Moving Vorties Interating with Periodi Pinning
GilsonCarneiro
InstitutodeFsia
UniversidadeFederaldoRio deJaneiro
C.P.68528, 21945-970,RiodeJaneiro-RJ, Brasil
Reeivedon28February,2002
Reentresultsobtainedbytheauthorforthedynamialphasediagramsforvortiesinleanlms,
drivenbyanuniformfore,andinteratingwithperiodipinningresultingfromaolumnardefet
lattiearedisussed. Usingnumerial simulationsofasimplemodelandotheronsiderations,the
dynamialphasediagramsareobtainedasafuntionofthedrivingforemagnitudeanddiretion,
the temperature, and the vortex density. The following dynamial phases and dynamial phase
transitionsarefound. Movingvortexlattiesat lowtemperatures, withspatialorderthat anbe
ommensurateor inommensuratewith the periodi pinning, moving vortex liquids and moving
smetis. Dynamialmelting ofmovingvortexlattiesintomovingvortexliquidstakesplaeand
transverse pinning ofmovingommensuratevortex lattiesand smetisours. Itis foundthat
thedynamialphase diagramsinthe theoretiallimit ofinnite drivingforemagnitudes play a
entral roleindetermining thewhole dynamialphasediagram: eahdynamialphaseoriginates
fromaninnite-drivelimit phasewith thesame spatialsymmetry thatevolvesontinuouslyinto
nite-driveregionsof thedynamialphasediagram. It isarguedthatthis onlusionalsoapplies
foralargelassofperiodipinningpotentials.
I Introdution
A problem of urrent interest is the study of
mov-ing vorties interating with arrays of pinning
en-ters. Random arrays have reeived the most
atten-tion. However,periodi pinning arrays are also of
in-terest. One reason is that they provide examples of
dynamialphasesand phasetransitions,asubjetnot
yetfullyunderstood,wheretheoretialpreditionsan
betestedinsuperondutinglmswithartiialdefet
latties[1℄, and in Josephson juntion arrays(JJA)[2℄.
Earlierworksonthissubjetareasfollows.
Vortex dynamis issimpler when thedriving fore
magnitude is large, in whih ase the veloity of the
enter of mass (CM) of the vortex array is alsolarge,
beausethemovingvortiesaveragethepinning
poten-tial. Intheaseofmovingvortiesinteratingwith
ran-dompinningarraysthelargedrivingforelimitwasrst
onsidered by Shmid and Hauger[3℄. They suggested
that fast movingvortiesaverage outthe pinning
po-tential and, onsequently, order in atriangularlattie
at low temperatures. In experiments [4℄ and
numeri-al simulations[5℄ nearlytriangular lattiesare indeed
observed at largeCM veloity. Koshlev and Vinokur
[6℄analyzedtheeetsofrandompinningatlargeCM
veloitiesonvortexdisplaementfromthelattie
equi-librium positions. They argued that it is equivalent
to a "shaking"temperature, inversely proportional to
perature,andpreditedthat dynamialmeltingof the
movingvortexlattietakesplae whenthe CM
velo-ity is suh that theombined temperature equals the
equilibrium meltingtemperature. Reently, Giamarhi
and Le Doussal [7℄ pointed out that this piture for
thelargeCM veloitiesbehaviorbreaksdown,beause
the vortiesorder in a movingglass, rather thanin a
movingtriangularlattie. Thereasonisthat averaging
of thepinning potentialbythe fastmovingvortiesis
onlypartial. Astatirandompinningpotentialremains
atingonthevortiesintheframeofreferenemoving
withtheCM veloity(CM frame),andhasnon-trivial
onsequenes on vortex order. These authors predit
that the vortiesmovealong stati hannels, on
aver-ageparalleltothediretionofdrive,andarepinnedon
themwithrespettoatransverseforeatzero
temper-ature. Thisisreferredtoastransversepinning. These
preditionsareinagreementwithexperimentaland
nu-merialresults[4℄.
