Glassy Freezing and Long-Range Order in the 3D Random Field $XY$ Model

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by  Ronald Fisch

Glassy Freezing and Long-Range Order in the 3D Random Field XY Model
Ronald Fisch∗
382 Willowbrook Dr.
North Brunswick, NJ 08902
(Dated: January 19, 2010)
MonteCarlo studiesofthe3D random fieldXY modelon simple cubiclattices of size 643,using
two different isotropic random-field probability distributions of moderate strength, show a glassy
freezing behavior above Tc, and long-range order below Tc, consistent with our earlier results for
weaker and stronger random fields. This model should describe random pinning in vortex lattices
0
in type-IIsuperconducting alloys, charge-density wave materials, and decagonal quasicrystals.
1
0
2
The random-fieldXYmodel(RFXYM) wasfirststud- twisted at a boundary scales as Ld−2. This was later

n
iedfortyyearsago,[1]asatoymodelforrandompinning justified[7,8]byshowingthatwithinaperturbativeǫex-
a
effects in the vortex lattice of a type-II superconductor pansiononefindsthephenomenonof“dimensionalreduc-
J
in an external magnetic field. For fixed-length classical tion”. TermbytermforT >T ,thecriticalexponentsof
9 c
1 spins the Hamiltonian of the RFXYM is anyd-dimensionalO(n)random-fieldmodelappeartobe
identicaltothoseofanordinaryO(n)modelofdimension

n] H = −JXcos(φi−φj) − Xhicos(φi−θi). (1) d−2.
n hiji i However, reservations about the correctness of apply-
- ing the n = ∞ results for finite n, i.e. the substitution
s Eachφi isadynamicalvariablewhichtakesonvaluesbe- of the harmonic potential for the exchange term, were
i
d tween0and2π. Thehijiindicateshereasumovernear-
soon expressed by Grinstein.[6] For the Ising (n = 1)
. est neighbors on a simple cubic lattice of size L×L×L.
t case, dimensional reduction was shown rigorously to be
a We choose each θ to be an independent identically dis-
m i incorrect.[9, 10]
tributed quenched random variable, with the flat proba-
One problem with dimensional reduction for n > 1 is
- bility distribution
d that the perturbation theory can only be calculated for
n T > T , while the twist energy of the order parameter
P(θ ) = 1/2π (2) c
o i
is only defined for T < T . Another, more interesting
c
c
for θ between 0 and 2π. The probability distribu- problem for n > 1 is associated with the question of
[ i
tion P(hi) is independent of θi. This Hamiltonian is properly defining the order parameter. As discussed, for
1
closely relatedto models ofincommensurate charge den- example, by Griffiths,[11] there is more than one way to
v
sity waves,[2, 3] which include decagonaland other axial do this. For the random-field model, there are several
7
9 quasicrystals[4]as specialcases. Inallofthese cases,the different types of two-point correlation functions.[6]
3 physicaloriginof the “randomfield” is the fact that any When n=2, one may use the vector order parameter
3 realsample contains a finite density of point defects, i.e. M~ , defined by
.
01 aidteoamlsstsriuttcitnugrei.n places where they would not be in the M~ = N−1Xcos(φi)xˆ + sin(φi)yˆ, (3)
0 Wewillusetheterm“sample”todenoteaHamiltonian i
1 withaparticularsetoftheh andθ variables. Athermal where N =Ld is the number ofspins. Alternatively, one
v: average of some quantity oiver a siingle sample will be can choose the scalar order parameter M2, defined by
i
X denoted here by angle brackets, h i, and an average over M2 = N−2Xcos(φi)cos(φj) + sin(φi)sin(φj). (4)
all samples will be denoted by square brackets, [ ].
r i,j
a Larkin[1] replaced the spin-exchange term of the
Hamiltonian with a harmonic potential, so that each φ It is very awkward to calculate M~ , since its value will
i
is no longer restricted to lie in a compact interval. This varywildly fromsample tosample. Thus we expectthat
approximation,which neglects mode coupling, is strictly [M~ ] will be zero, even though hM~ i for each sample may
valid only in the limit n → ∞, where n is the number becomelargeatlowtemperature. Ontheotherhand,we
of components for each spin. Larkin then argued that, anticipate that for weak disorder the probability distri-
for any probability distribution P(h ) which allows non- bution for M2 will become narrow as we take L → ∞.
i
zero values for h , this model has no long-range ordered Thus it is the quantity hM2i, andnot the quantity hM~ i,
i
phase on a lattice whose spatial dimension d is less than which is the natural carrier of long-range order in the
or equal to four. presence of the random field.
