The problem of determining the ground state of a $d$-dimensional interface
embedded in a $(d+1)$-dimensional random medium is treated numerically. Using a
minimum-cut algorithm, the exact ground states can be found for a number of
problems for which other numerical methods are inexact and slow. In particular,
results are presented for the roughness exponents and ground-state energy
fluctuations in a random bond Ising model. It is found that the roughness
exponent $\zeta = 0.41 \pm 0.01, 0.22 \pm 0.01$, with the related energy
exponent being $\theta = 0.84 \pm 0.03, 1.45 \pm 0.04$, in $d = 2, 3$,
respectively. These results are compared with previous analytical and numerical
estimates.Comment: 10 pages, REVTEX3.0; 3 ps files (separate:tar/gzip/uuencoded) for
  figure

A. Alan Middleton

A. J. Bray

B. M. Forrest

D. A. Huse

D. S. Fisher

J. Villain

M. A. Fisher

M. Kardar

T. Halpin-Healy

T. Hwa

T. T. Ogielski

Y. Shapir

Y.-C. Zhang

English

arXiv

Crossref

Numerical results for the ground-state interface in a random medium

The problem of determining the ground state of a $d$-dimensional interface embedded in a $(d+1)$-dimensional random medium is treated numerically. Using a minimum-cut algorithm, the exact ground states can be found for a number of problems for which other numerical methods are inexact and slow. In particular, results are presented for the roughness exponents and ground-state energy fluctuations in a random bond Ising model. It is found that the roughness exponent $\zeta = 0.41 \pm 0.01, 0.22 \pm 0.01$, with the related energy exponent being $\theta = 0.84 \pm 0.03, 1.45 \pm 0.04$, in $d = 2, 3$, respectively. These results are compared with previous analytical and numerical estimates