Intheaseofvortiesinteratingwithperiodi
pin-ning, it is also expeted that at large CM veloities
averageofthepinningpotentialtakesplae. However,
itsonsequenesforthedynamialphasediagramhave
onlybeenexploredreently[8,9℄. Previoustheoretial
Exten-in systems with olumnar-defet(CD) latties, arried
out by several workers [5℄, nd a rather omplex
be-havior,withmanydynamialphasesandphase
transi-tions. A simpler nite-temperature behavioris found
in JJA and lms. Maroni and Domimguez [10℄
ar-riedoutnumerialsimulationsofsquareJJA,withthe
driving fore along the[1,0℄ (or[0,1℄) diretion. They
nd three dynamial phases: two moving vortex
lat-tiesandamovingliquid. Inallthree,theCMveloity
is along the diretion of drive. The latties have the
same spatial order, but dier from eah other by the
response to a fore transverse to the diretion of
mo-tion. One lattie, named transversely pinned vortex
lattie, is pinned (transverse pinning), and the other,
named oating solid, is unpinned. Dynamial melting
of theoatingsolidinto theliquid,and dynamial
de-pinningfromthepinnedlattietotheoatingsolidare
reported. Furtherwork[11℄preditstransversepinning
onlyfordrivingforesalong[1,0℄(or[0,1℄),andverifyit
experimentally. Finite-temperaturework onlms was
arriedoutbyReihhardtandZimanyi[12℄forvorties
interating with a square CD-lattie driven along the
[1,0℄,or[0,1℄,diretions,andforthevortexdensityfor
whih there is onevortexperCD. These authors nd
three dynamial phases: amoving lattie,
ommensu-rate with the pinning potential periodiity along the
diretion transverse to motion, a moving smeti and
a moving liquid. Dynamial transitions between the
ommensuratelattieandthesmetiandbetweenthe
smeti and the liquidare found, driveneither by the
temperatureorbyvortexmotion.
Theremainingofthispaperdesribestheapproah
proposed by the author [8, 9℄ to onstrut the
nite-temperaturedynamial phase diagram forvorties
in-teratingwithaCD-lattieasafuntionofthe
magni-tudeanddiretionofthedrivingfore,thetemperature,
andthevortexdensity.
This approah starts by pointing out that in the
limit of very large CM veloities the moving vorties
averagetheperiodipinningpotentialonlyinthe
dire-tion of motion,and that thedynamial phasesredue
totheequilibriumonesforvortiesinteratingwiththe
averagedpinningpotential[8,9℄. Thenitonsidersthe
limit of very large CM veloities, referred to here as
theinnite-drivelimit,foragenerimodelforvorties
in lean lms interating with periodi pinning (Se.
II). Itisshownthatin thislimitthedynamialphases
redue, in the CM frame, to the equilibrium ones for
vortiesinteratingwiththeperiodipinningpotential
averagedin thediretionof motion. These phases,
re-ferredtohereasinnite-drivephases,stablishtheexat
asymptoti behaviorof the dynamial phases at large
values of the driving fore magnitude, for a given
di-retion of motionand temperature. The innite-drive
phases depend also on the details of the pinning
po-tentialand onthe vortexdensity. Thenextstep isto
onsider a simple model that keeps the most
impor-tant features of the problem ( Se. III). This model
desribesvortiesin lmsinterating with square
lat-tieofolumnardefets(Fig.1). Firsttheinnite-drive
limitforthemodelisonsidered.Thepinningpotential
(Fig. 2.a) averaged in the diretion of motion is
non-trivialonly forthe[0,1℄(or[1,0℄) and[-1,1℄(or[1,-1℄),
where itis a washboard periodi in the perpendiular
diretion (Fig. 2.b and). In otherdiretions itis
ei-theraonstantoraveryshallowwashboard(Fig.2.d).
The washboards for [0,1℄ (or [1,0℄) and [-1,1℄ (or
[1,-1℄)stabilize movingvortexlattiesatlowtemperature
thatanbeommensurateorinommensuratewiththe
washboard's periodiity (Fig. 3). The innite-drive
phasediagramsarethen obtainedasafuntion ofthe
diretionofmotionandtemperaturefortwotypial
vor-tex densities byequilibrium Monte Carlo simulations.
Next, numerial simulations of Langevin's stohasti
dynamialequationsforthemodelarearriedoutin
or-dertodeterminethedynamialphasediagramatnite
drives (Se. IV). The most important result obtained
fromthese simulationsisthat, asfarasthespatial
or-deris onerned, theinnite-drivephasesexhaust the
whole dynamial phase diagram. In other words, all
dynamial phases originate from innite-drive phases
with the same spatial symmetry that evolve
ontinu-ously into regions of nite drives. It is argued that,
withafewexeptions,thisisalsotrueforalargelass
ofperiodipinningpotentials(Se.V).
II Innite-drive limit
Here the innite-drive limit is disussed for a generi
model for vorties interating with a CD-lattie. By
onsideringLangevin'sstohasti dynamialequations
forthismodelinthelimitwherethedrivingfore
mag-nitudeapproahesinnity,itisshownthatthese
equa-tions,writtenintheCMframe,desribevorties
inter-ating between themselves and with a stati eetive
pinningpotential,equaltothe average oftheperiodi
pinningpotentialinthediretionofmotion. Asa
onse-quene,thedynamialphasesreduetothe
orrespond-ingthermalequilibriumones,theinnite-drivephases,
thatanbedistinguishedfrom eah other onlybythe
spatialsymmetry.
