2H, H 2 +½cH3(H g + H 2 ) .
Since the different fixed lines ( H ! = H 2 = 0, H 3 = a n y t h i n g ) or ( H ! = H 3 = 0,
H 2 = a n y t h i n g ) are supposed to be equivalent, eqs. (A.6) should be symmetrical
u n d e r the i n t e r c h a n g e of the indices 1, 2 a n d 3. F o r this r e a s o n choose a = 1, c = 2.
T h e n c e for small Y4, the flow e q u a t i o n s read
dY = _Y(2X+X2+
dl
y2)~_Y(2eX_2+
_d_X = - 2 ( y 2 + X y 2 ) ~
-2y2e x,
dl
dr,
___
dl
_y4(_2X+X
y2),
2_ y2)
- Y4(2e - x - 2 - y 2 ) .
These are the equations used i n the text.
References
[1] DJ. Amit, Y.Y. Goldschmidt and G. Grinstein, J. Phys. AI3 (1980) 585
[2] J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys. Rev. Bl6 (1977) 1217
[3] L.P. Kadanoff, Phys. Rev. B22 (1980) 1405
[4] L.P. Kadanoff, Ann. of Phys. 120 (1979) 39
[5] J.M. Kostcrlitz,J. Phys. C7 (1974) I046
[6] J.M. Kosteflitzand DJ. Thouless, J. Phys. C6 (1973) I181
[7] M. Nauenberg and D. Scalapino, Phys. Rev. Lett.44 (1980) 837
[8] A.M.M. Pruisken and L.P. Kadanoff, Phys. Rev. B (1980), to be published
[9] J. Rudaick and D.R. Nelson, Phys. Rev. B13 (1976) 2208
[I0] A.B. Zisook, J. Phys. AI3 (1980) 2451
[II] A.B. Zisook and L.P. Kadanoff, J. Phys. A 0980), to be published
(A .9)