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10-30-2009
Renormalized Field Theory of Collapsing Directed
Randomly Branched Polymers
Hans-Karl Janssen
Institut für Theoretische Physik III
Frank Wevelsiep
Institut für Theoretische Physik III
Olaf Stenull
University of Pennsylvania, [email protected]
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Janssen, H., Wevelsiep, F., & Stenull, O. (2009). Renormalized Field Theory of Collapsing Directed Randomly Branched Polymers.
Retrieved from http://repository.upenn.edu/physics_papers/58
Suggested Citation:
Janssen, H., Wevelsiep, F. and Stenull, O. (2009). "Renormalized field theory of collapsing directed randomly branched polymers." Physical Review E.
80, 041809.
© 2009 The American Physical Society
http://dx.doi.org/10.1103/PhysRevE.80.041809
This paper is posted at ScholarlyCommons. http://repository.upenn.edu/physics_papers/58
For more information, please contact [email protected].
Renormalized Field Theory of Collapsing Directed Randomly Branched
Polymers
Abstract
We present a dynamical field theory for directed randomly branched polymers and in particular their collapse
transition. We develop a phenomenological model in the form of a stochastic response functional that allows
us to address several interesting problems such as the scaling behavior of the swollen phase and the collapse
transition. For the swollen phase, we find that by choosing model parameters appropriately, our stochastic
functional reduces to the one describing the relaxation dynamics near the Yang-Lee singularity edge. This
corroborates that the scaling behavior of swollen branched polymers is governed by the Yang-Lee universality
class as has been known for a long time. The main focus of our paper lies on the collapse transition of directed
branched polymers. We show to arbitrary order in renormalized perturbation theory with ε expansion that
this transition belongs to the same universality class as directed percolation.
Disciplines
Physical Sciences and Mathematics | Physics
Comments
Suggested Citation:
Janssen, H., Wevelsiep, F. and Stenull, O. (2009). "Renormalized field theory of collapsing directed randomly
branched polymers." Physical Review E. 80, 041809.
© 2009 The American Physical Society
http://dx.doi.org/10.1103/PhysRevE.80.041809
This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/58
PHYSICAL REVIEW E 80, 041809 共2009兲
Renormalized field theory of collapsing directed randomly branched polymers
Hans-Karl Janssen and Frank Wevelsiep
Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
Olaf Stenull
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
共Received 20 July 2009; published 30 October 2009兲
We present a dynamical field theory for directed randomly branched polymers and in particular their collapse
transition. We develop a phenomenological model in the form of a stochastic response functional that allows us
to address several interesting problems such as the scaling behavior of the swollen phase and the collapse
transition. For the swollen phase, we find that by choosing model parameters appropriately, our stochastic
functional reduces to the one describing the relaxation dynamics near the Yang-Lee singularity edge. This
corroborates that the scaling behavior of swollen branched polymers is governed by the Yang-Lee universality
class as has been known for a long time. The main focus of our paper lies on the collapse transition of directed
branched polymers. We show to arbitrary order in renormalized perturbation theory with ␧ expansion that this
transition belongs to the same universality class as directed percolation.
DOI: 10.1103/PhysRevE.80.041809
PACS number共s兲: 64.60.ae, 05.40.⫺a, 64.60.Ht, 64.60.Kw
I. INTRODUCTION
It is well known that a single linear 共nonbranched兲 polymer consisting of N monomers in a diluted solution with a
good solvent where it is in a swollen conformation with gyration 共Flory兲 radius RN ⬃ N␯SAW, ␯SAW ⱖ 1 / 2, undergoes a
phase transition to a collapsed “globule” with Flory radius
RN ⬃ N1/d 共where d is the dimensionality of space兲 if the
solvent quality deteriorates as it usually does when the temperature of the solution is lowered below the so-called ␪
point. Many theoretical tools have been successfully applied
to linear polymers, and considerable progress has been made
in understanding this collapse transition 关1兴.
In contrast, much less is known about collapse transition
of randomly branched polymers, which are usually assumed
being macroscopically either isotropic or directed. As far as
the isotropic case is concerned, a number of numerical studies have been performed over the last 20 years or so 关2–7兴.
The picture that arises from these studies is much more complex than that for the collapse of linear polymers. The basic
reason for this added complexity is the possibility for introducing more than one fugacity to drive the collapse. It appears that the line of collapse transitions in the phase diagram consists of two qualitatively different parts. One part of
the line, called the line of ␪ transitions, corresponds to continuous transitions with universal critical exponents from
swollen polymer configurations with treelike character to
compact coil-like configurations. The other part of the transition line, the line of ␪⬘ transitions, corresponds to the collapse of swollen foamlike or spongelike polymers with many
cycles of bonds between the monomers to vesiclelike compact structures. If this transition is assumed to be continuous,
one finds nonuniversal exponents 关2兴. The two different parts
of the collapse-transition line are separated by a higher multicritical point which belongs to the isotropic percolation universality class. One of the open questions regarding the
phase diagram is the existence of a possible further transition
line between the configurations of the collapsed polymers.
As far as directed branched polymers 共DBPs兲 and their
collapse 共CDBP兲 are concerned, the theoretical picture
1539-3755/2009/80共4兲/041809共17兲
关4,8–11兴 is somewhat clearer than in the isotropic case, in
particular in 1 + 1 dimensions 共one transversal and one longitudinal; in the following, we will denote the full space
dimension as D = d + 1兲. Noting that branched polymers, lattice animals, and lattice trees belong to the same universality
class, one can use the results of Dhar 关9兴 on collapsing directed strongly embedded or site animals. This way, Henkel
and Seno 关4兴 have shown that there is only one type of ␪
transitions which describes the collapse of directed branched
polymers in D = 2 dimensions. This collapse transition belongs to the directed percolation 共DP兲 universality class. Furthermore, Dhar has shown in Ref. 关9兴 that the collapse transition is also of directed percolation type in D = 3 if a special
relation between the potentials 共fugacities兲 of directed site
animals holds. Flory theory was applied by Redner and
Coniglio 关8兴. They correctly derived the upper critical dimension dc = 4, but they found critical exponents for the
CDBP transition that are different from the DP exponents,
which, however, is incorrect as we shall see.
Whereas a comprehensive field theory for swollen 共extended兲 polymers has been existing for quite some time, the
field theory for the collapse of branched polymers is much
less developed. As far as we know, there exists to date no
field theory for the directed case. For the isotropic case, there
is the seminal work by Lubensky and Isaacson 关12兴 and Harris and Lubensky 关13兴. However, it turns out that these papers, as far as they consider the collapse transition, contain a
fundamental error in the renormalization procedure 共they
overlook a required renormalization兲, and as a consequence
the long-standing one-loop results for the collapse transition
are not entirely correct 关14兴. In an upshot, one may say that
there are important open questions and unresolved issues in
the theory for the collapse of directed and isotropic branched
polymers.
Here and in a following publication 关14兴, we develop dynamical field theories for the collapse of directed and isotropic branched polymers, respectively. The idea behind these
theories is to start out with stochastic activation processes
which lead to percolation clusters and then to exploit the
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©2009 The American Physical Society
PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
well know connection between branched polymers and lattice animals, i.e., cluster of a given mass. We concentrate on
large animals below the percolation point, and we equip the
theory with enough parameters to allow for triciticality
which corresponds to the collapse of large branched polymers. As alluded to above, the directed case has the benefit
of being simpler than the isotropic case. It allows us to learn
about the fundamental structure of dynamical field theories
for the collapse of branched polymers and thereby to sharpen
our tools for the more complicated isotropic case. The most
important concrete result of the present paper is that we show
to arbitrary order in renormalized perturbation theory with ␧
expansion that the CDBP transition is of DP type.
The outline of the remainder of this paper is as follows. In
Sec. II, we develop our field theoretic model. In Sec. II B we
analyze our model in mean-field theory. We determine the
mean-field phase diagram of DBP and discuss their scaling
behavior at the collapse transition ignoring fluctuations. In
Sec. III, we briefly discuss swollen DBP by making contact
to established theories. In Sec. IV, we present the core of our
field theoretic analysis of the collapse transition. We discuss
the scaling invariances of our model at this transition and
their consequences. We calculate the counterterms required
to renormalize our theory in a one-loop calculation and by
using Ward identities. We set up and solve renormalization
group 共RG兲 equations that provide us with the scaling form
of the equation of state and correlation lengths. In Sec. V, we
translate our field theoretic results for the collapse transition
into a form that is more commonly used in polymer theory,
and we compare them to numerical simulations. In Sec. VI,
we give a few concluding remarks. There are tow appendixes. In Appendix A, we present some details of our oneloop calculation, and in Appendix B, we discuss essential
singularities.
II. TOWARD A DYNAMICAL FIELD THEORY OF
DIRECTED BRANCHED POLYMERS
A. Generalized directed percolation process
In this section, we develop a field theoretic stochastic
functional 关15–18兴 for DBP based on very general arguments
alluding to the universal properties of a corresponding
Markoffian stochastic percolation process. We consider polymers as clusters in d transversal and one longitudinal 共time兲
directions. Below the percolation threshold all of the clusters
generated by such a process are finite. The distribution function for the large clusters decays exponentially with their
mass 共number of monomers兲. It is well known 关19兴 that untypical very large clusters with linear sizes essentially larger
than the correlation lengths are generated as rare events. The
statistical properties of these fractal clusters belong to the
universality class of lattice animals. Thus, they also form the
prototype model for single randomly branched polymers in a
dilute solution. Allowing for a further mechanism in the percolation process which favors contacts of the monomers, we
introduce a possible tricritical instability that leads to the
collapse of the large clusters describing the branched polymers.
A Langevin equation which describes directed percolation
processes allowing for tricritical behavior has been well
known for many years 关18,20–27兴. It is based on the fundamental phenomenological principles of absorbing processes
in conjunction with a density and gradient expansion. The
basic variable or field is the density n共x , t兲 of agents 共infected
individuals兲, which models the fractal monomer density of
the polymer generated from a given source h̃共x , t兲. This
choice of field is possible because the stochastic process generates clusters such that the numbers of monomers, branching, and end points are all of the same order which is in
contrast to the case of linear polymers. In other words, the
density is a proper field for the current problem because
branched polymers are really “fur-bearing animals.” The
Langevin equation in Ito interpretation is given by
␭−1ṅ = R共n兲n + D1共n兲ⵜ2n + D2共n兲共ⵜn兲2 + ¯ + h̃ + ␨ ,
共2.1a兲
where ␭ is a kinetic coefficient and the noise correlation
reads
␨共x,t兲␨共x⬘,t⬘兲 = ␭−1关Q共n兲n共x,t兲 + ¯兴␦共x − x⬘兲␦共t − t⬘兲.
共2.1b兲
We assume Gaussian noise which we can for our purposes
without loss of generality because higher order noise does
not change the universal properties of the process. The leading terms of the expansions are D1共n兲 = 1 + c1n + ¯, D2共n兲 =
−c2 + ¯, R共n兲 = −r − g⬘n / 2 − f ⬘n2 / 6 + ¯, and Q共n兲 = g + ¯.