The motion of N
v
two-dimensional vorties is
as-sumedtobegovernedbyLangevinequationsfor
mass-lesspartiles,whihforthel-thvortexreads[13℄,
dr
l
dt =F
v v
l +F
v dl
l
+F
d +
l
; (1)
where isthefritionoeÆient,
F v v
l =
Nv
X
j6=l=1 r
l U
v v
(r
l r
j
); (2)
istheforeof interationwith othervorties,U v v
(r)
being the vortex-vortex interation potential in
two-dimensions,
F v dl
= r
l U
v dl
(r
l
is the fore of interation with the CD-lattie,
U v dl
(r),beingtherespetivepotential,givenby
U v dl (r)= X R U v d
(r R); (4)
whereRdenotestheCD-lattiepositionsandU v d
(r)
istheinterationpotentialbetweenavortexanda
sin-gle CD, F
d
is the driving fore, and
l
is the
ran-dom foreappropriate fortemperature T. Intermsof
FouriertransformsU v dl
l
anbewrittenas
U v dl (r)= X Q U v d (Q)e iQr ; (5)
where Q denotes the CD-lattie reiproal lattie
vetors, and U v d
(Q) is the Fourier transform of
U v d
(r).
It is onvenient to onsider vortex motion in the
frame movingwith theCM. Let r 0
l (t) =r
l
(t) V
m t
denotethe l-th vortexpositionin theCM frame,V
m
beingtheCMveloity,tobedenedshortly. IntheCM
frame the vortex-CD lattieinteration, denoted here
byF v dl
l
(t), dependsexpliitlyontime,namely
F v dl l (t)= X Q ( iQ)U v d (Q)e iQr 0 l e iQVmt ; (6)
Theequation of motionfor thel-thvortexin the CM
framereads, dr 0 l dt =F v v l +F v dl l
(t)+F
d V m + l ; (7)
The CM veloity is dened by the ondition
P N v 1 hdr 0 l (t)=dt)i t
=0, wherehi
t
denotesaverageover
therandomforedistributionandovertime. Itfollows
fromEq.(7) that
V m =F d + 1 N v Nv X j=1 hF v dl l (t)i t : (8)
InthelimitF
d
!1,itfollowsfromEqs.(6)and(8)
thatV
m =F
d
=,sothat V
m
!1also. Inthisase,
allFourieromponentsinF v dl
l
(t),Eq.(6),forwhih
QV
m
6= 0 osillate fast, having a negligible eet
on the vortex trajetory [3℄, and F v dl
l
(t) redues to
thestatiforeobtainedbysummingtheFourier
om-ponents in F (v dl)
l
(t) with QV
m
=0, orQ ?F
d , namely F v dl l (t)! X Q?F d ( iQ)U v d (Q)e iQr 0 l (9)
Thisfore derivesfrom thestatieetivepinning
po-tential U eff (r 0 ? )= X Q?F U (v d) (Q)e iQr 0 ? ; (10) wherer 0 ?
istheCMframeoordinateperpendiularto
thediretion of F
d
, andthe orientationofF
d . This
potentialisequaltotheaverageofU v dl
inthe
dire-tionof motion(anddrive). Bydenition,U eff (r 0 ? )is
one-dimensional and periodi in thediretion
perpen-diularto F
d , ifF
d
isorientedalongoneofthesquare
CD-lattie diretions and a onstant otherwise, sine
no Q is perpendiular to F
d
. The equations of
mo-tion in the CM frame, Eqs. (7), redue then to those
desribingvortiesinteratingbetweenthemselvesand
withU eff (r 0 ?
)attemperatureT. Theirlong-time
solu-tions are theorresponding equilibrium phases, alled
innite-drivephasesinthispaper.
Theinnite-drivelimitestablishestheexat
asymp-totibehaviorF
d
!1ofthedynamialphaseforgiven
F
d
orientation and T. For nite F
d
, the dynamial
phase is expeted to be lose to the innite-drive one
with the same spatial symmetry for suÆiently large
F
d
. Theonditionforthistohappenisthatthevortex
displaementsin theCM frameaused bythepinning
fore osillationsin time,Eq. (7), during atime
inter-valoftheorderofone-halfperiod,aresmallompared
with thevortexmeanseparationa
v
. This meansthat
foreveryQ,suhthat QF
d 6=0,
QjU (v d)
(Q)j
QF
d a
v
: (11)
III Model and numeris
Hereasimplemodelthatallowsexpliitresultsforthe
thedynamialphasediagramstobeobtainedis
onsid-ered[8℄. Themodel onsiders vortiesand CD plaed
on a square lattie (spae lattie), subjeted to
peri-odiboundaryonditionswith 256256square
prim-itiveunit ellsof dimensions dd. The CD-lattieis
square, orientedparallel tospae lattieand
ommen-surate withit. TheCD-lattie, the oordinate system
and theanglesdening theorientationsof F
d
()and
V
m
()areshownin Fig. 1.
