C. Reichhardt and C.J. Olson Reichhardt- Moving Vortex Phases, Dynamical Symmetry Breaking, and...

8/3/2019 C. Reichhardt and C.J. Olson Reichhardt- Moving Vortex Phases, Dynamical Symmetry Breaking, and Jamming for V

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arXiv:0807.2671

v1

[cond-mat.supr-con]16Jul2008

Moving Vortex Phases, Dynamical Symmetry Breaking, and Jamming for Vortices in

Honeycomb Pinning Arrays

C. Reichhardt and C.J. Olson ReichhardtTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Dated: July 16, 2008)

We show using numerical simulations that vortices in honeycomb pinning arrays can exhibit aremarkable variety of dynamical phases that are distinct from those found for triangular and squarepinning arrays. In the honeycomb arrays, it is possible for the interstitial vortices to form dimeror higher n-mer states which have an additional orientational degree of freedom that can lead tothe formation of vortex molecular crystals. For filling fractions where dimer states appear, a noveldynamical symmetry breaking can occur when the dimers flow in one of two possible alignmentdirections. This leads to transport in the direction transverse to the applied drive. We show thatdimerization produces distinct types of moving phases which depend on the direction of the drivingforce with respect to the pinning lattice symmetry. When the dimers are driven along certaindirections, a reorientation of the dimers can produce a jamming phenomenon which results in astrong enhancement in the critical depinning force. The jamming can also cause unusual effectssuch as an increase in the critical depinning force when the size of the pinning sites is reduced.

PACS numbers: 74.25.Qt

I. INTRODUCTION

Vortex matter in type-II superconductors has been ex-tensively studied as a unique system of many interactingparticles in which nonequilibrium phase transitions canbe accessed readily [1, 2, 3, 4, 5, 6, 7, 8]. In the ab-sence of driving or quenched disorder, the vortex-vortexinteractions favor a triangular crystalline ordering. If thesample contains sufficiently strong quenched disorder inthe form of randomly placed pinning sites, the vortex lat-tice ordering can be lost as the vortices adjust their posi-tions to accommodate to the pinning landscape [1, 2, 3].Under an applied drive such as the Lorentz force from

a current, the vortices remain immobile or pinned forlow driving forces; however, there is a threshold appliedforce above which the vortices begin to move over thequenched disorder. For strong disorder, the initial mov-ing state is highly inhomogeneous with the vortices flow-ing in meandering and fluctuating channels, and there isa coexistence between pinned vortices and flowing vor-tices [1, 2]. At higher drives the vortices move morerapidly, the effectiveness of the quenched disorder is re-duced, and the fluctuations experienced by the vorticesbecome anisotropic due to the directionality of the ex-ternal drive [4]. The vortex-vortex interactions becomemore important at the higher drives when the quencheddisorder becomes ineffective, and a dynamical transitioncan occur into a moving smectic state where the vorticesregain partial order in one direction [5, 6, 7]. Here, thesystem has crystalline order in the direction transverse tothe vortex motion and liquid-like order in the direction ofvortex motion. Depending on the dimensionality and thestrength of the pinning, it is also possible for the vorticesto reorganize in both directions at high drives to forma moving anisotropic crystal [4, 5, 6, 7, 8, 9]. The ex-istence of these different phases and transitions betweenthe phases can be inferred from signatures in transport

[3] and noise fluctuations [10, 11], and the moving phaseshave also been imaged directly using various techniques[7, 8].

In addition to the naturally occurring randomly placedpinning sites, it is also possible for artificial pinning sitesto be created in a periodic structure [12]. Recent ad-vances in nanostructuring permit the creation of a widevariety of periodic pinning landscapes where the period-icity, shape, size and density of the pinning sites can bewell controlled. Distinct types of pinning arrays such assquare [13, 14, 15, 16, 17, 18, 19, 20] triangular [21, 22],rectangular [23, 24], honeycomb [25, 26], kagome [25],quasicrystalline [27], and partially ordered [28] structures

have been created. In these arrays the type of vortexstructure that forms is determined by whether the vor-tex lattice is commensurate with the underlying pinningarray. Commensurate arrangements appear at integermultiples of the matching field B, which is the mag-netic field at which the vortex density matches the pin-ning density, and in general, ordered vortex states oc-cur at matching or rational fractional values of B/B[13, 14, 15, 29, 30, 31, 32]. In samples where only onevortex can be captured by each pinning site, the vor-tices that appear above the first matching field sit in theinterstitial regions between the pinning sites, and theseinterstitial vortices can adopt a variety of crystalline con-figurations [14, 15, 16, 17, 19, 24, 29, 30, 32].

Since a number of distinct ordered and partially or-dered vortex states can be created in periodic pinningarrays, a much richer variety of dynamical vortex behav-iors occur for periodic pinning than for random pinningarrays [15, 16, 17, 33, 34, 35, 36, 37, 38, 39, 40, 41,42, 43, 44, 45, 46, 47, 48, 49]. Several of the dynami-cal phases occur due to the existence of highly mobileinterstitial vortices which channel between the pinnedvortices [15, 17, 35, 36, 37, 39, 40, 42, 43, 45, 49].As a function of applied drive, various types of mov-
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x(a)

y

x(b)

y

x(c)

y

x(d)

y

FIG. 1: (a) Pinning site locations (open circles) for a triangu-lar pinning array. (b) Pinning site locations for a honeycombpinning array constructed from the triangular array in (a) byremoving 1/3 of the pinning sites. (c) The pinning site lo-cations and vortex positions (dots) for a honeycomb pinningarray at B/B = 1.5. The overall vortex lattice order is tri-angular. (d) The pinning site locations and vortex p ositionsfor a honeycomb pinning array at B/B = 2.0, where twovortices are captured at the large interstitial sites and the re-sulting dimers all have the same orientation. Here Fp = 0.85,Rp = 0.35, and for the honeycomb array np = 0.3125/

2.

ing phases occur, including interstitial vortices movingcoherently between the pinning sites in one-dimensionalpaths [15, 17, 33, 34, 35, 36, 39, 43] or periodically mod-ulated winding paths [33, 35, 44, 45], disordered regimeswhere the vortex motion is liquidlike [33, 35, 39, 43],and regimes where vortices flow along the pinning rows[33, 38, 46, 47, 48]. Other dynamical effects, such asrectification of mixtures of pinned and interstitial vor-tices, can be realized when the periodic pinning arraysare asymmetric [49].

