Renormalization group approaches to polymers in disordered media

Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.

Renormalization group approaches to polymers in disordered media V. Blavats'ka^, C. von Ferber^, R. Folk^ and Yu. Holovatch^'^^ ^Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitski Str. Lviv, UA-79011 Ukraine ^Theoretical Polymer Physics, Hermann-Herder-Str. 3, 79104 Freiburg University, Germany ^Institut fiir Theoretische Physik, Johannes Kepler Universitat Linz, Linz, A-4040, Austria ^Ivan Franko National University of Lviv, 12 Dragomanov Str. UA-79005 Lviv, Ukraine This chapter focuses on the universal properties of the static critical behaviour of polymer systems with different types of complex structural disorder, in particular long-rangecorrelated defects of random orientation and defects defined by bond percolation. Universal properties of long flexible polymer chains in a solvent are well described by self-avoiding walks (SAWs) on a regular lattice. In the same way SAWs on disordered lattices serve as a model for polymer solutions in disordered media. To approach the universal content the field-theoretical renormalization group (RG) has been extraordinarily successful. We elaborate how the RG is extended to disordered systems focussing on uncorrelated and long-range correlated disorder as well as disorder that acts to confine the polymer to a percolation system. We review different field theoretical approaches as well as real space renormalization^ group results. 1. INTRODUCTION AND PHENOMENA The transfer of ideas from the theory of critical phenomena to polymer science has led to considerable progress in modelling polymers and understanding their universal properties since R G. de Gennes seminal work (in 1972) [1] provided a first link. In the theory of critical phenomena one is interested in the peculiarities of the behavior in the vicinity of the critical temperature Tc > 0. The critical temperature essentially depends on microscopic details of the model. However, asymptotically close to Tc the thermodynamic and correlation functions for very different systems may depend on the thermodynamic parameters by the same power-like functions primarily characterized by their critical exponents (scaling laws). In contrast to the critical temperature itself, the values of the critical exponents do not depend on any of the microscopic details of the system and are determined only by the global properties of the model such as the (lattice) dimension d, the dimension m and other symmetry properties of the order parameter. Due to the fact that the critical exponents may be identical for systems with very different 103

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y. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

microscopic nature, the critical exponents are called universal. Subsequently, if the critical properties of two systems are described by the same set of scaling functions and an identical set of critical exponents they are said to belong to the same universality class. The concept of universality has found its reflection in polymers, which appeared to belong to the universality class of the 0(m)-symetrical spin model in the formal limit m —^ 0. Furthermore, if one is interested in the behavior of polymers influenced by structural disorder, one of the central questions that arises concerns the possible change of the universality class under the influence of disorder. As far as scaling properties of polymers are well described by the formalism of critical phenomena, it is natural to study them by the renormalization group (RG) approach to get a quantitative description. Thefield-theoreticalRG method has originally been developed to investigate quantum field theory problems [2]. The application of the field-theoretical RG in the theory of critical phenomena is based on the formal similarity of statistical averages at the critical point with the expectation values in quantum field theory; this is especially apparent for the perturbation theory expansion using the Feynman diagram technique [3-6]. This powerful approach is successfully applied in the study of polymer models [7]. The complementary real-space RG approach has been developed in parallel starting from the famous Kadanoff's block transformation [8] and decimation procedure. Here, we are concerned with long flexible polymer chains dissolved in a good solvent. A good solvent is defined as one, in which it is energetically more favourable for the monomers of the polymer to be surrounded by molecules of the solvent than by other monomers. As a consequence, there exists around each monomer a region (the excluded volume) in which the chance of finding another monomer is very small. It turns out, that such diluted polymer solutions have physical properties that are independent of most of the details of the chemical microstructure of the chains and of the solvent. It is well established that the universal scaling properties of longflexiblepolymer chains in a good solvent are perfectly described within a model of self-avoiding walks (SAWs) on a regular lattice [9]. In this model, the monomer size is represented by the lattice constant, and the size of the polymer chain by the walk length. The trajectory consists of N steps, performed towards the nearest neighbor site, taking into account that the trajectory is not allowed to cross itself. The impossibility of the trajectory to cross itself reflects the excluded volume effect. As an example, the mean square end-to-end distance R and the number of diff"erent space configurations ZN of a SAW with N steps obey in the asymptotic limit of long chains N -^ oo the scaling behavior [1,7,9]: {R'')r.N^-,

ZN^Z''N^-\

(1)

where z is non-universal fugacity. Although the microscopic structure of the bonds is different for different types of lattices, the exponents v and 7 are universal, depending only on the space dimensionality d. In three dimensions c? = 3 the exponents read [11] V = 0.5882 lb 0.0011 and 7 = 1.1596 ± 0.0020. A surprisingly good approximation for the critical exponent v of SAW in general dimension d is given by the Flory formula [10]:

Renormalization group approaches to polymers in disordered media

105

At d — 1 one has a completely stretched chain with u = 1. At d = 2 the exact result {u = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has {u = 0.5882±0.0011 [11]) and Monte Carlo simulation gives {u = 0.592 ib 0.003 [12]), both values being in a good agreement. A question of great interest is the influence of disorder in the medium on the universality class of dissolved flexible polymers, namely: are the universal exponents (1) in this case the same as in the pure case? The question of how linear polymers behave in disordered media is not only interesting from a theoretical point of view, but is also relevant for understanding transport properties of polymer chains in porous media, such as an oil recovery, gel electrophoresis, gel permeation chromatography, etc. [14]. Let us note, that there are two classes of problems, dealing with disorder, namely those with annealed and quenched disorder. In the first case, the diff'erent realizations of disorder are averaged simultaneously with a thermodynamical averaging over different conformations of the SAW. In the present review, we however focus on the case of quenched disorder [15], which is introduced such that it is not in thermodynamic equilibrium with the unperturbed system. The quantities of physical interest must then first be calculated for a particular configuration of disorder, followed by the average over all configurations of disorder. The question of the change in scaling behavior of SAWs in disordered medium can be reformulated in the frames of the field theory in the form: how does the disorder influence the critical behavior of the m-component model in the limit m ^ 0? Numerous MC simulations [16-21], exact enumerations [22-28], and theoretical studies [29-39] which have been published since the early 80-ies [14], lead to the conclusion that there are the following regimes for the scaling of a SAW on a disordered lattice: (i) weak uncorrelated disorder, when the concentration p of bonds allowed for the random walker is higher than the percolation concentration pc (ii) weak, but long-range correlated disorder, and (iii) strong disorder, directly at p = Pc- By further diluting the lattice to p < Pc no macroscopically connected cluster, "percolation cluster", remains and the lattice becomes disconnected. In regime (i) the scaling law (1) is valid with the same exponent u for the diluted lattice independent of p, whereas in cases (ii) and (iii) the scaling law (1) holds with a new exponent v^i^v. For magnetic systems, the effect of weak quenched uncorrelated point-like disorder on the critical behavior is usually predicted by the Harris criterion [40]: disorder changes the critical exponents only if the specific heat critical exponent ot-pure of the pure (undiluted) system is positive:

Upure being the correlation length critical exponent of the pure system. This was confirmed by numerous theoretical and experimental studies (see [41,42]). The straightforward application of the Harris criterion to the statics of SAWs indicates, that the critical exponents should be modified in the presence of any amount of lattice

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disorder, as far as a for SAW on the pure lattice is positive. However, the renormalization group study [29] contradicts this prediction. The Harris criterion was then modified [43] for the quenched disordered m-vector model in the m —>• 0 limit and leads to the conclusion, that the critical behavior of SAWs is not affected by weak uncorrelated lattice disorder. Since this point is of central interest in this review, let us consider it in more details. Let 7 denote a SAW configuration of N steps starting at some site i and R^ the square of the end-to-end vector. Then for the mean square end-to-end distance of the trajectory one has: ^ R^ (i?2) = ^ ~ i V 2 ' ' .

(4)

7

Following the analytic considerations of Harris [43]. we denote by [...]p the average over all random configurations of the lattice, where every site is occupied with probability p and empty with probability I—p. The SAWs are allowed to have their steps only on the occupies sites. For every such a configuration C, let us denote 7 € C for configurations 7 of AT-step SAWs that start at a given site i and exists on C. The mean square end-to-end distance of the SAWs that start at point i reads: /R2\ =

^ '

Tg<^

[E]p

_

g

7€C

EP{C)Zi

_

7

C:-,€C

E E Pic)

....

^^

here P{C) is the probability of the realization of the configuration C. In case the distribution of occupied sites is uncorrelated, every SAW with N steps can be realized with probability p^. So, the last equality can be rewritten:

7

7

and the expression obtained is the same as for the pure (non-diluted) lattice. The only effect of the disorder lies in the change of the possibility of formation of each SAW with N steps, whereas the scaling properties of the realized SAW are not influenced. In the present review, we turn our attention to the three different classes of quenched (non-equilibrium) disorder, which from one side can be treated within the RG scheme and from the other side allow for a physical realization: 1) non-correlated point-like defects, 2) long-range correlated defects with a correlation function, governed by a power law ~ r~" at large distances, where a is some constant \ 3) the case when the concentration of dilution is exactly at the percolation threshold. While the influence of short- and longrange correlated disorder can be studied by related field-theoretical models with an upper critical dimension dupper = 4, the percolation problem leads to a different field theory ^This type of disorder allows to describe so-called extended impurities: a = d — 2 corresponds to the presence in the system of straight lines of impurities of random direction, whereas a = d—1 gives random planes of defects.

Renormalization group approaches to polymers in disordered media

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with dupper = 6. Let us note, that in the case of m-component systems with long-rangecorrelated disorder the Harris criterion is modified: for a < d the disorder is relevant, if the correlation length critical exponent of the pure system obeys u < 2/a. For a>d the original Harris criterion (3) remains valid. Again, this criterion cannot be applied directly to the polymer case, due to the peculiarities of the polymer limit. The set-up of the paper is as follows. In the following section we present the fieldtheoretical description of the polymer model, introduce different types of structural disorder into this model and present an introduction to real space renormalization. Section 3 reviews different treatments of these systems by field theoretical and real space RG approaches to analyze the scaling properties and to estimate the critical exponents. In the present review we focus on static properties of polymers. RG treatments of the impact of disorder on polymer dynamics may be found e.g. in [44-48]. 2. MODELING THE SYSTEM 2.1. Polymers: Field theory with zero-components Let us consider the model that we use to describe a single polymer chain in solution. In a first discrete version we describe a configuration of the polymer by a set of positions of segment endpoints r^: Configuration{ri,..., r^v} G iR^""^. Its statistical weight (Boltzmann factor) with the Hamiltonian H divided by the product of Boltzmann constant kB and temperature T will be given by

The first term describes the chain connectivity, the parameter 4 governs the mean square segment length. The second term describes the excluded volume interaction forbidding two segment end points to take the same position in space. The parameter P = 1/kT gives the strength of this interaction. The third parameter in our model is the chain length or number of segments N. The partition function Z is then calculated as an integral over all configurations of the polymer divided by the system volume f2, thus dividing out identical configurations just translated in space: ^ ( ^ ) = ^ / n d'-i e x p [ - - ^ F { n } ] .

