Evidences of conformal invariance in 2d rigidity percolation Nina Javerzat SISSA and INFN Sezione di Trieste via Bonomea 265 34136 Trieste Italy

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Evidences of conformal invariance in 2d rigidity percolation

Nina Javerzat∗

SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy

Mehdi Bouzid†

Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000, Grenoble, France

(Dated: October 13, 2022)

The rigidity transition occurs when, as the density of microscopic components is increased, a

disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to

the appearance of a space-spanning rigid connected component, or cluster. As a continuous phase

transition it exhibits a scale invariant critical point, at which the rigid clusters are random fractals.

We show, using numerical analysis, that these clusters are also conformally invariant, and we use

conformal ﬁeld theory to predict the form of universal ﬁnite size eﬀects. Furthermore, although

connectivity and rigidity percolation are usually though to belong to diﬀerent universality classes and

thus be of fundamentally diﬀerent natures, we provide evidence of unexpected similarities between

the statistical properties of their random clusters at criticality. Our work opens a new research

avenue through the application of the powerful 2D conformal ﬁeld theory tools to understand the

critical behavior of a wide range of physical and biological materials exhibiting such a mechanical

transition.

Introduction – Symmetries are the cornerstone to un-

derstand and to model physical phenomena [1], and their

identiﬁcation, a powerful guiding principle for deriving

physical laws. Indeed, the compatibility between sym-

metries often results in constraints on the physical prop-

erties of the system: for example the compatibility of dis-

crete translations and rotations in crystals leads to the

crystallographic restriction theorem, which classiﬁes all

patterns of periodic discrete lattices one can encounter

in nature [2]. But symmetries are not only deterministic:

second order phase transitions are a paradigmatic exam-

ple of systems possessing a symmetry of random nature,

where the long-range statistical ﬂuctuations are invariant

in law under change of scale. For a host of systems ex-

hibiting critical behaviour –as diverse as linear polymers

[3], graphene membranes [4], disordered systems [5], a

larger symmetry emerges and ﬂuctuations are also invari-

ant under local rescalings i.e. under all geometrical trans-

formations that preserve angles and rescale distances,

called conformal transformations [6]. The emergence of

this enhanced symmetry is a powerful tool: exploiting

the compatibility constraints on the physical observables

allows to understand and predict the universal features

of phase transitions [7], and even in some cases to fully

characterise the scaling limit [8]. The origin of conformal

symmetry is however still not systematically understood

[9], even in two dimensions. Indeed, while in 2d unitary

systems conformal invariance is automatically implied by

scale invariance [10, 11] this is not anymore true for non-

unitary phenomena, of which percolation is maybe the

most representative and versatile example. Still, percola-

tion in its various forms is believed (in some cases proven)

to be conformally invariant, for instance: uncorrelated

∗njaverza@sissa.it

†mehdi.bouzid@univ-grenoble-alpes.fr

(Bernoulli) percolation [12], the random Q−states Potts

model [13], percolation of random surfaces [4, 14], and to

our knowledge there is no equilibrium percolation model

which has been shown to be scale but not conformal in-

variant.

In this context, rigidity percolation (RP) is an ideal

model to study the possible emergence of conformal sym-

metry. On the one hand, establishing the conformal in-

variance of this phase transition, of prominent impor-

tance in soft matter, may allow to better characterise its

still poorly known universality class. On the other hand,

it is the ﬁrst time that conformal invariance is studied in

a percolation phenomenon of mechanical nature (a priori

distinct from the ”connectivity percolation” (CP) mod-

els mentioned above), and this might shed some light on

which features of a percolation model make its scaling

limit conformally invariant.

Rigidity percolation in central force random springs

models provides a generic theoretical and simple frame-

work to study how a system transitions from a liquid to

a solid phase, where the underlying building blocks as-

semble into a percolating cluster that is able to transmit

stresses to the boundary and sustain external loads. It

has been successfully used to highlight the structural and

mechanical properties of many soft materials such as liv-

ing tissues [15], biopolymers networks [16, 17], molecular

glasses [18], stability of granular packings [19–21] or col-

loidal gelation [22–24]. Several critical exponents, char-

acterising the long-distance critical behaviour, have been

numerically determined, such as the correlation length

exponent ν= 1.21 ±0.06 and the order parameter expo-

nent β= 0.18±0.02, deﬁning an a priori new universality

class [25]. Hyperscaling relations also give the fractal di-

mension of the rigid cluster as df= 2−β/ν = 1.86±0.02,

a value which was conﬁrmed by direct measurement [22].

In this article, we show that the rigidity percolation

clusters exhibit conformal invariance at the critical point

arXiv:2210.06271v1 [cond-mat.stat-mech] 12 Oct 2022

and, interestingly, that the ﬁne statistical properties of

the RP clusters and of the CP clusters share surpris-

ing similarities, despite belonging to distinct universality

classes.

Model and Methods – We perform three independent

numerical tests of conformal invariance, based on the

study of a geometrical property of the random rigid clus-

ters, their so-called n−point connectivity [26]:

p12···n(z1,··· , zn)def

= Prob [z1,··· , zn∈ RC].(1)

ziare points in the two-dimensional space and RC de-

notes a rigid cluster. (1) gives therefore the probability

that npoints are connected by paths inside the same rigid

cluster. These quantities have been very useful to un-

derstand connectivity percolation [27–31]. We make the

central assumption that, in the scaling limit, the connec-

tivities (1) can be described by a ﬁeld theory, and more

precisely that they are given by correlation functions of a

scaling ﬁeld that we denote Φc, of scaling dimension ∆c:

p12···n(z1,··· , zn)scaling

→

lim.a(n)

0hΦc(z1)···Φc(zn)i(2)

where a(n)

0is a non-universal constant that depends on

the microscopic details of the model.