Middleton, Alan

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 1995 Numerical Results for the Ground-State Interface in a Random Medium Alan Middleton Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Middleton, Alan, "Numerical Results for the Ground-State Interface in a Random Medium" (1995). Physics. 19. https://surface.syr.edu/phy/19 This Working Paper 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/9507130v1  29 Jul 1995Numerical Results for the Ground-State Interface in a RandomMediumA. Alan MiddletonDepartment of Physics, Syracuse University, Syracuse, NY 13244(February 1, 2008)AbstractThe problem of determining the ground state of a d-dimensional interfaceembedded in a (d + 1)-dimensional random medium is treated numerically.Using a minimum-cut algorithm, the exact ground states can be found fora number of problems for which other numerical methods are inexact andslow. In particular, results are presented for the roughness exponents andground-state energy fluctuations in a random bond Ising model. It is foundthat the roughness exponent ζ = 0.41 ± 0.01, 0.22 ± 0.01, with the relatedenergy exponent being θ = 0.84 ± 0.03, 1.45 ± 0.04, in d = 2, 3, respectively.These results are compared with previous analytical and numerical estimates.Typeset using REVTEX1In many physical systems, the equilibrium and non-equilibrium behavior of interfacesdetermine the thermodynamic and dynamic properties of the system. Examples of suchinterfaces include the phase boundary between spin-up and spin-down domains in an Isingmagnet and the fluid-vapor interface in porous media. A number of problems in statisticalmechanics can be mapped onto questions about surfaces in a higher dimensional system; forexample, the Potts models can be cast as “height models” [1]. The study of interfaces hasalso been applied to the fracture in solids, where the fracture location relative to the originalwhole body is treated as an interface [2]. Quenched disorder greatly affects the equilibriumand dynamical properties of interfaces and phase boundaries [3]. An interface is distortedby the random forces or potential caused by the disorder, so that its shape is much rougherthan in a pure system.The problem considered here is that of finding the ground state, or zero-temperatureconfiguration, of a d-dimensional interface in a (d + 1)-dimensional medium with quencheddisorder, i.e., spatially varying couplings that are fixed in time. Given a method for findingthe minimal energy configuration of an interface for a particular realization of the disorder,statistical properties can be found by averaging over many realizations. One can thendetermine, for example, how the perpendicular width W of the interface and the fluctuationsin the ground state energy depend on the linear size L of the interface. These quantitiescapture many of the features of the low energy state and are relevant to thermodynamicproperties of the interface at low temperatures. This information can also be used to placelower bounds on the barriers between low energy configurations [3,4], to study the sensitivityof the location of the interface to perturbations [5], and to help to analyze the motion of aninterface subject to an external drive [4].The approach to finding the ground state used in this paper applies a minimum-cut (or“mincut”) algorithm [6] to find the exact ground state. Algorithms for finding the mincuttake a time bounded by a polynomial in the volume of the system; finding the “best”algorithm is still a matter of active research in combinatorial optimization. The polynomialtime bound means, however, that this algorithm is extremely fast compared to annealing2techniques, exhaustive search techniques, or transfer-matrix algorithms [7] that take a timeexponential in the size of the system. Given such an algorithm, quantities of interest can bemeasured and averaged over many realizations of the quenched disorder. A type of mincutalgorithm has been applied previously to the random field Ising magnet [8]. The mincutalgorithm is applicable to many other physical problems, however. In this paper, I describethe general technique and summarize the results for the interface in the random bond Isingmodel.The problem of finding the ground state of a random bond Ising model (RBIM) in zerofield is that of minimizing the total energyE = −∑〈ij〉Jijsisj, (1)where the spin variables si take on the values ±1 and the sum is over nearest neighbor pairs〈ij〉 on a (d + 1)-dimensional lattice. The Jij ≥ 0 are independent identically-distributedrandom variables; in this paper, the Jij are taken to be uniformly distributed in the range[0, 1). I consider here a lattice of dimensions H × Ld, with coordinates {y, x1, . . . , xd},0 ≤ y < H , and 0 ≤ xα < L for 1 ≤ α ≤ d. Boundary conditions must be chosen to inducean interface; this is done by setting the spins si to have the value +1 on the surface y = 0and to be equal to −1 on the surface y = H − 1. Nearest neighbors are defined so that theconstant x0 surfaces represent slices of a “cubic” lattice taken along the [11 . . . 1] direction[9]. The RBIM is used to describe a ferromagnet with random variation in the couplingstrength due to spatial disorder of the spins or impurities. This model has also been usedto describe fracture in materials, where the Jij represents the local force needed to breakthe material [2] and it is assumed the fracture occurs along the surface of minimum totalrupture force.The problem of the random bond Ising magnet was treated numerically by Huse andHenley [10] for d = 1, in the case of the directed polymer, where the interface is restrictedto be described by a single valued function y(x1). The algorithm used, an iterative transfer-matrix method, will find the lowest energy state subject to this restriction. Though the3algorithm used here need not be subject to this restriction, the single-valued-y approximationis convenient for analytical work and is believed to not affect the long wavelength behavior.Huse and Henley found that the transverse widthW = {∑{xα}y2(xα) − [∑{xα}y(xα)]2}1/2 (2)scaled with the length L of the interface as W ζ , with ζ ≈ 0.66. It was also found thatthe sample-to-sample rms fluctuations in the ground state energy varied as ∆E ∼ Lθ, withθ ≈ 0.33. Subsequently the d = 1 case was solved analytically by Huse, Henley and Fisher[10], who showed that ζ = 2/3 and χ = 1/3 in d = 1, consistent with the numerical workof Huse and Henley. Numerical work for interface dimension d = 2 was done by Kardarand Zhang [7], using a transfer matrix which operates on the space of paths; this algorithm,though exact, requires a time exponential in one of the dimensions of the system; they wereconsequently restricted to lattices with linear dimension L ≤ 16. From their simulations,they found ζ = 0.50 ± 0.08 and θ = 1.10 ± 0.05 in d = 2. Analytical arguments have beengiven for the roughness exponents in higher dimensions [2,11,12].The algorithm used to find the ground state is based upon a maximum network flowalgorithm. The network flow problem is defined on a graph with given “capacities” indicatingthe rate at which fluid can flow from one node of the graph to another along a directed edgethat connects the two nodes. The goal is to determine the maximum amount of flow that canbe sustained between two given points, the source and the sink, given current conservationat each of the non-terminal nodes. It is straightforward to show that the maximum flowvalue is equivalent to the value for the minimum “cut”, which is defined as the weight of theset of edges with minimum total capacity that, when cut, disconnect the source and the sink.This minimum cut can be imagined as describing a minimal flow surface that separates thesource and the sink by intercepting the flows through the removed edges. The equivalenceof the network flow and the minimum cut is due to the fact that the minimum cut is the“bottleneck” through which all flow must pass.To apply this algorithm to the RBIM, a graph is constructed whose minimum cut directly4reflects the minimum energy interface in the spin lattice. This is done by attaching twofictitious spins to the lattice, the source and the sink, which can be seen as setting theboundary conditions for the spin in the Ising magnet. The source is connected by an edgeto each y = 0 site; each y = H − 1 site is connected to the sink. The capacity on eachof these auxiliary edges is set to be a very large value (2d times the maximum Jij) so thatthe minimum cut surface does not pass through these edges and instead lies in the bulk.One then has the choice of whether or not to implement the single-valued-y approximation,according to the choice of the capacities of the edges connecting the bulk nodes. If for eachpair of nearest neighbors 〈ij〉, two edges are created, one from i to j and the other from j toi, each with capacity 2Jij, the maximum flow algorithm will give the interface and energyof the unrestricted interface problem. The cost of the minimum cut, i.e., the minimum of2∑Jij over all surfaces which separate the source and the sink, is then equal to the minimalenergy of an interface, above the ground state (which has all spins equal). See Fig. (1) fora diagram of this construction.For d = 2, 3, I simulated the restricted, single-valued, problem by choosing the capacityof the edge connecting i to j to be Jij if y(i) < y(j) and to be 2d max(Jij) if y(i) > y(j).(Because of the choice of lattice alignment, y(i) 6= y(j) for nearest neighbors 〈ij〉.) Thehigh cost of the “backwards” edges, those connecting higher y-coordinate sites to those withsmaller y-coordinates, excludes them from the minimal cut [13]. In order to minimize theeffects of the boundary surfaces on the energy associated with the minimum cut, the edgecosts were increased to 2d max(Jij) on a single line of bonds from one boundary to the other,except on the center edge. The minimal cut surface will not intersect this line except at themidpoint, where the energy is not raised. This modification therefore “pins” the interfaceat one point, which is halfway between the boundaries.Once the corresponding graph is constructed, the max-flow/min-cut algorithm can bedirectly applied. Specifically, the algorithm and code developed by Cherkassky and Goldberg[14] was adapted to the class of graphs studied here; this code was approximately 2-3 timesfaster than other code applied to this problem. The computer time for the solution of a5particular realization on an IBM RS/6000-390 workstation was found to average 230s ford = 2, L = 120, H = 50 and 146s for d = 3, L = 30, H = 20, with the time for smallersystems being nearly proportional to the volume of the sample. The program gives both theminimum-cut cost, i.e., the ground state energy, and the location of the minimum-cut. Thewidth of the interface was computed by finding the rms fluctuation of the y-coordinate ofthe cut bonds; the cut can be visualized by drawing the plaquettes that are dual to the cutbonds.The results for the sample-averaged interface widths and energy fluctuations are shownin Fig. (2) and Fig. (3), for d = 2 and d = 3, respectively. The scaling plots assume a scalingform for the average widthW (L, H) = Lζw(H/Lζ), (3)with w(x) ∼ x for x → 0 and w(x) ∼ const for x → ∞, and a scaling form for the rmsenergy fluctuations of∆E(L, H) = Lθu(H/Lζ), (4)with u(x) ∼ xθ/ζ for x → 0 and w(x) ∼ c for x → ∞. The best scaling fits for the largest Lare obtained for the values ζ = 0.41 ± 0.01, 0.22 ± 0.01 and θ = 0.84 ± 0.03, 1.45 ± 0.04, ind = 2, 3, respectively. The statistical errors due to sample-to-sample fluctatuations in thewidth W are relatively small (for almost all sizes, 104 samples were generated; at least 103samples were generated in all cases). The statistical errors in the energy fluctuations ∆Eare somewhat larger, as can be seen from the scatter in the plots. The dominant sourceof errors in these fits are from the unknown corrections to scaling; the uncertainty in theexponents indicates the range of values over which the curves for the three largest L valuesagree to within statistical error when scaled. The larger