(8)

This will give us the 'number of configurations' of the polymer, which obey the scaling relation (1). We will do our investigations by mapping the polymer model to a renormalizable field theory making use of well developed formalisms (see [1,9,7] for example). To this end we make use of a continuous version of our model as proposed by Edwards [49,50]. The configuration of polymer is given by a path r{s) in d - dimensional space M^ parameterized by a variable 0 < 5 < 5 that has the dimension of a surface. The Hamiltonian H is then given by

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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

where UQ is the excluded volume parameter with densities p(r) = J^ ds6'^{r — r{s)) . In this formalism the partition function is calculated as a functional integral: Z{S) = I V[ris)] exp{-^H[r]}.

(10)

Here the symbol X>[r(5)] includes normalization such that Z{S) = 1 for n = 0. To make the exponential of (^-functions in (10) and the functional integral well-defined in the bare theory a cutoff 5o is introduced such that all simultaneous integrals of any variables s and s' on the same chain are cut off by | s — s' \> SQ. Let us note here that the continuous chain model (9) may be understood as a limit of a model of discrete self-avoiding walks, when the length of each step is decreasing £o -^ 0 while the number of steps A^ is increasing keeping the 'Gaussian surface' S = Nil fixed. The continuous chain model (10) can be mapped onto a corresponding field theory by a Laplace transform in the Gaussian surface variables S to conjugate chemical potential ("mass variables") /i [9]: noo

Z{fi) = / dSe-f'^ZiS). Jo

(11)

The Laplace-transformed partition function Z{/JL) can be expressed as the m = 0 limit of the functional integral over vector fields $ with m components {(/>i,..., 0r„} : i(/x) = lv[ct>{r)] exp{-nm

U^o-

(12)

The Landau-Ginzburg-Wilson Hamiltonian 'H[0] of this 0^-theory then reads 1^( \x\Xll( M2\ + I ^^(
(13)

here 0^(r) = YULii^i)^- Note that this theory is symmetric under 0{m) transformations of the m-component vectors (f). The limit m = 0 in (13) results in the cancellation of some special types of diagrams contributing to the perturbation theory expansions, which contains closed loops and thus are proportional to m. The same field-theoretical representation may be obtained starting from the quite different type of lattice model known as one of the basic models in the theory of magnetic systems. We present it here since for what follows it serves to introduce different types of disorder in the polymer system. Let us consider a simple (hyper) cubic lattice of dimension rf, and to each site prescribe a m-component vector S{r) with a fixed length (for convenience one usually sets |5p = m). Imposing a pair interaction with the energy proportional to the scalar products between pairs of spins, this defines the Stanley model (also known as the 0{m) symmetric model). The Hamiltonian of this model reads [77]: n = -J^JS{r).S{r'),

(14)



where the summation is over nearest neighbor sites r, r' and J parameterizes the interaction between spins S{r) and S'(r'), located at sites r and r'.

Renormalization group approaches to polymers in disordered media

109

The effective field-theoretical Hamiltonian of the model (14), obtained via special Stratonovich-Hubbard transformation, passing from the discrete system to the continuous field theory, is the same as (13). The relevant global properties of a microscopic model is represented by the structure of the effective Hamiltonian. In this case, //Q is a squared bare mass proportional to the temperature distance to the critical point, UQ > 0 is a bare coupling constant. Note that the effective Hamiltonian (13) preserves the 0{m) symmetry of the Stanley model (14). As far as the Stanley model is in this sense equivalent to the 0{m) symmetric 0^ theory, the analytic continuation m -> 0 of this model again leads to the polymer limit. 2.2. Randomness: symmetry consideration To introduce different types of disorder in the polymer system, we start from the mcomponent model (14). The presence of impurities can be modeled by a class of sitediluted models [51]. One considers some fraction of the sites to be occupied with some defects. The site-diluted Stanley model is introduced by the Hamiltonian: n = -Y,

JcrCr'S{r)S{r'),

(15)



where the occupation numbers Cr are introduced, which equal 1 when the site r is occupied and 0 when it is empty. 2.2.1. Uncorrelated disorder Let us consider the case when the values c^ in (15) are non-correlated random values distributed according to the probability distribution:

r

Cr

with:

Here p is the spin concentration and I — p the concentration of impurities. The pure non-diluted lattice corresponds to the case p = 1. To explain possible generalizations of the model (15) with site occupation distribution (16) let us note, that the first two moments of the distribution determine the critical behavior of the model. For the distribution (16) one gets: {Cr)=P, g{\f- r'l) = {crCr>) - {crY = p(l - p)5{r - r')

(18) (19)

where (...) means averaging with the distribution function (16), and 5{r — r') is Kronecker's delta. Let us introduce the notation VQ = —p(l — p) for the following. As far we consider the case of quenched disorder, the free energy of the system is obtained by averaging the logarithm of the partition function Z over the disorder distribution [15]; this amounts to use so-called replica trick [52] writing the logarithm in the

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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

form of the following limit: lnZ=liin

Z"-l

n-^O

.

(20)

n

While the powers of Z can only be evaluated for integer values of n, analytical continuation in n is assumed to perform the limit n —> 0. As a result one ends up [53] with the effective Hamiltonian:

Here, Greek indices denote replicas. Comparing this expression with the Hamiltonian of the pure model (13), we notice an additional interaction of the order parameter field: the last term presents the effective interaction between different replicas. The coupling constant VQ is proportional to the variance of the disorder distribution. The coupling UQ must be positive, otherwise the pure system undergoes a 1st order transition. For final results the replica limit n ^ 0 has to be taken. There exists a second way to obtain the effective Hamiltonian (21). A weak quenched disorder term can be introduced directly into the effective Hamiltonian (13). The presence of non-magnetic impurities in a microscopic model (15) manifests itself in fluctuations of the local temperature of the phase transition. Introducing -0 = ^(r) as the field of local critical temperature fiuctuations, one obtains the effective disordered Hamiltonian [53]: n4$)=ld'r{\

[|V^p + (/.^ + V)|^p] + | | 0 r } .

(22)

The Hamiltonian (22) depends on a number of macroscopic parameters that describe the specific configuration of the field '0(r). On the other hand, the observables should not depend on the specific realization of the random field ip and are to be averaged over the possible configurations of ^ [15]. In particular, the singular contribution to the free energy of the diluted quenched Stanley model (15) can be written in the form of a functional integral: F ^ ID[^P{r)]P[i;{r)]\nZ[^P{r)l

(23)

where the configuration-dependent partition function Z[ijj{r)] is the normalizing factor of the Gibbs distribution with effective Hamiltonian (22); P[ip] defines a probability distribution of the field '0(r). Introducing n replicas of the model (22) and taking that ip obeys a Gaussian distribution: P(V^) = - ^ e x p ( - t / ; V 4 a ; 2 )

(24)

with cj^ being the dispersion parameter, one again ends up with the effective Hamiltonian (21).

Renormalization group approaches to polymers in disordered media

111

The model (21) is interesting in the polymer limit m —^ 0. In this case it can be interpreted as a model for SAWs in disordered media. Note that such a limit is not trivial. As noted by Kim [62], once the double limit m, n -^ 0 has been taken, both UQ and T;O terms are of the same symmetry, and an effective Hamiltonian with one coupling UQ = UQ — VQ of the 0{mn = 0) symmetry results. This leads to the conclusion that weak quenched uncorrected disorder does not change the universal critical properties of SAWs. These results were confirmed by numerical [16-19,26,63,64] and analytical methods [38,39]. 2.2.2. Long-range correlated disorder In the above case of the diluted d-dimensional m-component spin model the disorder is correlated according to (16). Another type of disorder was proposed in the work of Weinrib and Halperin [65], with a pair correlation function of defects that decays at large distances f|*— r*| according to a power law: g{\r- fl) = {crCr.) - {Cr? ~ |f - r"|-«

(25)

where a is some constant. This model describes so called extended impurities in the system. In particular, the correlation function (25) with a = d — 1 describes randomly oriented impurity lines, while planar defects correspond to a = d — 2. In magnets, longrange-correlated disorder may be present in the form of continuously distributed dislocations and disclinations, so-called extended structural defects. These defects may have the form of lines or planes of random orientation [69] or may form some sponge-like fractal objects, which are considered as aggregation clusters [70]. The particular case of systems with dislocation lines or planes of parallel orientation, is investigated in Ref. [73]. The Fourier transform of the correlation function (25) at small k has the form [65]: g{k)r.vo-^Wok^-'^,

(26)

where VQ and WQ are some constants. Let us recall, that for the case of point-like noncorrelated defects the correlation function (19) reads g{\r — r'l) = vo6{\f — f*|), so its Fourier transform for small k behaves as: g{k) - vo.

(27)

Comparing (26) and (27), we can see that the case a = d describes point-Uke disorder. Applying the replica method in order to average the free energy over different configurations of quenched disorder one finds the effective Hamiltonian of the m-vector model with long-range-correlated disorder [65]:

a=l ^

'

a,^=l -^

Here, the replica interaction vertex g{x) is the correlation function with Fourier image (26). Passing to the-Fourier image in (28) and taking into account (26), an effective Hamiltonian results that contains three bare couplings UQ^VQ^WQ. For a > d the it;o-term is

