Ž .Physica C 332 2000 131–134www.elsevier.nlrlocaterphysc

Energetics of disorder-induced dislocations

Chen Zeng ), P.L. LeathDepartment of Physics, Rutgers UniÕersity, Piscataway, NJ 08854, USA

Abstract

Ž .A randomly pinned elastic medium in two dimensions is modeled by a disordered fully packed loop FPL model. Theenergetics of disorder-induced dislocations is studied using exact and polynomial algorithms from combinatorial optimiza-tion. Dislocations are found to become unbound at large scale, and the elastic phase is, thus, unstable giving evidence for theabsence of a Bragg glass in two dimensions. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Flux pinning; Glassy state; Magnetic susceptibility

1. Introduction

Randomly pinned elastic media are used to modelvarious condensed-matter systems with quencheddisorder including the vortex phase of dirty type-II

w xsuperconductors 1 . Much analytical progress onthese systems has been made within the elastic ap-proximation where dislocations are excluded by fiat.The intriguing possibility of spontaneous formation

Žof disorder-induced dislocations pairs and loops,.respectively, in two and three dimensions at large

scale, however, remains a challenging questionw x2a,2b,2c,2d . To address this issue at zero tempera-ture requires a detailed understanding of the energet-ics of dislocations in terms of their elastic-energycost and disorder-energy gain in the ground statew x2a,2b,2c,2d .

In this article, we briefly review our recent workw x4 on dislocations in 2 d randomly pinned elasticmedia, and show how the energetics of dislocationpairs can be studied numerically by applying thesepolynomial algorithms to a 2 d lattice model. The

) Corresponding author.

essential ingredients required of such a 2 d discreteŽ .model would be: 1 its large-scale fluctuations are

described by an elastic Hamiltonian with a quenchedrandom potential that reflects the periodicity intrinsic

Ž .to any elastic medium; 2 dislocations can be ‘‘con-Ž .veniently’’ generated; and 3 its ground-state ener-

gies with and without dislocations are amenable toexact numerical computations by these polynomialalgorithms.

2. Models

w xFortunately, recent works by Kondev et al. 5a 5bw xand Raghavan et al. 5c provided us with a large

class of such models whose degrees of freedom areŽ .described in terms of colors, tilings dimer and

loops, precisely the natural language for consideringnetwork flow optimizations. More importantly, these

Ž .models all permit a solid-on-solid SOS representa-tion whose large-scale height fluctuations are gov-erned by a few elastic constants and a locking poten-tial that is periodic in heights. As an illustration, we

Ž .consider here a fully packed loop FPL model de-fined on a honeycomb lattice. All configurations of

0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0921-4534 99 00655-3

( )C. Zeng, P.L. LeathrPhysica C 332 2000 131–134132

occupied bonds, which form closed loops and coverevery site exactly once, are allowed, as in the exam-

Ž .ple in Fig. 1 a . The corresponding SOS surface is aŽ .111 -interface of a simple cubic lattice constructedas follows: Define integer heights at the centers ofthe hexagons of this honeycomb lattice, then orientall bonds of the resulting triangular lattice connect-ing the centers such that elementary triangles point-ing upward are circled clockwise; assign q1 to thedifference of neighboring heights along the orientedbonds if a loop is crossed, and y2, otherwise. Thisyields single-valued heights up to an overall con-stant.

We introduce quenched disorder via random bondweights on the honeycomb lattice, chosen indepen-dently and uniformly from integers in the intervalw xyw,w with ws500. The total energy is the sumof the bond weights along all loops and strings. TheFPL model is shown to be equivalent to an array of

w xfluxlines confined in a plane 3 with the heightscorresponding to the displacement fields of the flux-lines. The SOS surface described above can beviewed as an elastic surface embedded in a 3drandom potential that is periodic in heights modulo3, since the smallest ‘‘step’’ of the surface is threew x5a,5b,5c . The coarse-grained effective Hamiltonian

w xbecomes 4,5a,5b,5c

K 2p2Hs d r =h r yucos h r yg rŽ . Ž . Ž .Ž .H ž /2 3

1Ž .

where the random bond weights enter as randomŽ .phase g r . Note also that both K and u depend on

Fig. 1. The FPL model with periodic boundary conditions. Theground states with and without a dislocation pair for one realiza-

Ž . Ž .tion of random bond weights are displayed in b and a , respec-Ž . Ž .tively. The dislocations solid dots in b are connected by an

Ž .open string thick line among the loops. The relevant physicalobject is, however, the domain wall which is induced by the

Ž .dislocations as shown in c . This domain wall represents the lineŽ . Ž .of all bond differences between the ground states a and b .

the disorder strength w, since it is the only energyscale in the problem. This is the well-studied model

Ž . w xfor charge-density waves CDW 6a,6b,6c .Dislocations are added to the FPL model by

‘‘violating’’ the fully packed constraint. One disloca-tion pair is an open string in an otherwise fully

Ž .packed system as shown in Fig. 1 b . The heightchange along any path encircling one end of thestring is the Burgers charge of "3 of a dislocation,so that the heights become multi-valued. Note thatthe configurations with and without a dislocationpair only differ along a domain ‘‘wall’’, as shown in

Ž .Fig. 1 c . Dislocations with higher Burgers chargesof "6 can also be created by introducing holesinstead of strings.

