Computer Physics Communications 121–122 (1999) 531–535www.elsevier.nl/locate/cpc

Application of polynomial algorithmsto a random elastic medium

Chen Zeng, P.L. Leath1

Department of Physics, Rutgers University, Piscataway, NJ 08854, USA

Abstract

A randomly pinned elastic medium in two dimensions is modeled by a disordered fully-packed loop model. The energetics ofdisorder-induced dislocations is studied using exact and polynomial algorithms from combinatorial optimization. Dislocationsare found to become unbound at large scale, and the elastic phase is thus unstable giving evidence for the absence of a Braggglass in two dimensions. 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Randomly pinned elastic media are used to modelvarious condensed-matter systems with quenched dis-order including the vortex phase of dirty type-II super-conductors [1]. Much analytical progress on these sys-tems has been made within the elastic approximationwhere dislocations are excluded by fiat. The intrigu-ing possibility of spontaneous formation of disorder-induced dislocations (pairs and loops, respectively intwo and three dimensions) at large scale, however,remains a challenging question [2]. To address thisissue at zero temperature requires a detailed under-standing of the energetics of dislocations in terms oftheir elastic-energy cost and disorder-energy gain inthe ground stateBG.

In recent years, we have witnessed a fruitful ex-ploration of novel algorithms in for complex disor-dered systems whose ground state itself is dominatedby random disorder. One class of these efficient al-gorithms is based on network flow optimization. Itincludes the min-cost-flow, max-flow and matching

1 E-mail: leath@physics.rutgers.edu.

algorithms which compute the exact ground state intime that grows only polynomially in the system size,an attribution of great practical importance. Some re-cent applications include studies of the roughness andtopography of random manifolds and 2D random elas-tic media by max-flow [3], matching [4] and min-cost-flow algorithms [5], the sensitivity exponents of therandom-field-Ising model by max-flow algorithms [6],the domain-wall energy in the gauge glass by min-cost-flow algorithms [7], the droplet excitations in dis-ordered systems by matching algorithms [8], and thecritical exponents of 2D generic rigidity percolationby matching algorithms [9].

In this paper we briefly review our recent work [10]on dislocations in 2D randomly pinned elastic mediaand show how the energetics of dislocation pairs canbe studied numerically by applying these polynomialalgorithms to a 2D lattice model. The essential ingre-dients required of such a 2D discrete model would be:(1) its large-scale fluctuations are described by an elas-tic Hamiltonian with a quenched random potential thatreflects the periodicity intrinsic to any elastic medium;(2) dislocations can be “conveniently” generated; and(3) its ground-state energies with and without dislo-

0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S0010-4655(99)00399-9

532 C. Zeng, P.L. Leath / Computer Physics Communications 121–122 (1999) 531–535

cations are amenable to exact numerical computationsby these polynomial algorithms.

2. Models

Fortunately, recent works of Henley, Kondev andtheir co-workers provided us with a large class of justsuch models whose degrees of freedom are describedin terms of colors, tilings (dimer) and loops [11],precisely the natural language for considering networkflow optimizations. More importantly, these modelsall permit a solid-on-solid (SOS) representation whoselarge-scale height fluctuations are governed by afew elastic constants and a locking potential that isperiodic in heights. As an illustration, we considerhere a fully-packed loop (FPL) model defined ona honeycomb lattice. All configurations of occupiedbonds which form closed loops and cover everysite exactly once are allowed, as in the example ofFig. 1(a). The corresponding SOS surface is a (111)-interface of a simple cubic lattice constructed asfollows. Define integer heights at the centers of thehexagons of this honeycomb lattice then orient allbonds of the resulting triangular lattice connecting thecenters such that elementary triangles pointing upwardare circled clockwise; assign+1 to the difference ofneighboring heights along the oriented bonds if a loopis crossed and−2 otherwise. This yields single-valuedheights up to an overall constant.

We introduce quenched disorder via random bondweights on the honeycomb lattice, chosen indepen-dently and uniformly from integers in the interval[−w,w] with w = 500. The total energy is the sumof the bond weights along all loops and strings. TheFPL model is shown to be equivalent to an array offlux lines confined in a plane [4] with the heights cor-responding to the displacement fields of the flux lines.The SOS surface described above can be viewed asan elastic surface embedded in a 3D random potentialthat is periodic in heights modulo 3 since the smallest“step” of the surface is three [11]. The coarse-grainedeffective Hamiltonian becomes [10–12]

H =∫dr[K

2

(∇h(r ))2− ucos

(2π

3h(r )− γ (r )

)], (1)

Fig. 1. The FPL model with periodic boundary conditions. Theground states with and without a dislocation pair for one realizationof random bond weights are displayed in (b) and (a), respectively.The dislocations (solid dots) in (b) are connected by an openstring (thick line) among the loops. The relevant physical objectis, however, the domain wall which is induced by the dislocationsas shown in (c). This domain wall represents the line of all bonddifferencesbetween the ground states (a) and (b).

where the random bond weights enter as random phaseγ (r ). Note also that bothK and u depend on thedisorder strengthw since it is the only energy scale inthe problem. This is the well studied model for charge-density waves (CDW) [13].