112

y. Blavats'ksi, C. von Ferber, R. Folk and Yu. Holovatch

irrelevant in the RG sense and one obtains an effective Hamiltonian of a quenched diluted (short-range correlated) m-vector model (21) with two couplings %, ^o- For a < d we have, in addition to the momentum-independent couplings, the momentum dependent one Wok^~^. Note that g{k) is positively definite being the Fourier image of the correlation function. From here one gets ?i;o > 0 at small k. A one-loop approximation for the model was given in Ref. [65] using an expansion in € = 4 — d,S = 4 — a. A new long-range-correlated fixed point was found with a correlationlength exponent u = 2/a and it was argued, that this new scaling relation is exact and also holds in higher order approximations. This result was questioned recently in Refs. [66,67], where the static and dynamic properties of 3d systems with long-range-correlated disorder were studied by means of the massive field-theoretical RG approach in a 2-loop approximation, for different fixed values of the correlation parameter, 2 < a < 3. The P and 7 functions in the two-loop approximation were calculated as an expansion series in renormalized vertices u, v and w. The study revealed the existence of a stable random fixed point with u* ^ 0, v* ^ 0, w* ^ 0 in the whole region of the parameter a. The obtained exponents u for various a however violate the supposed exact relation z/ = 2/a of ref. [65]. The authors assign difference to a more accurate field-theoretical description using higher order approximations for the 3d system directly together with methods of series summation. Recently, Ballesteros and Parisi [68] presented Monte-Carlo simulations of the diluted Ising model in three dimensions with extended defects in the form of lines of parallel orientation, confirming that the simulated critical exponents for the Ising model agree fairly well with theoretical predictions of Weinrib and Halperin. We are interested in the polymer limit m —> 0 of the model (28) interpreting it as a model for SAWs in disordered media. Note, that such a limit is not trivial, as explained in the last section. In Refs. [71,72] the asymptotic behavior of SAWs in long-range-correlated disorder of type (25) is investigated and found to be governed by a set of critical exponents which are different from that of the pure case. In ref. [48] an environment of a quenched configuration of a semi-dilute polymer solution is introduced as a special case of long-range correlated disorder for polymer dynamics. For the statics this environment is shown to be equivalent to an annealed one, i.e. without impact. 2.3. Geometry: Percolation system 2.3.1. Percolation clusters and the Potts model Like the SAW, the problem of percolation can also be treated as geometrical critical phenomenon [74]. To introduce the percolation problem, let us consider a regular hypercubic d-dimensional lattice, where either the sites or bonds are occupied with probability p. In percolation one asks questions concerning the connectivity of occupied bonds. Sets of mutually connected bonds form cluster. One can then ask what is the probability that there is a cluster spanning from the one end of the lattice to the opposite end; in the thermodynamic limit (the number of sites goes to infinity) this spanning cluster becomes the infinite cluster. An important quantity in percolation theory is the percolation probability P{p), which gives the probability that given site belongs to the infinite cluster. One can show that there exists a critical value Pc (also called the percolation threshold) such that P{p) is

Renormalization group approaches to polymers in disordered media

113

d = 2[99] d = 3[98] d = 4[98] c? = 5[98] d>6 df 91/49 2.51 lb 0.02 3.05 ± 0.05 3.69 =b 0.02 4 Table 1 Fractal dimension of percolation clusters in dimensions d = 2 , . . . , 6.

zero when p < p^ For p > pc^ it obeys the following scaling law, approaching Pc from above: i^(p)-b-Pc|^^^.

(29)

where l3pc is an universal critical exponent. Because of this property, we can consider P{p) is an order parameter. The correlation length of the percolation lattice (or connectedness length) is defined as follows:

ep' = ^ % — - ( P - P c ) - ^ ^ ^ ^

(30)

3

where pij is the probability for two sites i and j at the distance r^ = |fi - fj| to belong to the infinite cluster; vpc is the critical exponent. Let us note, that the infinite percolation cluster itself is a fractal with fractal dimension dependent on d\ df{d) = d—Ppc{d)/upc{d) [74]. The estimates for the fractal dimensions of the percolation cluster are given in Table 1. Like SAWs that are related to the 0(m)-model, percolation is connected to the qstate Potts model. The ^-state Potts model [86] is a spin model whose high temperature expansion (when g —> 1) corresponds to diagrams that are percolation configurations. In the Potts model to the each site x of a regular lattice a spin variable ax corresponds which can be in q possible states {q = 2 corresponds to the Ising case). Interactions favor nearest neighbors which are in the same state. The effective Hamiltonian for the model reads: H = JY1

(^'^"r' - 1)

(31)



with J an interaction constant and S the Kronecker symbol. The thermodynamical critical exponents of the magnetization P and the correlation length z/ of this model in the limit q -^ I correspond to the geometrical critical exponents of the percolation problem (29),(30). Consider the behavior of the m-component spin model on the disordered lattice with dilution near the percolation threshold and at low temperatures. As p decreases, also the critical temperature Tc{p) decreases and reaches zero at the percolation point Pc- At T = 0 the critical properties are determined by the properties of the connected spin clusters, i.e. this is a percolation problem. It has been argued [78] that the point p = p^^T = 0 should be viewed as an multicritical point, here the thermal correlation length and correlation length of the percolation cluster diverge simultaneously. The studies of Ising model [91]

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V. Blavats'ksL, C. von Ferber, R. Folk and Yu. Holovatch

and general xy-mode\ [100] near the percolation threshold have shown, that such systems are characterized by a multicritical spectrum of the critical exponents i^i, defined as i/j = J^pc/^ij where i/pc is the critical exponent of the correlation length of the percolation lattice (30) and (^j is a set of the crossover exponents [100]. 2.3.2. Modified Flory theories Let us turn our attention to the scaling of a SAW on a disordered lattice with strong disorder, directly at p = ppc- The Flory formula (2), giving a surprisingly good estimation for the critical exponent u of SAWs on pure lattice, has tempted a number of authors to search for generalizations to determine the exponent Up of SAWs on the percolation cluster [16]. The most direct one is:

'^ = 2h;-

^''^

Here, df is the fractal dimension of the percolation cluster. This first simple generalization is in a good agreement with results of Monte Carlo simulation [16-21] and exact enumerations [22-28] (see the Table 1). However, there exist a number of other attempts to derive Flory-like formulas for SAWs on fractals. For instance one may write the free energy F of the N-step SAW as the sum of energetic and entropic terms [34] as: F = a ^ + [-lnP{R,N)],

(33)

where R is the end-to-end distance for iV-step SAW, and P{R^ N) is the probability for a random walker to reach this distance after the N steps on the fractal. In general P{R, N) for large R has an exponential form: P{R, N) « exp[-b{R'^/N)%

(34)

where dyj is the random walk dimension on the fractal, C is some constant. The minimization of Eq. (33) with respect to R with Eq. (34) yields the Flory formula: - - '*' ' df + Cd^

(35)

Rammal et al. [31] suggested that C = !> which leads to the Flory formula: 3

(36)

Aharony [34] on the other hand proposed that (37)

where dmin is the fractal dimension of the minimum path on a fractal.

Renormalization group approaches to polymers in disordered media

115

Another kind of Flory-type formula is suggested in [30], where it was argued that the spectral dimension dg of the fractal percolation cluster must be an intrinsic property: ^

(38)

^^^

df{2 + dsy

In this problem there is one point to clarify. If one believes that the SAW can only live on the backbone of percolation cluster (otherwise it will be trapped at the dangling ends) [59,35,34,31], one should use in the expressions above the corresponding fractal characteristics {d?,df) for the backbone. A more sophisticated expression was proposed in [34]:

_ 4df - df ''^"df(2df-df + 2)'

^^^^

where df is called the spreading or the connectivity dimension of the backbone of the percolation fractal. A more general version of Eq. (35) with the additional condition (37) was derived in [36], assuming for the probability distribution P{NyR): P(iV,/?)-

l-N

<^:{^y*c.{jL-)']y

(40)

representing the requirement, that for small R the free energy of a SAW is dominated by a term [N/R^^Y^ which may be thought as being the repulsive energy between distant basic units, and for large i^ by a term (i^/iV^/^"^»")*^, which represents a configurational entropic term. Here, dmin is the fractal dimension of the shortest SAW on percolation cluster, a and 5 are unknown positive exponents. For the critical exponent Vp the following expression results: ''=dj{dB+\y

^''''^k = 5la.

(41)

The mean-field result for this ratio is given in terms of the random walk dimension d^: d^d

•

Another possible expression for Vp was proposed in [32], using geometrical considerations for the node-link picture of the backbone of the percolation cluster. For the diluted lattice just above the percolation threshold the lattice can be viewed as a collection of nodes which are connected by links, thought of as random paths. Two important lengths enter this picture. One is the distance between nodes which is of order of the percolation correlation length (30), the other important length is the length L of the random path between nodes which is given by a separate exponent C through: L(x\p-pc\-^.

(43)

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

116

If the number of steps N of SAWs is larger than L, a dilute lattice would behave as a normal (i-dimensional lattice. We call this the "normal" regime. In the normal regime, (R^) is still given by (1) with the critical exponent u. If N < L, the diluted lattice would behave anomalously like a fractal and we call this regime the anomalous one, here {R^) scales with new exponent Up. Considering the crossover for N ~ L,one obtains [32]: (44)

c

where upc is the correlation length critical exponent of the percolation transition. The Flory-like theories [30-37] offer good approximations for iyp{d) in a wide range of d (see Table 2).

d ^SAW

FL, [30] [31]

1

2

1

3/4 0.778 0.69(1)

[32] [33]

[34] 1

0.770 0.76 0.75-0.76 0.77

[35] [36] MC, [16] [17] - i^SAW [18] — ^SAW 0.77(1) [19] [20] 0.783(3) [21] EE, [22] 0.76(8) [23] 0.81(3) [23] 0.745(10) [24] [25] 0.745(20) [26] 0.770(5) [27] 0.778(15) [27] 0.787(10) 0.767 RS, [60] 0.778 [30] RG, [22] 0.595 0.785 1 (133)

3

1

4

1

5

0.5882(11) 1/2 1/2 0.662 0.593 0.543 0.57(2) 0.49(3) 0.70(3) 0.63 0.56 0.52 0.656 0.57 0.65 0.58 0.64-0.66 0.57-0.59 0.55-0.57 0.66 0.62 J 0.56 -2/3 0.612(10) 0.605(10) 1

0.62-0.63 0.67(4)

0.56-0.57 0.63(2)

0.54(2)

0.548 0.595

0.524 0.536

1 61 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

0.635(10) 0.65(1) 0.640(15) 0.660(5) 0.66(1) 0.662(6) 0.724 0.571 0.678

i

|l/2 |l/2

Table 2 The exponent Up for a SAW on a percolation cluster in several dimensions. FL: Flory-like theories, EE: exact enumerations, RS, RG: real-space and field-theoretic RG. The first line shows I^SAW for SAW on the regular lattice {d = 2 [13], d = 3 [H]).

Renormalization group approaches to polymers in disordered media

117

2.3.3. Monte Carlo analysis Kremer [16] was the first to perform Monte Carlo study of SAWs statistics on randomcite diamond 3-dimensional lattice. His study indicates no change in the exponent u for weak dilution, but for p close to Pc observes a higher value Up « 2/3. The data show a prefactor in the scaling law for the mean-square end-to-end distance (1) which increases with increasing dilution. Right at the percolation threshold Pc, a crossover to a larger exponent \ > Vp > v occurs. The trivial upper bound corresponds to the case of the absolutely stretched chain. This crossover (for large N) is described by the following scaling assumption: (i?Y/'ociV--./(iV|p-pen

(45)

where y is a crossover exponent, and the scaling function / has the form: a; —> oo;

^^^^

\ const,

For p> Pc the remaining lattice is built up by the "fractal" regions of extension ^ and the "normal" region. The extension f diverges in the percolation limit as: ^oc|p-pe|-"^^.