x
y
F
d
V
cm
Figure 1. Columnar defet-lattieand denitions of
oor-dinatesystem,CMveloity (V
m
)anddrivingfore(F
d )
orientations.
vortex has a ore of linear dimension d
v
= 4d (d
v
2(0)), so that eah vortex oupies 16 spae-lattie
sreened Coulomb one [14℄, periodi in the spae
lattie, and haraterized by the energy sale J =
( 2 0 d v =32 3 2
),whereisthepenetrationdepth. That
is, U v v
(r)isthelattieFouriertransformof
U v v
(k)=4 2 J e 2 2 2 + 2 ; where 2
= 4sin 2
(k
x
d=2)+4sin 2 (k y d=2), k x and k y
being the spae lattie reiproal lattie vetors
om-ponents,
= 2sin(d=2d
v
) is the vortex ore uto
in k-spae, and is the sreening length ( > ).
The square CD-lattie has N
d
= 88 sites, lattie
onstant a
d
= 32d. The interation potential
be-tween avortex and a singleCD is hosen with depth
U v d
(r = 0) = J, range R
d
= 12d and a spatial
dependenethatgivessquareequipotentialsanda
pin-ning fore ofonstantmodulusF
p =J=R
d
, asshown
in Fig.2.a. Thereiproal CD-lattievetorsare
Q=Q
1 (n
1 ^ x+n
2 ^ y); where Q 1 =2=a d , and n
1 ;n
2
= 4; 3;:::;3. It is
foundthatthelattieFouriertransformsU v d
(Q)are:
U v d
(0)= 0:19J,U v d
(Q
1 ^
x)=U v d
(Q
1 ^
y )=
0:10J,U v d
(Q
1
[^xy ℄)^ = 0:06,andessentially
neg-ligibleotherwise.
Themodelhasthesquarelattiesymmetry,sothat
thedynamialphasediagramsneedonlyto bestudied
forF d orientations0 o 45 o .
Thedynamial phasesspatial order is obtainedby
alulating the time-averaged density-density
orrela-tionfuntion,P(r),whih isproportionalto the
prob-ability that avortexis foundat r, giventhat thereis
oneat r=0,and its Fouriertransform, thestruture
funtion,S(k). Themotionofvortiesisharaterized
byalulatingtheCM veloity,and thetime-averaged
veloityofeahvortex.
Two types of numerial studies of the model are
arried out. Equilibrium Monte Carlo simulations to
obtain the innite-drive phases, desribed in detailin
Se. IV A. Numerial integration of Langevin's
equa-tionstoobtainthedynamialphasediagramasa
fun-tionofF
d
,andT fortwotypialvortexdensities: two
vorties per CD and ve vortiesperfour CD. These
orrespondsto the magnetiindutions B =2B
and
B = 1:25B
, respetively, where B
= 0 =a 2 d is the
mathing eld, for whih there is one vortex perCD.
The tehnialdetails of these simulations aregivenin
Refs. [8,9℄.
IV Results
Inthissetionthesimulationresultsforthedynamial
phasediagramsarereported. Firstintheinnite-drive
limit,thenfornitedrives. Inthegurespresentedhere
driving fore magnitudesare measured relative to the
singleCD pinningfore F
p
, Se.III;temperatures
rel-ativeto theinnite-drivemoving inommensurate
lat-tie meltingtemperatureT
m
, Se. IVA and enter of
mass veloity omponents relative to the omponents
ofV
d F
d =.
A.Innite-Drive Limit
The innite-drive phasediagrams are obtained by
equilibriumMonteCarlosimulationsofthelattie
Lon-don model with the pinning potential U eff
(r
? ). It
is found that for F
d
along [0,1℄ ( = 0 o
) and [-1,1℄
( =45 o
) the U eff
(r
?
)are the washboards shownin
Figs.2b)and). ForF
d
alongotherlattiediretions,
theU eff
(r
?
)arefoundtobeveryshallowwashboards,
beausetheU (v d)
(Q) (Eq.(10))areverysmall(Se.
III), and are onsidered as onstant potentialsin this
paper. Forexample, theU eff
(r
? ) forF
d
alongthe
[-1,2℄diretion(=26:6 o
),shownin Fig.2.dhasawell
depthmorethan oneorder ofmagnitude smallerthan
that for = 0 o
and 45 o
shown in Figs. 2b) and ).