Dislocations are added to the FPL model by “vi-olating” the fully-packed constraint. One dislocationpair is an open string in an otherwise fully-packed sys-tem as shown in Fig. 1(b). The height change alongany path encircling one end of the string is the Burg-ers charge±3 of a dislocation so that the heights be-come multi-valued. Note that the configurations withand without a dislocation pair only differ along a do-main “wall”, as shown in Fig. 1(c). Dislocations withhigher Burgers charges±6 can also be created by in-troducing holes instead of strings.

3. Algorithms

It turns out that the ground states of the disor-dered FPL model can be obtained via polynomial al-gorithms. A general description of such an optimiza-tion problem is given by the so-called linear program-ming which is to identify a set of variables minimiz-ing a linear objective function subject to a set of lin-ear constraints. Most physical problems are restrictedto integer-valued variables. Such an integer optimiza-tion problem in general is nondeterministic polyno-mial (NP) which implies that polynomial and exactalgorithms are unlikely to be found. However, for aspecial class of problems where the linear constraintsother than the upper and lower bounds on the vari-ables can be interpreted as “flow conservation” at the

C. Zeng, P.L. Leath / Computer Physics Communications 121–122 (1999) 531–535 533

nodes of a graph while the variables are identified withthe flows on the edges of the graph, the optimizationproblem is polynomial. Recent applications mentionedabove fall into this class. Since most textbooks [14]contain details on the proof of this result and exist-ing C++ codes for these polynomial algorithms canbe found in the LEDA library [15], we shall only dis-cuss how to transform the search for the ground statesinto an integer min-cost-flow problem on a suitablydesigned graph. The min-cost-flow problem is to findthe flow pattern of minimum total cost for sending aspecific amount of flow from a given nodes to an-other given nodet in a graphG in which the flowx onevery edge has an upper boundub and a lower boundlb (lb6 x 6 ub) as well as a unit costc. The total costis of course given by summingcx over all edges inG.

Suppose that thebipartite honeycomb lattice withperiodic boundary conditions contains 2N sites whichwe divide into two sublattices ofN A-sites andNB-sites. We can construct a graphG as follows. Inaddition to all sites and bonds of the honeycomblattice, this graph contains two extra sites, denoted ass

(the source) andt (the sink), and extra 2N bonds (theleads). All bonds of the honeycomb lattice are directedfrom A-sites to B-sites withlb = 0, ub = 1, and cbeing the corresponding random bond weight, whilethe remainingN in-leads are directed froms to A-sitesandN out-leads from B-sites tot with lb= 1, ub= 2,andc= 0 for all 2N leads. Therefore, the ground stateenergy of a loop configuration with or without defectsis equivalent to the minimum-cost-flow if loops andstrings are identified with bonds on the honeycomblattice that have flow (note that the flow value on thesebonds must be either zero or unity). Simple inspectionshows that this identification can indeed be made withthe above choice of bounds if the amount of flowsustained betweens andt , with flow conservation onall other nodes, is betweenN and 2N units.

Given the amount of flow sustained, the minimum-cost-flow algorithm establishes the flow pattern of theminimum cost. Various interesting physical situationscan be simulated by simple variations. 2N units offlow, for example, lead to the ground state of fully-packed loops (no dislocations). 2N − 1 units offlow, on the other hand, give the ground state withone dislocation pair withouta priori fixing the pairlocation. Keeping 2N−1 units of flow while changingub of a particular in-lead and out-lead from 2 to 1

simulates a fixed pair of dislocations with the Burgerscharges±3. If using 2N − 2 units of flow instead andchangingub of a particular in-lead and out-lead from2 to 0, we obtain a pair of dislocations (holes) withthe Burgers charges±6 at fixed locations. Clearly,dislocations of any desired density can be achievedby suitably varying the flow betweenN and 2Nunits. Moreover, introducing another extra link fromt back tos with a negative unit cost−Ec allows usto determine the optimal amount of flow sustained(thus the optimal dislocation density) withEc beingthe core energy. This last simple variation resultsin the min-cost-circulation problem in network flowoptimization.