(46)

If the end-to-end distance of a SAW is greater than the correlation length i? > ^, we find the behavior of a SAW on a normal lattice. On the other hand, if i^
Pel"''''' -^N'\p-

pc\^ ^ 1

Comparing this with (45) we find y = Ppcl^pIn Ref. [30] it was noted, that one expects to have two different critical exponents, depending on whether one averages the SAW configurations only on the infinite percolation cluster at Pc or on all clusters. If the SAW is averaged only on the infinite percolation cluster (infinite cluster, IC), we expect to have: {R^)^N^''^^,

(47)

whereas if the SAWs are averaged on the percolation system of all clusters (AC), we may anticipate that (^2)^iV2^Ac^

(48)

where ujc / I^AC stnd these exponents are universal. Moreover, we should have: I^IC > ^AC > Impure,

C? < 4,

(49)

since the fractal structure of the clusters should enhance the self-avoiding effect. The position-space renormalization-group (PSRG) approach was used in [30] to estimate vic in two and three dimensions.

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V. Blavats'ka^ C. von Ferher^ R. Folk and Yu. Holovatch

More recently, Kim [35] claimed, based on a scaling argument, that the inequality of the type (49) is invalid and ujc = VAC should hold. It was also argued, that that the critical behavior of the IC averages is also similar to that on the backbone (BB) of an infinite cluster, suggesting: ^ic = ^AC = J^BB,

d
(50)

Both equalities are based on the following argument. The infinite cluster generated near Pc in general contains many dead ends and "dangling loops" that are connected to the backbone via a single path. Since the walk that visits such bonds is eventually terminated after a finite number of steps, the walks that survive in the limit AT —>> oo are those confined to the backbone. With Eq. (50) and the inequality: dmin < —


(51)

where dmin is the dimension of the shortest SAWs between two points on the infinite cluster and df the fractal dimension of the backbone, Kim claimed that both UAC and UJC should be larger than Upure because otherwise the last inequality breaks down for d>4 since df < 2. The MC simulation of Kremer [16] has been referred to as the only numerical estimate for Up for a number of years, until Lee et al. [17] performed simulations for SAWs on square and cubic lattices at dilution very close to the percolation threshold. Their results for the SAWs critical exponent ujc rather surprisingly indicate that the exponent ujc even at pc is close to the pure result. Furthermore, numerical uncertainties in the [16] were indicated; namely, it was noted, that the chains used in that simulation are probably not long enough to estimate the critical exponents. The validity of either (49) or (50) has been tested numerically for the first time in [19]. Here, the critical behavior of the AC and BB averages of SAWs on two-dimensional site-percolation for a square lattice has been studied via Monte Carlo simulations. It was found, that UAC is very close to the pure lattice value, while UBB = 0,77 ± 0.01 which is about 3% larger than the full lattice value, concluding that the relation UBB > ^ic holds. In conclusion, it was claimed, that exponent z/p, governing the scaHng behavior of SAWs on the percolation cluster is close to the pure lattice value and, if any, the difference would be very small and virtually unobservable in Monte Carlo simulations. In Ref. [26] a detailed numerical study of enumerated SAWs on (Monte Carlo generated) random site diluted lattices in two and three dimensions was carried out. The nonsatisfactory results of MC simulations were explained in particular with difficulties in realizing the SAW trajectory on the percolation cluster. The percolation fractal contains self-similar regions connected by linearly or singly connected links, and MC diffusion of SAWs through these links is extremely difficult. This forces the MC generated SAWs on the percolation cluster to be confined (localized) practically to a some region of the percolation cluster of a size determined by the lattice size and the particular realization. As a result, this leads to incorrect estimates for the exponent of the mean-square end-toend distance of SAWs on the percolation cluster.

Renormalization group approaches to polymers in disordered media

119

2.3.4. Effective Hamiltonian The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension rfup = 6. The calculation of i/p was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that z/p c^ i/ in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. To obtain the effective Hamiltonian, describing the behavior of SAWs on the percolation cluster, we make use of results from [91,22,101]. Let us consider the m-vector spin model on a diluted lattice with bond randomness, described by the Hamiltonian:

H=-J2

J'^r'Sir)Sir'),

(52)



where the sum is over nearest neighbor pairs r, r' and Crr' are independent bivalued random variables, that indicate whether a given bond between the sites r and r' is present or not, and take on the value 1 and 0 with probabilities p and 1—p respectively. We are interested in the properties of the phase transition in such a model for bond concentration close to Pc and thus the critical temperature is very close to zero, this allows us to interpret Eq. (52) as a model for self-avoiding walks on the percolation cluster. To perform the average of the free energy over the quenched disorder, the replica method is applied and for the n-replicated partition function we obtain: Z^=Ttl[{l-p

+ pe' a?/"^'")^"''-'^) = TVe-^",

(53)



with Tr(...) = / r i L i rir dS''{r)S{\S''{r)\-y/m){...), tonian 1-L through -m

=

which defines an equivalent Hamil-

E {ln(l - p) + ln(l + ^e'^.P"^^^''^^. , \— P

Performing the expansion of the logarithm in this expression one finds after some transformations: C50

oo

l-O n

+E«'2'E 1=0

n

ai=ln=l m

E Sr:(r)S::{r')Sr^{r)Sr^{r')

ai ,a2=l ii ,12=1 ai
+ --- + E«'*' /=0

m

E

m

E

Qi,...,Q:t=lii,.-5«t=l ai<...
Sr:ir)St^{r')...Sr;{r)Sr;{r')) + ...}

(54)

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V. Blavats'ka, C. von Ferher, R. Folk and Yu. Holovatch

where the first three coefficients a^ read ao = - l n ( l - p ) ,

Introducing notations for the products of spin components on the same site, as proposed in [91]: S:^::^^{r)^S^{x)...S^{r),

a, < ... < a^

(56)

the Hamiltonian (54) can be written as a quadratic form: ^

n

m

n

-m = ^Yl^a, + M,) Y: t—l

E

E 5":TW5^:.:r(0

(57)

ti,...,it=l Qi,...,af=l ai<...
here, we have introduced the notation Mt = S S i ^«^*This expression can be converted to a continuous field theory using the standard Stratonovich-Hubbard transformation and introducing fields 0f/;i'*i^"(?'), conjugate to the products of spin components (56). The final field-theoretical effective Hamiltonian of this model then reads [101]:

+|f

E

E E E ^ * " ( 9 i ¥ * ' H 9 2 ) 0 ( ' H 9 3 ) 5 t e + 92 + 93)

91,92,^3 (t)

(/)

(58)

(s)

where the notations have been introduced (j)^^\q) for (l>^^JZ]i^^ {q) and m

^(...)standsfor^ (t)

t

^

n

J^

(...);

(59)

n,...,i<=lQ:i,...,at=l ai<...
rf = rQ-\- Mt, ro = 1 — ln(l — p), WQ = 1 is the bare coupling constant. We thus have a theory of interacting fields 0*(^) with diflFerent masses r^. The expression for the effective Hamiltonian (58) served as a basis of RG analysis, performed in the work [22] in the one-loop order and in [101] in two-loop order calculations. 2.4. Real space renormalization group approach Real-space RG (RSRG), sometimes also called position- or configuration- space RG allows to complement the usual momentum space RG by a more direct method. Generally, no Hamiltonian is needed and the method is especially suited for geometrically posed problems of connectivity as they are given for polymers and percolation [102,103].

Renormalization group approaches to polymers in disordered media

z

121

z z

^

z

r z

z

Figure 1. Elementary cell of a square lattice and its reduction.

We introduce its application to polymers following Family [104]. For SAWs on a square lattice one may divide the lattice into cells like the one shown in Fig.l. A SAW starting at the origin of this cell crosses the cell either horizontally or vertically. This fact may be described by a reduced cell. Assigning a fugacity z for each step of the SAW the renormalized fugacity z' on the bonds of the reduced cell is calculated by summation of the contributions of the corresponding SAW configurations: z' = R{z) = z^ + 2z^ + z^

(60)

This defines an RG transformation renormalizing polymer segments of extension 2L to monomers of extension L and weight z\ At the nontrivial fixed point z* of (60) we may then calculate the correlation length exponent v by V — IUAL/IUA^

(61)

where here AL = 2 is the rescaling factor of the lattice spacing and A^ = Mi(z) j^z\z^z*In two dimensions this results in ^* - 0.4656 and v = 0.7152

(62)

For the corresponding three-dimensional L^ cell the recursion relation is for L = 2 [104] z' = z'^+ Az^ + 8z^ + 12^^ + Uz^ + 16z^ + 10^^

(63)

with a fixed point z* = 0.2973 and u = 0.587. This result is surprisingly close to the Flory result of z/ = 0.6 and the high-order RG result of u = 0.588. However, there is no systematic way of improving this result. Increasing for instance the cell size to L = 3 the exponent becomes i/ = 0.5814 [105]. Its success and the simple way of implementation has led to a number of RSRG studies of SAWs on such lattices with disorder. The application of the RSRG method to the percolation problem may be illustrated by the renormalization of the same Kadanoff-cell given in Fig. 1. Following Ref.[102] we will now be interested in the probability of the left and right borders of the cell to be connected. As shown in Fig. 2 the cell is in this respect equivalent to a Wheatstone bridge configuration which represents the contribution of the backbone. Assigning a probability

122

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

Figure 2. Connectivity between peripheral sites of the elementary cell and equivalence to the Wheat stone bridge.

p to each bond the renormalization step for the probability of the endpoints i,j of the renormalized bond to be connected is described by [106]

i^(p) = 2p2+p3_5^4_^2p^

(64)

With p* = 1/2 we have Ap = 13/8. This approach now allows to study the behavior of the mean number of bonds in the minimal and maximal paths along the backbone. For the Wheatstone bridge these are [102]

Xmm(p) - V + 6p' - i V + 6 / ,

Xmax(p) = - V + 6p' - lOp^ + 3 / .

(65)

The scaling behavior of these quantities near the percolation threshold Pc is then described by an exponent Cx via = In\p-p^ — nJ^x X{p) =

(66)

With the renormalizations p' = R{p) and x' — \ x under the change of scale implied by the reduction one has Cx = lnA^/lnAp,

(67)

Here, we have x! = P* ^^ P = P* for both Xmin/max and thus the factors Amin = 17/8 and Amax = 17/8 which result in [102] Cmin = 1.55 and Cn

1.835.

(68)

To treat SAWs on the percolation cluster we will have to use recursion relations involving both the bond probability as well as a fugacity for the polymer. Work on this problem is reviewed in section 3.5.

Renormalization group approaches to polymers in disordered media

123

3. TREATING THE THEORY BY RENORMALIZATION GROUP METHODS 3.1. The general principles In this subsection, we give a brief account of the main relations of the field-theoretical renormalization group (RG) formalism that can be evaluated in different variants. The object of investigation are one particle irreducible vertex functions FQ , defined as averages of N order parameter fields 0 and L insertions of the type (jP as follows:

6{Eki+EPj)'^i'^''\{kh

M;Mg; {AO})= f ° e^i^^^^^^n)^

{(t>\n).. 4\ri)(j>{Rr).. 4{RN))^P'/^''RI^

. .d^i^^d^ri.. .d^r^.