relative errors in θ are also due to along crossover in the scaling plots; the widths W appear to converge more quickly, yet alsohave a slow convergence after a sharp crossover. The exponent values satisfy the scalingrelation 2ζ = θ + d − 2 to within numerical error [10]. Applying this scaling relation using6the numerical values of ζ gives θ = 0.82 ± 0.02, 1.44 ± 0.02 for d = 2, 3, respectively. Theexponents found here are in disagreement with the results of Ref. [7]; this is likeley dueto finite size effects in the smaller systems examined there. Given the current numericalaccuracy, it is difficult to differentiate between the analytical results of Fisher and thoseof Halpin-Healy, which are very close in value (ζ = 0.208(4 − d) and ζ = 2(4 − d)/9,respectively). The numerical results are in agreement with both. The constraint preventingcomputations for larger systems on workstations is memory ( > 256 MB will be required)rather than total CPU time.The max-flow algorithm for determining min-cuts has been applied to several problemsin the past, most notably the random field Ising magnet [8], but this is the first time thatit has been applied directly to the random bond Ising model. As has been noted, thistechnique is also applicable to other systems of current interest. This technique is notrestricted to single-valued height interfaces. For example, from simulations without thesingle-valuedness restriction, I find the roughness exponent ζ = 0.67 ± 0.02 for d = 1, inagreement with theory and numerical results for the case where overhangs are forbidden(the directed polymer problem) [3].I am pleased to acknowledge discussions with S. Rao, M. Kardar, and Y. Shapir. Themajority of the computational work was conducted using the resources of the NortheastParallel Architectures Center (NPAC) at Syracuse University.7REFERENCES[1] B. M. Forrest and L.-H. Tang, Jour. of Stat. Phys. 60, 181 (1990).[2] See, for example, M. Kardar in Disorder and Fracture, ed. J. C. Charmet, et al. (Plenum,New York, 1990).[3] J. Villain, Phys. Rev. Lett. 52, 1543 (1984); M. A. Fisher, D. S. Fisher, and D. A. Huse,Phys. Rev. B 43, 139 (1991). Phys. Rev. Lett. 63, 2303 (1989); Y. Shapir, Phys. Rev.Lett. 66, 1473 (1991); T. Hwa and D. S. Fisher, Phys. Rev. B 49, 3136 (1994).[4] M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur;L. Mikheev, B. Drossel, M. Kardar, to be published; A. A. Middleton, in preparation.[5] A. J. Bray and M. A. Moore, Phys. Rev. Lett. 58, 57 (1987); Y.-C. Zhang, Phys. Rev.Lett. 59, 2125 (1987).[6] See, for example, R. Sedgwick, Algorithms in C, (Addison-Wesley, New York, 1990); A.M. Tenenbaum, et al, Data Structures Using C, (Prentice-Hall, 1990).[7] M. Kardar and Y-C. Zhang, Europhys. Lett. 8, 233 (1989).[8] A. T. Ogielski, Phys. Rev. Lett. 57, 1251 (1986).[9] This choice of constant x0 surfaces (or, equivalently, boundary conditions) lowers theelasticity of the interface relative to the case where the boundaries are perpendicu-lar to the [00 . . . 1] direction; the former choice reduces lattice effects and is gener-ally used for the directed polymer problem in two dimensions. The neighbors of thesite at {y, x1, . . . , xd} have coordinates {y + 1, x1, . . . , xd}, {y + 1, x1 + 1, . . . , xd}, . . . ,{y + 1, x1, . . . , xd + 1}.[10] D. A. Huse, C. L. Henley, Phys. Rev. Lett. 54, 2708 (1985); D. A. Huse, C. L. Henley,and D. S. Fisher, Phys. Rev. Lett. 55, 2924 (1985); D. S. Fisher and D. A. Huse, Phys.Rev. B 43, 10728 (1991).8[11] D. S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).[12] T. Halpin-Healy, Phys. Rev. B 42, 711 (1990).[13] If the “backwards” edges are not included in the graph at all, one gets overhangs of anunusual type, as backwards facing edges will not have any cost.[14] B. V. Cherkassky and A. V. Goldberg, to be published.9FIGURESFIG. 1. Example of the construction of the graph whose minimum-cut corresponds to theground state of a random-bond Ising model with twisted boundary conditions. (a) The sites andbonds for an Ising problem, with dimensions L = 2 and H = 4 (in (d = 1) + 1 dimensions). Thestronger bonds (larger Jij) are indicated by thicker widths. The sites are labeled according to theirvalues in the ground state configuration, with sites fixed to take on the values − and + on theleft and right sides, respectively. (b) The corresponding graph for the minimum-cut problem. The“source” and “sink” nodes are auxiliary nodes added to the bulk lattice to define the boundaryconditions. Nodes are connected by directed edges, with the capacity of the edges indicated bythe width of the lines. The bulk forward edges have capacities Jij . “Forbidden” backwards edgeswith large weights (greatest thicknesses) both enforce boundary conditions and prevent overhangsin this example; see text for the case with overhangs. The minimum-cut surface separating thesink from the source is indicated by the broken line.FIG. 2. Scaling plots for average ground state quantities in 2 + 1 dimensions. Alllengths are measured in units of the lattice constant. (a) Scaling of the average sample widthW (L,H) = < y2(x) > − < y(x) >2. The scaled width W/Lζ is plotted vs. the scaled transversesample size H/Lζ for different L, using the value ζ = 0.41 ± 0.01. (b) Scaling plot for the sam-ple-to-sample fluctuations in the ground-state energy. The scale energy fluctuations ∆E/Lθ areplotted as a function of scaled transverse sample size H/Lζ for varying L, using θ = 0.84 ± 0.03.FIG. 3. Scaling plots for average ground state quantities in 3 + 1 dimensions. The plottedquantities are the same as in Fig. (2), with ζ = 0.22 ± 0.01 and θ = 1.45 ± 0.04.10(b)(a)+++++---source sinkFig. 1 - Middleton, Numerical Results for the Ground-State...0.11.0W/Lζ  L = 5 L = 10 L = 20 L = 40 L = 80 L = 1201 10H/Lζ1∆E/Lθ L = 5 L = 10 L = 20 L = 40 L = 80 L = 120(a)(b)d=2Fig. 2 - Middleton, Numerical Results for the Ground-State...1W/Lζ  L = 4 L = 5 L = 7 L = 10 L = 14 L = 20 L = 301 10H/Lζ1∆E/Lθ L = 4 L = 5 L = 7 L = 10 L = 14 L = 20 L = 30(a)(b)d=3Fig. 3 - Middleton, Numerical Results for the Ground-State...