Further terms in the expansions are possible but they turn out
being irrelevant. In mean-field theory, the percolation transition occurs at r = 0. As announced above, we restrict ourselves in the following to the region r ⬎ 0 in which only
nonpercolating directed branched clusters with a typical linear size ␰ ⬃ r−1/2 are generated from a time- and spacelocalized source. However, our primary interest is in the rare
events where the fractal clusters of linear size essentially
larger than r−1/2 are generated. As long as g⬘ ⬎ 0, the secondorder term f ⬘n2 of the rate R is irrelevant. We permit both
signs of g⬘ so that our model accounts for a tricritical instability. Consequently, we need the second order term with
f ⬘ ⬎ 0 for stabilization purposes, i.e., to limit the density n to
finite values. The parameters ci 共which are irrelevant for the
original percolation problem兲 are assumed to be positive to
smooth density fluctuations. An essential property of percolation processes is the linear dependence of the noise correlation 关Eq. 共2.1b兲兴 on the density if n → 0 for g ⬎ 0. This
property guarantees that the percolation process is really absorbing 关25兴 and guarantees the description by a density variable. Last but not the least, all the terms lead to ṅ = 0 if n
= 0 and h̃ = 0 on grounds of the absorbing state.
The approach that we are taking focuses on general principles for processes belonging to the same universality class
and is therefore necessarily phenomenological 关18兴. We devise our field theoretic model representing the universality
class using a purely mesoscopic stochastic formulation based
on the correct order parameters identified through physical
insight in the nature of the critical phenomenon. Hence, the
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PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
x−δ
stochastic response functional that we are about to derive
stays in full analogy to the Landau-Ginzburg-Wilson functional and provides a reliable starting point of the field theoretic method.
Alternatively, one might use the so-called “exact” approach which, as a central step, consists of reformulating a
microscopic master equation for chemical reactions as a
bosonic field theory on a lattice. For a recent excellent review article on this method see 关28兴. We choose not to use
this method as our main approach because we feel that it
treats universal properties not as transparently as the purely
mesoscopic formulation does. Nonetheless, we think that the
lattice reactions which would be the starting point for the
exact approach to the current problem have some pedagogical value as they can help to nurture ones intuition about the
epidemic process. Here, these reactions comprise the well
known reactions leading to DP,
␣
X共x兲→ X共x兲 + X共x + ␦兲,
2X共x兲→ X共x兲,
t + dt
t
β
t + dt
t
λ
t + dt
t
κ
共2.2b兲
t
␭
µ
共2.2c兲
where x denotes a lattice point, ␦ denotes a vector to a neighboring point, and X共x兲 stands for an agent 共particle兲 at x.
Moreover, because we are interested in tricriticality, we need
additional reactions that produce more compact clusters,
␬
X共x − ␦兲 + X共x + ␦兲→ X共x − ␦兲 + X共x兲 + X共x + ␦兲,
共2.2d兲
␮
3X共x兲→ X共x兲.
ddxdt␭ñ ␭−1⳵tn + 共r − ⵜ2兲n +
冎
g
− c1nⵜ2n + c2共ⵜn兲2 − ñn .
2
t + dt
FIG. 1. Reactions leading to tricritical DP. Open circles indicate
lattice sites. Solid dots symbolize agents occupying these sites.
ated by the stochastic process 关Eqs. 共2.1兲兴. Having this functional, one can calculate the average of any observable that is
polynomial in n and ñ, say O关n , ñ兴, by functional integration
with weight exp共−JDP兲,
共2.2e兲
For illustrations of the reactions defined in Eqs. 共2.2兲, see
Fig. 1 and Refs. 关26,29兴. Note that the single site back reactions 关Eqs. 共2.2b兲 and 共2.2e兲兴 act against multiple occupations of lattice sites and are thus reminiscent of an excludedvolume interaction. As mentioned above, we will not use the
exact approach in this paper. However, for the adherents of
the exact approach, we mention without presenting further
details that we have verified that the bosonic representation
of the reaction 关Eqs. 共2.2兲兴 leads to the same stochastic functional 共called an action in that approach兲 as the purely mesoscopic approach.
Returning to the latter approach, we now recast the
Langevin equations 关Eqs. 共2.1兲兴 as a stochastic response
functional 关15–18兴,
再
t
α
共2.2a兲
X共x兲→ 쏗 ,
冕
x+δ
t + dt
␤
JDP =
x
g⬘ 2 f ⬘ 3
n + n
2
6
共2.3兲
Here, we have neglected further higher order terms that will
not result in any relevant contributions to our final response
functional. JDP describes the statistics of DP clusters gener-
具O典DP =
冕
D关n,ñ兴O关n,ñ兴exp共− JDP兲
¬ Tr兵O关n,ñ兴exp共− JDP兲其.
共2.4兲
Here and in the following, path integrations are always interpreted in the sense of the 共mathematical save兲 Ito discretization. One has to be careful to set the range of the variables of
the functional integration correctly as there are some subtleties involved. Some authors 关28兴 used functional integrals of
the type featured in Eq. 共2.4兲 purely in a restricted sense as a
device for generating a perturbation expansion. This precautionary view arises from a narrow interpretation of the
bosonic functional integral after elimination of irrelevant
terms including higher order monomials of the fields which
are in general required to ensure the convergence when fields
become large. However these convergence problems can and
should be avoided by choosing the support for the functional
integration properly, as was shown, e.g., by Ciafaloni and
Onofri 关30兴 in the case of Reggeon field theory 共RFT兲 共which
is equivalent to DP兲. This type of consideration leads here to
the rule that n is to be integrated along the positive real axis
whereas the integration of ñ is performed along the full
imaginary axis. Of course, deviations in finite regions, as,
e.g., suggested by saddle points, are possible.
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PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
One of the most important observables in the current
problem is the mass or total number of monomers of a cluster,
M=␭
冕
ddxdtn共x,t兲,
共2.5兲
where we have included the kinetic coefficient ␭ into the
definition for later convenience 共if we were not including ␭
in the definition of M, we had to multiply M in subsequent
formulas by ␭ on dimensional grounds兲. Because we are
mainly interested in the scaling behavior of a single large
polymer with N Ⰷ 1 monomers, we will focus in the following on averages that are restricted to clusters of a given mass
N, and we assume without loss of generality that this cluster
emanates from a weak source h̃共x , t兲 = q␦共x兲␦共t兲 of agents at
the origin x = 0 at time t = 0. For our general observable O关n兴,
this leads to
具O典NP共N兲 = 具␦共N − M兲O关n兴exp关qñ共0,0兲兴典DP
⯝ q具␦共N − M兲O关n兴ñ共0,0兲典DP ,
P共N兲 ⯝
Note that P共N兲 is correctly normalized, 兰dNP共N兲
= 具exp共qñ兲典DP = 1, because 具F关ñ兴典DP = F关0兴 for any functional
F of ñ due to causality.
P共N兲 is asymptotically proportional 共up to nonuniversal
amplitudes and an exponential factor ␮N0 , where ␮0 is an
effective coordination number of the lattice兲 to the lattice
animal number AN which plays an important role in percolation theory. AN measures the number of directed clusters of
size N, weighted by fugacities for different cluster properties
such as contacts, loops, and so on. In terms of the number of
configurations C共N , b , c兲 consisting of N sites, b bonds, and c
not bounded contacts, AN can be expressed as 关13兴
AN = 兺 C共N,b,c兲⌳b1⌳c2 .
AN ⯝ ␮N0 P共N兲,
共2.9兲
共2.10兲
a relation that we shall use later on.
Now we turn to the signature of the collapse transition.
Chemically, in a diluted solution the transition occurs when
the hard core repulsion of the monomers is exactly compensated by their weak effective attraction, i.e., when the second
Virial coefficient A共2兲 in the expansion of the osmotic pressure in powers of the polymer density vanishes 关1兴. Up to a
minus sign this second Virial coefficient is proportional to
the space integral of the correlation function between two
polymers in the solution. Hence, in our formalism, the signature of the collapse transition is the vanishing of the spacetime integral of the two-polymer correlation,
C共2兲共N;x,t兲 = P共2兲共N;x,t兲 − P共2兲共N;⬁兲,
共2.11兲
P共2兲共N;x,t兲 ⯝ 具␦共N − M兲ñ共x,t兲ñ共0,0兲典DP
共2.12兲
where
is asymptotically the probability distribution of two large
clusters of total large mass N generated by two roots at 共0 , 0兲
and 共x , t兲. If the distance of the point 共x , t兲 from the origin is
substantially bigger than their linear sizes, the two clusters
decouple, and we have
P共2兲共N;x,t兲 ⯝ P共2兲共N;⬁兲 =
冕
⬁
dN⬘P共N⬘兲P共N − N⬘兲,
0
共2.13兲
共2兲
i.e., C 共N ; x , t兲 becomes a cumulant. The second Virial coefficient is proportional to the total measure of this correlation times the factor N1/2, which comes into play if one considers only clusters of the same mass N,
A共2兲 ⬃ − N1/2关P共N兲兴−1␭
⯝ − N1/2␭
冕
冕
ddxdtC共2兲共N;x,t兲q
ddxdt具ñ共x,t兲ñ共0,0兲典N共cum兲 .
共2.14兲
Instead of working with quantities discussed above as
functions of N, it is often more convenient to work with their
Laplace-transformed counterparts. For P共N兲 we then have
the representation by an inverse Laplace transformation,
共2.8兲
P共N兲 =
b,c
The fugacities ⌳1 and ⌳2 correspond to the parameters of the
stochastic functional JDP. The collapse transition occurs for
large N when ⌳2 becomes critical because the latter rewards
contact-rich animals. If we assume universality, the probability distribution P共N兲 and the cluster number should be related for N Ⰷ 1 via
,
and conversely
P共N兲 = 具␦共N − M兲exp关qñ共0,0兲兴典DP ⯝ q具␦共N − M兲ñ共0,0兲典DP .
共2.7兲
⬘
兺 AN⬘␮−N
0
N⬘
共2.6兲
where P共N兲 is the probability distribution for finding a cluster of given mass N with the last equation holding only asymptotically for large N and small q. Note that the zerothorder term in the Taylor expansion leading to the second line
of Eq. 共2.6兲 vanishes because 具F关n兴典DP = F关0兴 for any functional F of n because DP is an absorbing process. Note also
that the formalism allows with ease to ask for the probability
of clusters generated by several sources at different points
共ri , ti兲 by inserting more fields ñ共ri , ti兲 in the exponential of
the average 关Eq. 共2.6兲兴. Equation 共2.6兲 implies that the probability distribution for finding a cluster of mass N is given by
关25兴
AN␮−N
0
=
冕
冕
␴+i⬁
␴−i⬁
␴+i⬁
␴−i⬁
dz zN
e 具exp关− zM + qñ共0,0兲兴典DP
2␲i
dz
exp关zN + q⌽共z兲 + O共q2兲兴,
2␲i
共2.15兲
with ⌽共z兲 = 具ñ典z, where 具 ¯ 典z denotes averages taken with
respect to the new stochastic functional,
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PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
Jz = JDP + zM.
共2.16兲
Im z
Note that the normalization condition for P共N兲 results in
⌽共0兲 = 0. The function ⌽共z兲 corresponds to the generating
function of the animal numbers AN and plays in the following the role of an order parameter. Before moving on to
mean-field theory, we would like to mention that a shifted
version of the Laplace variable z will play later on a role akin
to that of an external field.
Re z
zc
B. Mean-field theory
In this section, we discuss various aspects of DBP in
mean-field theory. Our motivation for presenting this meanfield theory is twofold. 共i兲 The field theory that we are going
to present in the following sections is fairly involved. Meanfield theory, on the other hand, allows us to obtain results
with relative ease and may serve the reader as a warm up for
the more difficult sections to come. 共ii兲 At mean-field level,
various results for DBP are well known. By reproducing
these results, we will find that our model satisfies important
consistency checks.