Most of the studies of vortex ordering and dynamics inperiodic pinning arrays have been performed for squareand triangular arrays. Experiments with honeycomb andkagome pinning arrays revealed interesting anomalies inthe critical current at nonmatching fields which are aspronounced as the anomalies observed at matching fieldsin triangular pinning arrays [25, 26]. A honeycomb pin-ning array is constructed by removing every third pin-ning site from a triangular pinning array, producing aperiodic arrangement of triangular interstitial sites. InFigs. 1(a,b) we illustrate a triangular pinning array alongwith the honeycomb pinning array that results after theremoval of one third of the pinning sites. The match-

ing anomalies in the experiments coincide with fieldsB/B = m/2, with m an integer. At these fields, thevortex density would match with the regular triangularpinning array. At the matching anomalies for m > 2, aportion of the vortices are located in the large intersti-tial regions of the honeycomb lattice [26], as illustratedin Fig. 1(c) for B/B = 1.5. The overall vortex latticestructure is triangular and a strong peak in the depinning

force occurs at this field [50].Recently, we used numerical simulations to demon-

strate that vortices in honeycomb pinning arrays havea rich equilibrium phase diagram as a function of vor-tex density [50], with matching anomalies at integer andhalf integer matching fields that are in agreement withexperiments. The large interstitial sites created by themissing pinning sites can capture multiple interstitialvortices which form cluster states of n vortices. For1.5 B/B < 2.5, dimer states with n = 2 form, whilefor higher fields trimer and higher order n-mer statesform. At the integer and half-integer matching fields,the n-mer states can assume a global orientational or-

dering which may be of ferromagnetic or antiferromag-netic type; herringbone structures can also form, similarto those observed for colloidal particles on periodic sub-strates [51, 52, 53, 54, 55] and molecules on atomic sub-strates [56]. These orientationally ordered states havebeen termed vortex molecular crystals. Certain vortexmolecular crystals have ground states that are doubly ortriply degenerate, such as the dimer state illustrated inFig. 1(d) at B/B = 2.0 where the dimers align in oneof three equivalent directions [50]. As the temperatureis increased, the n-mers undergo a transition from an or-dered state to an orientationally disordered state in whichthe n-mers rotate randomly but remain confined to theinterstitial pinning sites. The rotating states have beentermed vortex plastic crystals. At matching fields wherevortex plastic crystals form, the anomalies in the criticalcurrent disappear [50]. The predictions from the simu-lations are in general agreement with the experimentalobservation of the loss of certain higher order matchinganomalies at higher temperatures [26]. The formation ofn-mers that can be aligned along degenerate symmetrydirections has also been predicted for kagome pinning ar-rays where every other pinning site is removed from everyother row of a triangular lattice [50, 57].

The formation of dimer states in the honeycomb pin-ning array leads to a variety of novel dynamical phases,including a spontaneous dynamical symmetry breakingeffect in which the moving vortices organize into one oftwo equivalent states which have a component of trans-lation perpendicular to the applied drive in either thepositive or negative direction [58]. The transverse re-sponse appears when the external driving force is appliedhalfway between the two symmetric directions of aligneddimer motion. The dynamical symmetry breaking oc-curs when the equilibrium ground states have no globalsymmetry breaking. At B/B = 2.0, the ground state issymmetry broken and the dynamical moving state has


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the same broken symmetry as the ground state. Forincommensurate fillings, when the dimer alignment isdisrupted, there is no global symmetry breaking in theground state, and instead a dynamical symmetry break-ing occurs due to the applied drive.

In this work we map the dynamical phase diagram forvortices in honeycomb arrays. We focus on the states1.5 < B/B < 2.5 to understand where dynamical sym-

metry breaking occurs and to examine what other typesof moving phases are possible. We study how the dy-namical phases change for driving along different axesof the pinning lattice. We find that very different kindsof dynamics occur when the driving direction is varied,and that the value of the depinning threshold is stronglydirectionally dependent. We also find that a novel jam-ming phenomenon can occur due to the formation of thedimer states. For certain directions of drive, the dimersare anti-aligned with the drive, causing the dimers to be-come blocked in the interstitial regions.

Although our results are specifically for vortices intype-II superconductors, the general features of this workshould also be relevant for other interacting particle sys-tems where a periodic substrate is present. Examples ofsuch systems include vortices on periodic substrates inBose-Einstein condensates (BEC), where different kindsof crystalline phases can occur which depend on thestrength of the substrate [59, 60]. It should be pos-sible to observe different types of vortex flow states inBEC systems [61]. Our results are also relevant for col-loids on periodic substrates, where an orientational or-dering of colloidal molecular crystals occurs which is verysimilar to that of the vortex molecular crystal states[51, 52, 53, 54, 55, 62]. Other related systems includecharged balls on periodic substrates [63] and models ofsliding friction [64].

II. SIMULATION

We use the same simulation employed in the previousstudy of vortex equilibrium states in honeycomb pinningarrays [50]. We consider a 2D system of dimensions Lx =L and Ly = L with periodic boundary conditions in the xand y directions. The sample contains Nv vortices, givinga vortex density of nv = Nv/L2 which is proportional tothe external magnetic field. In addition, there are Nppinning sites placed in a honeycomb arrangement with apinning density of np = Np/L

2. The field at which the

number of vortices equals the number of pinning sites isdefined to be the matching field B.

The dynamics of vortex i located at position Ri is gov-erned by the following overdamped equation of motion:

dRi

dt= Fvvi + F

vpi +FD + F

Ti . (1)

Here the damping constant is = 20d/22N, where

d is the thickness of the superconducting sample, isthe superconducting coherence length, N is the normal

state resistivity of the material, and 0 = h/2e is theelementary flux quantum. The vortex-vortex interactionforce is

Fvvi =

Nvj=i

f0K1

Rij

Rij (2)

where K1 is the modified Bessel function, is the Londonpenetration depth, f0 = 20/(20

3), Rij = |RiRj | isthe distance between vortex i and vortex j, and the unitvector Rij = (RiRj)/Rij. In this work all length scalesare measured in units of and forces in units of fo. Thevortex vortex interaction decreases sufficiently rapidly atlarge distances that a long range cutoff is placed on theinteraction force at Rij = 6 to permit more efficientcomputation times. We have found that the cutoff doesnot affect the results for the fields and forces we considerhere.

The pinning force Fvpi originates from individualnonoverlapping attractive parabolic traps of radius Rpwhich have a maximum strength of Fp. In this work we

consider the limit where only one vortex can be capturedper pinning site, with the majority of the results obtainedfor Rp = 0.35. The exact form of the pinning force is:

Fvpi =

Npk=1

f0

FpRp

R(p)ik

Rp R

(p)ik

R

(p)ik . (3)

Here, R(p)ik = |Ri R

(p)k |, R

(p)k is the location of pinning

site k, the unit vector R(p)ik = (RiR

(p)k )/R

(p)ik , and is

the Heaviside step function.The external drive FD = FDf0FD represents the

Lorentz force from an applied current J B which isperpendicular to the driving force and is applied uni-formly to all the vortices. We apply the drive at variousangles to the symmetry axes of the honeycomb pinningarray. The thermal force FTi originates from Langevinkicks with the properties FTi = 0 and F

Ti (t)F

Tj (t

) =2kBT ij(t t

). Unless otherwise noted, the ther-mal force is set to zero. The initial vortex configura-tions are obtained by simulated annealing, and the ex-ternal force is then applied gradually in increments ofFD = 0.0002 every 1000 simulation time steps. Forthe range of pinning forces used in this work, we findthat this force ramp rate is sufficiently slow that tran-sients in the vortex dynamics do not affect the overall

velocity-force curves. We obtain the velocity-force curvesby summing the velocities in the x (longitudinal) direc-

tion, Vx = N1v

Nvi=1 vi x, and the y (transverse) di-

rection, Vy = N1vNv

i vi y, where vi = dRi/dt.In Fig. 1(c,d) we illustrate the pinning sites and vortexconfigurations after simulated annealing for B/B = 1.5[Fig. 1(c)] and 2.0 [Fig. 1(d)]. Here Lx = Ly = 24 andnp = 0.3125/2. In our previous work, Ref. [58], thedrive was applied along the x-direction for the geometryin Fig. 1.


