Statistical topography of two-dimensional interfaces in the presence of
quenched disorder is studied utilizing combinatorial optimization algorithms.
Finite-size scaling is used to measure geometrical exponents associated with
contour loops and fully packed loops. We find that contour-loop exponents
depend on the type of disorder (periodic ``vs'' non-periodic) and they satisfy
scaling relations characteristic of self-affine rough surfaces. Fully packed
loops on the other hand are unaffected by disorder with geometrical exponents
that take on their pure values.Comment: 4 pages, REVTEX, 4 figures included. Further information can be
  obtained from chenz@physics.rutgers.ed

A. A. Middleton

B. Nienhuis

C. Zeng

Chen Zeng

D. McNamara

D. S. Fisher

D. Sherrington

G. Blatter

G. Grüner

H. Rieger

H. W. J. Blöte

J. Kondev

J. L. Cardy

J. Toner

Jané Kondev

M. B. Isichenko

M. T. Batchelor

U. Derigs

Y.-C. Tsai

English

arXiv

Statistical topography of two-dimensional interfaces in the presence of quenched disorder is studied utilizing combinatorial optimization algorithms. Finite-size scaling is used to measure geometrical exponents associated with contour loops and fully packed loops. We find that contour-loop exponents depend on the type of disorder (periodic ``vs\u27\u27 non-periodic) and they satisfy scaling relations characteristic of self-affine rough surfaces. Fully packed loops on the other hand are unaffected by disorder with geometrical exponents that take on their pure values

