Using an exact method, we numerically study the zero-temperature roughness of interfaces in the random bond, cubic lattice, Ising model (of size L3, with L<~80). Interfaces oriented along the {100} direction undergo a roughening transition from a weak disorder phase, which is almost flat, to a strong disorder phase with interface width w∼cL0.42 (c is a function of the disorder). For random dilution we find the roughening threshold p∗=0.89±0.01 and c∼p∗−p for p<~p∗ (p is the volume fraction of present bonds). In contrast {111} interfaces are algebraically rough for all disorder.Peer reviewe

Alava, Mikko J.

Duxbury, P. M.

Aaltodoc

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Alava, Mikko J. & Duxbury, P. M.
Title: Disorder-induced roughening in the three-dimensional Ising model
Year: 1996
Version: Final published version
Please cite the original version:
Alava, Mikko J. & Duxbury, P. M. 1996. Disorder-induced roughening in the
three-dimensional Ising model. Physical Review B. Volume 54, Issue 21. 14990-14993.
ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.54.14990.
Rights: © 1996 American Physical Society (APS). This is the accepted version of the following article: Alava, Mikko
J. & Duxbury, P. M. 1996. Disorder-induced roughening in the three-dimensional Ising model. Physical
Review B. Volume 54, Issue 21. 14990-14993. ISSN 1550-235X (electronic). DOI:
10.1103/physrevb.54.14990, which has been published in final form at
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.54.14990.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
ARTICLES
Disorder-induced roughening in the three-dimensional Ising model
M. J. Alava* and P. M. Duxbury
Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University,
East Lansing, Michigan 48824
~Received 19 August 1996!
Using an exact method, we numerically study the zero-temperature roughness of interfaces in the random
bond, cubic lattice, Ising model ~of size L3, with L<80!. Interfaces oriented along the $100% direction undergo
a roughening transition from a weak disorder phase, which is almost flat, to a strong disorder phase with
interface width w;cL0.42 ~c is a function of the disorder!. For random dilution we find the roughening
threshold p
*
50.8960.01 and c;p
*
2p for p<p
*
~p is the volume fraction of present bonds!. In contrast
$111% interfaces are algebraically rough for all disorder. @S0163-1829~96!02845-7#
The statistical mechanics of interfaces in random media is
a much-studied problem.1–3 Physical realizations thereof in-
clude domain walls in Ising magnets,4–13 fracture, and yield
surfaces in disordered materials14,15 and wetting
problems.16,1 The fracture surface problem has in particular
attracted a great deal of recent attention, with a continuing
debate about whether the fracture surface exponents are con-
trolled by disorder ~minimal surfaces!, by crack dynamics, or
by a combination of the two.17,18 Here we calculate the
roughness of minimal surfaces and hence show that, if dis-
order is dominant, then the average orientation of a fracture
surface is important in the analysis of its roughness. This
factor has not been assessed in the experiments so far.
In 111 dimensions, the random bond problem is related
~through the Burgers’ equation! to the celebrated Kardar-
Parisi-Zhang ~KPZ! model of kinetic growth.19 Although the
111 dimensional models ~if d is the bulk dimension, we
write ds115d where ds is the surface dimension! have been
intensely studied numerically, the more important 211 mod-
els have received little attention due to the numerical chal-
lenges involved in simulating equilibrium random surfaces in
three dimensions ~e.g., the usual problem with many meta-
stable minima!.20,21 Recently however,22 the use of the min-
cut/max-flow algorithms from graph theory have made simu-
lations of some 211 random surfaces relatively routine
~these methods find the global minimum of this problem and
avoid altogether getting stuck in metastable states!. Here we
use these methods to numerically study the transition from
the weak disorder phase to the strong disorder phase in ran-
dom magnets and minimal surfaces. We find that, for the
211 dimensional Ising model, this transition is at finite dis-
order for interfaces in the $100% orientation, but is at zero
disorder for interfaces in the $111% orientation. In the $100%
case, the prefactor c(p) ~see the abstract! approaches zero
continuously on approach to the transition.
Either disorder or thermal fluctuations immediately lead
to roughening ~i.e., w;Lz, with z.0! in 111 dimensions.
The roughening exponent in the thermal case z T11151/2,
while in the random-bond ~RB! case we have the KPZ value
zRB