3. Algorithms

It turns out that the ground states of the disor-dered FPL model can be obtained via polynomialalgorithms. A general description of such an opti-mization problem is given by the so-called linearprogramming, which is to identify a set of variablesminimizing a linear objective function subject to aset of linear constraints. Most physical problems arerestricted to integer-valued variables. Such an integeroptimization problem in general is nondeterministic

Ž .polynomial NP , which implies that polynomial andexact algorithms are unlikely to be found. However,for a special class of problems where the linearconstraints other than the upper and lower bounds onthe variables can be interpreted as ‘‘flow conserva-tion’’ at the nodes of a graph while the variables areidentified with the flows on the edges of the graph,the optimization problem is polynomial. Recent ap-plications mentioned above fall into this class. Since

w xmost textbooks 7 contain details on the proof ofthis result and existing Cqq codes for these poly-nomial algorithms can be found in the LEDA libraryw x8 , we shall only discuss how to transform thesearch for the ground states into an integer min-cost-flow problem on a suitably designed graph. Themin-cost-flow problem is to find the flow pattern ofminimum total cost for sending a specific amount offlow from a given node s to another given node t ina graph G in which the flow x on every edge has an

( )C. Zeng, P.L. LeathrPhysica C 332 2000 131–134 133

Ž .upper bound u and a lower bound l l FxFub b b b

as well as a unit cost c. The total cost is of coursegiven by summing cx over all edges in G.

Suppose that the bipartite honeycomb lattice withperiodic boundary conditions contains 2 N sites,which we divide into two sublattices of N A-sitesand N B-sites. We can construct a graph G asfollows. In addition to all sites and bonds of thehoneycomb lattice, this graph contains two extra

Ž . Ž .sites, denoted as s the source and t the sink , andŽ .extra 2 N bonds the leads . All bonds of the honey-

comb lattice are directed from A-sites to B-sites withl s0, u s1, and c being the corresponding ran-b b

dom bond weight, while the remaining N in-leadsare directed from s to A-sites and N out-leads fromB-sites to t with l s1, u s2, and cs0 for all 2 Nu b

leads. Therefore, the ground state energy of a loopconfiguration with or without defects is equivalent tothe minimum-cost flow if loops and strings are iden-tified with bonds on the honeycomb lattice that have

Žflow note that the flow value on these bonds must.be either zero or unity . Simple inspection shows that

this identification can indeed be made with the abovechoice of bounds if the amount of flow sustainedbetween s and t, with flow conservation on all othernodes, is between N and 2 N units.

Given the amount of flow sustained, the min-cost-flow algorithm establishes the flow pattern ofthe minimum cost. Various interesting physical situa-tions can be simulated by simple variations. 2 Nunits of flow, for example, lead to the ground state

Ž .of FPLs no dislocations . 2 Ny1 units of flow, onthe other hand, give the ground state with one dislo-cation pair without a priori fixing the pair location.Keeping 2 Ny1 units of flow while changing u ofb

a particular in-lead and out-lead from 2 to 1 simu-lates a fixed pair of dislocations with the Burgerscharges of "3. If using 2 Ny2 units of flow insteadand changing u of a particular in-lead and out-leadb

Ž .from 2 to 0, we obtain a pair of dislocations holeswith the Burgers charges of "6 at fixed locations.Clearly, dislocations of any desired density can beachieved by suitably varying the flow between Nand 2 N units. Moreover, introducing another extralink from t back to s with a negative unit cost yEc

allows us to determine the optimal amount of flowŽ .sustained thus, the optimal dislocation density with

E being the core energy. This last simple variationc

results in the min-cost-circulation problem in net-work flow optimization.

4. Numerical results

For a given disorder realization, two ground-stateenergies, E and E , were obtained, respectively, for1 0

cases with and without dislocations. The defect en-ergy E 'E yE was then determined. Variousd 1 0

L=L sample sizes with Ls12, 24, 48, 96, 192 andŽ .384 480 for optimized defects were simulated with

at least 104 disorder averages for each size.We first describe our results for a single disloca-

tion pair where the core energy E is set to zero. Thec

elastic constant of an elastic medium can be mea-sured in various ways by observing its response toperturbations. Here, we perturb the system with a

Žfixed dislocation pair large-scale topological excita-.tion . The defect energy E in this case is the elasticd

energy cost E , which according to the elasticela2 Ž .theory should scale as E ;Kb r2p ln L . This isela

indeed consistent with our numerical results shownin Fig. 2, and the elastic constant K is found to be

Ž . Ž .126 2 and 125 1 from dislocations pairs with theBurgers charges of "3 and "6, respectively. Whenthe dislocation pair is allowed to be placed opti-mally, E also contains the disorder energy gain Ed dis

in addition to E , i.e., E sE qE . As shownela d ela dis

Fig. 2. Energetics of a dislocation pair. Diamond and squaresymbols denote the defect energy E for a pair of fixed disloca-d

tions with the Burgers charges of "3 and "6, respectively. Solidlines are linear fits. Data denoted by circles are the defect energyE for a pair of optimized dislocations with the Burgers charges ofd

"3.