Middleton, Alan

Zeng, Chen

Kondev, Jane

McNamara, David

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 9-8-1997 Statistical Topography of Glassy Interfaces Alan Middleton Syracuse University Chen Zeng Rutgers University - New Brunswick/Piscataway Jane Kondev Brown University David McNamara Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Middleton, Alan; Zeng, Chen; Kondev, Jane; and McNamara, David, "Statistical Topography of Glassy Interfaces" (1997). Physics. 200. https://surface.syr.edu/phy/200 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:cond-mat/9709092v1  [cond-mat.dis-nn]  8 Sep 1997Statistical Topography of Glassy InterfacesChen ZengDepartment of Physics, Rutgers University, Piscataway, NJ 08855, USAJané KondevDepartment of Physics, Brown University, Providence, RI 02912, USAD. McNamara and A. A. MiddletonDepartment of Physics, Syracuse University, Syracuse, NY 13210, USA(February 1, 2008)Statistical topography of two-dimensional interfaces in the presence of quenched disorder isstudied utilizing combinatorial optimization algorithms. Finite-size scaling is used to measure geo-metrical exponents associated with contour loops and fully packed loops. We find that contour-loopexponents depend on the type of disorder (periodic “vs” non-periodic) and they satisfy scaling rela-tions characteristic of self-affine rough surfaces. Fully packed loops on the other hand are unaffectedby disorder with geometrical exponents that take on their pure values.Elastic manifolds in random media are used to modelvarious condensed-matter systems with quenched disor-der, including flux-line arrays in dirty type-II supercon-ductors [1], and charge density waves [2]. These dis-parate systems exhibit a common feature, namely a lowtemperature glassy phase in which static and dynamicproperties are dominated by the disorder. One of thesimplest models used to study this glassy phase is givenby a 2-dimensional isotropic interfaces embedded in a3-dimensional disordered medium. The interface Hamil-tonian in the continuum limit isH({h(x)}) =∫d2x[K2|∇h(x)|2 + V [x, h(x)]], (1)where h(x) is the height of the interface above a flatbasal plane indexed by a 2-dimensional vector x, and Kis the elastic stiffness constant. Depending on whetherthe otherwise uncorrelated random potential V [x, h(x)]is non-periodic or periodic along the height direction, theinterface is usually referred to as a random manifold or arandom elastic medium, respectively [3].Much of the analytical understanding of these glassyinterfaces comes from functional renormalization groupcalculations [4,5]. The random manifold is found to havea zero-temperature fixed point characterized by a rough-ness exponent ζ ≈ 0.41 [4], i.e., the width (W ) of theinterface, as measured by the square root of the heightvariance, scales with its lateral linear size (L) as W ∼ Lζ.The random elastic medium on the other hand exhibitsa low-temperature fixed line below a glass transition.Along this line the interface is super-rough (W ∼ ln(L)),while in the high temperature phase it is marginally rough(W ∼√ln(L)) [5]. Recent numerical studies of the exactground-state roughness based on combinatorial optimiza-tion algorithms strongly support the above picture [6].In this letter, we apply further refined implementationsof these polynomial-in-time algorithms to study the to-pography [7] of disordered interfaces at zero temperature.In contrast to the usual paradigm of a rugged landscape[8] where the surface morphology is described in terms ofheight-height correlations, we instead focus on extended,non-local features of the random geometry as expressedby contour loops (i.e., lines of constant height) and fully-packed loops (defined below). We measure different geo-metrical exponents for the two types of loops and checkvarious scaling relations among them (see Eq. (5)). Thisapproach has proven to be very useful in characterizingthe morphology of rough surfaces whenever the completeheight profile is available [9].From the results obtained for contour loops our mainconclusion is that the glassy interfaces studied are self-affine and rough with roughness exponents in good agree-ment with theoretical findings. The study of fullypacked loops on the other hand addresses the questionof the effect of quenched disorder on critical fluctuations.Namely, the discrete interface models we employ map todisordered fully packed loop models. In the absence ofdisorder this loop model is critical and its exponents havebeen calculated exactly using Bethe ansatz and Coulombgas techniques [10]. Unexpectedly, here we find that thevalues of the pure exponents are unaffected by disorder.a. Models and algorithms To simulate both randomand random-periodic interfaces, we consider a simple cu-bic lattice