Interested in a mean-field approximation in the spirit of
Landau theory as the starting point of the systematic perturbation expansion, we look for stationary and homogeneous
saddle points of Jz, which are determined by
␭−1
FIG. 2. Deformation of the contour of integration for the inverse
Laplace transformation. Originally, the integration is along the
imaginary axis. After deformation, it nestles to and leads around the
branch cut that terminates at the branch point zc on the negative real
axis.
disc ⌽ denotes the discontinuity of the function ⌽ at the
branch cut, we get
P共N兲 ⯝ qezcN+q⌽共zc兲
␦Jz
g
f⬘
= rñ + g⬘ñn + ñn2 − ñ2 + z = 0, 共2.17a兲
2
␦n
2
␭
−1
␦Jz
g⬘
f⬘
= rn + n2 + n3 − gñn = 0.
2
6
␦ñ
冕
冕
⬁
dx
0
⯝
qezcN+q⌽共zc兲
N
disc ⌽共zc − x兲 −xN
e
2␲i
⬁
dx
0
disc ⌽共zc − x/N兲 −x
e .
2␲i
共2.17b兲
共2.19兲
n兩SP = 0,
共2.18a兲
⌽共z兲 ª ñ兩SP = 关r − ␶共z兲兴/g,
共2.18b兲
The nonuniversal factor qezcN+q⌽共zc兲 is common to all the
quantities described by Eq. 共2.6兲 and cancels therefore in all
averages 具O典N. Note that for large N only the small neighborhood of zc matters. In mean-field theory, it is easy to
calculate P共N兲 without further approximation. We obtain
␶共z兲 = 冑2gz + r2 .
共2.18c兲
These equations are solved by
The correct sign of the square root is determined from the
normalization condition ⌽共0兲 = 0. The equation of state for
⌽共z兲 shows a branch point on the negative real axis at zc =
−r2 / 2g where ␶ = 0 becomes critical. Note that the presence
of ñ 兩SP in Eq. 共2.17b兲 leads effectively to the replacement of
the undercritical parameter r by the critical parameter ␶.
Hence, this equation leads to the usual mean-field equation
of tricritical DP 关18,22–27兴. It follows that the saddle-point
solution 关Eqs. 共2.18兲兴 is stable as long as g⬘ ⱖ 0. If g⬘ becomes negative, Eq. 共2.17b兲 develops continuously a nonzero positive solution n 兩SP = −3g⬘ / f ⬘ at the critical value ␶
= 0, signaling the collapse to a compact cluster as a continuous phase transition.
Qualitatively, it is the branch-point singularity that determines the form of P共N兲, and higher orders in perturbation
theory, though leading to quantitative improvement, do not
result in qualitative changes of that form. To get the
asymptotic expansion of P共N兲, we deform the contour of the
complex integral in Eq. 共2.15兲 as illustrated in Fig. 2. If
P共N兲 =
q
冑2␲g
冋
N−3/2 exp −
共rN − q兲2
2gN
册
⯝ qN−3/2 exp关− Nr2/2g兴.
共2.20兲
This distribution has its maximum near the characteristic
value N = N0 = q / r. As stated above, we are, however, mainly
interested in the rare events with N Ⰷ N0 where the
asymptotic forms are valid. Note that the second line which
gives the large-N behavior is in perfect agreement with the
共mf兲
well known asymptotic result P共mf兲共N兲 ⬃ N−␪ exp共
共mf兲
−N / N⬁ 兲 with the so-called entropic exponent given in
mean-field approximation by ␪共mf兲 = 3 / 2 and a nonuniversal
number N⬁共mf兲 = 2g / r2.
Gaussian fluctuations are governed by the second variations of the stochastic functional Jz. After Fourier transformation in space and time, we have
041809-5
⌫1,1共q, ␻兲mf =
冏
␦Jz关ñ,n兴
␦ñ␦n
冏
= i␻ + ␭共␶ + q2兲,
SP
共2.21a兲
PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
⌫0,2共q, ␻兲mf =
冏
␦Jz关ñ,n兴
␦n␦n
冏
ñ → ñ + ␤n,
= ␭共g⬘ + cq2兲⌽,
where c = c1 + c2. ⌫2,0;mf = −gn 兩SP is zero. The propagator follows from the inverse of ⌫1,1;mf in 共q , t兲 representation as
G1,1共q,t;z兲 = ␪共t兲exp兵− ␭关␶共z兲 + q2兴t其,
共2.22兲
ñ → ñ + ␥,
⌫0,2共0,0兲mf
g⬘关r − ␶共z兲兴
.
2 =−
兩⌫1,1共0,0兲mf兩
␭g␶共z兲2
共2.23兲
The second Virial coefficient follows from Eq. 共2.14兲 to leading order as
A共2兲 ⬃ N2g⬘r/g.
共2.24兲
It vanishes at the mean-field collapse transition, that is, for
g⬘r / g = 0.
Next, we calculate the average monomer density 具n共x , t兲典N
emanating from a source at 共0 , 0兲. For this calculation, we
recall Eq. 共2.6兲. When specialized to O关n , ñ兴 = n共x , t兲, the
second line of Eq. 共2.6兲 is equal to the Laplace transform of
具n共x , t兲ñ共0 , 0兲典z, the latter being the 共spacewise兲 Fourier
transform of G1,1共q , t ; z兲 关Eq. 共2.22兲兴. Collecting, we find
that 具n共q , t兲典N can be written as
冕
ñ → ñ +
g0 = g,
冋
册
共2.26兲
Form this result, we can read off directly that the longitudi共mf兲
nal and transversal radii of gyration scale are R储 ⬃ N␯A,储 and
共mf兲
R⬜ ⬃ N␯A,⬜ with the well known mean-field animal exponents
1
共mf兲
␯A,
储 = 2,
共mf兲
␯A,⬜
= 41 .
共2.27兲
ñ → ␣−1ñ,
共ii兲 a mixing of the fields
n → ␣n,
␶ˆ 0 = ␶,
g1 = g⬘ + 2cr,
共2.30兲
␶ˆ 1 = 共r − ␶兲共g⬘ − 2c␶兲/g,
ĥ = z + 共r2 − ␶2兲/2g.
共2.31兲
Note that the three control parameters ␶ˆ 0, ␶ˆ 1, and ĥ go to zero
in the mean-field theory of the collapse of large branched
polymers, i.e., near zc, and that, however, the coupling constant g1 stays finite.
The elimination of all terms that are at least irrelevant 共in
the sense of a naive scaling consideration兲 in comparison to
the retained ones reduces the stochastic functional to
Jz⬘ =
共2.28a兲
共2.29兲
and new control parameters,
C. Stochastic functional for directed branched polymers
The response functional Jz = JDP + zM 关Eq. 共2.16兲兴 is
form invariant under three continuous transformations. Thus,
three parameters of the functional are redundant. Here, we
exploit the symmetries to eliminate two redundant parameters and to define scale-invariant effective couplings.
The symmetry transformations are 共i兲 a rescaling of the
fields
共r − ␶兲
共1 − cn兲.
g
g2 = r共f ⬘ − 3cg⬘ − 3c2r兲/g,
共2.25兲
g␭t␪共t兲
x2 g共␭t兲2
−
exp
−
.
具n共x,t兲典N =
共4␲␭t兲d/2
4␭t
2N
共2.28c兲
This special shift is reminiscent of the saddle point 关Eq.
共2.18b兲兴 and eliminates the uncritical parameter r in favor of
␶. After this shift the fields and their expectation values are
small quantities for small ␶, and a perturbation expansion is
appropriate. The parameter ␶ is a free parameter at this place,
and, of course, we can set this parameter to zero. However, it
is more economical to use it in the renormalization program
as a small control parameter because it attains at least an
additive renormalization. The shift alone introduces besides
the bilinear gradient term ⬃ñⵜ2n a further quadratic gradient
term ⬃nⵜ2n that is thereafter eliminated by the special mixing in Eq. 共2.29兲. This term also attains renormalizations, and
to cure UV divergencies a counterterm ⬃nⵜ2n must be reintroduced via the renormalization scheme, which we will do
as we proceed.
The transformation 关Eq. 共2.29兲兴 suggests the definitions of
new coupling constants,
dz
G1,1共q,t;z兲exp共zN兲,
2␲i
where the integration path is taken appropriately as outlined
above for the probability distribution. The momentum integration back to the 共x , t兲 representation is straightforward,
and we obtain
n → n.
We eliminate the redundant parameters r and c by specializing the latter two, which do not transform the density field n,
by
and the ñ-correlation function integrated over space and time
is
具n共q,t兲典N ⯝ P共N兲−1q
共2.28b兲
and 共iii兲 a shift of the response field
SP
共2.21b兲
G0,2共q = 0, ␻ = 0; z兲 = −
n → n,
冕
再
ddxdt␭ ñ关␭−1⳵t + 共␶ˆ 0 − ⵜ2兲兴n +
冋
+ −
␶ˆ 1 2
n
2
册 冎
g0 2
g1
g2
ñ n + ñn2 + n3 + ĥn .
2
2
6
共2.32兲
At this place we remark that for g2 = 0 the stochastic functional Jz⬘ has the same form as the stochastic functional for
ordinary directed percolation with a quadratic rapidityreversal symmetry breaking ⬃␶ˆ 1. Thus, in this special case
the collapse transition belongs to the DP-universality class.
Hence, we want to find out whether g2 is a relevant perturbation or whether it goes to zero under renormalization.
041809-6
PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
The modifications of the dynamical functional including
the elimination of the irrelevant higher order couplings make
it worthwhile to discuss the support of the functional integration over the fields ñ and n anew. Like before, the integration
of n, the density of monomers, goes along the positive real
axes. We write the third order terms of the integrand of Jz as
冋
−
册 冋 冉
g0 2
g1
g2
n
g1
ñ n + ñn2 + n3 =
− g0 ñ −
n
2
2
6
2
2g0
+
冉
冊
2
III. BRIEF VIEW ON SWOLLEN DIRECTED BRANCHED
POLYMERS
冊册
g2 g21 2
+
n .
3 4g0
共2.33兲
This form shows that the support of integration of ñ must be
so that ñ − g1n / 2g0 is imaginary if this variable becomes
large. The convergence of the functional integral then requires
g0 ⱖ 0,
4g30g2 + 3共g0g1兲2 ⱖ 0
共2.34兲
to make sense beyond perturbation theory. If these conditions
are violated, higher order contributions become relevant for
stability reasons, and a first-order transition takes place.
Next, we consider the quadratic terms of Jz⬘. Writing these
terms as
ñ关␶ˆ 0 − ⵜ2兴n +
冉
冊
␶ˆ 1 2
g1
n = ñ −
n 关␶ˆ 0 − ⵜ2兴n
2
2g0
1
+ n
2
冋冉
␶ˆ 1 +
冊
terms do not alter the fixed point structure of RFT.
As it stands, the J is a truly proper field theoretic stochastic functional in the sense of being a minimal model. Nevertheless, it still contains that much generality that allows us to
study the statistics of swollen DBP and the CDBP transitions.
These will be the subjects of the following sections.
