Critical behavior of loops and biconnected clusters on fractals of dimension d<2.

Critical behavior of loops and biconnected clusters on fractals of dimension .

Dibyendu Das Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India    Supravat Dey Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India    Jesper Lykke Jacobsen Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75321 Paris cedex 05, France Institut de Physique Thorique, CEA Saclay, 91191 Gif sur Yvette, France    Deepak Dhar Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai - 400005, India
July 11, 2019

We solve the O() model, defined in terms of self- and mutually avoiding loops coexisting with voids, on a -simplex fractal lattice, using an exact real space renormalization group technique. As the density of voids is decreased, the model shows a critical point, and for even lower densities of voids, there is a dense phase showing power-law correlations, with critical exponents that depend on , but are independent of density. At on the dilute branch, a trivalent vertex defect acts as a marginal perturbation. We define a model of biconnected clusters which allows for a finite density of such vertices. As is varied, we get a line of critical points of this generalized model, emanating from the point of marginality in the original loop model. We also study another perturbation of adding local bending rigidity to the loop model, and find that it does not affect the universality class.

05.45.Df, 64.60.Ak, 05.50.+q, 64.60.Cn

I Introduction

The loop model is a very important model in statistical physics. It was defined originally in terms of the high temperature expansion of the -vector model Mukamel (); Nienhuis82 (); Baxter86 (). The cases of correspond to well-studied cases of self-avoiding polymers deGennes (), the critical Ising and the XY models Zinnbook (), respectively. The model has been studied quite extensively in dimensions, in several variants, including fully packed or dilute versions, and with loops of more than one type. One can determine the critical exponents of the model on the hexagonal and square lattices using the Bethe Ansatz technique, the Coulomb gas method, and numerical techniques involving exact diagonalization of transfer matrices of systems on finite width cylinders Nienhuis82 (); Baxter86 (); Saleur86 (); Batchelor88 (); DupSal87 (); Nienhuis92 (); Kondev98 (). The critical behavior of the model is also related to the dimer model, edge coloring model, compact polymer model (for ) Batchelor94 (); Kondev95 (); Kondev98 (), and the hard hexagon model (for ) guo (). In the presence of a staggered field, the model gets related to the critical Potts model das ().

The loop model to be studied here is defined by the partition function


where the summation is over configurations of self- and mutually-avoiding loops on a lattice. In the above, is the weight of a loop, is the weight of a vacancy (i.e., a lattice vertex not visited by any loop), denotes the number of loops in a given configuration, and the number of vacancies.

In Eq. (1), large values of correspond to a small average density of loops, and thus to the high-temperature phase of the -vector model. As is decreased, the average density of loops and the mean loop size increases, and diverges as tends to an -dependent critical value . The critical behavior of the loop model at gives the critical behavior of the -vector model at its critical point in . For and , the model shows a critical phase, which is called the dense phase. In , the critical exponents of the dense phase have been determined exactly Nienhuis82 (); Batchelor88 (); DupSal87 (); Nienhuis92 (). These vary with , but are independent of the precise value of .

The critical behavior of the loop model, and exponents of the dense phase have been less studied for . One expects that the dense phase of the loop model for would be related to the the low-temperature phase of the -vector model. The latter shows power-law correlations in the entire low-temperature phase because of the existence of gapless Goldstone modes. In , Jacobsen et. al. read () have shown that allowing loops to intersect or not leads to different critical behavior, and the dense phase of the loop model is different from the Goldstone phase.

Figure 1: The construction of the -simplex lattice is shown from level to and recursively for higher .

It therefore seems interesting to study the loop model in dimensions other than . In this paper, we study the loop model on fractal lattices with finite ramification index. We will take the -simplex lattice (see Fig. 1) as the simplest example of this type. The treatment is easily extended to other fractals. The loop model shows a nontrivial critical behavior for on this fractal, and we determine the critical behavior near the dilute critical point . We also study the critical properties of the dense phase. Further we define a generalization of the loop model, to be called the biconnected clusters model—or “bicon clusters model” for short—in which we allow the summation in the partition function to include all biconnected clusters, as shown in Fig. 2. A cluster is called bi-connected, iff between any two points of the cluster, there are at least two disjoint self-avoiding paths. Clearly all simple loops are biconnected. It turns out that this changes the critical behavior of the model, and we determine the new critical exponents.

Figure 2: (a) A typical configuration of the loop model. (b) A typical graph showing several biconnected clusters that contribute to the partition function of the bicon clusters model.

