Haug, N

Prellberg, T

Journal of Physics A Mathematical and Theoretical

English

Queen Mary Research Online

Multicritical Scaling in a Lattice Model ofVesiclesN. Haug1,2,3 and T. Prellberg11School of Mathematical SciencesQueen Mary University of London, United Kingdom2CeMSIIS, Section for Science of Complex SystemsMedical University of Vienna, Austria3Complexity Science Hub Vienna, AustriaMay 11, 2020AbstractVesicles, or closed fluctuating membranes, have been modelledin two dimensions by self-avoiding polygons, weighted with respectto their perimeter and enclosed area, with the simplest model givenby area-weighted excursions (Dyck paths). These models genericallyshow a tricritical phase transition between an inflated and a crumpledphase, with a scaling function given by the logarithmic derivative ofthe Airy function. Extending such a model, we find realisations ofmulticritical points of arbitrary order, with the associated multivari-ate scaling functions expressible in terms of generalised Airy integrals,as previously conjectured by John Cardy. This work therefore addsto the small list of models with a critical phase transition, for whichexponents and the associated scaling functions are explicitly known.1 IntroductionObtaining a thorough understanding of phase transitions is one of themain aims of statistical physics. For a continuous transition one would1like to know the critical exponents describing the singular power-law behaviour of thermodynamic quantities as the transition is ap-proached. Moreover, in the vicinity of such a transition it is generallybelieved that the thermodynamic quantities depend only on a suitablyscaled combination of the parameters in terms of a universal scalingfunction [1]. Most progress has been made in two dimensions withthe help of conformal invariance [2]. While these scaling functionscan be easily obtained numerically, there are only few instances forwhich one knows precise expressions, one classical example being thespin-spin correlations of the two-dimensional Ising model [3]. Basedon field-theoretic arguments, John Cardy postulated that by includingmany-body interactions in a model of vesicles, a hierarchy of scalingfunctions could be found, but cautioned that due to the technical lim-itations of the method used, “it is very difficult to say to what thesehigher multicritical points might correspond physically” [4]. In thisletter, we give explicit examples of a statistical mechanical model hav-ing precisely these scaling functions, thereby providing a resolution tothis problem.2 Vesicles and self-avoiding polygonsA vesicle consists of a closed membrane formed from a lipid bilayerinside a watery solution. Depending on parameters such as the tem-perature and the osmotic pressure difference between the outside andthe inside of the membrane, vesicles are found in different typicalconformations [5]. Subject to thermal fluctuations, a vesicle of fixedsurface area favours “crumpled” configurations with relatively smallvolume if there is a large net pressure acting onto the outside of themembrane. On the other hand, if there is a net pressure acting ontothe inside of the membrane, then the vesicle tends to appear in aninflated shape with larger volume.In [6], a two-dimensional model of vesicles was proposed in terms ofring polymers enclosing an area. In this case, the length of the polymerplays the role of the surface area of the vesicle, and the volume of thevesicle becomes the enclosed area. In [7, 8] the vesicles were modelledas self-avoiding polygons (SAP) on the square lattice – see Fig. 1 foran example. Note that any intrinsic property of the vesicle membranesuch as stiffness is neglected in that lattice model.In order to analyse the model of SAP, one defines the area-perimeter2Figure 1: A self-avoiding polygon (SAP) on Z2 of perimeter 52 and area 37.generating functionP (x, q) =∞∑m=0∞∑n=0pm,nxmqn, (1)where pm,n is the number of SAP with perimeter m and area n, withtwo SAP being considered identical if they are the same up to trans-lation. The qualitative behaviour of the radius of convergence qc(x)of G(x, q), seen as a series in q for fixed values of x, was discussed in[7, 8]. This quantity is closely related to the asymptotic growth rateof the partition function Zn(x) =∑∞m=0 