ForF
d
orientedinnon-lattiediretions,U eff
(r
? )isa
onstant.
a)
b)
d)
c)
U
v-cdl
U
eff
o
U
eff
o
U
eff
o
Figure2. a)Columnardefet-lattiepinningpotential.b-d)
Eetivepinningpotentialsformotionin: b)[0,1℄diretion
(=0 o
),)[-1,1℄diretion(=45 o
),andd)[-1,2℄diretion
(=26:6 o
). PotentialsinunitsofthesingleCDpotential
welldepth(J).
The orresponding low-T innite-drive phases are as
follows. For = 0 o
and 45 o
, they are vortex
lat-ties(VL) ommensurateorinommensuratewiththe
one-dimensionalperiodiity, depending onB. TheVL
for = 0 o
, are obtained here as a funtion of B, for
B B
. Their density-density orrelation funtions,
P(r), areshown in Fig.4. Theommensuratelatties
onsistofidentialvortexhainswithineahwashboard
hannel,withneighborhainsdisplaedwithrespetto
eahotherbyhalfahainperiod. Thereisasinglehain
forB
B B
1
, and twohains forB
1
<B B
2 .
B
1
< 1:125B
and 1:125B
< B
2
< 1:375B
. A
ommensurate-inommensurate transition takes plae
for B = B
i
(1:25B
< B
i
< 1:375B
). For
B>B
i
theVLisinommensurate,andnearly
trian-gular. For B =2B
andB =1:25B
theP(r)shown
in Fig. 4 orrespond, respetively, to thevortex
lat-ties labeled MIL and MCL0 in Fig. 3. Theeet of
the one-dimensional periodi potential on the
inom-mensurate lattie is to displae the vorties from the
triangular lattie positions by a distane small
om-paredtothelattieparameter[15℄. It isfoundthat, as
expeted, this eet isgreater forB in the viinityof
B
i
. ThisisevidenedbythebehaviorofP(r)shown
in Fig.4. ForB =1:375B
,1:5B
, and1:75B
, P(r)
hassmeared spotsenteredinanearlytriangulargrid,
withsmearinginreasingwithdistane. ForB>2B
,
P(r)hassharpspots,indiatingthatthedisplaements
arenegligible,andtheVLissharplydened.
B=2B
=45
o
B=2B
=0
o
B=1.25B
=45
o
B=1.25B
=0
o
a) MCL0
d) MCL
c) MIL
b) MCL45
Figure 3. Vortex positions in low-T innite-drive phases.
DashedgraylinesindiateminimaofU eff
showninFigs.2.b
and2.. Nomenlature: movingommensuratelatties: a)
MCL0, b) MCL45 and d) MCL. Moving inommensurate
latties: )MIL
The twoB values studied in this paper,B =2B
and B =1:25B
, are typialof parameter regionsfar
from ommensurate-inommensuratetransitions, both
for=0 o
and=45 o
, wheretheVLaresharply
de-ned. Theinnite-drivephasediagramsin these ases
are asfollows. For=0 o
and 45 o
the VLare shown
in Fig.3. These are referredto in this paper as
mov-ing inommensurate latties (MIL) and moving
om-mensuratelatties, withdistint vortex ongurations
labeled MCL,MCL0 and MCL45, as shownin Fig. 3.
For 0 o
< < 45 o
, the VL are inommensurate and
nearlytriangular. Itisfoundthattheinommensurate
tinguishedfromeahother. Hereafterall
inommensu-ratelattiesarereferredtoasmovinginommensurate
latties(MIL).
F
d
B=1.125 B
f
B=1.25 B
f
B=1.375 B
f
B=1.5B
f
B=1.75 B
f
B=2 B
f
Figure4. Density-densityorrelationfuntions,P(r)(r=0
is atthepanel'senter),for low-T innite-drivephases for
motioninthe[0,1℄(=0 o
)diretionasafuntionofB.
The T-dependene of the innite-drive phases (at
onstantisfoundtobeasfollows: themoving
inom-mensurate latties (MIL) melts into a moving vortex
liquid (MLQ)at k
B T
m
=0:09J,for bothB. ForB =
2B
themovingommensuratelattie(MCL) hanges
intoamovingsmeti(MSM)atT
ml =T
m
=1:4andthe
moving smeti (MSM) hanges into a moving liquid
(MLQ) at T
msm =T
m
=1:8. For B =1:25B
the
mov-ingommensuratelatties,MCL0andMCL45,hange
into amovingsmeti (MSM) at T
ml0 =T
m
=1:2 and
T
ml45 =T
m
=1:7,respetively.