A variant of the bicon model is known as the “net model” Fendley (). This model is relevant for the study of quantum models whose ground states are endowed with topological order. More generally, the constraint of -connectedness has also been imposed in classical models of clusters, such as percolation Zinn () and the Potts model Deng ().

There have been several studies of critical behavior of statistical mechanical systems on finitely ramified fractals kenevic (). The Ising model was the earliest to be studied on fractals nelson (). It was followed by a study of self-avoiding polymers on a -simplex lattice dhar2 (). Later self-interacting self-avoiding polymers klein (), and other trails Chang (), have been studied. The Lee Yang edge singularity for Ising model was considered in kenevic1 (). The collapse transition of branched polymers was studied in kenevic2 (). The distribution of sizes of erased loops for loop-erased random walks was studied in abhishek (). In sumedha (), it was studied how the number of self avoiding rings going through a site varies with the position of the site. Polymers with bending energy were studied on -simplex and other fractal lattices in maritan (). For a recent review, see dhar3 ().

The detailed plan of the paper is as follows. In Section II we find the critical branches of the loop model, and in section III the exponents are calculated. In section IV we study the generalized “bicon model”, as mentioned above. In section V we study the loop model with extra energies for local bending. In section VI we summarize our results.

Ii The critical branches of the loop model

The recursive construction of the -simplex fractal lattice, through a series of levels , is illustrated in Fig. 1. Let be the partition function at level , i.e. the sum of statistical weights of all configurations with loops contained within level , and denoted by an empty triangle in Fig. 3. We may have loops which close at levels higher than . Such a loop will come out of the corner vertices of an -th level triangle. The restricted partition function for configurations with such an open chain passing out of the corner vertices of a -th level triangle is denoted by and shown in Fig. 3.

Figure 3: The schematic representations of various restricted partition functions , , , and for an -th level triangle.

These functions and can be recursively related to their counterparts at the -th level by the following equations:


At the -th level the lattice has just a single site. In that case is the weight of it being empty, and is the weight that a loop passes through it. If we define , then the recursion relation for the latter is


At the -th level, . The total free energy on an infinite lattice is


Eq. (5) above may be rewritten, using , and after a little algebra we obtain


which is the usual form Cardy () of the recursion relation for the free energy under the real-space renormalization group (RG). Apart from , the finite-valued fixed points of the RG flow (4) are the roots of the following equation:


The special case corresponding to self-avoiding polymers was studied earlier dhar2 ().

Eq. (7) has three real solutions for , and only one real solution for the regions and (see Fig. 4). The three fixed points for the region are given by the formulae


where . The three solutions correspond to the following subdivision of the parameter range of :

  • For , runs from to , and from to . We shall refer to this line of fixed points as the dilute branch and denote it by . It is physical () for .

  • For , runs from to , and from to . We shall refer to this as the dense branch and denote it by . It is physical ( and ) for .

  • For , runs from to , and from to .

    This unphysical critical point at negative may be called the Yang Lee edge singularity in our problem. We shall denote it by . However, we note that the critical behavior at this ”Yang Lee edge singularity” is not independent of .

The three branches , and are shown in figure 4. Note that unlike the two-dimensional loop model, where the critical region between the dilute and dense branches extends up to , the domain of criticality of the present model is smaller. But here too, the upper branch is a repulsive fixed line. Starting from any the RG flows take one to while starting from any the lower branch serves as a line of attraction of the RG flow. We will refer to the region with as the dense phase.

For , the dilute branch can be extended backwards by replacing (with ), i.e replacing “” by “” and by in Eqs. (8) and (9). Although this regime is unphysical, we will show in section III.4 that the point on it is of some signficance.

For , the nontrivial fixed point other than is negative, and is obtained by replacing (with ), i.e. replacing “” by “” and by in Eqs. (8) and (9). There is no nontrivial critical point for positive . It can be shown supravat () that the loop model for can be exactly mapped to a weighted -vertex model, which also can have no nontrivial critical point for positive weights in this regime.

Figure 4: A schematic picture of the three lines of fixed point , and plotted against .

Iii The exponents of the loop model

In two dimensions, for a generic point lying on the dilute branch or in the dense phase, the loop model is known to have an infinite set of exponents characterizing its critical behaviour Kondev95 (); saleur (). The full set of principal exponents consists of an infinite number of correlation exponents associated with defects called the -string defects, plus one thermal exponent.

A -string defect has an integer number of “open chains” originating from a local neighborhood and terminating in an anti-defect very far away. Since chains are disallowed in the original loop model, such structures may be imagined to be created in isolation, as defects, upon application of suitable small external fields. A closed loop with two points marked on it is equivalent to a -string defect, and this observation can be used Kondev95 (); saleur (); das () to find the fractal dimension of the closed loops in the loop model. Although there is no upper bound on for ordinary Euclidean lattices, on fractals due to finite ramification, the exponent spectrum becomes finite. For the present model on the -simplex fractal lattice we can have such defects only with . [On the Sierpinski gasket, one could have .]