Numerical Results for the Ground-State Interface in a Random Medium

Syracuse University SURFACE Physics College of Arts and Sciences 7-29-1995 Numerical Results for the Ground-State Interface in a Random Medium Alan Middleton Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Middleton, Alan, "Numerical Results for the Ground-State Interface in a Random Medium" (1995). Physics. 203. https://surface.syr.edu/phy/203 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/9507130v1  29 Jul 1995Numerical Results for the Ground-State Interface in a RandomMediumA. Alan MiddletonDepartment of Physics, Syracuse University, Syracuse, NY 13244(February 1, 2008)AbstractThe problem of determining the ground state of a d-dimensional interfaceembedded in a (d + 1)-dimensional random medium is treated numerically.Using a minimum-cut algorithm, the exact ground states can be found fora number of problems for which other numerical methods are inexact andslow. In particular, results are presented for the roughness exponents andground-state energy fluctuations in a random bond Ising model. It is foundthat the roughness exponent ζ = 0.41 ± 0.01, 0.22 ± 0.01, with the relatedenergy exponent being θ = 0.84 ± 0.03, 1.45 ± 0.04, in d = 2, 3, respectively.These results are compared with previous analytical and numerical estimates.Typeset using REVTEX1In many physical systems, the equilibrium and non-equilibrium behavior of interfacesdetermine the thermodynamic and dynamic properties of the system. Examples of suchinterfaces include the phase boundary between spin-up and spin-down domains in an Isingmagnet and the fluid-vapor interface in porous media. A number of problems in statisticalmechanics can be mapped onto questions about surfaces in a higher dimensional system; forexample, the Potts models can be cast as “height models” [1]. The study of interfaces hasalso been applied to the fracture in solids, where the fracture location relative to the originalwhole body is treated as an interface [2]. Quenched disorder greatly affects the equilibriumand dynamical properties of interfaces and phase boundaries [3]. An interface is distortedby the random forces or potential caused by the disorder, so that its shape is much rougherthan in a pure system.The problem considered here is that of finding the ground state, or zero-temperatureconfiguration, of a d-dimensional interface in a (d + 1)-dimensional medium with quencheddisorder, i.e., spatially varying couplings that are fixed in time. Given a method for findingthe minimal energy configuration of an interface for a particular realization of the disorder,statistical properties can be found by averaging over many realizations. One can thendetermine, for example, how the perpendicular width W of the interface and the fluctuationsin the ground state energy depend on the linear size L of the interface. These quantitiescapture many of the features of the low energy state and are relevant to thermodynamicproperties of the interface at low temperatures. This information can also be used to placelower bounds on the barriers between low energy configurations [3,4], to study the sensitivityof the location of the interface to perturbations [5], and to help to analyze the motion of aninterface subject to an external drive [4].The approach to finding the ground state used in this paper applies a minimum-cut (or“mincut”) algorithm [6] to find the exact ground state. Algorithms for finding the mincuttake a time bounded by a polynomial in the volume of the system; finding the “best”algorithm is still a matter of active research in combinatorial optimization. The polynomialtime bound means, however, that this algorithm is extremely fast compared to annealing2techniques, exhaustive search techniques, or transfer-matrix algorithms [7] that take a timeexponential in the size of the system. Given such an algorithm, quantities of interest can bemeasured and averaged over many realizations of the quenched disorder. A type of mincutalgorithm has been applied previously to the random field Ising magnet [8]. The mincutalgorithm is applicable to many other physical problems, however. In this paper, I describethe general technique and summarize the results for the interface in the random bond Isingmodel.The problem of finding the ground state of a random bond Ising model (RBIM) in zerofield is that of minimizing the total energyE = −∑〈ij〉Jijsisj, (1)where the spin variables si take on the values ±1 and the sum is over nearest neighbor pairs〈ij〉 on a (d + 1)-dimensional lattice. The Jij ≥ 0 are independent identically-distributedrandom variables; in this paper, the Jij are taken to be uniformly distributed in the range[0, 1). I consider here a lattice of dimensions H × Ld, with coordinates {y, x1, . . . , xd},0 ≤ y < H , and 0 ≤ xα < L for 1 ≤ α ≤ d. Boundary conditions must be chosen to inducean interface; this is done by setting the spins si to have the value +1 on the surface y = 0and to be equal to −1 on the surface y = H − 1. Nearest neighbors are defined so that theconstant x0 surfaces represent slices of a “cubic” lattice taken along the [11 . . . 