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0 0.2 0.4 0.6F

D

-0.08

-0.06

-0.04

-0.02

0

0

0.1

0.2

0.3

0.4

0.5

B = 2B

P SB R ML

R MLSBP

(a)

(b)

FIG. 2: (a) The average velocity in the x-direction Vx vs

external driving force FD for the honeycomb pinning arrayfrom Fig. 1(d) at B/B = 2.0 with FD = FDx. (b) The cor-responding average velocity in the y-direction Vy vs FD. Weobserve four phases: the initial pinned phase (P), a symmetrybroken phase (SB), a random phase (R), and a moving lockedphase (ML).

III. DYNAMICS AND TRANSVERSE

RESPONSE FOR DRIVING IN THE

LONGITUDINAL DIRECTION

We first consider the case for driving in the x or lon-gitudinal direction, FD = F

Dx, for the system shown in

Fig. 1(d) with B/B = 2.0, Rp = 0.35, and Fp = 0.85.In Figs. 2(a,b) we plot Vx and Vy versus FD. At thisfilling there are four distinct dynamical phases, with thepinned (P) phase occurring at low FD. The depinningthreshold Fc occurs near FD = 0.14 when the interstitialvortices become depinned. For a system with randompinning and FD = FDx, there would be no transversevelocity response; the system would have Vy = 0 andonly Vx would be finite. In contrast, for the honeycombpinning array there is a finite velocity both in the posi-tive x direction and in either the +y or y direction. InFig. 2(b) the transverse response Vy is negative, indi-cating that the vortices are moving at a negative angle to

the x axis for 0.14 < FD < 0.37. Figure 3(a) illustratesthe vortex motion at FD = 0.25, where the vortices flowin one-dimensional paths oriented at 30 to the x axis.In Fig. 3(b) a snapshot of the vortex positions shows thatthe vortex lattice remains ordered in the moving phase,indicating that the vortices are flowing in a coherent man-ner. We term the phase shown in Fig. 3(a) the symmetrybroken (SB) phase, since the flow can be tilted in eitherthe positive or negative y-direction.

At B/B = 2.0 and FD = 0, the interstitial vortices

x(a)

y

x(c)

y

x(e)

y

x(b)

y

x(d)

y

x(f)

y

FIG. 3: The dynamics of the three moving phases fromFig. 2 for the honeycomb pinning array at B/B = 2.0 withFD = FDx. The vortex positions (filled circles), pinning sitelocations (open circles), and vortex trajectories (black lines)are shown in an 18 18 portion of the sample. (a) In thesymmetry broken (SB) phase at FD = 0.25, the interstitial

vortices move along a 30

angle to the x-axis while the vor-tices at the pinning sites remain immobile. (b) Vortex posi-tions only in the SB phase at FD = 0.25, showing the orderingpresent in the vortex lattice structure. (c) In the random (R)phase at FD = 0.42, the vortex motion is highly disorderedwith vortices pinning and repinning at random. (d) Vortexpositions only in the R phase at FD = 0.42 indicate that thevortex lattice is disordered. (e) In the moving locked (ML)phase at FD = 0.65, all the vortices channel along the pinningsites. (f) Vortex positions only in the ML phase at FD = 0.65reveal an anisotropic vortex lattice structure with differentnumbers of vortices in each row.

form an aligned dimer configuration with a three-fold de-generate ground state in which the dimers can be orientedalong the y-direction, as in Fig. 1(d), or along +30 or30 to the x-direction, as shown in previous work [58].When a driving force is applied to the +30 or 30

ground states, the vortices depin and flow along +30 or30, respectively. In these cases, the symmetry break-ing in the moving state is not dynamical in nature butreflects the symmetry breaking within the ground state.If the dimers are initially aligned along the y-directionin the ground state, an applied drive induces an instabil-ity in the pinned phase and causes the dimers to rotateinto the +30 or 30 directions, as we discuss in fur-ther detail below. In this case the symmetry breaking is

dynamical in origin.In Fig. 2(a,b) we find pronounced oscillations in both

Vx and Vy just above the depinning threshold Fc =0.14. These oscillations are not intrinsic features but aredue to the fact that at B/B = 2.0 the interstitial vor-tex lattice is perfectly ordered, so the interstitial vor-tices move in a coherent fashion as shown in Fig. 3(a).At depinning, the interstitial vortices are slowly mov-ing through a periodic potential created by vortices thatremain trapped at the pinning sites. This periodic po-


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0.2

0.4

Vy,V

x

0

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Vy,V

x

0.1

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Vy,V

x

(a)

(b)

(c)

FIG. 4: Time traces of vortex velocity at fixed FD. Uppercurves: Vx(t); lower curves: Vy(t). (a) The symmetry broken(SB) phase at FD = 0.25 from Fig. 3(a,b). Here pronouncedoscillations occur in both Vx and Vy as the vortices move ina coherent fashion. (b) The random (R) phase at FD = 0.42from Fig. 3(c,d). In this case the transverse motion is lostand Vy = 0. Additionally, there are no correlated oscilla-tions. (c) The moving locked (ML) phase at FD = 0.65 fromFig. 3(e,f). Vx has been shifted down for clarity. There is aweak oscillation in Vx due to the periodic substrate. Since theflow is strictly one-dimensional, as shown in Fig. 3(e), thereare no fluctuations in Vy.

tential causes the moving interstitial vortices to developan oscillating velocity. In Fig. 4(a), the instantaneoustime traces of the vortex velocity Vx and Vy at constantFD = 0.25 show strong velocity oscillations. The os-cillations are also visible in Fig. 2 at low drives due toour choice of averaging time spent at each value of thedriving current. If the averaging time is increased, theoscillations in Fig. 2 disappear. We note that the loca-tions of the boundaries between the different phases arenot affected by the value of the velocity averaging time.At incommensurate fields, there is enough dispersion inthe velocity of the moving interstitial vortices that thecoherent velocity oscillations are no longer distinguish-able.

As FD increases, the net vortex velocity in the SBphase increases linearly until FD = 0.365, where thereis an abrupt increase in Vx. Fig. 2(a,b) shows that thisincrease coincides with a jump in Vy to a zero aver-age, indicating that the vortices are moving only in thex-direction on average. In Fig. 3(c) we illustrate the dis-ordered vortex trajectories that occur in this phase atFD = 0.42. The vortices are continually depinning andbeing repinned, and the order in the vortex lattice is

lost, as shown in Fig. 3(d). We term this the random(R) phase. It resembles random dynamical phases thathave previously been observed for vortices in square pin-ning arrays when the interstitial vortices begin to depinvortices from the pinning sites [29]. Figure 2 shows thatthere are pronounced random fluctuations in Vx andVy in phase R, and also that Vx does not increaselinearly with FD but has a curvature consistent with

Vx = (FD FRc )

1/2

, where FRc = 0.365 is the thresh-old value for the SB-R transition. In the SB phase, the

number of moving vortices is constant and is equal to thenumber of interstitial vortices, while in the R phase thenumber of moving vortices increases with FD.