册
g1
g1
␶ˆ 0 − ⵜ2 n
g0
g0
In Sec. II C we have learned interalia that the shape of
P共N兲 for large N is controlled by the immediate vicinity of
the branch point zc and that correlation lengths are large
when the parameter ␶ defined in Eqs. 共2.18兲 is small. For
studying large clusters based on our functional J 关Eq. 共2.38兲兴
this means that we should focus on the limit of vanishing
control parameters ĥ and ␶ˆ 0. The collapse transition of these
large polymers occurs for vanishing ␶ˆ 1. This transition will
be treated in the later sections. Here we are interested in the
swollen phase of these polymers, and hence we consider a ␶ˆ 1
that is positive and finite, say of the order of 1. Then, ␶ˆ 1 can
be eliminated through a simple redefinition of the fields
共␶ˆ 1 / 2兲1/2˜␸ → ˜␸,
共␶ˆ 1 / 2兲−1/2␸ → ␸,
共␶ˆ 1 / 2兲1/2g0 → g0,
−1/2
−3/2
共␶ˆ 1 / 2兲 g1 → g1, 共␶ˆ 1 / 2兲 g2 → g2, or formally by setting
␶ˆ 1 = 2.
Now, we count engineering dimensions. As usual, we employ some inverse length scale ␮ which is convenient for this
task. Recalling that d denotes the number of transversal dimensions only, we have
␸ ⬃ ␮共d−2兲/2,
共2.35兲
␶ˆ 0 ⬃ ␮2,
we see that we have to require
␶ˆ 0 ⱖ 0,
g20␶ˆ 1
+ 共g0g1兲␶ˆ 0 ⱖ 0,
共g0g1兲 ⱖ 0
共2.36兲
for stability of the saddle point at ñ = n = 0 共for ĥ = ĥc = 0兲. If
these conditions are violated, a phase transition takes place.
Through the ␦-function condition in Eq. 共2.7兲 which selects from all the generated clusters only the clusters with
given mass N, the stochastic functional Jz⬘ has lost its typical
causal structure, whereas the absorbing state condition still
holds, Jz⬘关ñ , n = 0兴 = 0. To restore the causal structure, we perform the duality transformation,
n共t兲 = i˜␸共− t兲,
ñ共t兲 = − i␸共− t兲.
共2.37兲
These steps lead us from Eq. 共2.32兲 to
J=
冕
再
ddxdt␭ ˜␸关␭−1⳵t + 共␶ˆ 0 − ⵜ2兲兴␸ −
+i
冋
g1 ⬃ ␮共2−d兲/2,
册 冎
JYL =
Evidently, this new functional obeys causality, J关˜␸ = 0 , ␸兴
= 0, but the absorbing state property is satisfied only if h
= ␶1 = g2 = 0. If only g2 = 0 holds, it defines RFT with added
linear and quadratic symmetry breaking terms. In the following, two facts will play an important role: 共i兲 RFT is equivalent to ordinary DP and 共ii兲 the added symmetry breaking
g0 ⬃ ␮共6−d兲/2 ,
共3.1b兲
g2 ⬃ ␮−共d+2兲/2 .
共3.1c兲
冕
再冋
ddxdt ˜␸ ⳵t␸ + ␭
册 冎
␦HYL
− ␭˜␸2 ,
␦␸
共3.2兲
which is known to describe the relaxational dynamics at the
Yang-Lee singularity edge 关31兴. Here,
HYL =
共2.38兲
共3.1a兲
We see that the upper critical dimension dc is 6 and that g
ª g0 is the only relevant coupling constant below dc. Neglecting the irrelevant couplings, we obtain the response
functional
␶ˆ 1 2
˜␸
2
g0 2 g1 2
g
˜␸␸ + ˜␸ ␸ − 2 ˜␸3 + iĥ˜␸ .
2
2
6
˜␸ ⬃ ␮共d+2兲/2 ,
冕
d dx
再
1
g
␶
共ⵜ␸兲2 + ␸2 + i ␸3 + iĥ␸
2
6
2
冎
共3.3兲
is the Hamiltonian of an Ising-order parameter in an imaginary field near criticality 关32兴 with ␶ = ␶ˆ 0. It was shown
关33,34兴 that the entropic exponent of the probability P共N兲
˙
⬃ N−␪ exp共−N / N0兲 and the exponent of the transversal gyration radius R⬜ ⬃ N␯˙ A,⬜ are determined 共as conjectured by Day
and Lubensky 关35兴兲 by
041809-7
␪˙ = 1 + ␴YL ,
共3.4兲
PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
TABLE I. Values for the critical exponents ␪˙ and ␯˙ A,储 for various
transversal dimensions d.
d:
0
1
2
3
4
5
6
␪˙ :
␯˙ A,储:
0
1
1/2
0.79
5/6
0.69
1.08
0.63
1.26
0.58
1.40
0.54
3/2
1/2
renormalization theory see, e.g., Refs. 关40,41兴.
In the following, we use a ring ˚ to mark bare 共unrenormalized兲 quantities, i.e., we let ␸ → ␸˚ and so on. From here
on, quantities without a ring are understood as renormalized
quantities. To get rid of ␧ poles, we use the renormalization
scheme
␸˚ = Z1/2共␸ + K˜␸兲,
␯˙ A,⬜ = ␪˙ /d,
共3.5兲
␭˚ = Z−1Z␭␭,
where ␴YL is the Yang-Lee exponent which relates the order
parameter and the imaginary magnetic field via 共⌽ − ⌽c兲
⬃ 兩ĥ − ĥc兩␴YL. The 共␧ = 6 − d兲 expansion of the longitudinal
Flory exponent is 关31,33兴
␯˙ A,储 =
冋 冉
冊册
1
95 9 4 ␧ ␧
+ 1+
− ln
+ O共␧3兲.
2
108 2 3 6 24
IV. FIELD THEORY OF COLLAPSING DIRECTED
BRANCHED POLYMERS
␸ ⬃ ␮d/2,
␶ˆ 0 ⬃ ␶ˆ 1 ⬃ ␮2,
˜␸ ⬃ ␮d/2 ,
共4.1兲
ĥ ⬃ ␮共4+d兲/2 ,
共4.2兲
g0 ⬃ g1 ⬃ g2 ⬃ ␮共4−d兲/2 .
共4.3兲
共4.4b兲
兲
共4.4c兲
␧/2
g̊␣ = Z−1/2Z␭−1G−1/2
␧ ␮ 共u␣ + B␣兲,
共4.4d兲
where ␧ = 4 − d and G␧ = ⌫共1 + ␧ / 2兲 / 共4␲兲d/2. Here, we have
introduced the two-dimensional vector ␶គ̂ = 共␶ˆ 0 , ␶ˆ 1兲. Note that
the mixing-term proportional to K reintroduces the aforementioned gradient term 关see the discussion below Eq.
共2.29兲兴. In minimal renormalization with dimensional expan˚
sion the additive contributions ␶˚គ̂ c, ĥc, and Cគ̊ become formally
zero and the ␧ contents of the other counterterms are defined
by pure Laurent series,
⬁
Z.. − 1 ¬ Y .. = 兺
k=1
Our response functional J 关Eq. 共2.38兲兴 describes specifically the CDBP transition when both control parameters ␶ˆ 0
and ␶ˆ 1 become critical. In this case, they scale in terms of the
inverse length scale ␮ as ␶ˆ 0 ⬃ ␶ˆ 1 ⬃ ␮2. Then the engineering
dimensions of the fields and the remaining parameters follow
as
␶˚គ̂ = Z␭−1Z · ␶គ̂ + ␶˚គ̂ c ,
共
Now, we return to the CDBP transition as the main topic
of our paper.
A. Renormalization
共4.4a兲
˚
−␧/2
␶គ̂ · A · ␶គ̂ + ĥ˚c + Cគ̊ · ␶គ̂ ,
ĥ = Z1/2Z␭−1 ĥ + 21 G1/2
␧ ␮
共3.6兲
Table I compiles values for ␪˙ and ␯˙ A,储 based on exact results
for d = 0 , 1 and d = 2 关36兴 and the third order ␧-expansion
results of de Alcantara Bonfim et al. 关37兴. For corrections to
scaling and the derivation of the correction exponent in ␧
expansion see 关38,39兴.
˜␸˚ = Z1/2˜␸ ,
⬁
⬁
Y 共k兲
..
= 兺 ␧−k 兺 Y 共l,k兲
.. ,
␧k k=1
l=k
共4.5兲
where the counterterms Y 共l,k兲
.. , when determined at the looporder l, are homogeneous polynomials of the renormalized
dimensionless coupling constants u␣ of order 2l: Y 共l,k兲
.. 共su␣兲
共u
兲.
Corresponding
expansions
are
valid
for the
= s2lY 共l,k兲
␣
..
various other counterterms K, A, and B␣ which themselves
are homogeneous polynomials of orders 2l, 2l − 1, and 2l
+ 1, respectively, at each loop-order l. We calculate the various renormalization factors Z.. and additional counterterms
using a two-tiered approach, viz., we carry out an explicit
one-loop calculation and we employ Ward identities. The
latter will be discussed further below. Our one-loop calculation 关see Appendix A for details兴 produces
Now, the upper critical dimension is dc = 4. All three coupling
constants are relevant below dc. There are no further contributions that are relevant under the condition of causality
共originating from the absorbing condition of the process兲
which we have restored through the duality transformation
关Eq. 共2.37兲兴.
It is well known that a perturbation expansion of correlation and response functions 共generally called Green’s functions兲 based on a field theoretic model such as J produces
UV singularities which have to be regularized and renormalized by singular counterterms. Here, we use minimal renormalization, i.e., dimensional regularization followed by
minimal subtraction of ␧ poles, to cure the theory from UV
divergencies. For the general principles and methods of
041809-8
Z=1+
u 0u 1
+ ¯,
4␧
Z=1+
冉
冉
冉
u 0u 1
+ ¯,
8␧
冊
冉 冊
u0 2u1 − 2u0
+ ¯,
4␧ 3u2 4u1
冊
冊
u0
+ ¯ u 2,
8␧
A=
u 0u 1
+ ¯ u 0,
␧
B2 =
K= −
B0 =
Z␭ = 1 +
共4.6a兲
共4.6b兲
u0 0 1
+ ¯ , 共4.6c兲
2␧ 1 0
冉
冊
21u1
+ ¯ u 0u 2 ,
8␧
共4.6d兲
PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
B1 =
冉
冊
4u21 − 3u2u0
+ ¯ u0 ,
4␧
where ellipses symbolize terms of higher-loop order which
are regular expansions in the coupling constants u␣ but, of
course, singular in ␧. Note that in the case u2 = 0, where
rapidity inversion symmetry holds, K and B2 are zero to all
orders. In this case we have also B0 / u0 = B1 / u1. In the general case all renormalizations are trivial if u0 = 0 because then
only diagrams without loops are generated by the perturbation expansion.
B. Renormalization group
Next, we derive a renormalization group 共RG兲 in the usual
way by utilizing that the bare theory is independent of the
external arbitrary inverse length scale ␮ which enters
through the renormalization scheme. Because of this independence, the derivative ␮ ⳵ / ⳵␮Q̊ 兩0 共兩0 denotes derivatives
holding bare parameters fixed兲 vanishes for each unrenormalized quantity Q̊. Switching from bare to renormalized
quantities Q depending only on renormalized parameters, the
derivative ␮ ⳵ / ⳵␮ 兩0 changes to the RG differential operator,
D␮ = ␮⳵␮ + ␨␭⳵␭ + ␶គ̂ · k · ⳵␶គ̂ + 兺 ␤␣⳵u␣ + 关共␥/2 − ␨兲ĥ
␣
+
−␧/2
共␶គ̂
G1/2
␧ ␮
· a · ␶គ̂ 兲/2兴⳵hˆ ,
共4.7兲
which defines the infinitesimal generator of the RG, the
共half-兲group of scale changes of the length ␮−1. The application of D␮ to renormalized fields in averages produces the
normalization group equations 共RGEs兲
␥
D␮˜␸ = − ˜␸,
2
冉
␤ ␣ = ␮ ⳵ ␮u ␣兩 0 = −
共4.6e兲
␥
␥⬘
D␮␸ = − ␸ − ˜␸ .