1] direction[9]. The RBIM is used to describe a ferromagnet with random variation in the couplingstrength due to spatial disorder of the spins or impurities. This model has also been usedto describe fracture in materials, where the Jij represents the local force needed to breakthe material [2] and it is assumed the fracture occurs along the surface of minimum totalrupture force.The problem of the random bond Ising magnet was treated numerically by Huse andHenley [10] for d = 1, in the case of the directed polymer, where the interface is restrictedto be described by a single valued function y(x1). The algorithm used, an iterative transfer-matrix method, will find the lowest energy state subject to this restriction. Though the3algorithm used here need not be subject to this restriction, the single-valued-y approximationis convenient for analytical work and is believed to not affect the long wavelength behavior.Huse and Henley found that the transverse widthW = {∑{xα}y2(xα) − [∑{xα}y(xα)]2}1/2 (2)scaled with the length L of the interface as W ζ , with ζ ≈ 0.66. It was also found thatthe sample-to-sample rms fluctuations in the ground state energy varied as ∆E ∼ Lθ, withθ ≈ 0.33. Subsequently the d = 1 case was solved analytically by Huse, Henley and Fisher[10], who showed that ζ = 2/3 and χ = 1/3 in d = 1, consistent with the numerical workof Huse and Henley. Numerical work for interface dimension d = 2 was done by Kardarand Zhang [7], using a transfer matrix which operates on the space of paths; this algorithm,though exact, requires a time exponential in one of the dimensions of the system; they wereconsequently restricted to lattices with linear dimension L ≤ 16. From their simulations,they found ζ = 0.50 ± 0.08 and θ = 1.10 ± 0.05 in d = 2. Analytical arguments have beengiven for the roughness exponents in higher dimensions [2,11,12].The algorithm used to find the ground state is based upon a maximum network flowalgorithm. The network flow problem is defined on a graph with given “capacities” indicatingthe rate at which fluid can flow from one node of the graph to another along a directed edgethat connects the two nodes. The goal is to determine the maximum amount of flow that canbe sustained between two given points, the source and the sink, given current conservationat each of the non-terminal nodes. It is straightforward to show that the maximum flowvalue is equivalent to the value for the minimum “cut”, which is defined as the weight of theset of edges with minimum total capacity that, when cut, disconnect the source and the sink.This minimum cut can be imagined as describing a minimal flow surface that separates thesource and the sink by intercepting the flows through the removed edges. The equivalenceof the network flow and the minimum cut is due to the fact that the minimum cut is the“bottleneck” through which all flow must pass.To apply this algorithm to the RBIM, a graph is constructed whose minimum cut directly4reflects the minimum energy interface in the spin lattice. This is done by attaching twofictitious spins to the lattice, the source and the sink, which can be seen as setting theboundary conditions for the spin in the Ising magnet. The source is connected by an edgeto each y = 0 site; each y = H − 1 site is connected to the sink. The capacity on eachof these auxiliary edges is set to be a very large value (2d times the maximum Jij) so thatthe minimum cut surface does not pass through these edges and instead lies in the bulk.One then has the choice of whether or not to implement the single-valued-y approximation,according to the choice of the capacities of the edges connecting the bulk nodes. If for eachpair of nearest neighbors 〈ij〉, two edges are created, one from i to j and the other from j toi, each with capacity 2Jij, the maximum flow algorithm will give the interface and energyof the unrestricted interface problem. The cost of the minimum cut, i.e., the minimum of2∑Jij over all surfaces which separate the source and the sink, is then equal to the minimalenergy of an interface, above the ground state (which has all spins equal). See Fig. (1) fora diagram of this construction.For d = 2, 3, I simulated the restricted, single-valued, problem by choosing the capacityof the edge connecting i to j to be Jij if y(i) < y(j) and to be 2d max(Jij) if y(i) > y(j).