(69)

here {p} and {A:} are sets of external momenta, AQ is a cutoff and (...) stands for averaging with the corresponding effective Hamiltonian, {AQ} stands for the set of bare couplings of the effective Hamiltonian of the system: for instance, in the case of a model with point-like uncorrelated disorder, which is described by the effective Hamiltonian (21), one has two couplings: i^o and VQ, An intrinsic feature of the functions (69) is their divergence in the asymptotic limit AQ —> oo. In order to control the divergences and to map the divergent mathematical objects to convergent physical quantities one performs a rearrangement of the series for the vertex functions. Renormalization factors Z^ and ^^2 for the fields are introduced defining the renormalized function F ^ ' '\

^T\{kh

M ; /i'; {A}) = z,^.z;/^r(^'^'({fc}; W ;

M?.{AO})

(70)

Zfj, and Z^2 are constructed such that F^ ' ^ has no divergences order by order in perturbation theory. In this procedure the mass //Q and couplings {AQ} are renormaUzed to /i and {A}. Here we describe in more detail two effective schemes, that are used in the context of the model, considered in our review: the massive and minimal subtraction schemes. 3.1.1. Massive scheme In the massive scheme [80,81], the renormalization is performed at non non-zero mass and zero external momenta, passing from the initial parameters of the model //Q, {AQ} to the renormalized mass // and coupling constants {A}. The regularization scheme suggested in (70) is not unique. To define the regularization scheme completely one imposes renormalization conditions for the renormalized vertex functions [4,80]. For theories, where the interaction is represented by 0^-term in the Hamiltonian (e.g., (21)), these conditions read: ^f'(fc,-fc;/i^{A})b=o ^^f)(fc,-fc;/z^{A})U

= ^Ji\

(71)

= 1,

(72)

rK({^};/^''{^})l{'t.}=o = M ' - % , ^'«^•'*(p;fc,-fc;/x^{A})|fc=p=o = 1-

(73) (74)

124

V. Blavats'ka, C. von Ferber, R. Folk and Yii. Holovatch

In the case of long-range-correlated disorder we have another global parameter a along with dj and for the renormalization of the coupling WQ in Eq. (28) one imposes [67]:

^k°^'(W;/^^{A})U=o = M'-"t.

(75)

with a renormalized coupling w. For the theory, where the interaction is represented by (/>^-term in the Hamiltonian (e.g., (58)), instead of (73) one has:

rK(W;M',{A})l{M=o = /^'-%

(76)

The scaling properties of the system asymptotically close to the critical point are expressed by the homogeneous Callan-Symanzik equation [4,80] for the renormalized vertex function: {/^|I+E^A.({A})^-(f-L)7,({A})+Lv({A})}^k'''''({p};W;M^{A})-0,(77) i

with the coefficients defining the RG flow of the couplings (/^-functions) and the critical exponents (7-functions) 7^ and 7^2: o / r \ i \ ^^i I ^ ^ 0I ^^^ I PAi(|A|) = ^i^l{Ao},Mo. 7<^ = ^ri^l{^o},Mo. 7<^2 = -^]^l{Ao},itxo

/'7o\ (78)

where Z<^2 = Z^2Z^ and the sum over i spans over all couplings Aj. The fixed points {A*} of the RG transformation are given by the solutions of the system of equations: /^A.({A*}) = 0,

2 = 1,2,....

(79)

A stable fixed point, corresponding to the critical point of the^ystem, is defined as a fixed point where all the eigenvalues {CJJ} of the stability matrix:

B,A

(SO)

possess positive real parts. In such a point, the function 7^ determines the value of the pair correlation function critical exponent r\'. r) = lA{>^*}\

(81)

and the correlation length critical exponent v is determined as: i'-i=2-7^({A*})-7^.({A*}).

(82)

All other critical exponents may be obtained from familiar scaling relations. For example, the susceptibility exponent 7 is given by: 7 = K2 - ri).

(83)

Renormalization group approaches to polymers in disordered media

125

3.1.2. Minimal subtraction scheme In Refs. [82,83] the elegant method of renormalization at zero mass and non-zero external momenta was developed, which avoids the additional renormalization conditions. Here, the vertex functions are analytically continued in the dimensional parameter d leading to a so called dimensionally regularized theory, where the cutoff AQ in 69 can be removed. As one approaches the critical dimension {dc = 4 in the case of (^'^-theory and dc = ^ for (/)^-theory), from below, the terms in perturbation series for vertex functions develop poles in e = dc — d, which correspond to logarithmic ultraviolet divergences. In the dimensional renormalization scheme, the coupling constants of the given model {AQ} and renormalization factors Z^ and Z^2 are written as series in the renormalized couplings {A}, which for k couplings are defined as follows: OO

CX)

Aoi = A,[l + ^ . . . ^ a , „ . . . , , , A ^ . . A i ? ] , OO

OO

z^ = i+5;]...X:&^.,...,z.Ai^-..Ai? OO

h=l

OO

ffe=l

The coefficients af^,...,/^^, hi^^_^ij^^ and ci^^..^^^ are minimal Laurent series in e = dc ~" ^ which are constructed such as to cancel in the renormalized vertex functions

^fKidh

W; {A}) = Z^^.zf'-^r(^'^H{?}; M ; {Ao})

(85)

all poles in e of the bare functions FQ ' Mn every order of {A}. The renormalized vertex functions obey the renormalization group equation

{^|:+E^^<({^})^ - ( y - Lh,(W)+I^,.{{\))Y^'''\{q}-{p}-

{A}) ^ 0, (86)

i

Here, r is a rescaling parameter, which defines the scale of the external momenta in the minimal subtraction scheme. In the same way as in the Callan-Symanzik equation (77) the coefficients in (86) define the renormalization group functions:

The fixed points {A*} are again defined as common zeroes of all /^-functions, and the critical exponents are defined following Eqs. (81), (82). 3.1.3. Resummation of asymptotic series The fi- and 7-functions are calculated perturbatively as series in the couplings Aj. The order of the expansion corresponds to the number of loops in the diagrammatic Feynman representation of the vertex functions (69). However, due to the (asymptotic) divergence of the RG functions series [84], one cannot directly derive reliable physical information

126

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

from these expressions, the fact that is familiar to the theory of critical phenomena. If the nature of the divergence is such that the series are asymptotic, then the situation is, at least in principle, controllable: in this case a good estimate for the sum of the series is obtained by either keeping an optimally chosen number of low order terms ("optimal truncation") or by applying an appropriate resummation procedure. When resumming a series one should note that the values and the accuracy of the resummed results will depend on the procedure of resummation. Thus the analysis and appropriate selection of an adapted resummation procedure is crucial to obtain reliable results. However, there remains the principal question about the Borel summability of the perturbation theory series for a given model. Up to now a proof of summability exists only for the (j)^ theory with one coupling [85]. However, the field theoretical RG series for models with several couplings are generally analyzed as if they are asymptotically. Nevertheless, there exists strong evidence of possible Borel non-summability of the series obtained for disordered models [87-89]. The Pade-Borel resummation technique [90] is performed as follows: • Given the initial function in the form of truncated series in coupling u the Borelimage is constructed:

ta^u^^t'-^; i=l

(88)

i=l

• this Borel-image is extrapolated by a rational approximant [M/N] = [M/N] {ut); (89) here, [M/N] stands for the quotient of two polynomials in ut] M is the order of the numerator and N is that of the denominator; • the resummed function is then obtained by taking the inverse Borel image: /»oo

S'^'ix) = / dtexp{~t) [M/N] {ut). Jo

(90)

The Pade-Borel procedure can be optimized by introducing an additional fit parameter p to the Borel transformation. Substituting the factorial i\ by the Euler gamma-function r{i -\- p + 1) and inserting an additional factor t^ into the integral (90), one defines the Pade-Borel-Leroy resummation procedure. In order to suit the resummation procedure (88)-(90) for functions that depend on several variables one should change the first step (88): for example, for the two-variable case one defines the Borel image by [92]:

The rational approximation (89) can be performed then either in the dummy variable t or in the variables u, v. This defines two approaches that one can use to resum the functions

Renormalization group approaches to polymers in disordered media

127

of several variables. In the first case the resummation procedure is referred to as the PadeBorel resummation for resolvent series [94]. The application of the Chisholm approximants [93] which are the generalization of Pade-approximants to the many-variable case is necessary in the second case. A Chisholm approximant can be defined as a ratio of two polynomials both in variables u and v, of degree M and N such that the first terms of its expansion are equal to those of the function which is approximated. Again, the resummation is performed in eq. (90) replacing the Pade with the Chisholm approximant. This method will be referred below as the Chisholm-Borel resummation. It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. In the resummation schemes described above for series in several variables the variables are treated as "equal in rights". One may also relax this condition as in the recently proposed method of subsequent resummation, developed in the context of the d = 0dimensional disordered Ising model in Ref. [89]. Given the RG function f{u,v) as a power series in u^v (for the model (21), u is the original coupling of the undiluted system and v is the variance of the quenched disorder) one rewrites it as a series in the variable v:

with coefficients that are in turn series in the variable u and are to be resummed in advance by any one-variable method: An{u)

= J2Ck,nu'.