2
2
共4.8兲
Recalling the renormalization scheme 关Eqs. 共4.4兲兴, the form
of the renormalization factors Z.., and the additional counterterms discussed below it, we find
⬁
for the RG functions. Explicit one-loop results for these RG
functions follow readily from Eqs. 共4.6兲.
C. Ward identities, redundancy, and invariant renormalization
group equations
It has been noted since the early days of field theory that
each symmetry transformation 共form invariance兲 which
transforms not only fields but also parameters of a field theoretic functional implies the redundancy of one of its parameters and vice versa 关42兴. Redundancy means that a respective parameter is related to other parameters of the functional
through simple linear transformations of the fields. These
transformations, however, must not change the physical contents of the functional. Hence, in a proper field theory, redundant parameters can be and must be eliminated through
linear transformations of the fields.
As we have already noted, the response functional J 关Eq.
共2.38兲兴 is form invariant under the three continuous transformations 关Eq. 共2.38兲兴. The mixing and shift transformations
关Eq. 共2.29兲兴 were used to eliminate two of the corresponding
redundant parameters, r and c, and they introduced ␶ = ␶ˆ 0 as a
small free parameter. Now, we use this freedom to derive
Ward identities from the shift transformation 关Eq. 共2.28c兲兴,
which reads
␸ → ␸⬘ = ␸ + i␥,
J关˜␸, ␸ ; ␶គ̂ ,g␣,h兴 = J关˜␸⬘, ␸⬘ ; ␶គ̂ ⬘,g␣,h⬘兴,
⬁
共4.9c兲
⬁
1
b␤ = 关共u · ⳵u兲 − 1兴B␣共1兲 = 兺 lB␣共l,1兲 ,
2
l=1
共4.9d兲
where 共u · ⳵u兲 = 兺␣ u␣⳵u␣, as well as
␨ = ␮⳵␮ ln ␭兩0 = ␥ − ␥␭ ,
共4.10a兲
␶គ̂ · k = ␮⳵␮␶គ̂ 兩0 = 共␥␭ − ␥兲 · ␶គ̂ ,
共4.10b兲
共4.12兲
if we also change the control parameters as follows:
ĥ → ĥ⬘ = ĥ − ␥␶ˆ 0 −
⬁
共4.9b兲
共4.11兲
when expressed in terms of the fields ␸ and ˜␸. The response
functional J 关Eq. 共2.38兲兴 is invariant under this shift,
共4.9a兲
1
a = 关共u · ⳵u兲 + 1兴A共1兲 = 兺 lA共l,1兲 ,
2
l=1
˜␸ → ˜␸⬘ = ˜␸
␶គ̂ → ␶គ̂ ⬘ = ␶គ̂ + ␥ḡគ ,
l=1
共4.10c兲
−␧/2
␮⳵␮ĥ兩0 = 共␥␭ − ␥/2兲ĥ + G1/2
共␶គ̂ · a · ␶គ̂ 兲/2 共4.10d兲
␧ ␮
1
共l,1兲
␥.. = ␮⳵␮ ln Z..兩0 = − 共u · ⳵u兲Y 共1兲
.. = − 兺 lY .. ,
2
l=1
␥⬘ = − 共u · ⳵u兲K共1兲 = − 2 兺 lK共l,1兲 ,
冊
␧ ␥
+ + ␥␭ u␣ + b␣ ,
2 2
␥2
g0 .
2
共4.13a兲
共4.13b兲
Here, we have introduced the two-dimensional vector ḡគ
= 共g0 , −g1兲. Primarily, the shift is defined for unrenormalized
quantities. However, because it does not involve a transformation of the coupling constants and the renormalization
constants are only functions of the couplings, the shift invariance 关Eq. 共4.12兲兴 also holds for the renormalized quantities
␧/2
with g␣ ª G−1/2
␧ u ␣␮ .
Now, we compare the bare transformations with their
renormalized counterpart. We renormalize the bare form of
Eq. 共4.13a兲, ␶˚គ̂ ⬘ − ␶˚គ̂ = ␥˚ gគ̊ with ␥˚ = Z1/2␥,
␧/2
Z · 共␶គ̂ ⬘ − ␶គ̂ 兲 = ␥G−1/2
គ + B̄គ 兲,
␧ ␮ 共ū
共4.14兲
where ūគ = 共u0, − u1兲 and B̄គ = 共B0, − B1兲. Splitting this equation
in singular and nonsingular parts, we get
041809-9
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JANSSEN, WEVELSIEP, AND STENULL
␧/2
␶គ̂ ⬘ = ␶គ̂ + ␥G−1/2
គ,
␧ ␮ ū
共4.15a兲
B̄គ = Y · ūគ ,
共4.15b兲
with Y = Z − 1. The last equation, the first Ward identity, is
easily checked at one-loop order by using Eqs. 共4.6兲. We
perform the same procedure with Eq. 共4.13b兲 and obtain
ĥ⬘ = ĥ − ␥␶ˆ 0 −
␥2 −1/2 ␧/2
G ␮ u0 ,
2 ␧
共4.16a兲
with Yគ = 共Y 0,0 , Y 0,1兲. Also the second Ward identity 关Eq.
共4.16b兲兴 is easily checked at one-loop order by using Eqs.
共4.6兲.
The Ward identities 关Eqs. 共4.15b兲 and 共4.16b兲兴 imply that
there are further relations between the RG functions, viz.,
␥គ = ūគ · a,
共4.17兲
where ␥គ = 共␥00 , ␥01兲. These relations can be used interalia to
check the consistency of the one-loop results for the various
RG functions and the results 关Eqs. 共4.6兲兴 leading to them.
Our results fulfill this consistency check. More generally we
derive relations between different vertex functions. Note that
the vertex function −⌫˜k,k is diagrammatically defined as the
sum of all one-line irreducible diagrams with k̃ amputated
external ˜␸ line and k amputated external ␸ line 关41兴. Let
⌫关˜␸ , ␸ ; ␶គ̂ , g␣ , ĥ兴 be the generating functional of the vertex
functions, which we call the dynamic free-energy functional.
It has the same invariance property 关Eq. 共4.12兲兴 as the stochastic functional J, which is identical to the mf approximation of ⌫. Hence, we have
⌫关˜␸, ␸ ; ␶គ̂ ,g␣,ĥ兴 = ⌫关˜␸, ␸ ; ␶គ̂ ,g␣,0兴 + ␭ĥ
冕
␶ˆ 0 → ␶ˆ 0,
冉
冊
冉
冊
冊
⳵
⳵
− g1
⌫1,1共0兲,
⌫1,2共0兲 = i g0
⳵ ␶ˆ 0
⳵ ␶ˆ 1
共4.21兲
共4.22a兲
w = u30u2 ,
˜ = g−1˜␸,
␾
0
␶0 = ␶ˆ 0,
␾ = g 0␸ ,
␶1 = u20␶ˆ 1,
共4.22b兲
h = g0ĥ.
共4.22c兲
The RG differential operator D␮ applied only to scaling
invariant quantities reduces to
D␮ = ␮⳵␮ + ␨␭⳵␭ + ␶គ · ␬ · ⳵␶គ + ␤u⳵u + ␤w⳵w
共4.23兲
and its application to the invariant fields in averages produces
␥0
␥2 ˜
␾− ␾
,
2
2
˜ = − ␥1 ␾
˜.
D ␮␾
2
共4.24兲
Here, we have defined the invariant RG functions
␥ 0 = ␥ − ␧ − ␨ 0,
which is sometimes called equation of motion. This general
property of the dynamic free-energy energy functional holds
equally well in renormalized and unrenormalized forms and
is very helpful for higher-loop calculations. Especially, we
find by taking repeated functional derivatives
⳵
⳵
− g1
⌫1,0共0兲,
⳵ ␶ˆ 0
⳵ ␶ˆ 1
g 2 → ␣ 3g 2 .
and changing the fields and control parameters to their scaling invariant counterparts,
D ␮␾ = −
共4.19兲
冉
ĥ → ␣ĥ,
+ 关共␥1/2 − ␨兲h + 共␶គ · ␣ · ␶គ 兲兴⳵h ,
⳵
⳵
⳵
␦⌫
= i g0
− g1
− ␶ˆ 0
⌫,
␦␸共x,t兲
⳵ ␶ˆ 0
⳵ ␶ˆ 1
⳵ ĥ
⌫1,1共0兲 = ␶ˆ 0 + i g0
g 1 → ␣ g 1,
u = u 0u 1,
共4.18兲
ddxdt
共4.20c兲
This invariance holds likewise for the renormalized and unrenormalized theories. Hence, it would be absolutely wrong
to search at this place for fixed points, i.e., zeros of all three
Gell-Mann–Low functions, ␤␣ = 0, because the free scale
transformation poisons the renormalization flow. Before we
can search for fixed points, we first must eliminate the scaling redundancy through the definition of scale-invariant
quantities. We do that by defining the invariant dimensionless coupling constants,
d xdt˜␸共x,t兲
= ⌫关˜␸, ␸ + i␥ ; ␶គ̂ + ␥ḡគ ,g␣,ĥ − ␥␶ˆ 0 − ␥2g0/2兴.
冕
␶ˆ 1 → ␣2␶ˆ 1,
g0 → ␣−1g0,
d
Setting ␥ = 0 after differentiation leads to
冊
⳵
⳵
− g1
⌫2,0共0兲,
⳵ ␶ˆ 0
⳵ ␶ˆ 1
where the argument 0 means that the wave vectors and frequencies of the vertex functions are set to zero.
Up to this point we have not used the scaling transformation 关Eq. 共2.28a兲兴 for eliminating a further redundancy. Applying this rescaling of the fields, J remains invariant if we
change its parameters to
共4.16b兲
Y = − ūគ · A,
␤គ = 21 共− ␧ + ␥兲uគ + ūគ · k,
冉
⌫2,1共0兲 = i g0
␥ 1 = ␥ + ␧ + ␨ 0,
␥2 = u20␥⬘ , 共4.25兲
where
␨0 =
2␤0
.
u0
共4.26兲
The invariant matrices ␬ and ␣ follow from their respective
counterparts k and a as
共4.20a兲
␬00 = k00,
␬01 = u20k10,
共4.20b兲
041809-10
␣00 = u0a00,
␬10 = u−2
0 k10 ,
␬11 = k11 + ␨0 ,
␣10 = ␣01 = u−1
0 a10 ,
共4.27a兲
PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
␣11 = u−3
0 a10 .
From the above, we can straightforwardly collect the following explicit one loop for the invariant RG functions:
7u
+ ¯,
4
␥0 = −
␥2 =
␥1 =
冉 冊
and
␬=
5u
+ ¯,
4
共4.28a兲
u
+ ¯,
8
共4.28b兲
1
+ ¯ w,
4
␨=−
␨0 = − ␧ +
3u
+ ¯,
2
冉
3u/8
3w/4
− 1/2 19u/8 − ␧
␣=
冉
0
1/2
1/2
0
冊
冊
+ ¯,
+ ¯.