pm,nxm, and thus physicallyto the free energy per unit area in the thermodynamic limit of infinitearea. The model exhibits a phase transition at a value xc at whichqc(x) is not analytic. More precisely, it was shown that there existsa value xc > 0 such that for 0 ≤ x ≤ xc, qc(x) = 1; for x > xc,qc(x) is a continuous function of x, qc(x) < 1 and limx→∞ qc(x) = 0– see Fig. 2. For q < qc(x), polygons with relatively small area dom-inate the sum (1). This part of the (x, q)-plane is called the dropletphase. Analogously, the region q > 1 is labeled the inflated phase.The region where qc(x) < q < 1 is described as the ‘seaweed’ phase, inwhich the typical conformation consists of a space-filling, convolutedpolygon. Exact enumerations yield the estimate xc ' 0.379 [9]. Thepoint (x, q) = (xc, 1) is called a tricritical point [10].In [11, 12], exact enumeration data was used to analyse the singularbehaviour of the area-perimeter generating function of rooted SAP. Inrooted SAP, there exists one distinguished point on the perimeterof the SAP, therefore the number of rooted SAP with perimeter mand area n is mpm,n, and the area-perimeter generating function isR(x, q) = x ddxP (x, q). It was conjectured that in the vicinity of thepoint (x, q) = (xc, 1), the singular part of this function satisfies the31qxc x00θγφ  tricritical pointdroplet phaseinflated phaseseaweed phaseFigure 2: Qualitative picture of the phase diagram of the SAP model of vesicles. The lineqc(x) is the boundary of the droplet phase. The exponents γ, θ and φ characterise thesingular behaviour of the generating function P (x, q) around the tricritical point.scaling relationRsing(x, e−) ∼ θF ((xc − x)−φ) (→ 0+), (2)with the scaling function being, up to prefactors, given by the loga-rithmic derivative of the Airy function, which is defined for z ∈ C as[13]Ai(z) =12pii∫ ∞eipi/3∞e−ipi/3exp(u33− zu)du. (3)Physically, the parameter  = − ln(q) plays the role of an osmoticpressure acting onto the outside of the vesicle.In [4] it was argued via field theoretic methods that, upon intro-ducing further interactions into the SAP model, one should be ableto observe multicritical points of higher order, described by scalingfunctions of more than one variable. More precisely, upon introduc-ing `− 1 further interactions (wj)`j=2, there should exist multicriticalpoints in the vicinity of which the singular part of the multivariategenerating function of rooted SAP satisfies the scaling relationRsing(w2, . . . , w`, x, q) = θF (α1φ1 , α2φ2 , . . . , αφ`` ), (4)where the variables (αj)`j=1 depend on the parameters of the generat-ing function, the crossover exponents φj are given byφj = φj(`) =`+ 2− j`+ 2(1 ≤ j ≤ `), (5)4and θ = 1`+2 is a critical exponent. The scaling function F (s1, s2, . . . , s`)is expressible in terms of generalised, higher-order Airy integrals, de-fined asΘ`(s1, s2, . . . , s`) =12pii∫ ∞eipi/(`+2)∞e−ipi/(`+2)exp u`+2`+ 2−∑`j=1sjuj du, (6)However, no details of the interactions necessary to observe these mul-ticritical points were given in that reference.Progress in the study of SAPs can be made by considering directedsubclasses such as staircase polygons. In [20] it was shown that stair-case polygons show the same phase diagram as shown in Fig. 2, witha tricritical point around which the area-width generating functionsatisfies the same scaling behaviour as the one conjectured to hold forrooted SAPs. Dyck paths constitute an even more radical simplifica-tion of SAPs; their area-perimeter generating function has also beenshown to satisfy the same scaling relation [17]. This motivates us tosearch for a model of one-dimensional lattice paths with a multicriticalpoint as the one conjectured in [4] to hold for SAPs.3 The modelOne-dimensional lattice paths occur in many applications in proba-bility theory, combinatorics and statistical physics. For m ∈ N0 =N ∪ {0}, a one-dimensional lattice path of length m is a sequence(r0, r1, . . . , rm) of points of Z2, where for 0 < j ≤ m, rj − rj−1 ∈{1}×S, with S ⊆ Z [14]. One usually fixes r0 = (0, 0). The path thenstays in the right half-plane. Paths restricted further to stay in the up-per right quarter plane N0×N0 are called meanders, paths which endon the horizontal line N0 × {0} are called bridges, and paths whichare both meanders and bridges are called excursions.  