A more intuitive derivation of this result was given It is also necessary to consider the possibility that the
by Imry and Ma,[5] who assumed that the increase in thermalaveragehM2iforatypicalfinite samplemaybe-
the energy of an Ld lattice when the order parameter is come large as T is lowered, even though hM~ i remains
2
small. This will occur when the energy required for re-
orienting hM~ i is nonzero, but not large compared to T. 8(cid:13)
If this were to occur, then there would be no local order random-field Z(cid:13)
12(cid:13)
SC L=64(cid:13)
parameter, and thus no terms in the Landau-Ginsburg
hr= 1.5(cid:13)
free-energy functional which depended on gradients of 6(cid:13) 8 samples(cid:13)
hM~ i.
Monte Carlo calculations for Eqn. (1) on simple cubic
lHauttsiec[e2s] wanedreFpisecrhfo[3rm].eMd osoremreecteimntelya,gaoddbiytioGnianlgcraaslcualnad- S(k))](cid:13) 4(cid:13)
n(
tions were carried out.[12] The results of these studies, v[l
a
1.3125 (cid:13) 1.5(cid:13)
which used several different choices for the probability
1.34375 (cid:13) 1.5625(cid:13)
distribution P(hi), indicate that for typical samples the 2(cid:13) 1.375 (cid:13) 1.625(cid:13)
order parameter hM2i becomes positive at some posi- 1.40625 (cid:13) 1.6875(cid:13)
1.4375 (cid:13) 1.75(cid:13)
tive temperatureT , aslongasthe randomfields arenot
c 1.46875(cid:13)
too strong. However, the precise nature of what hap- 0(cid:13)
0.04(cid:13) 0.06(cid:13) 0.08(cid:13) 0.1(cid:13) 0.2(cid:13) 0.4(cid:13)
pens near T remained unclear. In this work we report
c
k(cid:13)
additionalstudies using new choices of P(h ), which dis-
i
playphenomenawhichwerenotseenclearlyintheearlier
studies.
FIG.1: (color online) Averageover8 samples of ln(S(k)) for
The computer program which was used was an en-
64×64×64 lattices with hr = 1.5 and p = 0 at various
hancedversionofthe oneusedbefore.[12]The datawere
temperatures. The error bars indicate approximately three
obtainedfromL×L×LsimplecubiclatticeswithL=64
standard deviation statistical errors, and the x-axis is scaled
using periodic boundary conditions. Some preliminary logarithmically.
studies for smaller values of L were also done. The
program approximates O(2) with Z12, a 12-state clock
model. It was modified to enable the study of random- initialconditionsfollowedbyslowwarming. FortheType
field probability distributions of the form Asamplesthreedifferentcoldstartinitialconditionswere
used, equally spaced around the circle. For the Type B
P(hi) = (1−p)δ(hi−hr) + pδ(hi). (5) samples, where the average random field is somewhat
weaker, only two cold start initial conditions were used.
This modification makes the program run a few percent
In Fig. 1 we show data for the average over our eight
slower than before. In this work we study two cases.
Type A samples of ln(S(k)) for T/J between 1.3125and
Type A samples have h = 1.5 with p = 0, and Type B
r 1.75. The data for T/J ≥1.4375 are taken from the hot
sampleshaveh =2withp=0.5. Forbothofthesecases
r startruns,andthedataforlowervaluesofT/J aretaken
we find T /J near 1.5, which was suggested by Gingras
c from the cold start run which gave the lowest average
and Huse[2] to be optimal for this sort of calculation.
value for the energy, hEi. When T = 1.4375 or more,
OnasimplecubiclatticeofsizeL×L×L,themagnetic
all runs for a given sample converge in a rather short
structure factor, S(~k), for n=2 spins is time to givesimilarvaluesofhEiandhM~ i. However,for
lower values of T/J the hot start runs remain stuck in
S(~k) = L−3Xcos(~k·~rij)hcos(φi−φj)i, (6) metastable states.
i,j
An attempt was made to drive the hot start runs
where~r is the vector on the lattice which starts at site into equilibrium by deep undercooling, followed by slow
ij
i and ends at site j. Thus, setting ~k to zero yields warming. This thermal cycling technique had proven
successful earlier,[12] for the weak random field case of
S(0)/L3 = hM2(L)i. (7) hr =1.0withp=0. Forthecasesstudiedhere,however,
asingleundercoolingwasonlypartiallysuccessfulindriv-
As we will see, the extrapolation of hM2(L)i to L = ∞ ingthe samplesto equilibrium. The authorbelievesthat
is not trivial. repeating cycling would have worked successfully. How-
In this work, we present results for the average over ever,sinceadetailedstudy ofthe nonequilibriumbehav-
angles of S(~k), which we write as S(k). Eight different ior was not the purpose of the work described here, this
L=64samplesoftherandomfieldsθ wereusedforeach was not attempted.