11152/3,6 and for random fields ~RF! zRF11151.4,5 In
d5ds11 dimensions, the random-field exponent appears to
be consistent with the Imry-Ma argument4,5 zRF
ds115(4
2ds)/3, while for random bonds, the best results are from
the functional renormalization group ~RG! calculations,
which give zRB
ds11;b(42ds), with b50.2083 ~Ref. 10! ~see
also, Ref. 23!. ds is the surface dimension. Only recently has
it been possible to test the ds.1 results to high precision.22
The current numerical result zRB21150.4160.01 is close to the
RG result for random bonds and clearly distinct from the
~ds52! random-field exponent ~2/3!. These simulations22
were done for a $111% orientation system, for a fixed continu-
ous distribution of bond strengths, with the intent of numeri-
cally testing the predictions for zRB . As we now show, when
the disorder is varied, and in particular in the case of $100%
interfaces, there is a roughening threshold at finite disorder.
That is, the exponents quoted above only apply for suffi-
ciently strong disorder.
Consider a nearest-neighbor, random-bond Ising model
on a cubic lattice, with Hamiltonian,
E52(
i j
J i js is j , ~1!
where the spin variables have the values 61. In the case of
the diluted Ising model, bonds are present ~Ji j51! with the
probability p and absent ~Ji j50! with probability 12p . The
spins on two opposite cube faces are aligned in opposite
directions, forcing a domain wall somewhere in the middle
of the system. The absent bonds act as local pins, and the
surface reduces its energy by wandering to take advantage of
these missing bonds. In the simple Imry-Ma arguments,4,5
one takes the energy cost of deforming the interface,
E ec;(w/L)2Ld21 ~for a continuum! and compares it to the
energy gain from the random bonds or fields. In the case of
random fields Ep
RF;(wLd21)1/2, while a first approximation
to the random-bond case is Ep
RB;L (d21)/2. The resulting
‘‘strong disorder’’ exponent is correct for random fields, but
incorrect for random bonds ~see the previous paragraph!. For
PHYSICAL REVIEW B 1 DECEMBER 1996-IVOLUME 54, NUMBER 21
540163-1829/96/54~21!/14990~4!/$10.00 14 990 © 1996 The American Physical Society
a $100% Ising interface, the energy cost of deforming the
interface can be higher than in the continuum due to lattice
effects, so that El;G1(w/L)Ld21, where G1 is a step-free
energy. Comparing this energy with the random-bond and
random-field energies, we find that for d.3, there is a flat
phase at weak disorder. d53 is the marginal dimension, and
this suggests that in d53, the interface width may scale
logarithmically11 for weak disorder. Note that in the $111%
direction the lattice effects are ‘‘washed out’’ by the high
degeneracy of the ground-state interface of the pure system.
In that case, we expect the ‘‘strong disorder’’ phase to apply
for any finite disorder. Now we turn to direct numerical tests
of the transition between the weak disorder phase and the
strong disorder ~algebraic! phase in the $100% direction.
As described by Middleton,22 following related work by
Ogielski24 and work on critical current in high-Tc
superconductors,25 the interface problem in spin-1/2 ferro-
magnets with nearest-neighbor interactions is exactly the
same as the min-cut/max-flow problem of operations re-
search. In the latter problem one imagines injecting a flow, I ,
into a network whose bonds each have a different maximum
allowed current ~usually called a ‘‘capacity’’ in the opera-
tions research literature!.26 The problem is to find the
‘‘bottleneck’’ or cross section in the network which sets a
limit on the amount of flow allowed through the system. This
cross section is called the ‘‘minimum cut,’’ and once the
‘‘flow’’ capacity across it is saturated, ‘‘maximum-flow’’ is
achieved. These problems are of vast importance in commu-
nications and transportation networks. In the diluted Ising
system, the interface which minimizes the interface energy is
the same as the min-cut. The interface energy is the maxi-
mum flow. The random-bond strengths map to the random
capacities. The analogy fails if the magnetic model becomes
a spin glass. The advantage of using this mapping is that the
min-cut/max-flow problem can be solved exactly in polyno-
mial time. We implemented a standard augmenting path
algorithm25,26 to solve this problem, and with it we were able
to find the exact minimum interface in a diluted Ising model
of linear dimension L in polynomial time. For example for
an 803 lattice at p50.9, we found the exact lowest energy
interface in 15 min using a 233 MHZ DEC Alpha worksta-
tion. However, the algorithm scales as L5, so it is expensive
at significantly larger system sizes. A ‘‘push-relabel’’ min-