Lukasiewiczpaths, which encode rooted ordered trees [15], are excursions withS = {k ∈ Z | k ≥ −1}. Excursions with S = {−1} ∪ {j | k ≤ j ≤ `},where k, ` ∈ N0, are called (k, `)- Lukasiewicz paths [16]. Figure 3shows a (1, 3)- Lukasiewicz path of length 13, with the horizontal andvertical axes of Z2 labelled by x and y, respectively. A step in thedirection (1,−1) is called a down-step and a step in direction (1, k),where k ≥ 1 is called an up-step of length k. The height of a pointis its distance from the x-axis, and the height of a step is the heightof its starting point. The heights of the up-steps in Fig. 3 are marked5by red dotted lines. Special subclasses of  Lukasiewicz paths are Dyckand Motzkin paths, corresponding to (k, `) = (1, 1) and (k, `) = (0, 1),respectively.In this letter we consider the generating functionG`(w, x, q) =∑m,n,j1,...,j`c(j1, . . . , j`,m, n)wj11 . . . wj`` xmqn, (7)where we abbreviate w1, . . . , w` = w and where c(j1, . . . , j`,m, n) isthe number of (1, `)- Lukasiewicz paths with m up-steps, of which j1have length one, j2 have length two etc., and the sum of the heightsof all the up-steps, which is an area-like quantity, is n. For example,the path shown in Fig. 3 has the weight w31w2w3x5q5 in the generatingfunction G3(w1, w2, w3, x, q).yxFigure 3: A (1, 3)- Lukasiewicz path of length 13.One can set w1 = 1 without loss of generality. In the following wetherefore write G`(1, w2, . . . , w`, x, q) ≡ G`(w2, . . . , w`, x, q).4 Previous resultsIn [17], the asymptotic behaviour of G1(x, q) in the limit q → 1−was analysed, despite the parameters being interpreted slightly dif-ferently there. It was shown that, in the vicinity of the tricriticalpoint (x, q) = (xc, 1) with xc =14 , the singular part of the generatingfunction satisfies the scaling relationGsing1(x, e−)= θF ((xc − x)−φ), (8)where θ = 13 , φ =23 and F (s) = b0dds ln(Ai(b1s)), with positive con-stants b0 and b1. Consistently with the solution G1(x, 1) =12x(1 −√1− 4x), Eq. (8) implies with the asymptotic expansions of the Airyfunction and its derivative [13] that Gsing1 (x, 1) ∼ 2(xc − x)γ , where6γ = θφ =12 . Up to different constants b0 and b1, the scaling relation(8) is identical to the one in Eq. (2) that was conjectured to hold forrooted self-avoiding polygons.5 ResultIn [18], it was shown rigorously that in the vicinity of the multicriticalpoint (w2, x, q) = (−19 , 13 , 1), the singular part of G2(w2, x, q) satisfiesthe scaling relationGsing2 (w2, x, q) = θF (α1−φ1 , α2−φ2), (9)where the scaling variables α1 and α2 are analytic functions of w2 andx, θ = 14 , φ1 =34 , and φ2 =12 . The scaling function F is expressiblevia Θ2(s1, s2), where the definition of Θ2 is given by Eq. (6).This result is generalised in this letter. More specifically, we showthat in the model of (1, `)- Lukasiewicz paths for arbitrary ` ≥ 2, thereexists a multicritical point (w2, . . . , w`, x, q) , with x = (`+1)−1, q = 1andwk =2`(`+ 1)( −1`+ 1)k−1(`+ 1k + 1)(2 ≤ k ≤ `), (10)in the vicinity of which the generating function G`(w2, . . . , w`, x, q)satisfies a scaling relation of the form of Eq. (4) with the same scalingfunction and the same critical exponents as predicted in [4]. We thuspresent an exactly solvable model representing a concrete realisationof the multicritical scaling postulated in that reference.6 MethodTo obtain an asymptotic expression for G`(w2, . . . , w`, x, q) in the limitq → 1− for arbitrary ` ≥ 2, one proceeds analogously to [17, 18].From a simple factorisation argument [16] one obtains the functionalequationG`(w, x, q) = 1 + xG`(w, x, q)∑`k=1wkk∏j=1G`(w, qjx, q) . (11)We linearise Eq. (11) by using the ansatzG`(w2, . . . , w`, x, q) =Φ(qx)Φ(x), (12)7where Φ(x) ≡ Φ(w2, . . . , w`, x, q). The solution of the linearised equa-tion is then given by the q-hypergeometric series [19]Φ(x) =∞∑n=0∏`−1j=1(ωj ; q)n(q; q)n(−x)nqn2−n, (13)where (z; q)n =∏n−1j=0 (1 − zqj) is the q-Pochhammer symbol and theparameters (ωk)`−1k=1 satisfywk = (−1)k−1`−1∑j1=1`−1∑j2=j1+1· · ·`−1∑jk−1=jk−2+1k−1∏p=1ωjp (2 ≤ k ≤ `). (14)Using the identity(−1)n+1q(n2)(q; q)n(q; q)∞= Res[(z; q)−1∞ ; z = q−n] , (15)we obtain for k ∈ Z the integral expressionΦ(qkx) =A2pii∫Cz12(logq(z)+1)−logq(x)zk(∏`−1j=1(ωj/z; q)∞)(z; q)∞dz, (16)where