共4.28c兲
H = 2h − 2␶␴ − ␴2 ,
共4.34b兲
y = 共␶គ · vគ 兲 = ␶ + ␴ ,
共4.34c兲
where ␴ = ␶1 / u = g0␶ˆ 1 / g1. Note that H is linearly related to
the Laplace-variable z.
In the following we are mainly interested in the average
具S典 = i共g0⌽ + ␴兲 ¬ −iM since this quantity is linearly related
to ⌽ = 具ñ典DP. We recall that the critical part of ⌽ yields the
asymptotic behavior of the probability P共N兲 by an inverse
Laplace transformation. Applying Eq. 共4.24兲 together with
˜ 典 = 0 and using the relations 关Eqs. 共4.31兲 and 共4.32兲兴 be具␾
tween the RG functions, we obtain the RGEs,
共4.29b兲
␤u = 共− ␧ + 23 u兲u − 43 w + ¯ ,
共4.30a兲
␤w = 共− 2␧ + 378 u + ¯兲w.
共4.30b兲
D␮y = ␬1y,
共4.35a兲
D␮M = ␬0M − u−1␬01y,
共4.35b兲
D␮H = 共␬0 + ␥␭兲H + 共␣00 − 2u−1␬01兲y 2 .
共4.35c兲
D. Fixed points and critical exponents
Next, we search for the stable fixed points of the invariant
RGE 关Eq. 共4.24兲兴 with the RG generator as given in Eq.
共4.23兲. The fixed point equations ␤uⴱ = ␤wⴱ = 0 possess the
following solutions: 共i兲 the trivial fixed point uⴱ = wⴱ = 0
which is unstable below four transversal dimensions, 共ii兲 a
fully stable fixed point
Once again, the Ward identities lead to relations between the
RG functions, namely,
␬10 = − 共v̄គ · ␣兲1
共4.31兲
and
␤u/u = − v̄គ · ␬ · vគ ,
共4.34a兲
共4.29a兲
In the DP case, w = ␶1 = 0, the RGE has to have the usual form
known from DP 关18,21兴. It follows that ␬01 and ␣00 have to
be zero to all loop orders for w = 0. For the invariant GellMann–Low functions we find
␬00 = 共␥ − ␨兲 − 共v̄គ · ␣兲0,
S = ␾ + i␴ ,
共4.27b兲
共v̄គ · ␬兲0 = − ␥0/2,
共4.32兲
where we have introduced the two orthogonal vectors vគ
= 共1 , u−1兲 and v̄គ = 共1 , −u兲. We define
uⴱ =
␬1 = 共␬ · vគ 兲0 = ␬00 + ␬01u−1 = ␬11 + ␬10u − ␤u/u.
共4.33b兲
Hence, if ␤u = 0, especially at a fixed point, it follows from
Eq. 共4.32兲 that v̄គ and vគ are a left and a right eigenvectors of
␬, respectively, since the two vectors are orthogonal. In this
case, we find the two eigenvalues ␬0 = −␥0 / 2 and ␬1. Of
course, our one-loop results for the invariant RG functions
satisfy these relations holding to all loop orders.
To determine the critical behavior, we need to extract
those quantities that are invariant under all the symmetry
transformations of the theory. Applying the shift transformation the dynamic free-energy functional 关Eq. 共4.18兲兴, such a
˜ and the combinations
set of quantities is given by ␾
wⴱ = 0,
共4.36兲
with stability 共Wegner兲 exponents ␭u = ␧ + ¯ and ␭w
= 13␧ / 12+ ¯ 共which correspond to the two leading
correction-to-scaling exponents兲, and 共iii兲 a fixed point with
one stable and one unstable directions,
uⴱ =
␬0 = 共v̄គ · ␬兲0 = ␬00 − u␬10 = ␬11 − u−1␬01 − ␤u/u,
共4.33a兲
2␧
+ ¯,
3
16␧
+ ¯,
37
wⴱ = −
冉 冊
13 8␧
3 37
2
+ ¯ =−
13 2
u + ¯,
12 ⴱ
共4.37兲
which lies on a separatrix between the region of attraction of
the fully stable fixed point and a region where the renormalization flow tends to infinity, signaling possibly a discontinuous collapse transition. However, this fixed point lies in the
unstable region and is thus not accessible by our model. We
conclude, therefore, that the renormalization group flow
reaches asymptotically the fully stable fixed point. The line
w = 0 in the two-dimensional space of the invariant coupling
constants has to be a fixed line of the renormalization flow
because this line corresponds to the RFT model, Eq. 共2.38兲
with g2 = 0, with its rapidity-reversal invariance. Thus, wⴱ
= 0 holds to all loop orders. We conclude that as long as the
collapse transition is continuous, it belongs to the universality class of directed percolation.
We would like to underscore that this result, which is the
main result of our paper, holds to arbitrary order in pertur-
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JANSSEN, WEVELSIEP, AND STENULL
bation theory. The pivotal point is that for w → 0, which is
the case at the stable fixed point, J reduces to RFT and thus,
the universal behavior governed by this fixed point is that of
the DP-universality class. The explicit one-loop calculation
was necessary to show the stability of this fixed point.
Higher orders in perturbation theory modify the two stability
exponents quantitatively and thus affect corrections to scaling. However, they do not change the stability of the fixed
point qualitatively since in ␧ expansion, ␧ is virtually an
infinitesimal quantity. This does not answer, as always for
the ␧ expansion, the question about a possible dimension d
⬍ dc at which the stability breaks down.
We conclude this section by collecting the critical exponents of the CDBP transition. The anomalous scaling dimensions of the invariant fields are defined by
␩ = ␥0ⴱ + ␧,
˜␩ = ␥1ⴱ − ␧.
共4.38兲
At the stable fixed point they coincide because ␨0
= 共2␤0 / u0兲ⴱ = 共␤u / u兲ⴱ = 0. Also ␥2ⴱ = 0 for wⴱ = 0. Thus, the
coinciding exponents are the same as for DP, ␩ = ˜␩. The two
eigenvalues ␬0ⴱ = −␥0ⴱ / 2 = 共␧ − ␩兲 / 2 and ␬1ⴱ define the order
parameter and the correlation length exponents of DP, respectively,
␤/␯ = 共2 − ␬0ⴱ兲 = 共d + ␩兲/2,
共4.39a兲
1/␯ = 共2 − ␬1ⴱ兲.
共4.39b兲
H ~ z−zc
y
IV
共4.39c兲
共4.39d兲
which is related to the scaling behavior of H.
Next, we determine the equation of state. To this end, we
introduce the dimensionless quantities M̂ = M / ␮2, Ĥ = H / ␮4,
and ŷ = y / ␮2 and seek M̂ as a function of Ĥ and ŷ,
共4.40兲
At the stable DP fixed point, this relation between the three
quantities must have scaling form. Using the critical DP exponents 关Eqs. 共4.39兲兴 and exploiting that ␬01 = ␣00 = 0 at the
fixed line w = 0 which contains the stable fixed point, we
obtain from Eqs. 共4.35兲 the fixed point flow,
1
d
ŷ共ᐉ兲 = − ŷ共ᐉ兲,
dᐉ
␯
共4.41a兲
d
␤
M̂共ᐉ兲 = − M̂共ᐉ兲,
dᐉ
␯
共4.41b兲
ᐉ
ᐉ
d
⌬
Ĥ共ᐉ兲 = − Ĥ共ᐉ兲,
dᐉ
␯
共4.41c兲
with the RG flow-parameter ᐉ defined by ␮共ᐉ兲 = ᐉ␮. The
scaling form of the equation of state follows now from the
solutions of the flow equations, and M̂共ᐉ兲 = F关Ĥ共ᐉ兲 , ŷ共ᐉ兲兴.
After elimination of the flow parameter ᐉ, the redefinitions
M̂ → M, etc., we get the equivalent scaling forms
M = 兩H兩␤/⌬ f̄ ⫾共y/兩H兩1/⌬兲 = y ␤ f共H/y ⌬兲,
E. Scaling form of the equation of state and correlation
lengths
M̂ = F共Ĥ,ŷ兲.
Hc(y)
FIG. 3. Phase diagram with 共y , H兲 as independent variables.
Regions I and III are the physical parameter regions for a DP process. Region III is excluded for the parameters of the statistics of
directed branched polymers. The axes marked “red” correspond to
some redundant variable such as g1 / g0, ␴, or c. The solid dot marks
the DP-transition point corresponding to the collapse of large DBP.
The line labeled Hc共y兲 ⬍ 0 with y ⬎ 0, where the order parameter
⌽共z兲 shows a branching point singularity, corresponds to large
swollen DBP. The shading indicates region IV, where the order
parameter ⌽共z兲 becomes a complex function. At the straight line
Hc共y兲 = 0 with y ⬍ 0 between regions III and IV, the order parameter
shows essential singularities. The lines become surfaces if a redundant variable is admitted.
ᐉ
As is customary, we furthermore introduce the exponent
⌬ = ␤ + 共z − ␩兲␯ ,
II
red.
␥␭ⴱ defines the dynamical exponent
z = 2 + ␨ⴱ = 2 + ␩ − ␥␭ⴱ .
I
III
共4.42兲
where the subscript ⫾ indicates the sign of H. Note that this
field theoretic result is fully consistent with the mean-field
equation of state as stated in Eqs. 共2.18兲.
Written in its scaling forms 关Eq. 共4.42兲兴, the equation of
state leads to the following observations 共see Fig. 3兲: the
region of the phase diagram corresponding to DP is given by
M ⱖ 0 and H ⱖ 0, where the primary control parameter y can
take both signs 关43兴. Note, however, that the stability condition 关Eq. 共2.36兲兴 restricts the variable y in our polymer problem to y ⱖ 0 and y = 0 at the collapse transition, coinciding
with the DP-transition point. The primary control parameter
here is the external field H ⬃ z − zc. The DP-transition point
becomes now a tricritical point given by Hc = 0 and y c = 0.
The control parameter y constitutes here the crossover variable which drives the critical behavior from the DPuniversality class to the critical point of the dynamical YangLee model that we have discussed in Sec. III. Hence, for y
⬎ 0 we expect a critical line H = Hc共y兲 ⬍ 0 with M ⬍ 0 in a
region of the phase diagram not accessible for DP itself.
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To discuss the crossover in some more detail, we note that
in mean-field theory 共where ␤ = 1 and ⌬ = 2兲, the scaling
functions defined by Eq. 共4.42兲 are simply given by
f共x兲 = 冑x + 1 − 1,
共4.43a兲
f̄ ⫾共x̄兲 = 冑x̄2 ⫾ 1 − x̄.
共4.43b兲
These equations show the tricritical point at x = 0 and x̄ = ⬁.
The critical line is described by the branch-point singularity
at x = −1 or x̄ = 1, respectively. However, mean-field theory
for the crossover is applicable only for d ⬎ 6, where both
types of polymers, collapsing and swollen, have mean-field
behavior.
To determine the general form of the crossover, we assume as usual that f共x兲 has in either case a singularity at
same negative value xⴱ of x 关44兴. To be specific, we assume
that
˙
f共x兲 = f 0共x兲 + 共x − xⴱ兲␪−1 f 1共x兲.