cut/max-flow algorithm is also available.27,22 The prelimi-
nary comparisons we have done between the two methods
indicates that for this problem, the push-relabel method has
the better scaling behavior. Note that the min-cut/max-flow
method is not restricted to ‘‘directed’’ surfaces, and so is
able to find minimal energy interfaces which have over-
hangs. We calculated the interface width defined by,
w25^h2&2^h&2 and the interface energy E and its fluctua-
tions ^E2&2^E&2.
Figure 1 shows an example of the minimum energy con-
figuration for an L520, $100% Ising interface. In Fig. 2, we
present a scaled plot of the interface width ~w/L0.42! of such
interfaces, where 0.42 is chosen to give the best data col-
lapse. Thus, the strong disorder ‘‘minimal surface’’ exponent
in 211 dimensions is 0.4260.02 in agreement with RG
calculations10,21 and with numerical simulations.22 For the
$100% case @triangles in Fig. 2~a!#, there is a threshold above
which the scaled interface width is zero. An analysis of this
threshold behavior is presented in Fig. 2~b!. The solid dots
FIG. 1. $100% interface in the diluted L520, cubic lattice Ising
model with p50.70.
FIG. 2. Scaled interface width w/L0.42 as a function of bond
dilution p1 with J51. ~a! Solid symbols are for $111% interfaces,
while open symbols are for $100% interfaces. Calculations are for L
~number of configurations!510 ~2000! $circles%, 20 ~1500! $tri-
angles%, 30 ~1000! $inverted triangles%, 40 ~200! $diamonds%. There
is also data on L560 ~100 configurations! and L580 ~40 configu-
rations! for $100% interfaces near p50.90, but the data collapse
obscures the symbols on this plot. ~b! A closer look at the value of
p at which the $100% scaled width approaches zero. The $solid
circles% indicate data found by using a least-squares fit to the size-
dependent widths and by assuming an exponent of 0.42 as found in
~a!. Using the solid line to extrapolate, we deduce that
p
*
50.8960.01.
54 14 991DISORDER-INDUCED ROUGHENING IN THE THREE- . . .
are found from a least-squares fit to the size-dependent
widths, assuming the minimal surface exponent 0.42 as
found in Fig. 2~a!. It is seen from this figure that for
pc!p,p* , w;c(p)L
0.42
, with c(p);p
*
2p and
p
*
50.8960.01. At smaller p , the scaled width increases and
in fact, w/L0.42 diverges on approach to the percolation
threshold pc @see Fig. 2~a!#. This is expected because at pc ,
w;L due to the isotropic fractal character of the infinite
cluster. We also assessed whether the data may indicate a
continuously varying exponent rather than the threshold be-
havior extracted above. However if we fit the data in that
way, there is a sharp transition in the ‘‘effective exponent’’
near p
*
50.89. Note that the interfaces studied here scale in
a manner consistent with conventional scaling in contrast to
recent suggestions that there may be a breakdown of scaling
in the bulk properties of the 3d diluted Ising model.28 Finally
there is another replica-type continuum field theory29 which
indicates that $100% interfaces remain strongly disordered for
all p , but that one must go to length scales L@Lc;exp~1/a!
in order to reach the strong disorder regime. Here a is a
disorder parameter which approaches zero as p!1. Ques-
tions remain about whether this result is relevant to the lat-
tice interfaces studied here.
In the $111% direction @see Fig. 2~a!, solid symbols#, there
is no threshold behavior at finite p , and the divergence at pc
still occurs. However there is an additional interesting behav-
ior very close to p51. To understand this effect, consider the
many degenerate interface configurations which occur in a
$111% interface at p51. Now imagine putting one defect in
the lattice. This selects from the degenerate set those which
pass through the defect. Now add two defects to the lattice.
This selects from the ground-state ensemble of ‘‘pure limit’’
interfaces those which have a roughness which is roughly
proportional to the vertical separation between the two de-
fects. Since the two defects are equally likely to be separated
by distances in the interval ~0,L!, the average roughness
scales as L . Thus there is a singular behavior near the pure
limit for the $111% interface. Note however, that once the
deviation from the pure limit 12p.1/L0.42, the pure limit
ground-state ensemble is no longer relevant, and the behav-
ior returns to the strong disorder minimal surface behavior.
As is expected, the interface energy E is smooth for all p
~see Fig. 3! and approaches zero as p!pc . That is, over the
entire range p.pc , the leading term in the interface energy
scales as E5a(p)Lds. In the pure limit ~p51! a(p)53 for
the $111% orientation and a(p)51 for the $100% orientation,