( )C. Zeng, P.L. LeathrPhysica C 332 2000 131–134134

Fig. 3. Optimal dislocation density r as a function of the coreŽenergy E . Shown in the inset are the elastic-energy costs E sc d

.E to a pair of fixed dislocations with E s0 and the Burgersela c

charges of "3 injected into a state with already optimal numberŽ .of pre-existing dislocations square as well as into a state with no

Ž .pre-existing dislocations circle . Data denoted by circles are thesame as those denoted by diamonds in Fig. 2, and are shown herefor comparison.

clearly in Fig. 2, E dominates over E , resultingdis ela

in the negative E and, moreover, E drops fasterd disŽ .than ln L . Detailed analysis showed that the numer-

ical results are consistent with the theoretical predic-3r2Ž . w xtion E ;yln L 4 , a result independent ofdis

the disorder strength w. Therefore, the elastic phaseof large systems is unstable to dislocation pairs. Withno restrictions on their number, dislocations willproliferate thereby driving the elastic constant K tozero.

We now discuss our results on multiple disloca-tions, which are summarized in Fig. 3. It is indeedclear from the inset that the elastic energy cost Eela

for introducing a pair of fixed dislocations into thestate where the number of dislocations is alreadyoptimal is independent of the separation of the fixedpair implying a zero elastic constant K. This is

w xconsistent with the result on a related model 9 . Therelation between the optimal dislocation density r

and the core energy E is, however, found to bec

r;eyŽ Ec r E0 .a

2Ž .Ž .with as0.74 3 . Such a stretched exponential ex-

ponent can be readily understood as follows: Since

the elastic interactions are vanishingly small in thepresence of optimal density of defects, the ener-getic balance between the core energy cost per de-fect ;E and the disorder energy gain per defectc

3r2Ž . Ž . 3r2Ž .; ln d s 1 2 ln r with d being the aver-age separation of defects will lead to an exponent2r3s0.67. We studied the energetics of dislocationpairs in a 2 d random elastic medium by applyingpolynomial algorithms to 2 d-disordered FPL modeland found the elastic phase is unstable against theproliferation of dislocations, and, thus, providingevidence against the formation of a Bragg glass intwo dimensions.

Acknowledgements

Part of this work is done in collaboration withD.S. Fisher, which is also gratefully acknowledged.

References

w x Ž .1 G. Blatter et al., Rev. Mod. Phys. 66 1994 1125.w x Ž .2a T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 72 1994

1530, and references therein.w x Ž .2b M. Aizenman, J. Wehr, Phys. Rev. Lett. 62 1989 2503, and

references therein.w x Ž .2c J. Kierfeld, T. Natterman, T. Hwa, Phys. Rev. B 55 1997

626, and references therein.w x Ž .2d D.S. Fisher, Phys. Rev. Lett. 78 1997 1964, and references

therein.w x3 C. Zeng, A.A. Middleton, Y. Shapir, Phys. Rev. Lett. 77

Ž .1996 3204.w x Ž .4 C. Zeng, P.L. Leath, D.S. Fisher, Phys. Rev. Lett. 82 1999

1935.w x Ž .5a J. Kondev, C.L. Henley, Phys. Rev. Lett. 73 1994 2786.w x Ž .5b J. Kondev, J. de Gier, B. Nienhuis, J. Phys. A 29 1996

6489.w x5c R. Raghavan, C.L. Henley, S.L. Arouh, J. Stat. Phys. 86

Ž .1997 517.w x Ž .6a J.L. Cardy, S. Ostlund, Phys. Rev. B 25 1982 6899.w x Ž .6b J. Toner, D.P. DiVincenzo, Phys. Rev. B 41 1990 632.w x Ž .6c T. Hwa, D.S. Fisher, Phys. Rev. Lett. 72 1994 2466.w x Ž .7 G.L. Nemhauser Ed. , Optimization, New York, NY, USA,

1989.w x8 S. Naher, C. Uhrig et al., The LEDA User Manual, Martin-¨

Luther Universitat Halle-Wittenberg, Germany, 1996.¨http:rrwww.mpisb.mpg.derLEDArleda.html.

w x Ž .9 A.A. Middleton, Cond-matr9807374 1998 .