with random bond weights (energies) and itsdirected (111)-interface. This interface, to be precise, isdefined on the dual lattice with each elementary plaque-tte (a square) intersecting a single bond of the simplecubic lattice. The cost of such an interface is definedto be the sum of the weights of all the bonds that itcuts. In the case of the random manifold the integer-valued weight of each bond is chosen independently and1uniformly in the interval [0, 1000]. When simulating arandom elastic medium the random weights are sampledin the same way with the important difference that theweights of all the bonds directly above one another alongthe (111) direction are set equal. In other words, thebond disorder is uncorrelated for the former and peri-odic along the height direction for the latter, reflectingthe intrinsic periodicity of the random elastic medium[3].The problem of finding the ground state of an interfacein the presence of quenched disorder is that of minimiz-ing its total energy cost. Employing max-flow-min-cutand minimum-cost perfect-matching algorithms, for thecase of random disorder and random-periodic disorderrespectively, enables us to determine the exact groundstate interface in time that grows only polynomialy withL. This allows us to simulate relatively large system sizesfor many disorder realizations.The max-flow-min-cut algorithm interprets the simplecubic lattice as a directed flow network. Each bond of thelattice is directed along the (111) direction, and its ran-dom weight specifies the maximum amount of flow thatcan be accommodated in this bond. The algorithm findsthe interface of minimum cost by searching instead forthe maximum flow (with flow conservation) that can besustained between the bottom and top layers with the in-terface sandwiched in between; see Fig.1. The algorithmworks because the interface of minimum cost is the bot-tleneck through which all flow must pass.Fully-Packed Loops (FPL)Contour Loops (CL)FIG. 1. (111) interface of a simple cubic lattice and its twoloop representations. An example of a ground-state interfaceconfined between two flat (111) layers is shown. The level setat mean height consists of closed contour loops due to peri-odic boundary conditions. Also shown is the fully-packed-loop(FPL) representation of the same interface (see text for detailsof construction).Next we note that the (111) projection of the interfacegives a rhombus tiling of the plane, which is also equiva-lent to a complete (perfect) dimer covering of a hexagonallattice. The hexagonal lattice (L) in question is the dualof the triangular lattice formed by the vertices of therhombi, while the dimers lie perpendicular to their shortdiagonals [11]. This mapping is purely geometrical in na-ture, but when the disorder is periodic it facilitates thetask of finding the interface of minimum cost. Namely,the periodic disorder can be represented entirely by ran-dom bonds on L, and the minimization problem becomesone of finding the perfect matching (dimer covering) withminimum cost. Detailed descriptions of both algorithmscan be found in Ref. [12].b. Geometric exponents and scaling Given the exactshape of the (111)-interface its topography is completelycharacterized by a contour plot with the level spacingequal to a single step of the the discrete height. Thecontour plot consists of contour loops which live alongthe bonds of the hexagonal lattice L. The contours areclosed due to periodic boundary conditions which we im-pose in both lateral directions. For example, in Fig.1we have drawn all the contour loops at mean height, forthe (111) interface shown. The union of all the contourloops for different realizations of disorder is the contourloop ensemble.Apart from this natural contour-loop characterization,there exists yet another interesting loop representation ofthe (111)-interface, the fully-packed loops (FPL). Theseloops owe their existence to the one-to-one mapping be-tween a (111)-interface and a complete dimer covering ofthe hexagonal lattice L. By removing the bonds of L thatcoincide with the dimers, we are left with a configurationof fully-packed loops, as shown in Fig.1, where every siteof L belongs to one and only one loop. A physical realiza-tion of FPL are the magnetic domain walls in the groundstate of the Ising antiferromagnet on the triangular lat-tice [11]. FPL models of general loop fugacity in theabsence of disorder have been studied recently [10] andwere shown to be critical for values of the loop fugacitythat does not exceed two. The interface-FPL mappingthus allows us to consider the effect of quenched disorderon the critical FPL model on the honeycomb lattice withfugacity equal to one.Following Ref. [9], we consider the fractal dimensionof a loop D, the loop distribution exponent