共4.44兲
f 0共x兲 is the analytic part of f共x兲 at x = xⴱ, in particular,
f 0共xⴱ兲 = −A ⬍ 0. f 1共x兲 may have further singularities at x = xⴱ
but it is assumed that f 1共xⴱ兲 = B exists. ␪˙ denotes the entropic
exponent of the swollen polymers 共cf. Sec. III兲. With these
assumptions, we find in the vicinity of the singular curve
H = Hc共y兲 = xⴱy ⌬ that
˙
functions with the subscript ⫾ denoting again the sign of H.
This scaling form implies the following scaling forms for the
correlation lengths in the transversal and longitudinal directions:
˙
M ⬇ − Ay ␤ + By 共␪−␪兲⌬共H − xⴱy ⌬兲␪−1 ,
共4.45兲
␪ = 1 + ␤/⌬
共4.46兲
where
is the entropic exponent of the collapsing polymers. To conclude our discussion of the crossover, we emphasize that the
problem involves two distinct kinds of nontrivial critical behavior with different upper critical dimensions. Thus, one
cannot simply extract via analytical continuation
␧-expansion results for f共x兲 from the known results for the
DP equation of state 关43兴. A glance at the two-loop part of
the latter shows, indeed, nonsummable singularities at
x = −1.
Now, we turn to the scaling form of the correlation
lengths, i.e., the diverging length scales in transversal and
longitudinal directions. To this end, we consider the scaling
behavior of the correlation functions Gk,k˜ of k fields ␾ and k̃
˜ . The RGE for these correlation functions follows
fields ␾
readily from the general RGE 关see Eqs. 共4.23兲 and 共4.24兲兴. At
the stable DP fixed point with ␥0ⴱ = ␥1ⴱ = ␥ⴱ = ␩, ␥2ⴱ = 0, this
RGE reads
␰⬜ = 兩z − zc兩−␯/⌬ f̂ ⬜共y/兩z − zc兩1/⌬兲,
共4.49a兲
␰储 = 兩z − zc兩−z␯/⌬ f̂ 储共y/兩z − zc兩1/⌬兲,
共4.49b兲
with scaling functions f̂ ⬜ and f̂ 储 with a crossover behavior
analogous to that of the equation of state.
V. ASYMPTOTIC PROPERTIES OF COLLAPSING
DIRECTED BRANCHED POLYMERS
Our field theory of Sec. IV provided us with a number of
results, in particular the scaling form of the equation of state
and the correlation lengths. Though certainly interesting in
their own right as they stand, they are not yet in a form or
language that is customary in polymer theory. Here, we
translate our results into results for the animal numbers and
the radii of gyration.
As discussed in Sec. II, the number of different polymer
configurations A共N兲 for large N is up to a nonuniversal exponential factor ␮N0 proportional to P共N兲, and the relation of
the latter to ⌽共z兲 via inverse Laplace led asymptotically to
Eq. 共2.19兲. Now we recall that ⌽共z兲 is linearly related to M
and revisit our result 关Eq. 共4.42兲兴 for the equation of state
which we rewrite as
⌽共z兲 − ⌽共zc兲 ⬃ 兩z − zc兩␤/⌬ f̂共y/兩z − zc兩1/⌬兲.
共5.1兲
Inserting this into the inverse Laplace transformation 关Eq.
共2.19兲兴, we obtain for N Ⰷ 1 the following scaling form for
the animal numbers:
A共N,y兲 ⬃ N−␪ f共yN␾兲␮N0
共5.2兲
with the animal exponent ␪, which is the same as the entropic exponent defined in Eq. 共4.46兲, and the crossover exponent
␾ = 1/⌬.
共5.3兲
Via solving RGE 关Eq. 共4.47兲兴, we obtain the scaling forms
We see here at the instance of the animal numbers that the
transcription of the field theoretic scaling result to the corresponding scaling result in the polymer language asymptotically for N Ⰷ 1 via inverse Laplace transformation amounts
to simply replacing 兩z − zc兩 by N−1 关see also the second part of
Eq. 共2.19兲兴. In this way, after the inverse Laplace transformation of the correlation functions and space and time coordinates, which originally scale with the correlation lengths,
now scale with the appropriate powers of N. Consequently,
we obtain for the longitudinal and the transversal radii of
gyration as counterparts of the correlation lengths,
Gk,k˜共兵x,t其,y,H兲 = 兩H兩共k+k̃兲␤/⌬ f̄ ⫾;k,k̃共兵r/␰⬜,t/␰储其,y/兩H兩1/⌬兲
R储共N,y兲 ⬃ N␯A,储 f 储共yN␾兲,
共5.4a兲
R⬜共N,y兲 ⬃ N␯A,⬜ f ⬜共yN␾兲,
共5.4b兲
冋
D␮ +
␩
2
册
共k + k̃兲 Gk,k˜共兵x,t其,y,H兲 = 0.
= y 共k+k̃兲␤ f k,k˜共兵r/␰⬜,t/␰储其,H/y ⌬兲
共4.47兲
共4.48兲
of the correlation functions, where f ⫾;k,k̃ and f̄ k,k˜ are scaling
with critical exponents
041809-13
PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
t
TABLE II. Values of the CDBP exponents for various transversal dimensions d as produced by our field theory 共␧-exp.兲 and numerical simulations 关27,45兴.
d:
1
2
3
4
␪:
共␧-exp.兲
1.10825
1.074
1.27
1.255
1.40
1.397
1.5
3/2
␾:
共␧-exp.兲
0.39151
0.375
0.459
0.444
0.49
0.486
0.5
1/2
␯储:
共␧-exp.兲
0.69007
0.601
0.584
0.573
0.54
0.539
0.5
1/2
␯ ⬜:
共␧-exp.兲
0.451494
0.376
0.337
0.327
0.285
0.285
0.25
1/4
共5.5a兲
␯A,⬜ = ␯/⌬,
共5.5b兲
where z is once again the dynamical exponent of DP which
should not de confused with the variable z introduced
through the Laplace transformation.
It remains to state the ␧ expansions results for the polymer
exponents encountered in this section. Using the relations of
these exponents to the DP exponents, which are known to
second order in ␧ 关18,20,21兴, we obtain
冊 册
3 ␧
53
37
␪= −
+
ln共4/3兲 ␧ + ¯ , 共5.6a兲
1+
2 12
288 144
冋 冉
= −i λ g 0
1
1
共5.6b兲
冊 册
55
1 ␧
13
+
−
ln共4/3兲 ␧ + ¯ ,
1+
2 24
288 144
冋 冉
q
0 = C(q,t )
= −i λ g 1
= iλ g2
theory. As far as mean-field theory was concerned, our model
reproduced a number of well known results and, therefore,
passed important consistency checks. The same holds true
for the field theory of swollen directed branched polymers
which we extracted from our general model. Our main focus
laid on the collapse transition of directed branched polymers
for which, to our knowledge, no field theory existed hitherto.
We showed to arbitrary order in perturbation theory that this
transition belongs to the DP-universality class. Because we
used ␧ expansion, we can be sure about the stability of the
DP fixed point only for those dimensions for which the ␧
expansion is valid. Given that it is known from earlier work
that the collapse transition of directed branched polymers
belongs to the DP-universality class in 1 + 1 dimensions and
that our present work shows that this is also the case in the
vicinity of the upper critical dimension 4 + 1, we are led to
expect that this holds true in any dimension. We calculated
the scaling behavior of several quantities of potential experimental relevance such as gyration radii and the probability
distribution for finding a polymer consisting of N monomers.
ACKNOWLEDGMENT
This work was supported in part 共O.S.兲 by the National
Science Foundation under Grant No. DMR 0804900.
In this appendix we provide some details of our one-loop
calculation. The diagrammatic elements entering this calculation are the Gaussian propagator
G共q,t兲 = ⌰共t兲exp关− ␭共q2 + ␶ˆ 0兲t兴,
共5.6c兲
␯A,⬜ =
t
APPENDIX A: ONE-LOOP CALCULATION
1 ␧2
␾ = − 关1 + ¯兴,
2 18
␯A,储 =
0 = G(q,t)
FIG. 4. Diagrammatic elements.
␯A,储 = z␯/⌬,
冋 冉
q
冊 册
1 ␧
17
43
1+
+
−
ln共4/3兲 ␧ + ¯ .
4 32
288 144
共A1兲
the correlator
共5.6d兲
In Table II, we compare these ␧ expansions to results of
numerical simulations as listed, e.g., in the review article of
Hinrichsen 关45兴 or the recent book of Henkel et al. 关27兴,
which, of course, are obtained by simulations of the DP transition and not for the polymer problem itself. Down to d
= 2, the agreement between field theory and simulations is
remarkably good.
VI. CONCLUDING REMARKS
In summary, we have developed a model for directed
branched polymers in the framework of dynamical field
C共q,t兲 =
␶ˆ 1/2
exp关− ␭共q2 + ␶ˆ 0兲兩t兩兴,
q2 + ␶ˆ 0
共A2兲
and the three-leg vertices −i␭g0, −i␭g1, and i␭g2 共see Fig. 4兲.
As usual, we concentrate on those vertex functions that are
superficially divergent in the ultraviolet. These are ⌫1,1, ⌫2,0,
⌫1,2, ⌫2,1, ⌫3,0, and ⌫1,0. The diagrammatic contributions to
these vertex functions to one-loop order are collected in Fig.
5.
To give an example for the calculation of the various oneloop contributions, let us consider the diagram 关Fig. 3共c兲兴.
The mathematical expression for this diagram is
041809-14
PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
1 1
1 1
(a)
1 1
(b)
1
⌫˚ 2,0 = − ␭␶ˆ 1 − 3共c兲 − 3共d兲
1 1
(c)
1
1
再
1
1
(e)
= − ␭␶ˆ 1 + ␭␶ˆ −␧/2
0
(d)
⫻ g1␶ˆ 1 + g2 ␶ˆ 0 +
1
(f)
(g)
FIG. 5. One-loop contributions to the superficially divergent
vertex functions.
⫻
1
共2␲兲d
冕
冊冎
,
⬁
共A6d兲
3G␧
g 0g 1g 2 ,
⌫˚ 3,0 = − i␭g2 − 3共h兲 = − i␭g2 + i␭␶ˆ −␧/2
0
␧
dt exp关− 共i␻ + 2␭␶ˆ 0兲t兴
0
共A6e兲
G␧
g0␶ˆ 0␶ˆ 1 .
⌫˚ 1,0 = i␭ĥ − 3共i兲 = i␭ĥ − i␭␶ˆ −␧/2
0
2␧
共A3兲
where the last term indicates that this diagram can also be
drawn with the arrows on the propagators pointing to the
right which corresponds to the first term with ␻ replaced by
−␻. The momentum and time integrations are straightforward. They lead to
冋
q
␭g0g2 ⌫共␧/2 − 1兲 i␻
+ ␶ˆ 0 +
3共c兲 = −
4
4
共4␲兲d/2 2␭
册
⌫1,0 = Z1/2⌫˚ 1,0,
+ 共␻ → − ␻兲.
冋 册
G␧
q2
␶ˆ 0 +
,
4
␧
共A5兲
where we have omitted finite contributions that are inconsequential for our quest. Note that the contributions proportional to ␻ cancel as it should be the case. Note also that this
diagram contributes a term proportional to q2 to ⌫2,0. A similar situation occurs for the collapse transition of isotropic
randomly branched polymers. This and the renormalization
mandated by it were overlooked in previous work 关12,13兴
and hence led to incorrect results.