i
P(h ). The same samples of random fields were used for Somewhat surprisingly, the cold start runs were much
i
all values of T. more successful in finding a single free energy minimum
The Monte Carlo calculations were performed using at T/J between 1.3125 and 1.40625. In the majority
both hot start (i.e. random) initial conditions followed of samples, all of the cold start initial conditions were
by slow cooling, and also cold start (i.e. ferromagnetic) converging to the same free energy minimum. This was
3
8(cid:13) random-field Z(cid:13) 8(cid:13)
12(cid:13) random-field Z(cid:13)
SC L=64(cid:13) 12(cid:13)
SC L=64(cid:13)
hr= 2.0 dil= 0.5(cid:13)
8 samples(cid:13)
7(cid:13)
6(cid:13)
av[ln(S(k))](cid:13) 4(cid:13) v[ln(S(0.049))](cid:13) 56(cid:13)(cid:13)
1.4375 (cid:13) 1.75(cid:13) a
2(cid:13) 1.5 (cid:13) 1.8125(cid:13)
1.5625 (cid:13) 1.875(cid:13) 4(cid:13) hr= 1.5(cid:13)
1.625 (cid:13) 1.9375(cid:13) hr= 2.0 dil= 0.5(cid:13)
1.6875 (cid:13) 2.0(cid:13)
0(cid:13)
3(cid:13)
0.04(cid:13) 0.06(cid:13) 0.08(cid:13) 0.1(cid:13) 0.2(cid:13) 0.4(cid:13)
1.3(cid:13) 1.4(cid:13) 1.5(cid:13) 1.6(cid:13) 1.7(cid:13) 1.8(cid:13) 1.9(cid:13) 2.0(cid:13)
k(cid:13)
T/J(cid:13)
FIG. 2: (color online) Average over8samples of ln(S(k)) for
FIG.3: (color online) Averageover8samples of ln(S(π/64))
64×64×64 lattices with hr = 2 and p = 0.5 at various for 64 ×64×64 lattices as a function of T/J. The error
temperatures. The error bars indicate approximately three
bars indicate approximately three standard deviation statis-
standard deviation statistical errors, and the x-axis is scaled
tical errors.
logarithmically.
hysteresis which is associated with a normal first-order
no longer true for T/J < 1.3125, and thus results for
phase transition. As was discussed in somewhat more
these lower temperatures are not reported.
detail in the earlier work,[12] the author believes that
For each sample at each value of T/J, the data shown
this phase transition should be described by a theory of
inthefigurewereobtainedbyaveragingS(k)over16spin
the Anderson-Yuval[13]type, wherethe orderparameter
states takenat intervals of51,200Monte Carlosteps per
jumpstozerodiscontinuouslyatT . Aphasetransitionof
spin (MCS), after letting the sample equilibrate. Thus c
thistypeisalsobelievedtooccurinthek-corepercolation
the length of eachdata-gatheringrun was 768,000MCS.
model,[14] which has been suggested as model for the
ThevaluesshownforS(k)areobtainedbyaveragingover
all of the values of ~k which have length k. This multi- ordinary glass transition.
The qualitative behavior which we see in Fig. 1 and
plicity, which is not the same for all values of k, was not
Fig.2 is thatthe averageoversamplesof ln(S(k)) shows
taken into account in calculating the error bars shown
a peak at small k which grows in size as we lower T,
in the figures. Thus each error bar shown in the figures
until it reaches a maximum at T . The value of T /J is
representsapproximatelythreestandarddeviations. The c c
about1.40625fortheTypeAsamplesand1.5625forthe
procedure used for the Type B samples was essentially
Type B samples. Upon lowering T below T , the size of
identical. c
the peak becomes somewhat smaller. This is seen more
InFig.2weshowthecorrespondingdataforthe Type
clearly in Fig. 3, which shows the temperature depen-
B samples, for T/J between 1.4375 and 2.00. The data
denceofav[ln(S(π/64))]forbothtypesofsamples. (The
for T/J >1.5625 are taken from the hot start runs, and
value of π/64 is approximately 0.049.)
the data for lower values of T/J are taken from the cold
start run which gave the lower average value for the en- For T > Tc both types display a substantial range
ergy, hEi. It was also true for these Type B samples of T over which S(π/64) is increasing exponentially as
that the hot start samples were unable to equilibrate on T is lowered, and this increase slows very close to Tc.