as expected. However, there are singular corrections to the
interface energy, so that E;a(p)Lds1b(p)Lu. The interface
energy fluctuations ^E2&2^E&2 are also controlled by the
exponent u, and we find, ^E2&2^E&2;L2u with u50.82
60.05 for pc,p,p* . This is consistent with the scalinghypothesis u52z.
The presence of the roughening threshold for the $100%
orientation is not peculiar to the case of random dilution. For
example, a uniform distribution centered at J and with width
2dJ also shows this effect ~see Fig. 4, triangles!. Again in
the rough phase, the scaling exponent is 0.4260.01. In this
case, for the sample sizes studied, the surface is completely
flat for dJ,0.5. The case dJ51 corresponds to the distribu-
tion extending to the origin. In the $111% orientation ~circles
in Fig. 4!, the roughness exponent 0.42 applies for all values
of dJ . The anomalous behavior seen near the pure limit in
the case of random dilution does not occur for the continuous
distribution.
In summary, T50 interfaces in cubic lattice Ising models
exhibit a disorder-induced roughening transition for the
$100% orientation. The transition is between a weak disorder
‘‘almost flat’’ phase and a strong disorder phase which has
width w;L0.42. In contrast in the $111% orientation, the in-
terface is always strongly disordered. This has direct impli-
cations for recent measurements14,15,17,18 of the roughness of
fracture surfaces, where ‘‘quasistatic’’ cleavage along the
$100% orientation could have different roughness than frac-
ture occuring at other orientations, at least for weakly disor-
dered materials.
This work has been supported by DOE Grant No. DE-
FG02-090-ER45418 and by the Academy of Finland
~M.J.A.!. P.M.D. thanks A. Bovier, E. Bouchaud, J. P.
Bouchaud, and W. Selke for useful electronic mail corre-
spondence.
FIG. 3. The interface energy of the diluted 3d Ising model. The
values plotted are found by extrapolation using the same simulation
data as those used in Fig. 2~a!. Triangles are for the $111% orienta-
tion, while circles are for the $100% case.
FIG. 4. A plot of the scaled width of interfaces in the 3d ran-
dom bond Ising model with bond couplings drawn from a distribu-
tion of mean J51 and half-width dJ . The $111% data is for lattices
of size L510,20,30,40, while the $100% data is for lattices of size
L510,20,30.
14 992 54M. J. ALAVA AND P. M. DUXBURY
*Permanent address: Laboratory of Physics, Helsinki University of
Technology, Espoo, Finland.
1G. Forgacs, R. Lipowsky, and Th. M. Nieuwenhuizen, in
Phase Transitions and Critical Phenomena, edited by
C. Domb and J. L. Lebowitz ~Academic, San Diego, 1991!,
Vol. 14.
2A.-L. Baraba´si and H. E. Stanley, Fractal Concepts in Surface
Growth ~Cambridge University Press, Cambridge, 1995!.
3T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. 254, 215 ~1995!.
4Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 ~1975!.
5G. Grinstein and S. K. Ma, Phys. Rev. B 38, 2588 ~1983!.
6D. A. Huse and C. L. Henley, Phys. Rev. Lett. 54, 2708 ~1985!.
7D. A. Huse, C. L. Henley, and D. Fisher, Phys. Rev. Lett. 55,
2924 ~1985!.
8M. Kardar, Phys. Rev. Lett. 55, 2235 ~1985!.
9M. Kardar, Phys. Rev. Lett. 55, 2923 ~1985!.
10D. Fisher, Phys. Rev. Lett. 56, 1964 ~1986!.
11A. Bovier, J. Fro¨lich, and U. Glaus, Phys. Rev. B 34, 6409
~1986!.
12T. Nattermann and I. Vilfan, Phys. Rev. Lett. 61, 223 ~1988!.
13A. Bovier and C. Ku¨lske, Rev. Math. Phys. 6, 413 ~1994!; A.
Bovier and C. Ku¨lske ~unpublished!.
14E. Bouchaud and J. P. Bouchaud, Phys. Rev. B 50, 17 752
~1994!; P. Daguier, S. Henaux, E. Bouchaud, and F. Creuzet
~unpublished!.
15S. Roux and A. Hansen, J. Phys. ~France! III 2, 1007 ~1992!.
16M. Huang, M. E. Fisher, and R. Lipowsky, Phys. Rev. B 39, 2632
~1989!.
17V. Y. Milman, N. A. Stelmashenko, and R. Blumenfeld, Prog.
Mater. Sci. 38, 425 ~1994!.
18E. Bouchard and S. Nave´os, J. Phys. ~France! I 5, 547 ~1995!.
19M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889
~1986!.
20M. Kardar and Y.-C. Zhang, Europhys. Lett. 8, 232 ~1989!.
21Z. Jiang and C. Ebner, Phys. Rev. B 41, 9121 ~1990!.
22A. A. Middleton, Phys. Rev. E 52, R3337 ~1995!.
23T. Halpin-Healy, Phys. Rev. B 42, 711 ~1990!.
24T. T. Ogielski, Phys. Rev. Lett. 57, 1251 ~1986!.
25R. Riedinger, Cryogenics 30, 465 ~1990!.
26L. R. Ford and D. R. Fulkerson, Flows in Networks ~Princeton
University Press, Princeton, NJ, 1962!. See also any standard
book on graph theory and flow problems.
27A. V. Goldberg and R. E. Tarjan, J. Assoc. Comput. Mach. 35,
921 ~1988!.
28V. Dotsenko, A. B. Harris, D. Sherrington, and R. B. Stinch-
combe, J. Phys. A 28, 3093 ~1995!.
29 J. P. Bouchard and A. Georges, Phys. Rev. Lett. 68, 3908 ~1992!.
54 14 993DISORDER-INDUCED ROUGHENING IN THE THREE- . . .