4. Numerical results

For a given disorder realization, two ground-stateenergies,E1 andE0, were obtained respectively forcases with and without dislocations. The defect energyEd ≡ E1 − E0 was then determined. VariousL × Lsample sizes withL = 12,24,48,96,192, and 384(480 for optimized defects) were simulated with atleast 104 disorder averages for each size.

We first describe our results for a single disloca-tion pair where the core energyEc is set to zero. Theelastic constant of an elastic medium can be measuredin various ways by observing its response to pertur-bations. Here we perturb the system with a fixed dis-location pair (large-scale topological excitation). Thedefect energyEd in this case is the elastic energycostEela which according to the elastic theory shouldscale asEela∼Kb2/2π ln(L). This is indeed consis-tent with our numerical results shown in Fig. 2, and theelastic constantK is found to be 126(2) and 125(1)from dislocations pairs with the the Burgers charges±3 and±6, respectively. When the dislocation pairis allowed to be placed optimally,Ed also containsthe disorder energy gainEdis in addition toEela, i.e.,Ed = Eela+ Edis. As shown clearly in Fig. 2,Edisdominates overEela resulting in the negativeEd, andmoreover,Edis drops faster than ln(L). Detailed analy-sis showed that the numerical results are consistentwith the theoretical predictionEdis∼− ln3/2(L) [10],a result independent of the disorder strengthw. There-fore the elastic phase of large systems is unstable todislocation pairs. With no restrictions on their number,

534 C. Zeng, P.L. Leath / Computer Physics Communications 121–122 (1999) 531–535

Fig. 2. Energetics of a dislocation pair. Diamond and square symbolsdenote the defect energyEd for a pair of fixed dislocations withthe Burgers charges±3 and±6, respectively. Solid lines are linearfits. Data denoted by circles are the defect energyEd for a pair ofoptimizeddislocations with the Burgers charges±3.

Fig. 3. Optimal dislocation densityρ as a function of the core energyEc. Shown in the inset are the elastic-energy costs (Ed =Eela) to apair of fixed dislocations withEc = 0 and the Burgers charges±3injected into a state with already optimal number of pre-existingdislocations (square) as well as into a state with no pre-existingdislocations (circle). Data denoted by circles are the same as thosedenoted by diamonds in Fig. 2 and are shown here for comparison.

dislocations will proliferate thereby driving the elasticconstantK to zero.

We now discuss our results on multiple dislocationswhich are summarized in Fig. 3. It is indeed clearfrom the inset that the elastic energy costEela forintroducing a pair of fixed dislocations into the statewhere the number of dislocations is already optimal

is independent of the separation of the fixed pairimplying a zero elastic constantK. This is consistentwith the result on a related model [16]. The relationbetween the optimal dislocation densityρ and the coreenergyEc is, however, found to be

ρ ∼ e−(Ec/E0)α

(2)

with α = 0.74(3). This exponent remains elusive to usat the present.

5. Conclusion and outlook

In conclusion, we studied the energetics of disloca-tion pairs in a 2D random elastic medium by applyingpolynomial algorithms to 2D disordered FPL modeland found the elastic phase is unstable against the pro-liferation of dislocations, and thus providing evidenceagainst the formation of a Bragg glass in two dimen-sions.

Further exploration of these disordered 2D latticemodels with SOS representations will certainly helpto address the fundamental issue of how non-randomcritical systems are affected by quenched disordersince most of these non-random models are criti-cal [11]. For example, the non-random FPL modelflows to the densely-packed loop (DPL) fixed point ofthe O(n) model upon the perturbation of holes [17].How this DPL fixed point get modified by bond ran-domness (if at all) can now be examined by using non-bipartite matching algorithms. Another exciting exten-sion of these polynomial algorithms to compute theenergetics of dislocation loops in 3D random elasticmedia is now also feasible. Compared with the pastfew years in which polynomial andexactalgorithmshave been productively explored, it is fair to say thatthe next few years will see a rapid closing-in on aclass of even NP-hard disordered systems which allowpolynomial andapproximatedalgorithms with near-optimal solutions of guaranteed bounds.

Acknowledgment

We thank J. Kondev, C.L. Henley and A.A. Middle-ton for useful discussions. Part of this work was donein collaboration with D.S. Fisher which is also grate-fully acknowledged.

C. Zeng, P.L. Leath / Computer Physics Communications 121–122 (1999) 531–535 535

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