(93)

The main result of Ref. [89] is that the expansions of the coefficients (93) and the resulting series (92) at fixed u are Borel summable (for a c? = 0 dimensional system, however). This suggests to analyze the RG expansions of the d = 3 disordered models by first performing a Pade-Borel resummations of the corresponding series for the coefficients in one coupling and then, using the computed coefficients, resumming the series in the other subsequent coupling (s). 3.2. Unperturbed case Let us illustrate, how the renormalization scheme works, considering the m-component model in the limit m -> 0 on the pure (undiluted) lattice with the effective Hamiltonian (13). Here, we have only one coupling constant UQ, and the situation is rather simple. In the so-called one-loop approximation, corresponding to the first order of perturbation

128

y. Blavats'ka, C. von Ferher, R. Folk and Yu. Holovatch

theory in the renormalized coupling u, the massive scheme RG functions read [4]: /^uW

=

7^(w) = 0

-£uil--uli], ,702=s-w/i,

(94)

(95)

where e = 4 — d is the deviation from the upper critical dimension and the one-loop integral reads:

The next step is to find and analyze the stable fixed points (see (79) for definition) of the given y^-function. One gets: • The Gaussian fixed point u* = 0, which is stable at d > 4; • the pure SAW: u* ^ 0, stable at d < 4. The critical exponent u at Gaussian fixed point reads u = 1/2, which corresponds to the fact that SAWs at dimensions above the upper critical one behave like RWs (simple random walks). There are essentially two ways to proceed in order to obtain the qualitative characteristics of the critical behavior of the model. The first is to substitute the loop integral in equations (94), (95) by its ^-expansion:

/> = H^-|)'

(^^)

which finally leads to obtaining the expressions for critical exponents (via Eq. (81), (82)) as expansions in parameter 6. Another scheme is to consider the polynomials in Eq. (94), (95) and evaluate the integral for fixed dimension d. In the first case, one finds for SAWs on a pure regular lattice: Impure = 2 ^ 16'

^^^^

Currently, the RG function of the pure m-component model are known in the 6-loop approximation in fixed d-scheme [96], leading in three dimensions to the estimate [11] Impure = 0.5882 ± 0.0011. Critical exponents have also been calculated in the form of 6:-expansions up to five loops [97]. The estimate for the exponent i/ in this case in three dimensions reads [11]: Upure = 0.5878 ± 0.0011. 3.3. Uncorrelated disorder Let us consider the properties of the critical behavior of the model (21). Following the renormalization procedure the expressions for the RG functions are obtained as series in the renormalized couplings u and v. The massive RG functions of the model with uncorrelated disorder are known up to six-loop expansions [95] up to date. However, already the one-loop analysis of the model in the polymer limit has led to contradictory results concerning the existence of the stable fixed point in the system and thus of a

Renormalization group approaches to polymers in disordered media

129

phase transition [29]. Let us analyze the fixed point structure of the ^-functions of the m = 0-model, obtained in the massive scheme within the one-loop approximation. They read: 0u{u,v) = -e (u-

(-u^-^2uv]li]

,

(99)

I3y{u,v) = -6(v-

(3^^+3^^j^i) '

(100)

with /i given by (96). One obtains 4 fixed points, see Fig. 3,a: • Gaussian (G): u = O^v = 0 corresponds to ideal model without interaction; • polymer (Heisenberg) point(H): u^ 0,v = 0 turns out to be unstable; • unphysical (U): ix = 0, i; 7^ 0 is stable; • mixed (random) point (R): u^O,v

^0 unphysical and unstable.

a)

Figure 3. a) The lines of fixed points of the SAWs on a lattice with point-like uncorrelated disorder (see the text). b) Fixed points in a 4 + e expansion for d > 4 according to Le Doussal and Machta [39]

Let us recall, that only the region u> O^v > 0 is physically accessible in this problem. Thus, the situation is uncertain concerning the critical behavior of the system - there no physical stable fixed point, governing the universality class of the system. In Ref.[39] it was argued that while the upper critical dimension for the problem of polymers in disorder may be larger than dc = 4 one may analytically continue the eexpansion from d = 4 — 6 t o a 4 - h €-expansion. For the fixed points this has a dramatic effect (see Fig.3b). In the 4 -h e case the unphysical fixed point (U) is mapped to V in the physical region. It is interpreted by these authors as a strong disorder fixed point. Also

130

V. Blavats'kdy C. von Ferber, R. Folk and Yu. Holovatch

the random fixed point R is mapped to R in the physical region where it is interpreted as a multicritical fixed point. However, as it has been noticed in Ref. [62], in the double limit m, n ^ 0 the terms with UQ and VQ become of the same symmetry, joining these terms one passes to an effective Hamiltonian with only one coupling constant of the 0{mn = 0)-symmetry. This can be proved for the expressions of the one-particle irreducible vertex functions T^^'^\ Following Kim [62] we consider the so-called faithful representation of the familiar Feynman diagrams [4], representing the vertexes with coupling constants ito, VQ, which describe the interaction of the fields diagrammatically, as is shown in Fig. 4. Here

la

"JP

^

^

ia^

^JP

Figure 4. Interacting vertexes in diagrammatic representation. The Latin indexes denote field components, the Greek indexes denote replicas.

z, j = 1 , . . . , m are the field component indices. The diagrammatic representation of the two-point function T^^ (k) of the one-loop contributions is shown in Fig.5. Here, a solid line represents a propagator A:^ + Mo? loops correspond to the integration over an internal moment. We call disconnected any loop that is not connected to external lines only via propagator lines. If the diagram has a disconnected loop, the corresponding replica and field indexes are summed over and give contributions that are proportional to m and n, as it is shown in the figure. The analytical expression for the function Tia^^^k) corresponding to Fig.5, in the massive scheme reads: rfJik) = k' + ,,l+

(4m| + 8 | + 4nm^

+ 8 ^ )

/(MO),

(101)

here /(/xo) is the one-loop integral:

In the double limit m, n —>> 0 the diagrams with closed disconnected loops have zero contributions, and the expression (101) has the form: rl^' (k) = e+ nl + ^{uo-

vo)lM.

(103)

Similarly, in all orders of the expansion the vertex function diagrams, that contain closed disconnected loops with contributions proportional to m and mn, disappear in the

Renormalization group approaches to polymers in disordered media

0 J.

,

VQ

I

+

O vo

131

UQ-VQ

la (2)/M Figure 5. The one-loop contributions to the one-particle irreducible vertex function Ha (A:) in the diagrammatical representation.

double limit rriyn -^ 0. Thus the corresponding contributions become symmetric in UQ and VQ at all orders. In this way one can turn to the problem with only one coupling tto = UQ — -^o already at the level of the effective Hamiltonian (21). These considerations [62] solve the question of universality class of SAWs in media with uncorrected weak disorder. Passing to the problem with one coupling u one finds that fixed points are replaced by lines (Fig. 3). One has the line of the stable fixed points (containing points H and U) and non-stable (containing G and R). The universal behavior of the system is defined by the fixed point of the pure model H. In this way the RG confirms, that weak point-like uncorrelated disorder does not change the universality class of polymers. 3.4. Long-range-correlated disorder The effective Hamiltonian (29) is the starting point for a study of the polymer limit m —>^ 0 of the weakly diluted Stanley model with long-range-correlated disorder. Imposing the renormalization conditions of the massive scheme in the one-loop approximation leads to the following expressions for the RG functions [72]:

h'''

Pu = -e

I r ^{26-e)-wH^, 62uw r 72+3/4

+ e- [wuli — ^%],

(105)

o

72

"^ T

x'^

W

T

(104)

(106)

Here, besides the one-loop integral /i given by (96) we get three additional one-loop integrals, that depend on the space dimension d and the correlation parameter a: = /

(^qq',a-d

{q^+ir

I

o«2(a-d) dqq (92 + 1 ) 2

l4

=

_d_ f_dc,dqq',a-d

J [q +kf + l\ * 2 = 0

(107)

Note that in contrast to the usual (/>^ theory the 7^ function in Eq. (106) is nonzero already at the one-loop level. This is due to the /c-dependence of the integral I4 in Eq. (107). Similarly as explained in subsection 3.2, one can proceed either by considering the polynomials in Eqs. (104), (105) for fixed a, d and look for the solution of the fixed

132

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

point equations (it is easy to check that these one-loop equations do not have any stable accessible fixed points for d < 4), or by evaluating these equations in a double expansion in £ = 4 — d and J = 4 — a as proposed by Weinrib and Halperin [65]. Substituting the loop integrals in Eqs. (104)-(106) by their expansion in e = A- d and S = 4- a:

'-H'-l)''-K'-0''-^('-^)''-K^)' one obtains the 3 fixed points given in the table 3.4. The following conclusions may be Fixed Point u* W* UJi ^2 Gaussian (G) 0 0 -6 -6 Pure SAW (P) 6 0 6 6/2-5 2(J2 6{e-25) - 4e5 + 8J2} Long-range (LR) (e-5) l{£-45±y/£^(e-6) Table 3 Fixed points and stability matrix eigenvalues in the first order of the e^6 - expansion [72].

drawn from these first order results: Three distinct accessible fixed points are found to be stable in diflFerent regions of the a, cZ-plane: the Gaussian (G) fixed point, the pure (P) SAW fixed point and the long-range (LR) disorder SAW fixed point. The corresponding regions in the a, d-plane are marked by I, II and III in Fig. 6. In the region IV no stable fixed point is accessible. For the correlation length critical exponent of the SAW, one finds distinct values z/pure for the pure fixed point and I/LR for the long-range fixed point. Taking into account that the accessible values of the couplings are w > 0, ty > 0, one finds that the long-range stable fixed point is accessible only ioi 6 < e < 25^ ox d < a < 2-\- d/2, a region where power counting shows that the disorder is irrelevant. In this sense the region III for the stability of the LR fixed point is unphysical. Formally, the first order results for d < 4 read: \

= 1/2 + 5/16, 1/2 + J / 8 ,

2^LR =

6
. ^^""^^

The first line in (108) recovers (98), whereas the second line brings about a new scaling law. Thus, in this linear approximation the asymptotic behavior of polymers is governed by a distinct exponent I/LR in the region III of the parameter plane a, d. However, the region where the LR fixed point is found appears to be unphysical. Something similar happens if the e, (J-expansion is applied to study models of m-vector magnets with long-range-correlated quenched disorder [65,71]: also in the case of magnets, as well as for polymers the first order e, (^-expansion leads to a controversial phase diagram. In order to obtain a clear picture and more reliable information, one should proceed to higher order calculations. Here we pass to the analysis performed for fixed values of parameters d = 3, a. To investigate the peculiarities of the critical behavior in the 2-loop approximation one may

Renormalization group approaches to polymers in disordered media

1

2

3

133

4

Figure 6. The critical behavior of a polymer in a medium with long-range-correlated disorder in different regions of the d, a-plane as predicted by the first order 6, (5-expansion [72]. Region I corresponds to the Gaussian random walk behavior, in the region II scaling behavior is the same as in the medium without disorder, in region III the "long-range" fixed point LR is stable and the scaling laws for polymers are altered, in region IV no accessible stable fixed points appear; this may be interpreted as the collapse of the chain.

make use of the m —>> 0 limit of the appropriate m-vector model (28), investigated recently [66,67]. Starting from the two-loop expressions of Ref. [67] for the RG functions of the Stanley model with long-range-correlated disorder and making use of the symmetry arguments [62,72] in the polymer limit m = 0 one finds the following expressions for the d = 3 RG functions of the model described by the effective Hamiltonian (28) [71]: 95 1 3u{u, w) = -u + u^ - (3/i(a) - f2ia))uw - —u^ + -b2{a)u'^w (63(a) - -bG{a))uw'^ + h{a)w'^ + h{a)w^,

(109)

WW

P^{u, w) = - ( 4 - a)w - (/i(a) - f2{a))w'^ "^ "2" "^ ho{a)w^ 23 1 zlo 4

(110)

J4,{u,w) = -f2{a)w + —u^ -f- ci{a)w'^ - -C2{a)uw,

(111)

1 1 1 1 %2{u,w) = -u- -fi{a)w - —u^ - cs{a)w'^ + -C4{a)uw. (112) 4 z lo 4 Here, the coefficients fi{a) are expressed in terms of the one-loop integrals in Eq. (107), bi{a) and Ci{a) originate from the two-loop integrals and are tabulated in Ref. [67] for

134

V. Blavats^ka, C. von Ferber, R. Folk and Yu. Holovatch

LR^

|G

y

p..