τ , the loopcorrelation exponent xl, and finally the interface rough-ness exponent ζ. These geometrical exponents are used tocharacterize the morphology of an interface which is sta-tistically invariant under the rescaling h(x) → b−ζh(bx),where b > 1 is an arbitrary rescaling parameter; suchinterfaces are termed self-affine.The natural quantities associated with a loop are itslength s and the radius of gyration R, both measured herein units of the lattice spacing. For the ensemble of loopswe define n(s, R), the number density of loops of lengths and radius R. Contour loops of a self-affine interfacehave no characteristic length scale and we anticipate ascaling form for the number density:n(s, R) = s−(1/D+τ)f(s/RD) . (2)2Integrating n(s, R) over all radii gives the number densityof loops of length s,n(s) ∼ s−τ , (3)which we use to extract the exponents τ and D (see thefollowing section).Yet another measure that was introduced in Ref. [9] isthe loop correlation function g(r) which gives the proba-bility that two points separated by distance r belong tothe same loop. Just like in the case of the number densityof loops we anticipate a power lawg(r) ∼ r−2xl , (4)where xl is the loop correlation exponent.In Ref. [9] scaling relations were derived for the loopexponents τ , D, and xl assuming a self-affine interface ofroughness ζ:D = 2 − xl − ζ/2τ = 1 + (2 − ζ)/D . (5)These results follow from sum rules for the average looplength and the average number of large loops (i.e., thosewith radii comparable to ρ) inside an area of linear sizeρ. Eq. 5 tells us that there are only two independent ex-ponents so measuring all four and checking the validityof the scaling relations provides an important consistencycheck on the assumed self-affine nature of the rough inter-face. Furthermore, the second scaling relation suggestsa method for measuring the roughness exponent directlyfrom the loop data. Namely, integrating n(s, R) over alls and making use of D(τ − 1) = 2 − ζ, we findn(R) ∼ Rζ−3 (6)for the number density of loops of radius R.c. Numerical results We first describe our results forthe random elastic medium. Four different sample sizes,L = 72, 120, 240, and 480 were simulated with 104 disor-der realizations for each size.The fractal dimension D and loop exponent τ canbe simultaneously extracted from n(s) using a finite-size scaling (FSS) form n(s) = s−τfn(s/LD), whichfollows from Eq. 3. In order to minimize the statisti-cal noise at large s, we consider instead the cumulativenumber density N(s) ≡∫s̃>sn(s̃)ds̃, with the scalingform N(s) = s1−τfN(s/LD). Results are summarized inFig 2. The power-law scaling regime is evident over threedecades in loop length (40 < s < 40000). The deviationsfrom scaling at small and large s are attributed to the lat-tice cutoff and finite lattice size, respectively. The bestdata collapses yield D = 1.46± 0.01 and τ = 2.32± 0.01for CL, and D = 1.75±0.01 and τ = 2.15±0.01 for FPL.100102104106s10−910−610−3100N(s)CL (L=480)FPL (L=480)0.0 2.0 4.0 6.0s/LD0.010.020.0N(s)/s1−τL=72L=120L=240L=480D=1.46τ=2.32D=1.75τ=2.15FPLCLFIG. 2. Cumulative number density N(s) (normalized bytotal loop number), for CL and FPL; L = 480. FSS analysisfor L = 72, 120, 240, and 480, is shown in the inset and usedto determine exponents D and τ (see text). N(s) is binnedin intervals of 0.08s at successive s to estimate the systematicerrors which are smaller than the symbols shown.We have also computed the loop correlation functiong(r), from which the exponent xl is extracted using ascaling form g(r) = r−2xlfg(r/L). The best data col-lapses, shown in the inset of Fig. 3, yield xl = 0.50±0.01and xl = 0.25 ± 0.01 for CL and FPL, respectively. Anexact value (xexactl = 1/2) of the loop correlation expo-nent for CL was proposed for self-affine rough surfacesindependent of the roughness [9]. Our results (see Ta-ble I) therefore provide further evidence of the universalnature of xl for contour loops.100101102103r10−810−610−410−2100g(r)CL (L=480)FPL (L=480)0.0 1.0 2.0 3.0r/L 0.00.20.50.81.0g(r)/r−2x lL=72L=120L=240L=480FPL (2xl=0.50)CL (2xl=1.00)FIG. 3. Loop correlation function g(r) for both CL andFPL; L = 480. FSS plots are shown in the inset; systematicerrors estimated as in Fig. 2 are smaller then the symbolsshown.From Eq. 6 it follows that the roughness exponentζ can be obtained from n(R). Once again we considerthe cumulative number density N(R) for which we pro-pose the scaling form: N(R) = Rζ−2fN(R/L). The re-sults of the FSS analysis are given in Fig 4 which yieldsζ = 0.08 ± 0.01 and ζ = 0.00 ± 0.01 for CL and FPL,respectively.3100101102103R10−910−610−3100N(R)CL (L=480)FPL (L=480)0.0 0.5 1.0 1.5 2.0R/L −0.53.57.511.515.5N(R)/Rζ−2L=72L=120L=240L=480FPL (ζ=0.00)CL (ζ=0.08)FIG. 4. Cumulative number density N(R) (normalizedby