When present, conformal symmetry constrains the

form of correlations, hence of the connectivities, in a pre-

cise way. In this work we use a lattice model of rigid-

ity percolation to measure numerically certain rigid clus-

ter connectivities on speciﬁc geometries. Using (2) gives

the corresponding CFT predictions for these probabili-

ties, which we can compare with the measurements. The

Geometr ic percolation Rigidity percolation

(a) (b)

Rigid cluster

Liquid Solid

(c)

FIG. 1. Examples of site-diluted triangular lattice conﬁgura-

tions showing connectivity percolation transition (a) at pCP

for which the system is macroscopically liquid. (b) Rigid clus-

ter decomposition where red particles belong to the largest

rigid cluster obtained via constraints counting analysis (peb-

ble game) and (c) macroscopic rigidity percolation transition

at pRP

cexhibiting a percolating rigid cluster in the two direc-

tions able to sustain external loads.

model is a site-diluted triangular lattice with local spatial

correlations. It has been recently introduced to model

the rigidity percolation of soft solids [22]. At each step,

particles are drawn randomly one by one to populate a

doubly-periodic triangular lattice of size L1×L2, accord-

ing to the following probability p= (1 −c)6−Nn, where

c∈[0,1[ represents the degree of correlation and Nnis

the number of nearest ﬁlled sites varying between 0 to

6 for fully occupied neighboring sites. Since the ﬁlling

probability depends only on the degree of occupation of

the ﬁrst neighbors, the introduced correlations are local

and in the limit of c= 0 we recover the classical uncorre-

lated random percolation where all particles has the same

ﬁlling probability. In practice, the larger cthe smallest is

the critical probability threshold pRP

c(equivalently crit-

ical volume fraction) which yields to macroscopic ﬁnite

elasticity. These correlations are irrelevant and the large-

scale behaviour is unaﬀected by the value of c, so that

the transition still belongs to the same universality class

as classical uncorrelated RP [22]. In practice we used

c= 0.3 at which pRP

c(c= 0.3) ∼0.66.

To identify rigid clusters on a discrete lattice, we

use the so-called ’Pebble game’, a fast combinatory

algorithm[25, 32]. It is based on Laman’s theorem

for graphs’ rigidity, which uses Maxwell’s constraint

counting argument for each subgraph to detect over-

constrained clusters highlighting rigidity [33]. Figure 1

shows an example of cluster decomposition while increas-

ing p. Connectivity percolation arises at pCP

cand is

characterized by a space-spanning percolating cluster (in

blue). The system is macroscopically liquid and cannot

sustain external loads. Figure 1b and 1c show the largest

rigid cluster (in red) that percolates at pRP

c> pCP

c, lead-

ing to macroscopic elasticity.

In the following, we analyse the statistical properties of

the rigid clusters at the critical point. We ﬁrst obtain

a direct measurement of the anomalous dimension expo-

nent η, then move on to test conformal invariance, using

the 3-point and 2-point connectivities. Finally we high-

light the similarities with CP in the structure of these

functions.

Anomalous dimension – We measure the 2-point con-

nectivity p12(r, θ) on the lattice, ie the probability that

points (i, j) and (i+rcos(θ+π/3), j +rsin(θ+π/3))

are in the same rigid cluster. θis the angle wrt the

short cycle of the doubly-periodic lattice, and rthe dis-

tance between the two points. We use translation in-

variance to average over the L1×L2positions (i, j), as

well as symmetry by reﬂection about θ= 0, so that p12

is an average over 2L1L2Nmeasurements with Nthe

number of samples (N= 1200 for the largest sizes).

The inset in ﬁgure 3 shows the data points in log-log

scale which follow a power law in the scaling region

1rL2/2. This is expected from scale invari-

ance, namely that for points separation 1 z12 L2,

the 2-point connectivity decays as p12(z1, z2)∼ |z12|−η,

where ηis the so-called anomalous dimension, satisfy-

ing the hyperscaling relations η= 2β/ν = 4 −2df[34].

Using assumption (2) and that hΦc(z1)Φc(z2)i=z−2∆c

[35], gives the scaling dimension of Φcas ∆c=η/2.

Expected deviations in the region r∼L2/2 are due

to universal ﬁnite size eﬀects coming from the doubly-

periodic boundary conditions. Fitting the data points

corresponding to the angle that minimises such eﬀects

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Evidencesofconformalinvariancein2drigiditypercolationNinaJaverzatSISSAandINFNSezionediTrieste,viaBonomea265,34136,Trieste,ItalyMehdiBouzidyUniv.GrenobleAlpes,CNRS,GrenobleINP,3SR,F-38000,Grenoble,France(Dated:October13,2022)Therigiditytransitionoccurswhen,asthedensityofmicroscopiccomponentsisincrea...

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Evidences of conformal invariance in 2d rigidity percolation Nina Javerzat SISSA and INFN Sezione di Trieste via Bonomea 265 34136 Trieste Italy

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