(Because of the choice of lattice alignment, y(i) 6= y(j) for nearest neighbors 〈ij〉.) Thehigh cost of the “backwards” edges, those connecting higher y-coordinate sites to those withsmaller y-coordinates, excludes them from the minimal cut [13]. In order to minimize theeffects of the boundary surfaces on the energy associated with the minimum cut, the edgecosts were increased to 2d max(Jij) on a single line of bonds from one boundary to the other,except on the center edge. The minimal cut surface will not intersect this line except at themidpoint, where the energy is not raised. This modification therefore “pins” the interfaceat one point, which is halfway between the boundaries.Once the corresponding graph is constructed, the max-flow/min-cut algorithm can bedirectly applied. Specifically, the algorithm and code developed by Cherkassky and Goldberg[14] was adapted to the class of graphs studied here; this code was approximately 2-3 timesfaster than other code applied to this problem. The computer time for the solution of a5particular realization on an IBM RS/6000-390 workstation was found to average 230s ford = 2, L = 120, H = 50 and 146s for d = 3, L = 30, H = 20, with the time for smallersystems being nearly proportional to the volume of the sample. The program gives both theminimum-cut cost, i.e., the ground state energy, and the location of the minimum-cut. Thewidth of the interface was computed by finding the rms fluctuation of the y-coordinate ofthe cut bonds; the cut can be visualized by drawing the plaquettes that are dual to the cutbonds.The results for the sample-averaged interface widths and energy fluctuations are shownin Fig. (2) and Fig. (3), for d = 2 and d = 3, respectively. The scaling plots assume a scalingform for the average widthW (L, H) = Lζw(H/Lζ), (3)with w(x) ∼ x for x → 0 and w(x) ∼ const for x → ∞, and a scaling form for the rmsenergy fluctuations of∆E(L, H) = Lθu(H/Lζ), (4)with u(x) ∼ xθ/ζ for x → 0 and w(x) ∼ c for x → ∞. The best scaling fits for the largest Lare obtained for the values ζ = 0.41 ± 0.01, 0.22 ± 0.01 and θ = 0.84 ± 0.03, 1.45 ± 0.04, ind = 2, 3, respectively. The statistical errors due to sample-to-sample fluctatuations in thewidth W are relatively small (for almost all sizes, 104 samples were generated; at least 103samples were generated in all cases). The statistical errors in the energy fluctuations ∆Eare somewhat larger, as can be seen from the scatter in the plots. The dominant sourceof errors in these fits are from the unknown corrections to scaling; the uncertainty in theexponents indicates the range of values over which the curves for the three largest L valuesagree to within statistical error when scaled. The larger relative errors in θ are also due to along crossover in the scaling plots; the widths W appear to converge more quickly, yet alsohave a slow convergence after a sharp crossover. The exponent values satisfy the scalingrelation 2ζ = θ + d − 2 to within numerical error [10]. Applying this scaling relation using6the numerical values of ζ gives θ = 0.82 ± 0.02, 1.44 ± 0.02 for d = 2, 3, respectively. Theexponents found here are in disagreement with the results of Ref. [7]; this is likeley dueto finite size effects in the smaller systems examined there. Given the current numericalaccuracy, it is difficult to differentiate between the analytical results of Fisher and thoseof Halpin-Healy, which are very close in value (ζ = 0.208(4 − d) and ζ = 2(4 − d)/9,respectively). The numerical results are in agreement with both. The constraint preventingcomputations for larger systems on workstations is memory ( > 256 MB will be required)rather than total CPU time.The max-flow algorithm for determining min-cuts has been applied to several problemsin the past, most notably the random field Ising magnet [8], but this is the first time thatit has been applied directly to the random bond Ising model. As has been noted, thistechnique is also applicable to other systems of current interest. This technique is notrestricted to single-valued height interfaces. For example, from simulations without thesingle-valuedness restriction, I find the roughness exponent ζ = 0.67 ± 0.02 for d = 1, inagreement with theory and numerical results for the case where overhangs are forbidden(the directed polymer problem) [3].I am pleased to acknowledge discussions with S. Rao, M. Kardar, and Y. Shapir. Themajority of the computational work was conducted using the resources of the NortheastParallel Architectures Center (NPAC) at Syracuse University.7REFERENCES[1] B. M. Forrest and L.