Apart from the -string defects, the loop model also has the thermal defect and different exponents associated with it, namely the specific heat exponent, the correlation length exponent, etc. Among the latter only one is independent and others can be related to it by identities. Thus there are four independent exponents for the loop model on the -simplex lattice, three -string exponents plus one thermal exponent, and we calculate them in the following subsections III.1-III.4.

iii.1 The exponents related to thermal excitation

Let us define as the “thermal distance” from the critical point . Starting from , under succesive iteration of Eq. (4), the separation increases with increasing . We note that . Using Eqs. (4) and (7), we easily find that with


The thermal exponent is defined as usual by the fact that at the -th step of RG transformation, the thermal distance . Thus comparing the two expressions for we obtain


The specific heat exponent may be defined via the scaling behavior of the non-analytic part of the free energy, namely . Substituting the latter scaling form into Eq. (6) and comparing the singular terms, a straightforward calculation yields


where is given by Eq. (10).

Thermal fluctuations create thermal defects and the correlation function for two such defects separated by distance is defined for as


The correlation length is finite for and the scaling relation defines the correlation length exponent . Under RG, if , since it follows immediately that


For , in Eq. (13), and , where is called the thermal correlation exponent at criticality. From standard RG arguments Cardy () it can be shown that , where


is the box dimension of the -simplex fractal lattice. From Eqs. (12) and (14) we see that the hyperscaling relation holds.

Apart from the exponents of thermal origin discussed above, there are several other exponents caracterizing the behavior of other physically interesting observables, which can be expressed in terms of the exponents described above:

  • The fractal dimension of the loops given by relates the length and radius of a loop as . The exponent is distinct for the two critical regions (dilute branch) and (dense phase). We show in section III.3 that they can be easily related to the -string correlation exponent .

  • For , the distribution of loop sizes has an exponential cutoff . The cutoff length , where is the finite correlation length in the disordered phase. One can find as in the derivation of above.

  • For the critical regimes and , the probability distribution of loop size is an unbounded power law . The exponent is distinct for and , and it is easy to express the exponent in terms of the exponents and the fractal dimension of the loops (see section III.3).

The exponents like and can all be expressed in terms of the “-string defect” exponents which we derive in the following three subsections.

iii.2 A -string defect

Let the application of a small external magnetic field create an open chain, or a -string defect, with the magnetic scaling exponent associated with it. Assuming for the system is critical, a finite introduces a finite correlation length given by


Eq. (16) defines the scaling exponent . In real space RG, upon coarse-graining from level to level , increases by a factor of , while increases by a factor of , i.e. . Below, we will calculate , and using this with Eq. (16), one gets


The probability saleur () of strings originating at and ending at in the critical phase is given by


¿From standard RG analysis it is known Cardy () that the correlation exponent is related to via the relation:


Thus for our -string defect, by using Eq. (17) and box dimension (15), we get


Now we proceed to obtain the scale factor . First we note that an -string defect may have its one endpoint inside an -th level triangle. The corresponding restricted partition functions and are represented by diagrams and in Fig. 3. The recursion relations for , , and are as follows:


We can linearize Eqs. (21)–(22) around the fixed point , and we are left with the matrix


with . The largest eigenvalue of the matrix is and is given by


Note that in Eq. (25), must be replaced by or in order to describe the dilute or dense branch, respectively. Using Eq. (25), we may thus write . It is clear that is proportional to the small external field , and thus using Eq. (25), we may read off the values of and in Eqs. (17) and (20).

Another exponent of interest is associated with the approach from above of the dilute critical branch. It is defined via the scaling behavior of the average open chain length . In general,


where is the number of distinct configurations each with an open chain length in a lattice of size , and is the relative weight factor for such a configuration normalized by the partition function. On our fractal lattice, this becomes


where the statistical weight of fully containing an open chain at the th level is . For a fixed and small distance from the critical point , since is finite, the sum in Eq. (27) is sharply cut off at some finite level . We note that for , , and , and (see Eq. (10)). It immediately follows from the scaling relation that


As a check of consistency, we may verify the following exponent equality saleur (), using Eqs. (28), (20) and (14),


iii.3 A -string defect

Let a triangle at the -th level which has endpoints of -strings on two neighboring sites inside it, be defined to have a statistical weight . Note that this weight is different from (see section III.3 above), since the latter puts no restriction on the location of the endpoints. Further defining , we easily see that its recursion equation is