the prefactor A is independent of k, and the contour C connectsthe points −i∞ and i∞ such that all zeros of (z; q)∞ lie to the right ofthe contour and all zeros of zk(∏`−1j=1(ωj/z; q)∞)lie to the left of thecontour. Substituting an asymptotic expression for the q-Pochhammersymbol [20], the above integral satisfiesΦ(qkx) ∼ A2pii∫Cexp(1f(z))g(z)zkdz (q = e− → 1−), (17)where the functions aref(z) = log(x) log(z)− log(z)22+ Li2(z) +`−1∑j=1Li2(ωjz),g(z) =(z`(1− z)∏`−1j=1(z − ωj)) 12,8and where Li2(z) is the Euler dilogarithm [13]. The saddle points off(z) are the zeros of the polynomialχ(z) = x`−1∏j=1z − ωj− z`(1− z)= z`+1(1− 1z+xz2+xz∑`k=2wkzk)(z 6= 0).Comparing the bracket in the last expression with Eq. (11), we seethat the solution of Eq. (11) for q = 1 is equal to the inverse of asaddle point of f(z). If we set the weights for 2 ≤ k ≤ ` to theones given in Eq. (10), then ` + 1 saddle points coalesce in the pointz = (`+1)−1 for x = (`+1)−1. To obtain an asympotic expression forΦ(qkx), we apply a method devised in [21], based on a theorem from[22], from which it follows that if the parameters of the function f(z)are close to the critical values given in Eq. (10), then there exists amapping T : u 7→ z(u), which is analytic and bijective in the vicinityof the point z = (`+ 1)−1, such thatf(z(u)) =u`+2`+ 2−∑`j=0αjuj = p(u). (18)Moreover, the coefficients (αj)`j=0 are analytic functions of (wj)`j=2 andx in the region around the point of coalescence of the saddle points.Using the transformation T, the integral in Eq. (17) can be rewrittenasΦ(qkx) =A2pii∫C′exp1 u`+2`+ 2−∑`j=0αjujSk(u)du , (19)where q = e− → 1−, C ′ is the image of the contour C under T−1 andSk(u) =g(z(u))z(u)kdzdu . Now one writesSk(u) =∑`j=0P(k)j uk + p′(u)H(u), (20)where the (P(k)j )`j=0 are analytic functions of the (wj)`j=2 and x, andH(u) is some analytic function of u. Substituting Eq. (20) into Eq. (19),9one arrives at the asymptotic expressionΦ(qkx) = A∑`j=0P(k)j j`+2 Θ(j)` (α1− `+1`+2 , α2− ``+2 , . . . , α`− 2`+2 ), (21)where Θ(0)` = Θ` and Θ(j)` = − ∂∂sj Θ` for 1 ≤ j ≤ `; see Eq. (6) forthe definition of Θ`. By substituting Eq. (21) into Eq. (12) for k = 0and k = 1, we obtain an asymptotic expression for G`(w2, . . . , w`, x, q)which is valid uniformly with respect to the parameters (wj)`j=2 andx, in particular in the vicinity of the multicritical point at which `+ 1saddle points of the function f(z) coalesce. Close to the multicriti-cal point, the singular part of G`(w2, . . . , w`, x, q) satisfies the scalingrelationGsing` (w2, . . . , w`, x, e−) = θF (α1−φ1 , . . . , α`−φ`), (22)where F is expressible via Θ`(s1, . . . , s`), θ =1`+2 , and the φj ≡ φj(`)are given by Eq. (5).References[1] J. Cardy, Scaling and renormalization in statistical physics, Cam-bridge Lecture Notes in Physics (Cambridge University Press,1996).[2] M. Henkel, Conformal invariance and critical phenomena(Springer Science & Business Media, 2013).[3] T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Phys.Rev. B 13, 316 (1976).[4] J. Cardy, Journal of Physics A: Mathematical and General 34,L665 (2001).[5] U. Seifert, Advances in Physics 46, 13 (1997).[6] S. Leibler, R. R. P. Singh, and M. E. Fisher, Physical ReviewLetters 59, 1989 (1987).[7] J. R. Banavar, A. Maritan, and A. Stella, Physical Review A 43,5752 (1991).[8] M. E. Fisher, A. J. Guttmann, and S. G. Whittington, Journalof Physics A: Mathematical and General 24, 3095 (1991).10[9] I. Jensen and A. J. Guttmann, Journal of Physics A: Mathemat-ical and General 32, 4867 (1999).[10] I. D. Lawrie and S. Sarbach, in Phase Transitions and CriticalPhenomena, Vol. 9, edited by C. Domb and J. L. 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Rahman, Basic Hypergeometric Series, Ency-clopedia of Mathematics and its Applications, Vol. 96 (CambridgeUniversity Press, 1990) p. 428.[20] T. Prellberg, Journal of Physics A: Mathematical and General28, 1289 (1995).[21] F. Ursell, Mathematical Proceedings of the Cambridge Philosoph-ical Society 72, 49 (1972).[22] N. Levinson, Duke Mathematical Journal 28, 345 (1961).11

Multicritical scaling in a lattice model of vesicles

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Multicritical scaling in a lattice model of vesicles

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