At FD = 0.53, the system organizes into a one-dimensional flowing state where the vortex motion islocked along the pinning rows, as shown in Fig. 3(e,f)for FD = 0.65. The onset of this phase also coincideswith the decrease of fluctuations in Vx and the loss offluctuations in Vy, as shown in Fig. 4(c). For FD > 0.53,all of the vortices are mobile and Fig. 2(a) illustratesthat the Vx versus FD curve becomes linear again. We

term this the moving locked (ML) phase, since the vor-tex motion is effectively locked along the pinning sites.When the vortices are rapidly moving, the pinning siteshave the same effect as a flashing one-dimensional troughthat channels the vortices [33, 38]. The vortices assumea smectic structure in the ML phase, since different rowshave different numbers of vortices, resulting in the forma-tion of aligned dislocations. The ML phase is essentiallythe same state found in square pinning arrays at highdrives when B/B > 1.0 [33].

In previous studies of square pinning arrays withstrong pinning, the initial motion of the vortices forB/B > 1.0 occurred in the form of one-dimensional

channels between the vortices trapped at the pinningsites [33]. In the honeycomb pinning array, similar flowoccurs in the SB phase as shown in Fig. 3(a). ForB/B < 1.5 in the honeycomb array, the initial inter-stitial flow for depinning in the x direction occurs via theflow of individual vortices in a zig-zag pattern around thepinned vortices. Since there is no dimer ordering for thesefillings, no transverse response occurs for B/B < 1.5.For B/B 1.5, the interstitial vortices begin to formdimer states when two interstitial vortices are capturedin a single large interstitial site. The dimers can lowertheir orientational energy by aligning with each other inboth the ground state and the moving states. Dimerscan only remain aligned in the moving state if they arechanneling along one of the symmetry axes of the pin-ning lattice. If the dimers were to move strictly in thex-direction, they would be forced directly into the pinnedvortex in the pinning site to the right of each large in-terstitial site. This would destabilize the rodlike dimers.Instead, the dimers maintain their integrity by movingalong 30 to the x-axis. Within the moving state, if oneof the dimers were to move along +30 while the remain-ing dimers were moving along 30, the two interstitialvortices comprising the first dimer would be forced close


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together, destabilizing the dimer state due to the repul-sive vortex-vortex interactions. Instead, all of the dimersmove in the same direction.

The SB-R transition occurs when the combined forceson the pinned vortices from the external drive and themoving dimers are strong enough to depin the pinnedvortices. At the closest approach in the x direction be-tween a dimer and a pinned vortex, the frontmost dimer

vortex is a distance a0/2 from the pinned vortex andthe rear dimer vortex is a distance 3a0 from the pinnedvortex, where ao is the lattice constant of the undilutedtriangular pinning lattice. In addition to the force fromthe dimerized vortices, the pinned vortex experiences anopposing force from the neighboring pinned vortex a dis-tance a0 away. In a simple approximation, the drivingforce needed to depin a vortex at a pinning site is thusFD = Fp [(K1(a0/2) + K1(3a0/2)) K1(a0)]. Set-ting Fp = 0.85 gives FD = 0.41, close to the value ofFD = 0.37 for the SB-R transition in Fig. 2. Once thepinned vortices depin, the system enters the random (R)phase, and since FD is still considerably less than Fp, it

is possible for vortices to be pinned temporarily in phaseR.

Studies of square pinning arrays have shown that afterthe onset of a random dynamical phase, the vortices canorganize into a more ordered phase of solitonlike pulsemotion along the pinning rows, followed by a phase inwhich all of the vortices channel along the pinning rows[33]. At the transition to the one-dimensional pulse likemotion, a larger fraction of the vortices are pinned com-pared to the random phase, so a drop in Vx with in-creasing FD occurs, giving a negative differential con-ductivity. In the honeycomb pinning arrays for the pa-rameters we have chosen here, we do not observe one-

dimensional pulse motion or negative differential con-ductivity for driving along the x-direction. For the one-dimensional pulse motion or the ML phase motion seenin Fig. 3(e,f) to occur, the vortices must be moving ata sufficiently high velocity for the pinning sites to actlike a flashing trough. When the vortices move alongthe pinning rows, the vortex lattice structure adopts ahighly anisotropic configuration which would be unsta-ble at FD = 0. During the period of time when a vor-tex passes through a pinning site, the vortex is pulledtoward the center of the pinning row, which stabilizesthe one-dimensional motion. When the vortex is mov-ing between the pinning sites, it can drift away from theone-dimensional path until it encounters another pinningsite. In Ref. [33], it was shown that for square pinningarrays, increasing the pinning radius Rp stabilized theone-dimensional flow down to lower values of FD. In thehoneycomb pinning array, the one-dimensional flow is lessstable due to the fact that the vortices must move overthe much wider large interstitial site, giving the vorticesmore time to drift away from the pinning row. Since thismeans that a larger value of FD is required to stabilizethe one-dimensional motion, it should be more difficultin general to observe the onset of one-dimensional soli-

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(b)

(c)

FIG. 5: The power spectra S() of the x component of thevelocity Vx(t) for the three phases in Fig. 4. (a) The SB phaseat FD = 0.25 shows a pronounced narrow band noise signa-ture. (b) The R phase at FD = 0.42 has a broad band noisesignature. (c) In the ML phase at FD = 0.65, a number ofdifferent frequencies are present due to the fact that differentrows of the vortices move at different velocities.

tonlike motion or negative differential conductivity in thehoneycomb pinning arrays than in the square pinning ar-rays.

A. Fluctuations and Noise Characteristics

In order to characterize the moving phases more quan-titatively, in Fig. 4 we show time traces Vx(t) and Vy(t) ofthe vortex velocities at fixed FD for the different phasesfor the system in Fig. 2. In the symmetry broken (SB)phase at FD = 0.25, shown in Fig. 4(a), Vx is greater than

|Vy| by tan(30) or about 1.7. Here both components ofthe velocity show a pronounced oscillation which ariseswhen the interstitial vortices move in a coherent fashionover the periodic potential substrate created by the im-mobile vortices in the pinning sites. In Fig. 5(a), we plotthe corresponding power spectrum S() of Vx obtainedfrom

S() =

Vx(t)e2itdt

2

. (4)


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7

There is a pronounced peak in S() at the frequency ofthe velocity oscillation in the SB phase, indicating thatmode locking effects could appear at B/B = 2.0 whenthe symmetry breaking flow occurs. In square pinningarrays, experiments [17] and simulations [36] revealedShapiro step-like mode locking of interstitial moving vor-tices at B/B = 2.0. In the honeycomb lattice, sincethere is also a strong oscillation in Vy in the SB phase, we

expect that transverse mode locking could occur if an ad-ditional ac drive is applied in the y-direction. Such modelocking would appear as steps in both Vx and Vy ver-sus FD in the SB phase. Transverse phase locking, whichproduces steps that are distinct from Shapiro steps, hasbeen observed for the motion of vortices in square arrays[37]. In general, if the vortices already have an intrin-sic velocity oscillation in the transverse direction, thenpronounced transverse phase locking is possible.