1. Drivealong =0 o
and =45 o
Dynamial phase diagrams (F
d
vs. T) are shown
in Fig. 5. For B = 2B
, =0 o
, onlytwodynamial
phases exist: amovinginommensurate lattie(MIL)
andamovingliquid(MLQ),separatedbyadynamial
meltingline, asshowninFig.5. Thedynamialphase
diagramsforbothB,=45 o
,andforB=1:25B
,=
0 o
,aresimilartooneanother,ontainingthreephases:
a moving ommensurate lattie (MCL for B = 2B
,
MCL0 andMCL45forB =1:25B
)amovingsmeti
(MSM)andamovingliquid. Onlythedynamialphase
diagrams for B = 2B
, = 45 o
, and B = 1:25B
,
= 0 o
are shown in Fig. 5. That for B = 1:25B
,
=45 o
, is similar. Inthe temperaturerange overed
P(r) at large F
d =F
p
for the above desribed phases
are shown. It is found that the dynamial phases at
large F
d =F
p
essentially oinide with the
orrespond-ing innite-driveones. Thedynamialphasediagrams
in Fig.5showlearlythateahdynamialphase
origi-nates fromtheinnite-driveonewiththesamespatial
symmetrythatevolvesontinuouslyintonitedrive
re-gions.
As disussed next, the ommensurate VL (MCL,
MCL0 and MCL45) show transverse pinning at low
T. This may be expeted beause for = 0 o
and
= 45 o
, V
m
remains parallel to F
d
. In all these
ases, aording to Eq. (6), the eetive pinning
po-tential(U eff
(r
?
)) ats on the moving vorties for all
F
d
,notonlyin theF
d
!1limit. Theommensurate
phasesarepinnedbyU eff
(r
?
)withrespettoasmall
fore alongthediretiondened byr
? .
0.0
0.5
1.0
0.0
0.5
1.0
1.5
0.0
2.0
4.0
0.5
1.0
MIL
MLQ
T/
T
m
MSM
MCL
T/
T
m
B=2B
=45
o
B=2B
=0
o
B=1.25B
=0
o
F
d
/ F
p
MSM
MCL0
T/
T
m
Figure 5. Dynamialphasediagramsfor drivesalong[0,1℄
( = 0 o
) and [-1,1℄ ( = 45 o
). Panels: density-density
orrelation funtions P(r) at high drives (r = 0 is at
the panel's enter). Nomenlature: MLQ=moving liquid,
MSM=moving smeti. Others as in Fig.3. Dotted lines
in thepannels for B =2B indiatewhere vortexmotion
stops.
2. Drives along 0 o
<<45 o
Thedynamialphases
found in this range of driving-fore orientations have
thesamespatialsymmetriesasthosedesribedinSes.
IV Aand IVA 1. andare referredto in whatfollows
bythesamenomenlature.
The F
d
vs. dynamial phase diagrams at low-T
are shown in Fig. 6. Both have two dynamial
tran-sitionlines. One, referred to hereas dynamial
melt-ing line, separating a moving inommensurate lattie
(MIL)andamovingliquid(MLQ).Anotherseparating
amovingommensuratelattie(MCLforB=2B
and
MCL45for B = 1:25B
) oramovingsmeti (MSM)
for B = 2B
and a moving liquid. Within the
mov-ing ommensurate lattie and moving smeti regions
transverse pinning ours, with the vorties moving
along the [-1,1℄ diretion. The transition line in the
B = 2B
dynamial phase diagram from the moving
ommensurate lattie(MCL) and the moving smeti
(MSM)ourswiththevortiesmovinginthe[-1,1℄
di-retion,andisessentiallyidentialtothatfor=45 o
disussedinSe.IVA 1.
Theevideneleadingtotheonstrutionofthe
dy-namialphasediagramsin Fig.6isdisussed indetail
inRefs.[8,9℄
1.0
3.0
5.0
0
15
30
45
MSM
a) B=2B
φ
T/T
m
=0.83
MIL
MLQ
MCL
α
o
F
d
/ F
p
1.0
3.0
5.0
0
15
30
45
MSM
MLQ
MCL0
b) B=1.25B
φ
T/T
m
=0.89
MIL
MLQ
MCL45
α
o
F
d
/ F
p
Figure6.DynamialphasediagramsforT =onstant.
Dy-namialphasesnamedasinFigs. 3and4.
Thedetailed propertiesof thedynamialphase
di-agramsshowninFig.6areasfollows.
i)DynamialMelting:Thedynamialmeltinglines
ex-tendfrom=0 o
to =45 o
. ForB =2B
ittouhes
the = 0 o
-axisat the F
d
value where the dynamial
meltingtakesplaefor =0 o
andT=T
m
=0:83(Fig.
5). It doesnottouhthe=45 o
axisforbothB and
the= 0 o
axisfor B = 1:25B
, beause the
dynam-ialphases in theseaxes arenotmoving
inommensu-rate latties, as disussed in Se. IV A. These result
of (Fig. 6) showthat, for bothB, themoving
inom-mensurateandommensuratelattiesareseparatedby
a moving liquid, at least up to the highest F
d
stud-iedhere. However,in theinnite-drivephase diagram
there isno movingliquid at these temperatures. It is
unlearhowthedisappearaneofthemovingliquidas
F
d
!1takesplae.