-H. Tang, Jour. of Stat. Phys. 60, 181 (1990).[2] See, for example, M. Kardar in Disorder and Fracture, ed. J. C. Charmet, et al. (Plenum,New York, 1990).[3] J. Villain, Phys. Rev. Lett. 52, 1543 (1984); M. A. Fisher, D. S. Fisher, and D. A. Huse,Phys. Rev. B 43, 139 (1991). Phys. Rev. Lett. 63, 2303 (1989); Y. Shapir, Phys. Rev.Lett. 66, 1473 (1991); T. Hwa and D. S. Fisher, Phys. Rev. B 49, 3136 (1994).[4] M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur;L. Mikheev, B. Drossel, M. Kardar, to be published; A. A. Middleton, in preparation.[5] A. J. Bray and M. A. Moore, Phys. Rev. Lett. 58, 57 (1987); Y.-C. Zhang, Phys. Rev.Lett. 59, 2125 (1987).[6] See, for example, R. Sedgwick, Algorithms in C, (Addison-Wesley, New York, 1990); A.M. Tenenbaum, et al, Data Structures Using C, (Prentice-Hall, 1990).[7] M. Kardar and Y-C. Zhang, Europhys. Lett. 8, 233 (1989).[8] A. T. Ogielski, Phys. Rev. Lett. 57, 1251 (1986).[9] This choice of constant x0 surfaces (or, equivalently, boundary conditions) lowers theelasticity of the interface relative to the case where the boundaries are perpendicu-lar to the [00 . . . 1] direction; the former choice reduces lattice effects and is gener-ally used for the directed polymer problem in two dimensions. The neighbors of thesite at {y, x1, . . . , xd} have coordinates {y + 1, x1, . . . , xd}, {y + 1, x1 + 1, . . . , xd}, . . . ,{y + 1, x1, . . . , xd + 1}.[10] D. A. Huse, C. L. Henley, Phys. Rev. Lett. 54, 2708 (1985); D. A. Huse, C. L. Henley,and D. S. Fisher, Phys. Rev. Lett. 55, 2924 (1985); D. S. Fisher and D. A. Huse, Phys.Rev. B 43, 10728 (1991).8[11] D. S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).[12] T. Halpin-Healy, Phys. Rev. B 42, 711 (1990).[13] If the “backwards” edges are not included in the graph at all, one gets overhangs of anunusual type, as backwards facing edges will not have any cost.[14] B. V. Cherkassky and A. V. Goldberg, to be published.9FIGURESFIG. 1. Example of the construction of the graph whose minimum-cut corresponds to theground state of a random-bond Ising model with twisted boundary conditions. (a) The sites andbonds for an Ising problem, with dimensions L = 2 and H = 4 (in (d = 1) + 1 dimensions). Thestronger bonds (larger Jij) are indicated by thicker widths. The sites are labeled according to theirvalues in the ground state configuration, with sites fixed to take on the values − and + on theleft and right sides, respectively. (b) The corresponding graph for the minimum-cut problem. The“source” and “sink” nodes are auxiliary nodes added to the bulk lattice to define the boundaryconditions. Nodes are connected by directed edges, with the capacity of the edges indicated bythe width of the lines. The bulk forward edges have capacities Jij . “Forbidden” backwards edgeswith large weights (greatest thicknesses) both enforce boundary conditions and prevent overhangsin this example; see text for the case with overhangs. The minimum-cut surface separating thesink from the source is indicated by the broken line.FIG. 2. Scaling plots for average ground state quantities in 2 + 1 dimensions. Alllengths are measured in units of the lattice constant. (a) Scaling of the average sample widthW (L,H) = < y2(x) > − < y(x) >2. The scaled width W/Lζ is plotted vs. the scaled transversesample size H/Lζ for different L, using the value ζ = 0.41 ± 0.01. (b) Scaling plot for the sam-ple-to-sample fluctuations in the ground-state energy. The scale energy fluctuations ∆E/Lθ areplotted as a function of scaled transverse sample size H/Lζ for varying L, using θ = 0.84 ± 0.03.FIG. 3. Scaling plots for average ground state quantities in 3 + 1 dimensions. The plottedquantities are the same as in Fig. (2), with ζ = 0.22 ± 0.01 and θ = 1.45 ± 0.04.10(b)(a)+++++---source sinkFig. 1 - Middleton, Numerical Results for the Ground-State...0.11.0W/Lζ  L = 5 L = 10 L = 20 L = 40 L = 80 L = 1201 10H/Lζ1∆E/Lθ L = 5 L = 10 L = 20 L = 40 L = 80 L = 120(a)(b)d=2Fig. 2 - Middleton, Numerical Results for the Ground-State...1W/Lζ  L = 4 L = 5 L = 7 L = 10 L = 14 L = 20 L = 301 10H/Lζ1∆E/Lθ L = 4 L = 5 L = 7 L = 10 L = 14 L = 20 L = 30(a)(b)d=3Fig. 3 - Middleton, Numerical Results for the Ground-State...

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Numerical Results for the Ground-State Interface in a Random Medium

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