accessible time scales for T/J ≤1.5625. This kind of temperature dependence is well known
We can define T in a number of ways. For instance, phenomenologically[15] to be a characteristic of glassy
c
it can be defined as the highest temperature for which freezing behavior. However, the reader should keep in
we have been unable to equilibrate the hotstartruns. It mind that one expects the usual glass transition to cor-
is also the temperature for which the equilibration time respond to n= 3 spins, and not the n=2 spins studied
for the cold start runs becomes very long. The fact that here. If one generalizes Eqn. (1) to the n = 3 case, sub-
equilibrationtimes getverylongatthe same T for both stantially more computing resources would be required
c
thehotstartandthecoldstartinitialconditionsthatthe to make the equivalent calculation feasible.
behaviorofthis modelcannotbe explainedbythe usual Below T , S(π/64) first decreases, and then becomes
c
4
approximately constant, at least over the range of T peratures. In agreementwith prior results[12] on weaker
showninthefigure,wherethesamplesarebelievedtobe andstrongerrandomfields, wearguethat our resultsin-
equilibrated. This behavior below T is consistent with dicate a phase transition into a long-range ordered low
c
the existence of a δ-function peak in S at k = 0 in the temperature phase, with a jump in the order parameter
limit L → ∞, indicative of true long-range order. One at T . The behavior above T strongly resembles well
c c
would not expect to see this behavior if there were only known phenomenology for glassy phase transitions.[15]
power-law correlations in S(k), with no δ-function peak.
The reader should understand that we are not claim-
ing the existence of some mathematical error in the The author thanks the Physics Department of Prince-
analysis[16]oftheelasticglass,whichleadstothepower- ton University for providing use of computers on which
law correlated Bragg glass phase. That analysis begins the data were obtained.
with the LarkinHamiltonian,[1] which approximatesthe
spin-exchange term of Eqn. (1) with a harmonic interac-
tion. What we are saying is that the long-range behav-
ior of the Larkin Hamiltonian is different from that of ∗ [email protected]
Eqn. (1), at least for the types of P(hi) distributions we [1] A. I. Larkin, Zh. Eksp. Teor. Fiz. 58, 1466 (1970) [Sov.
have studied. Phys. JETP 31, 784 (1970)].
If we fit the data for av[ln(S(k))] at Tc over the range [2] M.J.P.GingrasandD.A.Huse,Phys.Rev.B53,15193
π/64 ≤ k ≤ π/8, as shown in Fig. 1 and Fig. 2, we (1996).
findgoodstraightline fits,withslopesof-3.20(3)forthe [3] R. Fisch, Phys. Rev.B 55, 8211 (1997).
[4] P. J. Steinhart et al.,Nature 396, 55 (1998).
Type A samples and -3.10(2) for the Type B samples.
[5] Y.ImryandS.-K.Ma,Phys.Rev.Lett.35,1399(1975).
Because both of these slopes are more negative than -3,
[6] G. Grinstein, Phys. Rev.Lett. 37, 944 (1976).
itis clearlynotcorrecttoextrapolatethese fits tok =0.
[7] A.Aharony,Y.ImryandS.-K.Ma,Phys.Rev.Lett.36,
S(k) satisfies a sum rule, and thus nonintegrable singu-
1364 (1976).
larities cannot occur. Thus if we were able to obtain [8] G.Parisi and N.Sourlas, Nucl.Phys.B206, 321(1982).
data using these P(hi) distributions for even larger val- [9] J. Z. Imbrie,Phys. Rev.Lett. 53, 1747 (1984).
ues of L, we would expect to find that the behavior of [10] J.BricmontandA.Kupiainen,Phys.Rev.Lett.59,1829
av[ln(S(k))]atT forsmallenoughnonzerok wouldlook (1987).
c
[11] R. B. Griffiths, Phys.Rev.152, 240 (1966).
like the data for the case h =2 with p=0 found in the
r [12] R. Fisch, Phys. Rev.B 76, 214435 (2007).
earlier work.[12]
[13] P.W.AndersonandG. Yuval,J. Phys.C4, 607(1971).
In this work we have performed Monte Carlo studies
[14] J. M. Schwarz, A. J. Liu and L. Q. Chayes, Europhys.
of the 3D RFXYM on L=64 simple cubic lattices, with Lett. 73, 560 (2006).
isotropic random field distributions of two types: hr = [15] M. L. Williams, R. F. Landel and J. D. Ferry, J. Am.
1.5 with p = 0, and h = 2 with p = 0.5. We present re- Chem. Soc. 77, 3701 (1955).
r
sultsforthe structurefactor,S(k),atasequenceoftem- [16] D. S. Fisher, Phys. Rev.Lett. 78, 1964 (1997).