The vs. F
d
urves in Figs. 7 and 8 show that
in the moving inommensurate lattie (MIL) the
di-retionsofdriveandvortexmotionareapproximately
equal ( ' ), and that the melting transition is
thishange issharp,withinreasingrapidlytowards
=45 o
asF
d
dereases. ForB =1:25B
thesameis
true,aslongas&20 o
,whileforsmallerthemoving
liquidslowlyapproahes=0 o
asF
d
dereases.
ii)TransversePinning. TheresultsshowninFigs.
7and 8 indiate that in the regionswhere the spatial
order is that of a moving ommensurate lattie, or a
moving smeti for B = 2B
, transverse pinning
o-urs, and vortex motion is restrited to the [-1,1℄
di-retion. This is seenin Figs. 7and 8, where V
? = 0
and =45 o
in theF
d
rangeofmovingommensurate
lattiesorsmetis, andthe distributionof individual
vortiesdiretion of motion is sharply peakedaround
=45 o
(Fig.7,inset). Theseresultsalsoshowthatthe
movingommensurateand smetiregionsevolve
on-tinuouslyfromtheorrespondinginnite-drivephases.
The temperature dependene of the dynamial
phase diagrams is as follows. For temperatures lower
thanthoseofFig.6thedynamialphasediagramsare
similar. The dynamial melting lines are shifted to
lowerF
d
, similarly to what happens forB =2B
and
=0 o
(Fig.5). Transversepinninginthemoving
om-mensuratelattiesalsoours,asevidenedbytheV
?
vs. F
d
urvesshowninFig.9. Forhighertemperatures
several hanges are observed. The dynamial melting
lines also exist, shifted to larger F
d
, provided T does
not exeedthe innite-drive melting temperature T
m .
WhenT >T
m
themovinginommensuratelattiesdo
notexist,onlymovingliquids.
0
2
4
6
10
20
30
40
50
0
2
4
0.0
0.4
0.8
44.5 45.0 45.5
0
20
40
60
80
B=2B
φ
T / T
m
=0.83
θ
o
F
d
/F
p
θ
α
=35
o
α
=35
o
α
=22.5
o
α
=15
o
α
=10
o
V
]
/ V
d
]
F
d
/ F
p
Figure7.Centerofmassveloitydiretion()and
ompo-nent perpendiular tothe[-1,1℄ diretion,V
?
,vs. driving
foremagnitudealonglinesofonstantinFig.5.a. Inset:
histogramforthedistributionofvortiesdiretionofmotion
()for=35 o
andFd=Fp=1:5.
ommensurate lattiesorsmetis only exists,stritly
speaking, at T = 0. At nite T thermalexitation of
lattiedefets leadto uxmotionawayfromthe[-1,1℄
diretion, giving rise to a nite V
?
. The regions of
transversepinningfoundinthepresentsimulations
o-urbeausethethermallyexitedV
?
issmallerthanthe
simulationresolution. Inexperiments, wherethe
reso-lution isalso nite,these regionsmayalsobepresent.
The T dependene of the V
? vs F
d
urves are shown
in Fig.9. ForB =2B
transverse pinningis foundin
themovingommensuratelattieandmovingsmetis
at temperaturesaboveT
m
, whereasforB =1:25B
it
disappearsatlowertemperatures.
0
2
4
6
0.0
0.4
0.8
0
2
4
6
0
10
20
30
40
v
]
/ v
d
]
F
d
/ F
p
α
=42.5
o
α
=40
o
α
=22.5
o
B=1.25B
φ
T/T
m
=0.89
θ
o
F
d
/F
p
α
=40
o
α
=42.5
o
α
=22.5
o
α
=5
o
α
=39
o
Figure8. Centerofmassveloitydiretion()and
ompo-nent perpendiular to the [-1,1℄diretion ,V?, vs. driving
foremagnitudealonglinesofonstantinFig.5.b.
V Disussion
The resultsdesribedabove reveal the importantrole
playedbytheinnite-drivephases. Theyshowthat all
dynamial phases originate from innite-drive phases
with the same spatial symmetry that evolve
ontinu-ouslyintoregionsofnitedrives.