English

Disorder-induced roughening in the three-dimensional Ising model

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Alava, Mikko J. & Duxbury, P. M.Title: Disorder-induced roughening in the three-dimensional Ising modelYear: 1996Version: Final published versionPlease cite the original version:Alava, Mikko J. & Duxbury, P. M. 1996. Disorder-induced roughening in thethree-dimensional Ising model. Physical Review B. Volume 54, Issue 21. 14990-14993.ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.54.14990.Rights: © 1996 American Physical Society (APS). This is the accepted version of the following article: Alava, MikkoJ. & Duxbury, P. M. 1996. Disorder-induced roughening in the three-dimensional Ising model. PhysicalReview B. Volume 54, Issue 21. 14990-14993. ISSN 1550-235X (electronic). DOI:10.1103/physrevb.54.14990, which has been published in final form athttp://journals.aps.org/prb/abstract/10.1103/PhysRevB.54.14990.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)ARTICLESDisorder-induced roughening in the three-dimensional Ising modelM. J. Alava* and P. M. DuxburyDepartment of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University,East Lansing, Michigan 48824~Received 19 August 1996!Using an exact method, we numerically study the zero-temperature roughness of interfaces in the randombond, cubic lattice, Ising model ~of size L3, with L<80!. Interfaces oriented along the $100% direction undergoa roughening transition from a weak disorder phase, which is almost flat, to a strong disorder phase withinterface width w;cL0.42 ~c is a function of the disorder!. For random dilution we find the rougheningthreshold p*50.8960.01 and c;p*2p for p<p*~p is the volume fraction of present bonds!. In contrast$111% interfaces are algebraically rough for all disorder. @S0163-1829~96!02845-7#The statistical mechanics of interfaces in random media isa much-studied problem.1–3 Physical realizations thereof in-clude domain walls in Ising magnets,4–13 fracture, and yieldsurfaces in disordered materials14,15 and wettingproblems.16,1 The fracture surface problem has in particularattracted a great deal of recent attention, with a continuingdebate about whether the fracture surface exponents are con-trolled by disorder ~minimal surfaces!, by crack dynamics, orby a combination of the two.17,18 Here we calculate theroughness of minimal surfaces and hence show that, if dis-order is dominant, then the average orientation of a fracturesurface is important in the analysis of its roughness. Thisfactor has not been assessed in the experiments so far.In 111 dimensions, the random bond problem is related~through the Burgers’ equation! to the celebrated Kardar-Parisi-Zhang ~KPZ! model of kinetic growth.19 Although the111 dimensional models ~if d is the bulk dimension, wewrite ds115d where ds is the surface dimension! have beenintensely studied numerically, the more important 211 mod-els have received little attention due to the numerical chal-lenges involved in simulating equilibrium random surfaces inthree dimensions ~e.g., the usual problem with many meta-stable minima!.20,21 Recently however,22 the use of the min-cut/max-flow algorithms from graph theory have made simu-lations of some 211 random surfaces relatively routine~these methods find the global minimum of this problem andavoid altogether getting stuck in metastable states!. Here weuse these methods to numerically study the transition fromthe weak disorder phase to the strong disorder phase in ran-dom magnets and minimal surfaces. We find that, for the211 dimensional Ising model, this transition is at finite dis-order for interfaces in the $100% orientation, but is at zerodisorder for interfaces in the $111% orientation. In the $100%case, the prefactor c(p) ~see the abstract! approaches zerocontinuously on approach to the transition.Either disorder or thermal fluctuations immediately leadto roughening ~i.e., w;Lz, with z.0! in 111 dimensions.The roughening exponent in the thermal case z T11151/2,while in the random-bond ~RB! case we have the KPZ valuezRB11152/3,6 and for random fields ~RF! zRF11151.4,5 Ind5ds11 dimensions, the random-field exponent appears tobe consistent with the Imry-Ma argument4,5 zRFds115(42ds)/3, while for random bonds, the best results are fromthe functional renormalization group ~RG! calculations,which give zRBds11;b(42ds), with b50.2083 ~Ref. 10! ~seealso, Ref. 23!. ds is the surface dimension. Only recently hasit been possible to test the ds.1 results to high precision.22The current numerical result