total loop number) for both CL and FPL; L = 480. FSSplots are shown in the inset; systematic errors estimated asin Fig. 2 are smaller then the symbols shown.Given the numerical values of the geometrical expo-nents in the case of the random elastic medium, which aresummarized in Table I, it is straightforward to check thatthe scaling relations, given by Eq. 5, hold for both CL andFPL. Although the small non-zero roughness ζ = 0.08appears to be in disagreement with the super-rough sce-nario (ζ = 0), it can in fact be understood as an effectiveexponent due to the extra log-divergence of the interfacewidth. We therefore expect this effective roughness toapproach zero with increasing system size as 1/ lnL.We also carried out numerical simulations for the ran-dom manifold. Our results for the geometrical exponentsare summarized in Table I. They were obtained in theexact same fashion as in the case of the random elasticmedium using a FSS analysis for L = 64, 72, 96, and150, with the interfaces confined between two boundarylayers separated by H = 55, 58, 65, and 78, respectively;see Fig. 1. For each system size 6 × 103 disorder re-alizations were simulated. Our three main conclusionsare: First, the roughness exponent ζ = 0.40(2) obtainedfrom N(R) is in excellent agreement with that obtainedfrom a more traditional approach that relies on measur-ing height fluctuations [6]. Second, as in the case of therandom elastic medium, geometric exponents of both CLand FPL satisfy scaling relations given in Eq. (5) givingstrong support to the claim that this glassy interface isself-affine. Finally, and quite surprisingly, the geometricexponents for FPL remain unchanged from the random-periodic case. Moreover, their values agree, within sta-tistical errors, with those obtained in the absence of dis-order, where the interface is marginally rough due to en-tropic fluctuations [10].In conclusion, we have measured the geometrical expo-nents of contour loops and fully packed loops associatedwith the ground states of interfaces in the presence ofquenched random and periodically-random disorder. Thecontour loop results are consistent with the ground stateinterfaces being self-affine and rough, with roughness ex-ponents in agreement with renormalization group calcu-lations. The geometrical exponents for the fully packedloops were found to be unaffected by the disorder. Thisdiscovery is rather unexpected in light of the fact thatthere is a 1-1 mapping between the two loop types, and itcalls for a more detailed study of disordered fully packedloop models with general loop weights.We thank J. Cardy, C.L. Henley, J. Jacobsen, P. Leath,C. Marchetti, and V. Pasquier for useful discussions. Weacknowledge support by the NSF through grants DMR-9214943 (CZ), DMR-9357613 (JK), and DMR-9702242and Alfred P. Sloan Fellowship (AAM).TABLE I. Geometric exponents of both contour loops (CL)and fully-packed loops (FPL). Rational numbers are the pro-posed exponents.Random Elastic MediumContour Loops (CL) Fully-Packed Loops (FPL)D = 1.46 ± 0.01 (3/2) D = 1.75± 0.01 (7/4)τ = 2.32 ± 0.01 (7/3) τ = 2.15± 0.01 (15/7)xl = 0.50 ± 0.01 (1/2) xl = 0.25 ± 0.01 (1/4)ζ = 0.08± 0.01 (0) ζ = 0.00 ± 0.01 (0)Random ManifoldContour Loops (CL) Fully-Packed Loops (FPL)D = 1.31 ± 0.02 (?) D = 1.74± 0.01 (7/4)τ = 2.19 ± 0.02 (?) τ = 2.15± 0.01 (15/7)xl = 0.49 ± 0.02 (1/2) xl = 0.25 ± 0.01 (1/4)ζ = 0.40± 0.02 (?) ζ = 0.01 ± 0.01 (0)[1] G. Blatter et al, Rev. Mod. Phys. 66, 1125 (1994).[2] G. Grüner, Rev. Mod. Phys. 60, 1129 (1988).[3] L. Balents, Lecture Notes (unpublished).[4] D.S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).[5] J.L. Cardy and S. Ostlund, Phys. Rev. B 25, 6899 (1982);J. Toner and D. P. DiVincenzo, Phys. Rev. B 41, 632(1990); Y.-C. Tsai and Y. Shapir, Phys. Rev. Lett. 69,1773 (1992).[6] A.A. Middleton, Phys. Rev. E. 52, R3337 (1995); C. Zenget al, Phys. Rev. Lett. 77, 3204 (1996); H. Rieger and U.Blasum, Phys. Rev. B. 55, R7394 (1997).[7] For a review of statistical topography see M.B. Isichenko,Rev. Mod. Phys. 64, 961 (1992).[8] D. Sherrington, to be published in Physica D (1997).[9] J. Kondev and C.L. Henley, Phys. Rev. Lett. 74, 4580(1995); J. Kondev, D.G. Salinas, and C.L. Henley, un-published.[10] H.W.J. Blöte and B. Nienhuis, Phys. Rev. Lett. 72, 1372(1994); M.T. Batchelor et al, Phys. Rev. Lett. 73, 2646(1994); J. Kondev et al, J. Phys. A 29, 6489 (1996).[11] H W J Blöte, and H J Hilhorst, J. Phys. A: Math. Gen.15, L631 (1982). B. Nienhuis et al, J. Phys. A: Math.Gen. 17, 3559 (1984); C. Zeng and C.L. Henley, Phys.Rev. B. 55, 14935 (1997).[12] U. Derigs, Programming in Networks and Graphs,Springer-Verlag, New York (1988); H. Rieger, LectureNotes in Physics, Springer-Verlag, New York (1997).4

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