In Fig. 4(b) we plot the time trace of Vx and Vy forthe random (R) phase at FD = 0.42. In this case Vy =0, and although both Vx and Vy show fluctuations, nooscillations or washboard frequencies appear. In Fig. 5(b)

we show the corresponding S() for Vx, where we finda broad band noise feature consistent with disorderedplastic flow [3, 6, 11]. Since there are no coherent velocityoscillations, mode locking should be absent in the random(R) phase.

In Fig. 4(c) we plot Vx and Vy in the ML phase atFD = 0.65, where Vx has been shifted down by a factorof 3 for clarity. There are no visible fluctuations in Vy dueto the one-dimensional nature of the flow, but there aresmall periodic oscillations in Vx generated by the motionof the vortices over the periodic substrate. Due to thefact that different one-dimensional rows contain differentnumbers of vortices, producing dispersion in the vortexvelocities, the oscillation in Vx is not as pronounced as

in the SB phase. The corresponding power spectrum inFig. 5(c) contains a rich variety of peaks due to the widerange of frequencies present in this phase. The main peakis smaller in magnitude than that found for the SB phase.As FD increases, the frequency at which the first peakoccurs also increases. It should be possible to generatephase locking in the ML phase; however, it would likelynot be as pronounced as in the SB phase. These resultssuggest that noise fluctuations can be a useful techniquefor exploring the presence of different dynamical phasesin periodic pinning arrays.

B. Dynamical Symmetry Breaking in the PinnedPhase

As previously noted, the ground state at B/B = 2.0is three-fold degenerate. When the dimers are aligned ateither +30 or 30 to the x-axis in the ground state, thesubsequent SB flow is aligned in the same direction as theground state. It is also possible for the dimers to align inthe y-direction, as shown in Fig. 6(a). At FD = 0.09, thedimers and the vortices in the pinning sites are shifted

x(a)

y

x(b)

y

x(c)

y

FIG. 6: Vortex positions (filled circles), pinning site locations(open circles), and trajectories (black lines) for a system withB/B = 2.0 and Fp = 0.85 which started in the y-alignedground state. (a) The pinned phase at FD = 0.09. Here thedimers and the vortices in the pinning sites have all shiftedslightly to the right compared to the ground state due to theapplied drive. (b) A rotational instability occurs at FD 0.11, when the vortices move in a manner that allows thedimers to align along 30 to the x-axis. There is also asmall shift of the vortices in the pinning sites. (c) The pinnedstate at FD = 0.12 where the dimers are aligned in the new30 direction.

0.06 0.08 0.1 0.12 0.14 0.16F

D

-0.01

0

0.01

,

Vx

Vy

P Pr

FIG. 7: The velocity Vx (upper curve) and Vy (lowercurve) vs FD for the system in Fig. 6. The rotational insta-bility seen in Fig. 6(b) appears as a positive peak in Vx anda negative peak in Vy just above FD = 0.11. The systemremains pinned until around FD = 0.14.

slightly to the right due to the applied drive. As FD isfurther increased, a symmetry breaking transition occurswithin the pinned phase. For FD < 0.11 the dimers re-main aligned the y-direction; however, at FD 0.11, therotational instability illustrated in Fig. 6(b) occurs. Thedimers rotate in such a way that they end up aligned inthe 30 direction. The interstitial vortex at the bottomof the dimer moves in the +x direction and by a smalleramount in the +y direction, while the vortex at the topof the dimer moves in the y direction and by a smalleramount in the x direction. There is also a slight shiftof the vortices in the pinning sites that are closest to thebottom of each dimer. In Fig. 6(c) the rotation process


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8

x(a)

y

x(b)

y

FIG. 8: Vortex positions (filled circles) and pinning site loca-tions (open circles) for the system in Fig. 2 at (a) B/B = 1.91and (b) B/B = 2.08. At these fillings, the long range orien-tational ordering of the dimer state is lost.

is completed and the dimers are aligned in a new direc-tion, 30. The vortices remain pinned until FD = 0.14,at which point the system enters the SB phase. At finitetemperatures, the dimer realignment occurs at even lowervalues ofFD. The rearrangement can also be observed asa jump in Vx and Vy as shown in Fig. 7, where thereis a positive spike in Vx and a negative spike in Vynear FD = 0.11, in agreement with the motion shown inFig. 6(b). We term this a dimer polarization effect sincethe driving force induces an alignment of the dimers. Inruns with slightly different initial conditions, the dimersmay align along the +30 direction with 50% probability.

C. Dynamics for 1.5 B/B < 2.5

We next consider the effect of changing the vortex den-sity for fillings where interstitial dimers are present andthe SB phase occurs. In Fig. 8(a) we illustrate the vortexpositions for B/B = 1.91, where a mixture of monomersand dimers appear in the large interstitial sites. At thisfilling, the overall orientational ordering of the dimers islost in the ground state, and the dimers are oriented onlyin local patches. For B/B > 2.0, a mixture of intersti-tial dimers and trimers is present, as shown in Fig. 8(b)for B/B = 2.08, and the orientational ordering is againlost. In Ref. [58] it was shown that the SB state stilloccurs at incommensurate fields as long as some dimersare present. If FD is suddenly increased from zero to afinite value at which only the interstitial vortices depin,

the moving state for the incommensurate fields organizesinto a dynamically symmetry broken state where all ofthe dimers flow along +30 or 30. At the incommensu-rate fields, only the dimers undergo dynamical symmetrybreaking; the monomers and trimers continue to move inthe direction of the drive, with some fluctuations in thetransverse direction.

In Fig. 9 we plot Vy for the system in Fig. 2 atB/B = 1.89, 1.94, and 2.5. In Figs. 9(a,b), the samefour phases described above are labeled. The SB phase

0 0.2 0.4 0.6 0.8F

D

-0.2

0

0.2

-0.08

-0.04

0

0

0.04

P

SB

R ML

P

SB

R ML

(a)

(b)

(c)

FIG. 9: Vy versus FD for (a) B/B = 1.89, (b) B/B =1.94, and (c) B/B = 2.5 for a system with the same param-eters as in Fig. 2. P: pinned phase; SB: symmetry broken

phase; R: random phase; ML: moving locked phase.

has opposite sign in Fig. 9(a) and Fig. 9(b); the dy-namical symmetry breaking can occur in either directionsince there is no symmetry breaking in the ground state.If slightly different initial conditions are used, such as bychanging the initial annealing procedure, the dynamicalsymmetry breaking has equal probability to occur in thepositive or negative direction, as shown previously [58].In Figs. 9(a) and (b) the initial portion of the SB phasehas fluctuations in Vy due to the fact that we are in-creasing FD at a finite rate and there is a transient timefor the moving state to fully organize into the SB state,as studied previously [58]. The transient time increasesas |B/B2.0| increases. If we decrease FD, the fluctu-ations at depinning are reduced; however, the boundarybetween the phases does not shift.