zRB21150.4160.01 is close to theRG result for random bonds and clearly distinct from the~ds52! random-field exponent ~2/3!. These simulations22were done for a $111% orientation system, for a fixed continu-ous distribution of bond strengths, with the intent of numeri-cally testing the predictions for zRB . As we now show, whenthe disorder is varied, and in particular in the case of $100%interfaces, there is a roughening threshold at finite disorder.That is, the exponents quoted above only apply for suffi-ciently strong disorder.Consider a nearest-neighbor, random-bond Ising modelon a cubic lattice, with Hamiltonian,E52(i jJ i js is j , ~1!where the spin variables have the values 61. In the case ofthe diluted Ising model, bonds are present ~Ji j51! with theprobability p and absent ~Ji j50! with probability 12p . Thespins on two opposite cube faces are aligned in oppositedirections, forcing a domain wall somewhere in the middleof the system. The absent bonds act as local pins, and thesurface reduces its energy by wandering to take advantage ofthese missing bonds. In the simple Imry-Ma arguments,4,5one takes the energy cost of deforming the interface,E ec;(w/L)2Ld21 ~for a continuum! and compares it to theenergy gain from the random bonds or fields. In the case ofrandom fields EpRF;(wLd21)1/2, while a first approximationto the random-bond case is EpRB;L (d21)/2. The resulting‘‘strong disorder’’ exponent is correct for random fields, butincorrect for random bonds ~see the previous paragraph!. ForPHYSICAL REVIEW B 1 DECEMBER 1996-IVOLUME 54, NUMBER 21540163-1829/96/54~21!/14990~4!/$10.00 14 990 © 1996 The American Physical Societya $100% Ising interface, the energy cost of deforming theinterface can be higher than in the continuum due to latticeeffects, so that El;G1(w/L)Ld21, where G1 is a step-freeenergy. Comparing this energy with the random-bond andrandom-field energies, we find that for d.3, there is a flatphase at weak disorder. d53 is the marginal dimension, andthis suggests that in d53, the interface width may scalelogarithmically11 for weak disorder. Note that in the $111%direction the lattice effects are ‘‘washed out’’ by the highdegeneracy of the ground-state interface of the pure system.In that case, we expect the ‘‘strong disorder’’ phase to applyfor any finite disorder. Now we turn to direct numerical testsof the transition between the weak disorder phase and thestrong disorder ~algebraic! phase in the $100% direction.As described by Middleton,22 following related work byOgielski24 and work on critical current in high-Tcsuperconductors,25 the interface problem in spin-1/2 ferro-magnets with nearest-neighbor interactions is exactly thesame as the min-cut/max-flow problem of operations re-search. In the latter problem one imagines injecting a flow, I ,into a network whose bonds each have a different maximumallowed current ~usually called a ‘‘capacity’’ in the opera-tions research literature!.26 The problem is to find the‘‘bottleneck’’ or cross section in the network which sets alimit on the amount of flow allowed through the system. Thiscross section is called the ‘‘minimum cut,’’ and once the‘‘flow’’ capacity across it is saturated, ‘‘maximum-flow’’ isachieved. These problems are of vast importance in commu-nications and transportation networks. In the diluted Isingsystem, the interface which minimizes the interface energy isthe same as the min-cut. The interface energy is the maxi-mum flow. The random-bond strengths map to the randomcapacities. The analogy fails if the magnetic model becomesa spin glass. The advantage of using this mapping is that themin-cut/max-flow problem can be solved exactly in polyno-mial time. We implemented a standard augmenting pathalgorithm25,26 to solve this problem, and with it we were ableto find the exact minimum interface in a diluted Ising modelof linear dimension L in polynomial time. For example foran 803 lattice at p50.9, we found the exact lowest energyinterface in 15 min using a 233 MHZ DEC Alpha worksta-tion. However, the algorithm scales as L5, so it is expensiveat significantly larger system sizes. A ‘‘push-relabel’’ min-cut/max-flow algorithm is also available.27,22 The prelimi-nary comparisons we have done between the two methodsindicates that for this problem, the push-relabel method hasthe better scaling behavior. Note that the min-cut/max-flowmethod is not restricted to ‘‘directed’’ surfaces, and so isable to find minimal energy interfaces which have over-hangs. We calculated the interface width defined by,w25^h2&2^h&2 and the interface energy E and its fluctua-tions ^E2&2^E&2.Figure 1 shows an example of the minimum energy con-figuration for an L520, $100% Ising interface. In Fig. 2, wepresent a scaled plot of the interface width ~w/L0.42! of suchinterfaces, where 0.42 is chosen to give the best data col-lapse. Thus, the strong disorder ‘‘minimal surface’’ exponentin 211 dimensions is 0.4260.02 in agreement with RGcalculations10,21 and with numerical simulations.22 For the$100% case @triangles in Fig. 2~a!