Replacing with the fixed point value in the above equation, we get , with


where we have used Eq. (7). Note again that in Eq. (31) for we have to use for the upper branch, and for the lower branch. If a small field creates a -string defect and the finite correlation length arising due to it is , we find that the scaling exponent for a -string defect is


¿From Eq. (19) we conclude that the corresponding correlation exponent is given by


It is important to note that is also the fractal dimension of the loops. This is explicitly seen as follows. Let be the typical length of a segment of a loop that goes from one corner vertex to another in one -type triangle (see Fig. 3) of order . Then the typical length of a loop which closes at the -th level is . The recursion for is:


The above Eq. (34) follows from the fact that at level , a typical loop length can be made of two segments with statistical weight , or three segments with statistical weight vani_df (). On the other hand, the typical diameter of such a segment of length is given by . Combining these two results and using the definition of given by the scaling behavior , we conclude that


We note that , through , is distinct for (dilute loops) and (dense loops). For and in the dense loop phase, , and so coincides with the box dimension of Eq. (15). This was to be expected, since the limit of dense loops () in fact means that there is a single loop covering the entire lattice, i.e., the loop is Hamiltonian. For dilute loops, we have , whence and .

The probability distribution of loop size is an unbounded power law for both the dilute branch and the dense phase. The exponent can be related to and following a general derivation as in kondev_loop (). Let be the probability that two points separated by are on a loop of size . The expected scaling form of is:


Firstly, the sum of over all the space, i.e. , is nothing but number of points on the loop . This gives a relation . Secondly, by definition (see Eq. (18)) we have . The latter relation combined with the former gives


which is the desired result.

iii.4 A -string defect:

Let a triangle at the -th level which has endpoints of -strings on three neighbouring sites inside it, be defined to have a statistical weight . The recursion relation of is


Assuming that a small field creates a -string defect and the corresponding finite correlation length , we find that




The correlation exponent corresponding to is given by Eq. (19) with .

We now note something very interesting. If we extend the upper dilute critical branch to negative values of , we find that at the special point , we have and (see eqs. (39) and (40)). Thus although the -, - and -string defects are relevant for general on both the critical branches, at the point the -string defect becomes marginal. In the next section IV, we define a more general model called the “biconnected cluster model” which allows for vertices of degree 3, each occuring with a finite weight . We find that the latter model has a new critical line in its larger parameter space, which precisely meets with the line of critical points corresponding to at the point on the extended dilute branch.

Iv The biconnected cluster model

The finding in section III.4 that at a point in the parameter space, motivates our study of a model called the biconnected cluster model in which the allowed vertex degrees are 0, 2 and 3. In other words, -strings are allowed to emerge from any vertex (see Fig. 5(a)). The connected components are further required to be 2-connected, i.e., they cannot be disconnected upon cutting a single link (see Fig. 5(b))—note that this is a stronger requirement than simply disallowing vertices of degree 1. Henceforth we will refer to this model as the “bicon clusters model”, to distinguish it from the “loop model” studied this far in the paper. A similar model—with no requirement of biconnectedness—was studied in two dimensions in Fendley () under the name of “net model”. The partition sum for the bicon model is


where the summation is over configurations of any number of self-, and mutually-avoiding biconnected clusters. Here is the weight of a cluster of any size, is the weight of an empty vertex, and is the weight of a vertex of degree 3. Further, denotes the number of clusters, the number of vacancies, and the number of vertices of degree , in a given configuration. Note that by setting , the bicon clustes model reduces to the usual loop model. Some of the configurations that we exclude from the model (see Fig. 5(b) for an example) will be treated as defect configurations called “-defects” in section IV.2 below.

Figure 5: (a) The structure with thick dark lines is a possible cluster in the bicon clusters model. (b) This is a disallowed cluster as it can be disconnected into two pieces by cutting one link.
Figure 6: The three vertex weights for the bicon clusters model are shown in the top row. In the bottom row, the three schematic representations of the restricted partition functions are shown.