In Fig. 9(c) at B/B = 2.5, Vy = 0 since there areonly trimer states present. The large oscillations in Vyoccur when the system forms a completely ordered trimerground state [50] and the vortex motion is highly coher-ent, similar to the effect seen in Fig. 2. For the rate atwhich we sample and average Vy versus FD, the peri-odic fluctuating vortex velocity is visible. For FD > 0.5the system enters a partially moving locked phase wherea portion of the vortices move along the pinning rows.There are, however, too many vortices to form straightone-dimensional chains of the type shown in Fig. 3(e) forB/B = 2.0. A buckling instability of the chains oc-curs since the amount of anisotropy that would occur ifone-dimensional chains formed is too large for the vor-tex lattice to sustain. Instead, a partially moving locked(PML) phase forms with a disordered moving vortex lat-tice. This result is interesting since it indicates that mov-ing vortex phases do not always organize into ordered


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9

1.5 1.75 2 2.25 2.5B/B

0

0.5

1

1.5

FD

P

SB

R

ML PML

FIG. 10: The dynamic phase diagram for FD vs B/B high-lighting the different dynamical phases. P: pinned, SB: sym-metry broken, R: random, ML: moving locked. Here Fp =0.85, Rp = 0.35, and np = 0.3125/

2 . For B/B > 2.125, at

high FD the system forms a partially moving locked (PML)phase where not all of the vortices move along the pinningrows.

states. A time trace of a PML state at fixed FD showsmuch weaker velocity oscillations than those shown inFig. 4(c) for the ML state. This suggests that phaselocking with PML states will be very weak or absent. Inprevious work on phase locking for square arrays, it wasshown that the phase locking is most pronounced at com-mensurate fields where the moving vortex structures aremore ordered [17, 36].

By performing a series of simulations for varied B/B,measuring the features in the velocity force curves andobserving the vortex structures, we construct the dynam-ical phase diagram ofFD vs B/B shown in Fig. 10. Thedepinning force marking the end of the pinned (P) phaseshows peaks at B/B = 1.5, 2.0, and 2.5, correspondingto the commensurate and ordered ground states reportedpreviously [50]. The SB-R transition line is fairly flatas a function of B/B with an enhancement to highervalues of FD occurring near B/B = 2.0, while at theincommensurate fields, monomers or trimers create fluc-tuations that cause the vortices at the pinning sites todepin at slightly lower values of FD. For B/B < 2.1,upon increasing FD the random state organizes into aML state where all the vortices move along the pinningrows as shown in Fig. 3(e), while for B/B 2.1, therandom state organizes into the PML state. The widthof the random phase, as determined by the fluctuationsin the velocity, increases and persists to higher values ofFD for increasing B/B at B/B 2.1. For B/B < 1.5and B > 2.5, where dimers are no longer present, theSB phase is lost and a new set of dynamical phases ariseswhich we discuss in more detail below.

0 0.5 1 1.5 2 2.5F

p

0

0.05

0.1

0.15

0.2

Fc

PPP PHB

FIG. 11: The critical depinning force Fc vs Fp for a systemwith B/B = 2.0, Rp = 0.35, and np = 0.3125/

2. For Fp 0.2 we ob-serve the same phases illustrated in Fig. 15 at Fp = 0.85. Thecurves do not extend above Rp = 0.55 since for Rp > 0.55,multiple vortex pinning at individual pinning sites occurs. ForRp < 0.2, a partially pinned phase appears and the initialdepinning is into a new moving interstitial phase termed MI2,illustrated in Fig. 17(a). The upper dashed line separates theMC phase from the ML phase.


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12

x(a)

y

0 0.2 0.4 0.6 0.8F

D

0

0.2

0.4

0.6

0.8

PP

MI2

R

ML(b)

FIG. 17: (a) Vortex positions (filled circles), pinning sitelocations (open circles) and vortex trajectories (black lines)for the system in Fig. 16 showing the moving interstitial 2phase (MI2) at Rp = 0.15 and Fp = 0.2. (b) Vx vs FD forthe same system. A sharp transition from the MI2 phase tothe random (R) phase occurs.

E. Changing Rp and B

We next examine the effects of changing the pinning

radius in a system with fixed Fp = 0.85 and B/B = 2.0.In Fig. 16 we show the dynamic phase diagram for FDversus Rp obtained from a series of simulations. For0.2 < Rp 0.55, we find the same three movingphases, SB, R, and ML, as in Fig. 2. For Rp > 0.55, thepins are large enough to permit double vortex occupancyat the individual pinning sites; in this case, a new setof phases appears which we do not consider in this work.For Rp > 0.2, the R-ML transition decreases in FD withincreasing Rp since the larger pinning sites make it eas-ier for the vortices to localize along the pinning rows andflow in the one-dimensional motion of the ML phase. ForRp < 0.2 at low FD, we find the partially pinned PP

state illustrated in Fig. 12(a). The onset of the PP phasecoincides with a drop in Fc at Rp 0.2. Unlike the PPphase that occurs at low Fp in Fig. 15, which depins elas-tically, the PP phase at small Rp depins into a movinginterstitial phase that is distinct from the moving inter-stitial phase shown in Fig. 13(b). In Fig. 17(a) we illus-trate the vortex trajectories in the phase which we termthe moving interstitial phase 2 (MI2). The vortices flowin winding interstitial channels; however, unlike the MIphase, in the MI2 phase only half of the pinning sites areoccupied. In Fig. 17(b) we plot Vx vs FD for a systemwith Rp = 0.15. A clear two-step depinning transitionoccurs, with Vx increasing linearly with increasing FDin the MI2 phase. At high FD and Rp < 0.2, the pin-

ning sites are too small for the ML phase to occur, andinstead the vortices flow in the MC phase illustrated inFig. 13(a).

We next consider samples with fixed B/B = 2.0,Rp = 0.35, and Fp = 0.85, but vary the value of Bby changing the pinning density np. This alters the av-erage spacing between neighboring vortices. Up to thispoint we have used np = 0.3125/2. In Fig. 18 we illus-trate the dynamic phase diagram for FD versus np. Asnp increases, Fc increases since the depinning of the in-

0 0.2 0.4 0.6 0.8n

p

0

0.5

1

1.5

FD

P

SB

R

MI

ML

FIG. 18: The dynamic phase diagram FD versus pinning den-sity nv, which determines B, at B/B = 2.0, Fp = 0.85, andRp = 0.35. P: pinned dimer phase; MI: moving interstitialphase; SB: symmetry broken phase; R: random phase; ML:

moving locked phase. The transition to the ML phase shiftsto higher FD with decreasing np. The SB phase appears onlyfor intermediate values of np.

terstitial vortices is determined by the potential createdby the vortices located at the pinning sites, and as thevortex density increases, the depth of the interstitial pin-ning potential also increases. The R-ML transition shiftsto higher FD with decreasing np. Since the distance be-tween the pinning sites increases with decreasing np, themoving vortices spend less time in the pinning sites. Thisdestabilizes the ML phase and the vortices must move athigher velocities for the effective trough potential to beable to stabilize the ML phase. For np 0.78/2, the SBphase is lost, since at this pinning density the interactionsbetween the interstitial vortices and the pinned vorticesbecome sufficiently strong that the depinning of the in-terstitial vortices also causes the pinned vortices to depin.As a result, the system passes directly from the P phaseto the R phase with increasing FD. For np < 0.14/

2, thevortex-vortex interaction becomes weak enough that thesystem depins into the MI phase illustrated in Fig. 13(b).This also coincides with an increase in the value of FDat which the R phase appears, since the moving intersti-tial vortices in the MI phase do not approach the pinnedvortices as closely as they do in the SB phase.