#, there is a threshold abovewhich the scaled interface width is zero. An analysis of thisthreshold behavior is presented in Fig. 2~b!. The solid dotsFIG. 1. $100% interface in the diluted L520, cubic lattice Isingmodel with p50.70.FIG. 2. Scaled interface width w/L0.42 as a function of bonddilution p1 with J51. ~a! Solid symbols are for $111% interfaces,while open symbols are for $100% interfaces. Calculations are for L~number of configurations!510 ~2000! $circles%, 20 ~1500! $tri-angles%, 30 ~1000! $inverted triangles%, 40 ~200! $diamonds%. Thereis also data on L560 ~100 configurations! and L580 ~40 configu-rations! for $100% interfaces near p50.90, but the data collapseobscures the symbols on this plot. ~b! A closer look at the value ofp at which the $100% scaled width approaches zero. The $solidcircles% indicate data found by using a least-squares fit to the size-dependent widths and by assuming an exponent of 0.42 as found in~a!. Using the solid line to extrapolate, we deduce thatp*50.8960.01.54 14 991DISORDER-INDUCED ROUGHENING IN THE THREE- . . .are found from a least-squares fit to the size-dependentwidths, assuming the minimal surface exponent 0.42 asfound in Fig. 2~a!. It is seen from this figure that forpc!p,p* , w;c(p)L0.42, with c(p);p*2p andp*50.8960.01. At smaller p , the scaled width increases andin fact, w/L0.42 diverges on approach to the percolationthreshold pc @see Fig. 2~a!#. This is expected because at pc ,w;L due to the isotropic fractal character of the infinitecluster. We also assessed whether the data may indicate acontinuously varying exponent rather than the threshold be-havior extracted above. However if we fit the data in thatway, there is a sharp transition in the ‘‘effective exponent’’near p*50.89. Note that the interfaces studied here scale ina manner consistent with conventional scaling in contrast torecent suggestions that there may be a breakdown of scalingin the bulk properties of the 3d diluted Ising model.28 Finallythere is another replica-type continuum field theory29 whichindicates that $100% interfaces remain strongly disordered forall p , but that one must go to length scales L@Lc;exp~1/a!in order to reach the strong disorder regime. Here a is adisorder parameter which approaches zero as p!1. Ques-tions remain about whether this result is relevant to the lat-tice interfaces studied here.In the $111% direction @see Fig. 2~a!, solid symbols#, thereis no threshold behavior at finite p , and the divergence at pcstill occurs. However there is an additional interesting behav-ior very close to p51. To understand this effect, consider themany degenerate interface configurations which occur in a$111% interface at p51. Now imagine putting one defect inthe lattice. This selects from the degenerate set those whichpass through the defect. Now add two defects to the lattice.This selects from the ground-state ensemble of ‘‘pure limit’’interfaces those which have a roughness which is roughlyproportional to the vertical separation between the two de-fects. Since the two defects are equally likely to be separatedby distances in the interval ~0,L!, the average roughnessscales as L . Thus there is a singular behavior near the purelimit for the $111% interface. Note however, that once thedeviation from the pure limit 12p.1/L0.42, the pure limitground-state ensemble is no longer relevant, and the behav-ior returns to the strong disorder minimal surface behavior.As is expected, the interface energy E is smooth for all p~see Fig. 3! and approaches zero as p!pc . That is, over theentire range p.pc , the leading term in the interface energyscales as E5a(p)Lds. In the pure limit ~p51! a(p)53 forthe $111% orientation and a(p)51 for the $100% orientation,as expected. However, there are singular corrections to theinterface energy, so that E;a(p)Lds1b(p)Lu. The interfaceenergy fluctuations ^E2&2^E&2 are also controlled by theexponent u, and we find, ^E2&2^E&2;L2u with u50.8260.05 for pc,p,p* . This is consistent with the scalinghypothesis u52z.The presence of the roughening threshold for the $100%orientation is not peculiar to the case of random dilution. Forexample, a uniform distribution centered at J and with width2dJ also shows this effect ~see Fig. 4, triangles!. Again inthe rough phase, the scaling exponent is 0.4260.01. In thiscase, for the sample sizes studied, the surface is completelyflat for dJ,0.5. The case dJ51 corresponds to the distribu-tion extending to the origin. In the $111% orientation ~circlesin Fig. 4!, the roughness exponent 0.42 applies for all valuesof dJ . The anomalous behavior seen near the pure limit inthe case of random dilution does not occur for the continuousdistribution.In summary, T50 interfaces in cubic lattice Ising modelsexhibit a disorder-induced roughening transition for the$100% orientation. The transition is between a weak disorder‘‘almost flat’’ phase and a strong disorder phase which haswidth w;L0.42. In contrast in the $111% orientation, the in-terface is always strongly disordered. This has direct impli-cations for recent measurements14,15,17,18 of the roughness offracture surfaces, where ‘‘quasistatic’’ cleavage along the$100% orientation could have different roughness than frac-ture occuring at other orientations, at least for weakly disor-dered materials.This work has been supported by DOE Grant No. DE-FG02-090-ER45418 and by the Academy of Finland~M.J.A.!. P.M.D. thanks A. Bovier, E. Bouchaud, J. P.Bouchaud, and W. Selke for useful electronic mail corre-spondence.FIG. 3. The interface energy of the diluted 3d Ising model. Thevalues plotted are found by extrapolation using the same simulationdata as those used in Fig. 2~a!. Triangles are for the $111% orienta-tion, while circles are for the $100% case.FIG. 4. A plot of the scaled width of interfaces in the 3d ran-dom bond Ising model with bond couplings drawn from a distribu-tion of mean J51 and half-width dJ . The $111% data is for latticesof size L510,20,30,40, while the $100% data is for lattices of sizeL510,20,30.14 992 54M. J. ALAVA AND P. M. DUXBURY*Permanent address: Laboratory of Physics, Helsinki University ofTechnology, Espoo, Finland.1G. Forgacs, R. Lipowsky, and Th. M. Nieuwenhuizen, inPhase Transitions and Critical Phenomena, edited byC. Domb and J. L. Lebowitz ~Academic, San Diego, 1991!,Vol. 14.2A.-L. Baraba´si and H. E. Stanley, Fractal Concepts in SurfaceGrowth ~Cambridge University Press, Cambridge, 1995!.3T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. 254, 215 ~1995!.4Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 ~1975!.5G. Grinstein and S. K. Ma, Phys. Rev. B 38, 2588 ~1983!.6D. A. Huse and C. L. Henley, Phys. Rev. Lett. 54, 2708 ~1985!.7D. A. Huse, C. L. Henley, and D. Fisher, Phys. Rev. Lett. 55,2924 ~1985!.8M. Kardar, Phys. Rev. Lett. 55, 2235 ~1985!.9M. Kardar, Phys. Rev. Lett. 55, 2923 ~1985!.10D. Fisher, Phys. Rev. Lett. 56, 1964 ~1986!.11A. Bovier, J. Fro¨lich, and U. Glaus, Phys. Rev. B 34, 6409~1986!.12T. Nattermann and I. Vilfan, Phys. Rev. Lett. 61, 223 ~1988!.13A. Bovier and C. Ku¨lske, Rev. Math. Phys. 6, 413 ~1994!; A.Bovier and C. Ku¨lske ~unpublished!.14E. Bouchaud and J. P. Bouchaud, Phys. Rev. B 50, 17 752~1994!; P. Daguier, S. Henaux, E. Bouchaud, and F. Creuzet~unpublished!.15S. Roux and A. Hansen, J. Phys. ~France! III 2, 1007 ~1992!.16M. Huang, M. E. Fisher, and R. Lipowsky, Phys. Rev. B 39, 2632~1989!.17V. Y. Milman, N. A. Stelmashenko, and R. Blumenfeld, Prog.Mater. Sci. 38, 425 ~1994!.18E. Bouchard and S. Nave´os, J. Phys. ~France! I 5, 547 ~1995!.19M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889~1986!.20M. Kardar and Y.-C. Zhang, Europhys. Lett. 8, 232 ~1989!.21Z. Jiang and C. Ebner, Phys. Rev. B 41, 9121 ~1990!.22A. A. Middleton, Phys. Rev. E 52, R3337 ~1995!.23T. Halpin-Healy, Phys. Rev. B 42, 711 ~1990!.24T. T. Ogielski, Phys. Rev. Lett. 57, 1251 ~1986!.25R. Riedinger, Cryogenics 30, 465 ~1990!.26L. R. Ford and D. R. Fulkerson, Flows in Networks ~PrincetonUniversity Press, Princeton, NJ, 1962!. See also any standardbook on graph theory and flow problems.27A. V. Goldberg and R. E. Tarjan, J. Assoc. Comput. Mach. 35,921 ~1988!.28V. Dotsenko, A. B. Harris, D. Sherrington, and R. B. Stinch-combe, J. Phys. A 28, 3093 ~1995!.29 J. P. Bouchard and A. Georges, Phys. Rev. Lett. 68, 3908 ~1992!.54 14 993DISORDER-INDUCED ROUGHENING IN THE THREE- . . .

https://aaltodoc.aalto.fi/bitstream/handle/123456789/18943/A1_alava_mikko_j_1996.pdf?sequence=1&isAllowed=y

Disorder-induced roughening in the three-dimensional Ising model

Abstract

Similar works

Full text

Available Versions

Aaltodoc

Aaltodoc Publication Archive