[ ••• •;•

/ 2

3

5

6

Figure 7. The lines of zeroes of the 3d y^-functions (109), (110) resummed by the Chisholm-Borel method at a = 2.9. The dashed line corresponds to Pu — 0, the solid lines depict ^^ = 0. The intersections of the dashed and solid lines give three fixed points shown by filled circles at n* = 0,-^;* = 0 (G), tx* = 1.63, ly* = 0 (P), and u* = 4.13, w;* = 1.47 (LR). The fixed point LR is stable.

d = Z and different values of the parameter d in the range 2 ^ A ^ 3. The series are normalized by a standard change of variables u -> ^ / i , w -^ ^ / i , so that the coefficients of the terms it, iP' in ^^ become 1 in modulus. As discussed before, the question about the summability of the series in Eqs. (109) - (112) is open. In Ref. [71] various kinds of resummation techniques have been applied in order to obtain reliable quantitative results for the system under consideration and to check the stability of these results. First, a simple two-variable Chisholm-Borel resummation technique was employed, which turns out to be the most effective one for the given problem. In addition to the familiar fixed points G and P describing Gaussian chains and polymers (SAWs) on regular lattices, a stable long-range fixed point LR for polymers in long-range-correlated disorder was found. Fig. 7 depicts the lines of zeroes of the resummed /3-functions (109), (110) at a — 2.9 in the u, ly-plane in the region of interest are depicted. The intersections of these curves correspond to the fixed points. The corresponding values of the stable fixed point coordinates and the stability matrix eigenvalues for different values of the correlation parameter a < 3 are given in Table 3.4. To calculate the critical exponents the same resummation technique was applied. The numerical values for i/, 7 and 77 are listed in Table 3.4 for a — 2.3,..., 2.9. Note, that for a = 3, which corresponds to short-range-correlated point-like defects, the interactions u and w become of the same symmetry, so one can pass to one coupling {u — w) and reproduce the well-known values of the critical exponents for the pure SAW model. The numerical values corresponding to those listed in Table 3.4 in this case read: u* = 1.63, u = 0.59, 7 = 1.17, r; = 0.02, u = 0.64. As departing from the value a = 3 downward to 2 one notices a major increase of the value of the coupling li, so the results are more reliable for a close to 3. At some value a = amarg the LR fixed point becomes unstable. To verify the results, also the method of subsequent resummation was applied to the

Renormalization group approaches to polymers in disordered media

135

1/* a w* u a;i,2 ' rj 7 2.9 4.13 1.47 0.64 1.25 0.04 0.25 =b 0.62 i 2.8 4.73 1.68 0.64 1.26 0.04 0.22 ± 0.76 i 1 2.7 5.31 1.81 0.65 1.28 0.03 0.18 ± 0.89 i 2.6 5.89 1.87 0.66 1.29 0.03 0.15 lb 0.99 i 1 2.5 6.48 1.89 0.66 1.31 0.02 0.11 d= 1.09 i 2.4 7.10 1.87 0.67 1.33 0.01 0.07 ± 1.18 i 2.3 7.76 1.84 0.68 1.36 0.01 0.03 ± 1.26 i Table 4 Stable fixed point of the 3d 2-loop ^-functions, resummed by the Chisholm-Borel method, the corresponding critical exponents and the stability matrix eigenvalues at various values of a.

RG functions (109), (110). Here, the summation was carried out first in the coupling u and subsequently in w. Again, the presence of stable fixed point LR for a^ < a amarg) the polymer coil swells with increasing correlation of the disorder. The self avoiding path of the polymer has to take larger deviations to avoid the defects of the medium. 3.5. RG theory for SAWs on the percolation cluster In the following we elaborate the renormalization group scheme for the theory of SAWs on the percolation cluster as described by the effective Hamiltonian developed in section 2.4. Extending the ideas of Meir and Harris [22] in this respect we refer to this as the MH-model. The motivation for this model is to calculate the average of a logarithm - as usual for a quenched average-

F(p,/^)=.52[7ylogGy(if)]^

(113)

with logGij{K) the generating function of SAWs with a fugacity K per step between the sites i and j . The sites are connected if jij = 1 (else jij = 0). The average over bond occupation is denoted by [']p. As usual this can be done in terms of kth moments in the

136

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

limit A: -^ 0. MH derive that

G«(p,K) = E b « ^ « W ] p

(114)

J

= Xp + kF{p,K)-\-0{e)

(115)

where the susceptibility Xp is given by the k = 0 moment. In terms of a spin model this means that the MH model seeks to calculate the scaling behavior of the moments G^''\p,T)

= ^

[(7ii5i(i)5i(i))*]^

(116)

J

in the polymer limit with zero component spin fields S{i) averaging over the disorder at the critical concentration p = Pc and over spin configurations (•)r at temperature T. While the (^"^-theory discussed in section 3.3 does not provide such averages it is essential that these can be performed in the framework of the MH model. With the effective Hamiltonian derived in section 2.4 it turns out that the moments G^^"^ correspond to the propagators of this theory with masses r^ that reflect the fact that there is a distinct critical point associated to each moment, i.e. there is no multicritical point as in the spin models with finite numbers of components and as suggested by the d> 4 interpretation of the 0^ polymer theory [39] in sect. 3.3. In extracting the scaling behavior of the moments G^^^ or equivalently of the masses r^ the central quantities will be the terms linear in k in an expansion in k as suggested by Eq. (115). As far as the MH-model is a (j)^fieldtheory the diagrams and integrals that appear are identical to those of the standard 0^ field theory of the Q = 1-Potts model. However, we have to carefully deal with the overhead of spin component and replica indices and the algebra that results from their combinatorics. In the following we review how this is worked out in the two loop approximation in Ref. [107]. Only those 0^ cubic terms are allowed in the perturbative expansion of the theory for which any pair of indices (ckj, Pi) on the contributing factors appear exactly twice. In terms of the diagrams where each line carries the indices of the corresponding field (^ any pair of indices that "flows" into an interaction vertex must flow out on another line. Furthermore, the polymer limit of m = 0 components for the fi indices implies that diagrams in which some indices pi flow on closed loops i.e. do not appear on external lines, have vanishing contributions. As noted above we are interested in the k ^ 0 limit of the algebra that is generated by replica and component index summation together with the combinatorics of the mass parameters r^. It is a somewhat tedious but straightforward matter to show that with respect to the evaluation of the standard (j)^ theory (see [108]) there is no change to the contributions to the vertex functions T^^^ which controls the renormalization of the coupling WQ and the inverse propagator F^^^ in this limit. However, the contributions to the vertex functions r(^'2) with a mass insertion develops a characteristic behavior in linear and higher orders of k as we demonstrate in the following. Let us for simplicity consider the one loop contribution in detail. The one loop diagrams and their integrals contributing to F^^^ and F^^'^^ are labeled Ai and A[ respectively in Fig. 8. For any particular choice of the indices on the in- and outgoing lines of diagram Ai its contribution is the sum over all labellings of the inner lines that are compatible with the above rules. For k label pairs on the outer

Renormalization group approaches to polymers in disordered media

Q A'l

A2

A3

137

B3

B4

A;

X

K

A;

^K

Figure 8. Graphs of one- and two- loop contributions to 0^ - theory. Graphs Af. contributions to r(^\ Graphs Bi: contributions to r^^\ Graphs A[: contributions to P^^'^^.The insertion is marked by a dot. Graphs A' and B in the same column have identical contributions [108].

lines we have thus to sum over the possibilities to distribute those among the inner lines. Representing the masses Vk by an expansion rk = Yli^i^^^ ^^^^ combinatorics gives for the first order contribution to r(^^(p, r^) = 1 4- wlT^x^kiPt Ui) -\-...

^Mu.) = -\Y.^Qjd^ \{p- q)' + Ee uts'] [Q' + T,e ^k

- sY]

(117)

Evaluating this for renormalized critical mass Ui = 0 and taking the limit A: -> 0 one recovers the standard first order contribution to F^^^ The same applies to the mass insertion generated by the derivative ^ . However, we are interested in the behavior of terms of linear and possibly higher order in k. So we evaluate the general derivative

A|^^^,Ei,,(p, Ui) = i E Q (^' + (^ - ^)') / ^'^(P- ^~"^^' -.(1,2)

(118)

-^(1,2)

which defines the first order contribution to F ) ' \ We expand F^ ' ^ again in powers of k

r^'' = r ( f +
(119)

Then, with FQ Q we recover the standard mass insertion while the one loop approximation to Fi {^ is the term considered by MH. The generalized two loop contribution to the bare vertex function may be written as

T^lf' = l+wla^A[

+ wtZa'^]A'^ 9=2

(120)

138

V. Blavats'ka, C. von Ferber, R, Folk and Yu. Holovatch

with the integrals A'^ corresponding to the diagrams given in Fig. 8 and coefficients a^j derived from the combinatorics of distributing indices among the lines of the diagrams. This is calculated in terms of the sums [107]

f = IC)/

a,

(m,

,(2)

,(")

.(5)

J6)

=_ 5 £ (')!:(')£(';')('•+')'«'«».-'«»'-(.-«)

a^s) (127)

Here, the ^-factors with ^-^ = 1 for x > 0 else Ox = 0 control that only those terms contribute which have at least one index pair on each line. The A:-expansions of these series af^ = a[^Q + ^^ij + ^'^^£,2 + • • • then define the coefficients a^^J. For our considerations we need ao,o = (-1,1,4,1,1/2,1) ai,i = (-1/2,1/3,2,2/3,1/6,1/4)

(128) (129)

to calculate the two-loop approximation of FQ^Q ,ri^i . The minimal renormalization of these by ^^2-factors as explained in section 3.1.2

would the define exponents 7^^ of the percolation system for £ = 0 and of the polymer for ^ = 1. However, it turns out that such a multiplicative renormalization does not exist for Ti[\ We have thus to resort to an additive renormalization procedure by taking into account the coupling between the different orders in A;, i.e. the /.-linear contributions of r^^'^\ With a2,i = (1/4, -5/18, - 1 , -5/18, -5/36, -3/8)

(131)

we find that the k-lmear term of the two-loop linear combination

(ci,r(f+ c..ir(f)

(132)

Renormalization group approaches to polymers in disordered media

139

can now be renormalized multiplicatively by a Z-factor Z<^2^i if ci,i = 1 H- C2,i/2 and C2,i = 1/2 while preserving the one-loop result [101,107]. The /3-function and the 7^exponent are determined by the minimal subtraction scheme (see sect. 3.1.2) for the standard 0^-theory. With the renormalizing Z02^^-factors for the expressions in eqs.(130), (132) this the defines corresponding exponents 7^2 according to eq. (87). We find the standard result for 7V and [101] (133)

7^2^ = - 5 / 7 - 6 0 4 ^ 7 9 2 6 1 ,

Together with the standard result for 7^ and the scaling relation v ^ = 2 — 7^ — 7^2 this brings about the correlation length exponent for the polymer in the percolation system [101] •^^'^ = u„

(134)

1/2+ e/42 +1102721^

recovering the MH one-loop result and extending it to the second loop order. As far as the coefficients in this truncated series are small we may evaluate it without resummation. The numerical values of Up at fixed space dimension is then obtained by direct substitution Up{d = 2) = 0.785, i^p{d = 3) = 0.678, i/p{d = 4) = 0.595, which is in good agreement with results of MC simulations, exact enumerations and Flory-like formulae.