F. Effect of Finite Temperature

We next consider the effect of finite temperature onthe system in Fig. 2 with B/B = 2.0, Fp = 0.85, Rp =0.35, and np = 0.3125/

2. In Ref. [50], we showed thata transition can occur at finite temperature in which thevortex n-mer states lose their orientational ordering andbegin to rotate while remaining confined within the large


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13

0 0.5 1 1.5

FT

0

0.5

1

FD

0 0.4 0.8F

D

0

0.4

0.8

P

SB

R

ML

FIG. 19: The dynamic phase diagram of FD vs thermalforce FT for a system with B/B = 2.0, Fp = 0.85, np =0.3125/2, and Rp = 0.35. P: pinned dimer phase; SB:symmetry broken phase; R: random phase; and ML: movinglocked phase. Inset: Vx vs FD at F

T = 1.25 where a smoothdepinning transition occurs.

x(a)

y

x(b)

y

x(c)

y

x(d)

y

FIG. 20: Vortex positions (filled circles), pinning site loca-tions (open circles), and vortex trajectories (black lines) forthe system in Fig. 19 at FT = 1.25. (a) At FD = 0.1, thereis appreciable creep of the interstitial vortices and the dimershave lost their orientational ordering and are rotating withinthe large interstitial sites. (b) At FD = 0.2, the vortex flowis disordered and vortices are continually depinning and re-pinning. (c) At FD = 1.0, all of the vortices are moving butthere is still diffusion in the direction transverse to the drive.(c) At FD = 1.6 the vortices are about to enter the ML statewhere the motion is confined to one-dimensional channels.

interstitial sites. This state was termed a vortex plasticcrystal. In Ref. [58] we demonstrated that the SB phasedisappears in the vortex plastic crystal state. In Fig. 19we plot the dynamical phase diagram of FD vs F

T forthe same system in Fig. 2. In our units, the dimers meltat FT = 1.0. Above the melting temperature, there isappreciable creep of the interstitial vortices as they hopfrom one large interstitial site to another, as illustrated

in Fig. 20(a) for FD = 0.1 and FT

= 1.25. As FD isfurther increased the system enters the random (R) phaseshown in Fig. 20(b) for FD = 0.25. At higher drives, thevortices begin to localize along the pinning rows in one-dimensional channels; however, there is still appreciablehopping from one row to another as shown in Fig. 20(c)for FD = 1.0. At even higher drives, the ML is recoveredas illustrated in Fig. 20(d) for FD = 1.6. For F

T >1.35, the ML phase is lost and the high FD flow is inthe random (R) phase shown in Fig. 20(c). In the insetto Fig. 20 we demonstrate that at FT = 1.25, the sharpfeatures in the velocity force curve seen at FT = 0 inFig. 2 disappear.

IV. DYNAMICS FOR DRIVING IN THETRANSVERSE DIRECTION

We now consider the case where FD is applied alongthe y-direction, FD = FDy, for the same system asin Fig. 1(d) with Fp = 0.85, Rp = 0.35, and np =0.3125/2. A different set of dynamic phases appearthat are distinct from those found for driving in the x-direction. In particular, the SB phase is lost and thedimers align in the y-direction with the initial depinningoccurring in one-dimensional interstitial flow paths. In

Fig. 21 we plot the velocity force curves for B/B = 2.0,2.15, and 2.32.Figure 21(a) shows the three phases, pinned (P),

one-dimensional moving interstitial (1DMI), and movinglocked (ML) that occur at B/B = 2.0. In the P phase,if the ground state contains dimers which are alignedat either +30 or 30 to the x-axis, a polarization ef-fect is induced by the applied drive similar to the ef-fect discussed earlier. In this case, however, the dimersshift such that they are aligned in the y-direction. Theone-dimensional moving interstitial (1DMI) state whichappears above depinning is illustrated in Fig. 22(a) atFD = 0.2, where the interstitial vortices move betweenthe vortices in the pinning sites. Near FD = 0.55 thereis a sharp depinning transition for the vortices in thepinning sites. After this depinning transition occurs, thevortices very rapidly rearrange into a moving locked (ML)phase where the vortices move along the pinning sites, asshown in Fig. 22(b) for FD = 0.8. In the ML phasesfor driving along the x and the y directions, the vorticestravel in one-dimensional channels along a row or a col-umn of pinning sites, respectively. The pinning rows fol-lowed by the vortices for x-direction driving are evenlyspaced in the y-direction, so the vortices flow through


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14

0 0.5 1F

D

0

0.5

1

0 0.2 0.4 0.6 0.8F

D

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8

FD

0

0.2

0.4

0.6

0.8

P

P

P

ML

ML

ML

1DMI

1DMI

R

R

R

(a)

(b)

(c)

FIG. 21: The velocity force curves Vy vs FD for FD = FDyfor the system in Fig. 2 with Fp = 0.85, Rp = 0.35,and np = 0.3125/

2. P: pinned dimer phase; R: randomphase; 1DMI: one-dimensional moving interstitial phase, il-lustrated in Fig. 22(a); ML: moving locked phase, illustratedin Fig. 22(b). (a) B/B = 2.0. (b) B/B = 2.15, wherean additional random flow phase occurs at depinning due tothe presence of trimers. (c) B/B = 2.32, where there is no

longer a 1DMI phase.

x(a)

y

x(b)

y

FIG. 22: Vortex positions (filled circles), pinning site posi-tions (open circles), and vortex trajectories (black lines) forthe system in Fig. 21(a) at B/B = 2.0. (a) One-dimensionalmoving interstitial (1DMI) phase at FD = 0.2. (b) The mov-ing locked (ML) phase for FD = FDy at FD = 0.8.

1.6 1.8 2 2.2 2.4B/B

0

0.2

0.4

0.6

0.8

1

FD

ML

1DMI

R

P

FIG. 23: The dynamical phase diagram for FD vs B/B forFD = FDy in a sample with Fp = 0.85, Rp = 0.35, andnp = 0.3125/

2. P: pinned phase; 1DMI: one-dimensionalmoving interstitial phase; R: random phase; ML: movinglocked phase. For B/B > 2.3 the 1DMI flow is lost.

the centers of the pinning sites. In contrast, the pinningcolumns followed by the vortices for y-direction drivingare unevenly spaced in the x direction due to the sym-metry of the honeycomb lattice. As a result, for they-direction driving the vortices do not flow through thecenters of the pinning sites, but are instead shifted to theright and left of the pinning sites in alternate columns,as seen in Fig. 22(b). This produces a more even spacingbetween the columns of moving vortices. Since differentcolumns contain different numbers of vortices, the MLphase for y-direction driving has smectic type character-istics. The velocity-force curves for 1.5 < B/B < 2.0have the same general form as the curve in Fig. 21(a) andshow the same three phases.