The remaining diagrams in Fig. 5 can be calculated by
similar means. To save space, we merely list here the soobtained results for the as yet unrenormalized vertex functions,
⌫˚ 1,1 = i␻ + ␭共␶ˆ 0 + q 兲 − 3共a兲 − 3共b兲
2
= i␻ + ␭共␶ˆ 0 + q2兲 + ␭␶ˆ −␧/2
0
冉
⫻ g0␶ˆ 1 − g1 ␶ˆ 0 +
G␧
g0
2␧
i␻ q2
+
2␭ 4
冊冎
,
共A6a兲
⌫1,1 = Z⌫˚ 1,1 ,
⌫2,0 = Z共⌫˚ 2,0 + 2K⌫˚ 1,1兲,
共A4兲
3共c兲 = − ␭g0g2
共A6f兲
It can easily be checked that these results satisfy the Ward
identities discussed in Sec. IV C. Application of the renormalization scheme 关Eqs. 共4.4兲兴 leads to renormalized vertex
functions,
2 1−␧/2
␧ expansion finally produces
共A6c兲
G␧
g0兵g0g2 − g21其,
⌫˚ 2,1 = i␭g1 − 3共f兲 − 3共g兲 = i␭g1 + i␭␶ˆ −␧/2
0
␧
ddk exp兵− ␭关k2 + 共q + k兲2兴t其 + 共␻ → − ␻兲,
再
共A6b兲
G␧ 2
⌫˚ 1,2 = i␭g0 − 3共e兲 = i␭g0 − i␭␶ˆ −␧/2
g g1 ,
0
␧ 0
(i)
冕
q2
4
(h)
1 1
1
3共c兲 = 共− i␭g0兲共i␭g2兲
2
冉
G␧
g0
␧
⌫1,2 = Z3/2⌫˚ 1,2 ,
共A7a兲
共A7b兲
⌫2,1 = Z3/2共⌫˚ 2,1 + 2K⌫˚ 1,2兲,
共A7c兲
⌫3,0 = Z3/2共⌫˚ 3,0 + 3K⌫˚ 2,1 + 3K2⌫˚ 1,2兲,
共A7d兲
and then finally to the one-loop results for the renormalization factors and additive counterterms stated in Eqs. 共4.6兲.
APPENDIX B: ESSENTIAL SINGULARITIES
In this appendix, we briefly discuss essential singularities
appearing in the generating function ⌽共z兲 for y ⬍ 0 while
crossing the separation line between regions III and IV in the
phase diagram 共Fig. 3兲. These singularities produced by instantons are intimately related to the well known so-called
false-vacuum problem 关46兴. They lead to the behavior
P ⬃ N−␪⬘ exp关− b共兩y兩1/␾N兲␨兴
共B1兲
of the distribution function P with ␨ ⬍ 1 and a nonuniversal
amplitude b. Our discussion follows that of Harris and
Lubensky 关13兴 and that of Lubensky and McKane 关47兴 which
are based on Langers classical paper 关48兴 on the droplet
model.
We have shown in the main text that the universal critical
behavior of collapsing directed branched polymers is equivalent to that of DP. Being interested in essential singularities,
041809-15
PHYSICAL REVIEW E 80, 041809 共2009兲
JANSSEN, WEVELSIEP, AND STENULL
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1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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1
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1
1
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1
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1
1
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1
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10
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−r0
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1
1
1
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10
10
1
1
1
1
1
10
10
1
1
1
1
1
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
1
1
1
1
10
10
1
1
1
1
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
1
1
1
1
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
1
1
1
1
10
10
10
1
1
1
1
10
10
10
1
1
1
1
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10
1
1
1
1
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1
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1
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1
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1
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1
10
10
1
1
1
1
10
1
1
1
1
10
1
1
1
1
10
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
1
1
10 1 10
10
10
1 1010 1 1010
1 1100 1 1100
1 1100 1 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1100 10 1100
10 1010 10 1010
10 1010 10 1010
10 1010 10 1010
1
1
1
1
1
1
1
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1
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1
1
1
1
ϕ
r0 /v
1
1
1
1 110
10
1 1010
1 1100
1 1100
10 1100
10 1100
10 1100
10 1100
10 1010
10 1010
10 1010
10 1010
10 1100
10 1100
10 1100
10 1100
10 1100
10 1010
10 1010
10 1010
10 1010
10 1010
10 1100
10 1100
10 1100
10 1100
10 1100
10 1010
10 1010
10 1010
0
1
1
1
1 110
10
1 1010
1 1100
1 1100
10 1100
10 1100
10 1100
10 1100
10 1010
10 1010
10 1010
10 1010
10 1100
10 1100
10 1100
10 1100
10 1100
10 1010
10 1010
10 1010
10 1010
10 1010
10 1100
10 1100
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(1/3,−1/3)
01
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再冋
01
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(0,−1)
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r0
册 冎
g
ddxdt␭ s̃ ␭−1⳵t + ␶ − ⵜ2 + 共s − s̃兲 s − hs̃ .
2
共B2兲
We need to understand the behavior of functional integrals
with weight exp共−J关s , s̃兴兲 for ␶ = −兩␶兩 near the first-order line
h ⬇ 0. To this end, we seek for saddle points. If h ⬎ 0 near
zero, the true vacuum, i.e., the saddle point producing the
lowest value of J, is clearly the mean-field solution
共s = 2兩␶兩 / g , s̃ = 0兲 up to corrections proportional to h. However, if we cross the borderline at h = 0 and go to small h
⬍ 0, the mean-field solution becomes a false vacuum. At first
sight, the saddle-point solution 共s = 0 , s̃ = −2兩␶兩 / g兲 appears to
be the true vacuum. However, the response field has to satisfy the boundary condition s̃ → 0 when the space coordinate
or time tends to infinity. Thus, only nonstationary inhomogeneous solutions of the saddle-point equations, i.e., instantons
or droplets, are allowed 共see Fig. 6兲.
To search for droplet solutions, it is useful to switch to
dimensionless quantities. We introduce new fields
s = 2兩␶兩␸ / g, s̃ = 2兩␶兩˜␸ / g, h = 4␶2k / g and scale time and space
by ␭兩␶兩t = t⬘, 兩␶兩1/2xគ = xគ ⬘. In the following we will drop the
primes for notational simplicity. The stochastic functional
becomes
J=
A=
冕
4 2−d/2
兩␶兩
A,
g2
ddxdt兵˜␸␸˙ − H共˜␸, ␸兲其,
ϕ
01
we are hence led here to studying the behavior of the cluster
distribution function of finite DP clusters in the active phase
above the transition point 共where also an infinite cluster in
time and space is generated with finite probability兲. As the
basis for this study, we use the stochastic functional 关Eq.
共2.38兲兴 in its usual symmetrized real form with g0 = g1 = g,
g2 = 0, and with ␶ˆ 1 eliminated. We set ␶ˆ 0 = ␶ and work with
冕
(1,0)
01
FIG. 6. 共Color online兲 共d + 1兲-dimensional cut of a droplet configuration 共green inner part兲 of the true vacuum 共s = 0 , s̃ =
−2兩␶兩 / g兲 in the sea of the false vacuum 共s = 2兩␶兩 / g , s̃ = 0兲 and its
approximation for r0 Ⰷ 1 / 冑兩␶兩 by a 共d + 1兲-dimensional cone 共green
inner part and red outer part兲.
J=
(0,0)
01
共B3a兲
FIG. 7. 共Color online兲 The orbit of the droplet’s instanton configuration with 兩xគ 兩 = const between the false and the true vacua
共green inner orbit兲 and its approximation by kink solutions 共red
outer orbit兲.
H共˜␸, ␸兲 = − ⵜ˜␸ ⵜ ␸ + ˜␸共1 − ␸ + ˜␸兲␸ + k˜␸ .
The functional integral over exp共−J关s , s̃兴兲 is definitely
equal to 1. However, if we add a source term qs共xគ 0 , t0兲 to the
exponent, the saddle-point equations,
␸˙ = ⵜ2␸ + 共1 − ␸ + 2˜␸兲␸ + k,
共B4a兲
− ˜␸˙ = ⵜ2˜␸ + 共1 + ˜␸ − 2␸兲˜␸ ,
共B4b兲
generate a droplet of the true vacuum that has its tip at
共xគ 0 , t0兲 and faces backward in time 共see Fig. 6兲. We are interested in the critical droplet configuration whose lateral extension diverges as r = rc ⬃ 兩k兩−1 with 兩k兩 → 0. We approximate
this configuration by a critical spherical symmetric cone 共see
Fig. 6兲. Hence, we look for solitary waves ␸共␳兲 and ˜␸共␳兲
with ␳ = r + vt − rc as kink solutions of the saddle point 关Eqs.
共B4兲兴 which have their variations near ␳ = 0. By neglecting k
as well as 1 / 兩xគ 兩 in comparison to v, we obtain from Eqs. 共B4兲
the differential equations
␸兲␸ ,
v␸⬘ = ␸⬙ + 共1 − ␸ + 2˜
共B5a兲
− v˜␸⬘ = ˜␸⬙ + 共1 + ˜␸ − 2␸兲˜␸ ,
共B5b兲
where the stroke means differentiation with respect to ␳. The
properties of the kink solutions 共see Fig. 7兲 and the fluctuations around them are extensively discussed in the work of
Alessandrini et al. 关49兴 on classical kinks and their quantization in RFT. Also, in this work the appearance of the critical
velocities v = −⬁ 共the base of the cone兲 and v = 2 共the envelope of the cone兲 is shown.
Taking all things together, the action A 关Eq. 共B3b兲兴 of the
large droplet consists of the usual terms arising from the
volume ⬃rd+1 of the droplet times k˜␸, where ˜␸ takes its
value in the true vacuum, namely, −1, and a surface part ⬃rd
arising from the variations of the kink solutions 共in the limit
v → 2 this surface part comes only from the base of the
cone兲. We get, therefore,
A = − a兩k兩rd+1 + brd ,
共B3b兲
共B3c兲
共B6兲
where a and b are positive constants. Minimizing A with
respect to r, we obtain r = rc ⬃ 1 / 兩k兩 and finally
041809-16
PHYSICAL REVIEW E 80, 041809 共2009兲
RENORMALIZED FIELD THEORY OF COLLAPSING…
Considering the eigenvalues of the Gaussian fluctuations
about the critical instanton, we have 共i兲 one negative eigenvalue coming from the uniform variations of the extension r
of the critical droplet leading to the imaginary part of ⌽, 共ii兲
共d + 1兲 eigenvalues 0 coming from the translations in space
and time of the droplet and their compensation by the pinning of the droplet at its source, and 共iii兲 a series of positive
eigenvalues coming from the Goldstone modes of the broken
translation symmetries. For fully spherical symmetric droplets, the calculation of the latter eigenvalues is straightfor-
ward 关50兴. In the present problem, however, the droplets are
conical 共spherical symmetric in space but directed in time兲.
For this configuration, the Goldstone-mode eigenvalues have
not yet been calculated, as far as we know, and their calculation represents an interesting and challenging opportunity
for future work.
Finally, we return to the cluster distribution function P
关Eq. 共B1兲兴, which follows via inverse Laplace transformation
as discussed extensively in the main text. For the exponent ␨,
this leads to ␨ = d / 共d + 1兲. To determine ␪⬘, however, we first
need to know the Goldstone-mode eigenvalues touched upon
in the preceding paragraph. Thus, we here have to leave the
interesting problem of calculating ␪⬘ for future work.
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Jc =
4 2−d/2
兩␶兩
Ac ⬃ 兩h兩−d .
g2
共B7兲
041809-17