0.8

— 1 —

—'

—'—

—

1

—

—'—

^

—i—\—

•\

1

—'

\

0.75

^ \

0.7

\

• \

0.65

\

0.6

\ \

• \

\ H

0.55 0.5

—

2

2.5

3

3.5

4

4.5

5

-

1

—

-

5.5

--•11

6

d

Figure 9. The correlation exponent Up. Bold line: (133), thin line: one-loop result [22], filled boxes: Flory result Up = 3/(dpc + 2) with dpc froni [109]. Exponents for the shortest and longest SAW on percolation cluster [110] are shown by dotted lines.

Prom the physical point of view, this result for the exponent Up together with the data of EE and Flory-like theories predicts a swelling of a polymer coil on the percolation

140

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

cluster with respect to the pure lattice: z/p > I/SAW for rf = 2 — 5. At d = 3 the formula (133) predicts a 13% increase of z/p with respect to USAW which is larger than at d = 2 (5%) and should be more easily observed by current state-of-art simulations. Given that even at d = 2 we are in nice agreement with MC and EE data and the reliability of the perturbative RG results increases with d, this number calls for verification in MC experiments of similar accuracy. The validity of the MH model has been questioned in Ref.[39] by an argument which in our present formulation reads that the one loop contributions to Vff^ for higher i indicate the instability of the fixed point with respect to each parameter u^. These authors explain that it remains to be shown that the instability with respect to Ui with i > 1 does not influence the renormalization of the 9/9ui-insertion. To understand how this may be resolved in the present approach it is instructive to consider the insertion corresponding to a derivative in U2. For this case one finds in the same way as before a multiplicative renormalization for the A:^-term of [107]

^^+cs^'if) {C2A]

(135)

with coefficients C22 = 1 + C32/2 and C32 = 3/104 using the following results for the a-factors 0:2,2 = 0:3,2 =

(-3/4,11/18,3,17/18,11/36,5/8), (3/8,-19/36,-3/2,-11/36,-19/72,-9/16).

(136) (137)

This way we may define a corresponding exponent [101] 7(^2) = -3^/14 - 113443^71926288

(138)

related to the scaling behavior of higher order cumulants. The corresponding calculations for d/dui-inseTtions are straight forward. These lead to the following conclusions. With respect to multiplicative renormalization and thus to the RG flow we have instead of ^A; = uo-\-Uik -\-U2k'^ -{-... the generic expansion rk = uo-{- vi{cnk + C2ik^ + . . . ) + ^2(c22A:^ + C32k^ + ...)•

(139)

For this expansion one has multiplicative renormalization at the two-loop level for the 5/9i;£-insertions decoupling the flow of the v^ for different i. Explicitly,this means that the renormalization of the 9/9t;i-insertion is independent from that of d/dve for £ > 1. It may be speculated that this scheme can be continued to higher loop orders [107]. The different series parameterized by the V£ would then correspond to the fact that instead of a multicritical point we expect to find a series of critical points in this problem [113,22] generating a multifractal spectrum of exponents 7L • 3.6. Real space RG for polymers on diluted lattices Iterating the RSRG transformation e.g. for the Wheatstone bridge as described in section 2.4 one effectively operates on a hierarchical lattice, see Fig. 10. One may then investigate the influence of disorder on models defined on such lattices by RSRG methods.

Renormalization group approaches to polymers in disordered media

141

Figure 10. Iterating the Wheat stone bridge renormalization

Figure 11. Diamond lattice cell and generalization to b branches With respect to the Harris criterion it has been shown explicitly that while disorder is always relevant if the specific heat exponent is positive a > 0, it may also be relevant for some negative a [112,111], in particular for lattice cells in which not all bonds are equivalent. In a first attempt to study polymers on disordered lattices by RSRG methods [114] Derrida and Griffith constrain their investigation to directed polymers simplifying the lattice to the so-called diamond hierarchical lattice and its generalization to b branches, see Fig. 11. The latter model can be analytically continued to non-integer b and allows for a perturbative solution in terms of an expansion in e = 6 — 1. The authors calculate the ground state energy and the growth of its fluctuation with polymer size L which increases like L^ with an exponent oj. In Ref.[115] Cook and Derrida extend this study to finite temperature where the renormalization acts on energy or partition function distributions. The temperature enters only the initial distribution of the partition function. Using higher order approximations a high temperature phase is identified for 6 > 2 where the energy distribution approaches a delta function as the number n of renormalization iterations increases. In a separate low temperature phase the width of this distribution grows with n. However, for the transition temperature Tc between these phases only an upper bound could be constructed. The main advantage of the RSRG method for the problem of polymers or SAW on disordered lattices is that it provides means to study the crossover between different universal scaling behavior in terms of an RG flow. In the following we first introduce the RSRG flow for this problem as proposed by Meir and Harris [22] (MH). We then proceed to discuss the critique brought about on this approach by Le Doussal and Machta [39](DM). We follow [22,116] in defining the generating function G,,(i^)-^C;vi,(i^)ii:^

(140)

N

of SAWs between sites i and j on the lattice, where

CMIJ

denotes the number of SAWs

142

V. Blavats'ka, C. von Ferbery R. Folk and Yu. Holovatch

between i and j and K is the fugacity for each step. Indicating if sites i and j are connected by the diluted lattice or not by 7^ = 1 or 0, the expansion of the log-average of Gij{K) around K — KQ reads [7ylogGy(ir)]p «

[^^^\ogGii{K,)]^+^^^Nij{K,)

Wii\LNij{K) =

(141) (142)

Here, KQ is chosen such that the first term on the right hand side of eq.(141) vanishes. The correlation length exponent Up for the SAW on the diluted lattice is the defined by the power law scaling for the end-to-end distance Rij between the sites i and j : Nij ~ Rli"'. For the Wheatstone bridge log Gij of the renormalized bond is equal to logK. recursion relations for log K and p read [22] p' = 2p2 + 2 / - 5p^ + 2p^ p'logJ^' = 4p'^\ogK + 6pHogK-\-p^{\og2-U\ogK-{-4\og{K-\-l)) +p^6\ogK -S\og{K + 1))

(143) The (144) (145)

where the first relation is the standard one for the Wheatstone bridge [106]. The flow diagram that results from these recursion relations is depicted in Fig. 12. The fixed points marked A (p* = 1,K* = 0.366), B {p* = 1/2, K* = 0.788), and A {p* = 0,K* = oo) are interpreted to correspond to SAWs on the pure lattice (A), at the percolation threshold (B), and on lattice animals (C). The corresponding crossover exponents (/> together with the 2D-percolation correlation exponent upc = 4/3 give the SAW correlation exponent Up = upc/(l>. At the fixed point B this reads (j) = 1.682. The RG fiow in Fig. 12 also brings about two more fixed points D and E which are interpreted as corresponding to minimal and maximal SAWs with (j)jnm = 1.55 and ^max = 1835. An additional fixed point Ci {p* = 0,K* = 1) is found in Fig. 12 which apparently was overlooked by MH. In a similar approach on the same lattice DM derive recursive relations involving distributions instead of the constant fugacity K on each bond visited by a SAW which corresponds to a bimodal distribution. This relation for the distribution is Z' = Z1Z2 + Z3Z4 + Z1Z5Z4 + Z3Z5Z2.

(146)

Note that also eq. (145) can be decoupled from (144) in terms of the variable pK: p'K' = 2p''K^ + 2p^K^

(147)

corresponding to all Zi = pK. DM criticize the approach of MH which projects back to the bimodal distribution after each renormalization step. The full RG flow derived by DM from these equations is depicted in Fig. 12a. Here, there is one more parameter, namely the variance A(logZ) of the bond energy distribution which grows with the size of the SAW according to

Renormalization group approaches to polymers in disordered media

,D 1«

K

« « m ^ i k i k % t

(logZ) x/1+AlogZ

b)

<

i t r r^^rtf f t t f r f f f t \

0.8

r

ftftnttr ttttmtX trrrrrrtr

o.e

L

r

tftttfttW Uttftttf

Ci

143

IB

[ -

- . -

- .

i

>x^* * * ^ f t

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P-Pc

( k V ^ <»rvi^^'

%im m

a

\ \ \ \ ss

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1

Y

Y •,',*•, r r » , r r r .• .1

•

-

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1

•»••'

v

AlogZ.

•E

Figure 12. a)RSRG flow derived from Eqs. (144) and (145) [22]. b) Full RSRG flow derived from Eqs. (144) and (146); the critical manifold is shaded [39].

A(logZ) ~ L'^. It appears that the plane A(logZ) = 0 where the MH flow was defined is unstable. Instead two strong disorder fixed points Sp^ and Si appear at infinite A(logZ) and the fixed points D and E corresponding to the maximal and minimal SAWs in this picture are also shifted to infinite A (log Z). DM derive exact values for their corresponding C-exponents that are related to v via 1/u = l/C^—uj. These values, Cmin = log2/log(17/8) and Cmax = log 2/ log(39/16) are again in accordance with previous numerical results [117]. At the strong disorder fixed point Sp^ at percolation p ~ Pc however, DM find C = 0.850 which is very close to C = 0.848 found at the fixed point B of the MH flow diagram. We conclude that apparently the MH RSRG approach oversimplified the situation. However, a systematic scheme for the strong disorder fixed points that would include an upper critical dimension dc = 6 and in particular a field theoretic model that could reproduce the complete RG flow remain to be found. Acknowledgments: R.F. and Yu.H. acknowledge the Austrian Fonds zur Forderung wissenschaftlicher Forschung for support under project N 16574-PHY. REFERENCES 1. G.-R de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London, 1979. 2. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Wiley & Sons, New York, 1959. 3. E. Brezin, Le Guillou J C, and Zinn-Justin J, Field theoretical approach to critical phenomena, in Phase Transitions and Critical Phenomena, edited by C. Domb and

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