TheresultsobtainedbyReihhardtandZimanyi[12℄
for vorties in lms interating with a square
CD-lattie,driveninthe[1,0℄diretion,atB=B
,showa
similarbehavior. Theinnite-drivephasediagramwas
notonsideredinthispaper,butanbeidentiedwith
thereporteddynamialphasediagramsat highdrives.
fol-T, with single-hain struture similar to that shown
in Fig. 4 for B = 1:125B
rotated by 90 o
, a moving
smetiatintermediateT,andamovingliquidathigh
T. Thedynamial phasediagram reported in Ref.[12℄
shows that these three phases exist in ontinuous
re-gions of the driving fore vs. T phase diagrams that
extendfrom hightolowdrives. Thisphasediagramis
similartothosefoundhereforB =1:25B
,=0
o
and
45 o
,andforB=2B
=45
o
(Se. IVA1).
0
2
4
0.0
0.5
1.0
B=1.25B
φ
α
=40
o
V
]
/ V
d
]
F
d
/ F
p
T / T
m
=0.56
T / T
m
=0.89
T / T
m
=1.1
T / T
m
=1.4
0
2
4
0.0
0.5
1.0
α
=22.5
o
α
=35
0
B=2B
φ
V
]
/ V
d
]
F
d
/ F
p
T / T
m
=0.56
T / T
m
=0.83
T / T
m
=1.1
T / T
m
=0.83
T / T
m
=1.1
Figure 9. Component of CM veloity perpendiular to
[-1,1℄,V?,vs. drivingforemagnitudealonglinesofonstant
andatonstanttemperatures.
For vorties in JJA, driven in the [1,0℄diretion, the
dynamial phase diagram reported by Maroni and
Dominguez[10℄ alsoshowdynamial phasesexisting in
ontinuousregionsofthedrivingforevs. T planethat
extendfrom hightolowdrives. Howeverit isnot
pos-sibleto identifytheinnite-drivephasesfromthedata
reportedinthepaper.
Some studies of vorties interating with
CD-latties at T =0[5℄ nddynamial phasesand phase
transitions similar to the ones desribes here and in
Ref.[12℄. Examplesaretranversepinning,moving
om-mensurate and inommensurate latties and
dynami-al transitions between them. However, these
stud-iesprobedynamialbehaviordierent fromthe
nite-temperature ones disussed here. At nite
tempera-asteady state,identiedasthedynamialphase,that
is uniqueasfar asthe probabilitydistribution is
on-erned [16℄. Aordingly, dynamial phase hanges
aused by varying the driving fore, thetemperature,
orboth,arereversible. Thisisnotneessarilythease
atT =0. However,innumerialstudies,irreversibility
appearsduetoinsuÆientruntimestoreahthesteady
state,partiularlywhenrelaxationtimesareverylarge.
Thisoursin severalirumstanes,suhas low
tem-peratures, or viinity of dynamial phase hanges. In
thesimulationsreportedinthispaperlowtemperatures
areavoided. Somesimulationrunswhereperformed
y-lingthedrivingforemagnitudefromalargevalueto
asmall oneand bak. Small hysteresis is found near
dynamialphasetransitions.
In onlusion, the approah desried here to
on-strutthedynamialphasediagrams,startingfromthe
innite-driveones,providesasimplemethod, basedon
equilibriumstatistialmehanis,toidentifydynamial
phases spatial order, and to predit dynamial phase
transitions. Itisexpetedthatthis methodis
applia-ble to alarge lass of periodi pinning potentials and
vortexdensities. Thereasonsarethattheeetive
pin-ningpotentialsresultingfromaveragingphysially
rea-sonable periodi pinning potentials in the diretion of
motionare,asthosedisussedinSe.IVA,essentially
onstantformostdiretions, and one-dimensionaland
periodiforafewpartiularones. Thispreditsthe
ex-istene,forB B
,ofmovinglatties,ommensurate
orinommensuratewiththeeetivepinningpotential
periodiity, moving smetisand moving liquids,
simi-larto theones reported here,and of dynamialphase
transitionsbetweenthen. For B <B
themoving
lat-tiesmaybedierent,partiularlyatlowvortex
densi-ties,wheretheommensurate innite-drivephasesare
notexpetedtoretainthesimplehainstruturefound
here for the B B
ones. However, the lose
rela-tionship between the dynamial phase diagrams and
the innite-drive ones are still valid, as evidened by
theJJAresultsofRef.[10℄. Thedynamialphases
ob-tainedbythis method do not,in general, exhaust the
dynamialphasediagram. Oneknowntypeof
dynami-alphasethathasnoorrespondinginnite-drivephase
isthat in whih somevortiesare pinned by the
peri-odi potentialandothersaremoving. Thesearefound
at low drives in some T = 0 simulationsfor B >B
[5℄. Nosuhphasesarefoundinthepresentsimulations
norin Refs.[5, 10℄. Thedetails of how the dynamial
phases and dynamial phase transitions predited by
themethodproposedheretintothedynamialphase
diagram for eah partiular model and vortexdensity
dependsinaompliatedwayonthemodelparameters,
andhasto bedeterminedineahase.
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