The three possible vertex configurations with weights , and are shown in Fig. 6. Note that if is the number of vertices of degree , we have the simple topological identity , where is the total number of links in the configuration. Therefore is necessarily even. Accordingly, the bicon clusters model partition function is a function of . At the -th level, the real space RG closes for three restricted partition functions schematically shown as , and in Fig. 6. While is the partition function summing over configurations with no strings coming out of the corner vertices, and have two and three strings coming out of the corners, respectively. Note that the constraint of biconnectedness implies that it is not possible to have a configuration with one string coming out of the corner vertices. At the level ,


At the -th level, the recursion relations are:


Further defining , and , we get


Note that and . There are several fixed points of Eqs. (46)–(47). Apart from the trivial weak coupling fixed point and the fixed point corresponding to all bonds being fully covered, there are the dilute and dense fixed points of the loop model which we denote by and , respectively. But most importantly there is a nontrivial fixed point given by,


The latter defines a new critical line in the space, which terminates on one end at the point ; note that this is the point where we found to be marginal for the loop model in section III.4. The dense-phase fixed point is unstable to introduction of trivalent sites, and the limiting behaviour of the critical net model is governed by the fixed point .

If the starting value of and is near and , but a bit larger, it is easy to check that and diverge to infinity as and respectively. This implies that in the dense phase of the bicon clusters models, corner sites of triangles of high order belong to the infinite cluster with a large probability, and this probability tends to as tends to infinity.

iv.1 The fractal dimension of the biconnnected clusters

Just like in the loop model, we would like to find the fractal dimension of the clusters in the bicon clusters model at its critical point. Consider two marked points and on a cluster, free of tadpole overhangs. Imagine that these marks are “defects” created by some external field . In addition to and , we now define new functions and which are analogous to and , except that they represent configurations in which there is exactly one marked point. At level we have:


The recursion relations are as follows:


The largest eigenvalue of the matrix obtained by linearizing the above equations around the fixed point of Eq. (48) is


and the fractal dimension of clusters on the new critical line is


The scale factor and thus , the fractal dimension of the loops (see Eq. (35)), at the point , as expected.

iv.2 The -defects

What kind of defects are natural extensions of -string defects, appropriate to the net model? A possibility is what we call the “-defects” shown in Fig. 7. A -defect has multiple clusters connected in a series by strings, and the two dangling ends are marked; they are depicted in short-hand as filled black blobs (see Fig. 7). The obvious motivation for defining such defects is that as , they become our usual “-string” defects in the loop model. The restricted partition functions contributing to each -defect is shown alongside the defects in Fig. 7.

Figure 7: The -, - and -defects are shown along with their shorthand representations (filled black blobs). The restricted partition functions contributing to the three defects are shown schematically on the right.

-defect dimension: In Fig. 7, , , and represent the relevant restricted partition functions for the real space RG of a system containing a -defect. At the -th level, (where is an external field which gives rise to this defect) and . The recursion equations for , , and (scaled by as usual) are as follows:

Table 1: Some values of versus associated with the -defect.

Solving the matrix obtained by linearizing the above equations about the fixed points in Eq. (48), the largest eigenvalue , gives the scaling dimension associated with a -defect:


Some values of versus are given in Table . Note that upon approaching the point , we have , i.e., we recover the scaling exponent of the -string defect in the loop model, as expected.

-defect dimension: Again the relevant restricted partition functions are shown in Fig. 7 by representative symbols and . The recursions for the latter scaled by are as follows:


Assuming the scale factor and independent of . Putting the critical point value (Eq. (48)), we get:


-defect dimension: For this defect the relevant restricted partition function is shown in Fig. 7 and is represented by the symbol . The recursion relation for is:


and the corresponding scale factor on the critical line (Eq. (48)) is


Thus for , the scale factor and hence the -defect is irrelevant on the entire new critical line (Eq. (48)). This was certainly to be expected, since it is exactly this -defect that induces the flow from the line to the new critical line. At the point , we have as expected, since for the loop model at that point.

V The loop model with local bending energy

The problem of self-avoiding polymers with bending energy has been of long standing interest, and was first introduced by Flory Flory (). With high energy cost for bending, the polymer is in an ordered state (with minimal bending), while reducing the energy cost leads to a disordered (but critical) state. The nature of the phase transition separating the two phases was unclear for a long time, and finally it was shown recently that for a compact (i.e., space filling) polymer on a two-dimensional square lattice the transition is continuous kondev04a (); kondev04b (). The latter works actually dealt with the full loop model and obtained the relevant results for the polymer by taking the limit. In a similar spirit we would look at the loop model for general with local bending energy on -simplex fractal lattice; unlike kondev04a (); kondev04b () our loops are not compact. We note that the limit has already been studied earlier in maritan () where it was found that the bending energy is irrelevant and no new fixed points appear in the extended phase space. We show below that the same is true for general .

The model to be studied here is defined as


On a -simplex fractal lattice, at any vertex, a loop can go straight, or bend by , or (see Fig. 8). Accordingly we define local vertex weights , , respectively, for the three cases. Correspondingly, the number of vertices covered with loops having and