For B/B > 2.0 the appearance of trimer states dis-rupts the 1DMI flow since the trimers cannot align com-pletely in the y-direction. This produces random (R)vortex flow at depinning, as shown in Fig. 21(b), withdiffusive vortex motion occurring along the x-direction.There are more pronounced fluctuations in Vy in theR phase, and the velocity-force curve is nonlinear andlower than the extrapolated linear behavior in the 1DMIphase that begins near FD = 0.23. The trimers can blockthe one-dimensional channels of flow shown in Fig. 22(a),lowering the number of mobile vortices. At higher drives,the trimers depin, straighten into a linear configuration,and flow in the 1DMI phase. At FD 0.5, the vorticesin the pinning sites depin, resulting in a transition fromthe 1DMI phase to the R phase. For sufficiently highdrives, the ML phase forms. For driving along the x-axisat B/B > 2.0, we showed in Fig. 9(c) that the ML phaseis lost due to a buckling transition of the one-dimensionalchains of vortex motions, and that a partially movinglocked (PML) phase forms instead when a portion of the


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15

0 1 2 3F

D

0

1

2

3

0 0.5 1F

D

0

0.5

1

0 0.5 1

FD

0

0.5

1

1.5

(a)

(c)

(b)

MC

ML

1DMI

1DMI

P

PP

PHB

ML

FIG. 24: Vy vs FD for FD = FDy for B/B = 2.0,Rp = 0.35, and np = 0.3125/

2. (a) At Fp = 0.35, thereis a single step elastic depinning transition from the partiallypinned (PP) phase to the moving crystal (MC) phase. (b) AtFp = 1.25 we find the pinned (P), one-dimensional movinginterstitial (1DMI), and moving lattice (ML) phases. (c) AtFp = 2.25, there are sharp transitions between the pinned her-

ringbone (PHB), one-dimensional moving interstitial (1DMI),and moving locked (ML) phases.

vortices move through the interstitial regions. For driv-ing along the y-axis at B/B > 2.0, the ML state re-mains stable for much higher values of B/B than forthe x-axis case. A comparison of Fig. 22(b) and Fig. 3(e)shows that the interstitial region crossed by the vorticesin the moving channels is not as wide for y-axis driving asfor the x-axis driving, resulting in more stable y-axis MLflow. As B/B is further increased, the random regimegrows until the 1DMI phase is completely lost, as shownin Fig. 21(c) for B/B = 2.35. In Fig. 23 we plot the dy-namical phase diagram for FD versus B/B, highlightingthe onset of the different phases. The transition to theML phase shifts to higher values of FD with increasingB/B since the ML vortex channels become increasinglyanisotropic as the number of vortices in the sample in-creases.

0 1 2F

p

0

1

2

FD

PHB

1DMI

MLMC

PPP

FIG. 25: The dynamic phase diagram FD vs Fp for FD =FDy at B/B = 2.0, Rp = 0.35, and np = 0.3125/

2. PP:partially pinned phase; P: pinned dimer phase; PHB: pinned

herringbone phase; 1DMI: one-dimensional moving intersti-tial phase; MC: moving crystal phase; ML: moving lockedphase. A peak in the depinning threshold Fc occurs nearFp = 0.5 at the PP-P transition. At the P-PHB transition,Fc increases by a factor of three. The dashed line separatesthe MC phase from the ML phase.

A. Dynamics as a function of Fp and DimerJamming

We now consider the vortex dynamics in a system withfixed B/B = 2.0, Rp = 0.35, and np = 0.3125/

2

for varying Fp with FD = FDy. As noted above, forFp < 0.5 a partially pinned (PP) vortex lattice forms. InFig. 24(a), the velocity-force curve for Fp = 0.35 showsthat the depinning of the PP phase is elastic and occurs ina single step transition to a moving crystal (MC) phase,as seen earlier in Fig. 14(a) for driving along the x-axis.In the MC phase, half of the vortices move along thepinning sites. For 0.5 < Fp < 1.75, at low drive the sys-tem is in the pinned (P) phase of orientationally ordereddimers, and as the drive increases, Fig. 24(b) shows thatthe same 1DM1 and ML phases illustrated in Fig. 21(a)appear. The rapid rearrangement of the vortices fromthe 1DMI phase to the ML phase results in a small jumpnear FD = 0.8 which marks the 1DMI-ML transition.For strong pinning Fp 1.75, Fig. 24(c) indicates thatthe ground state forms the pinned herringbone (PHB)phase illustrated in Fig. 12(b). The same pinned state ap-pears for x-direction driving at strong pinning, as shownin Fig. 14(c). In Fig. 24(c), the velocity-force curve atFp = 2.25 shows the abrupt nature of the depinning tran-sition from the PHB phase to the 1DMI phase, which dif-fers from the smoother depinning transition that occursfrom the P phase to the 1DMI phase in Fig. 24(b). Thedepinning threshold increases markedly with increasing


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x(a)

y

x(b)

y

FIG. 26: Vortex positions (filled circles) and pinning sitelocations (open circles) for the system in Fig. 24(c) at B/B =2.0 and FD = 0.56, just before depinning. Under the influenceof the driving force which is applied in the y direction, thedimers align with the x direction and shift to the top of thelarge interstitial sites. Because the dimers are not alignedwith the direction of the drive, a jamming phenomenon occurswhich is responsible for the large increase in Fc seen in Fig. 25at Fp = 1.75. We call this the jammed state J. In (b) onlythe vortex positions are shown and it can more clearly be seen

that the dimers are shifted in the positive y-direction. Thevortex configuration in the jammed state is distinct from thepinned herringbone state.

Fp once the system enters the PHB state. In Fig. 25 weplot the dynamic phase diagram for Fd versus Fp. Nearthe transition from the PP to the P phase, there is a peakin Fc similar to the peak observed at the PP-P transitionfor driving in the x-direction in Fig. 11. For Fp > 0.175the strong enhancement of the depinning threshold in thePHB state can be seen clearly.

In Fig. 26(a) we illustrate the vortex positions just be-

fore depinning for Fp = 2.25. Even though the drive isapplied in the y-direction, the dimers have aligned withthe x-direction. When the dimers are oriented alongthe x-axis, they cannot fit through the easy-flow one-dimensional channel between the pinning sites, but in-stead are essentially jammed by the two pinned vorticesat the top edge of the large interstitial site. In the or-dered dimer pinned (P) phase, the dimers all reorient inthe same direction under an applied drive. In contrast, inthe pinned herringbone (PHB) phase, the dimers rotatein opposite directions under an applied drive, so whenthe drive is applied along the y-direction the dimers endup aligning in the x-direction. In Fig. 26(b) only thevortex positions from Fig. 26(a) are shown to indicatemore clearly the shift of the dimers in the positive y-direction. This vortex configuration, which we term the

jammed (J) state, has a structure that is distinct fromthat of the pinned herringbone phase shown in Fig. 12(b).The jammed state configuration exists only in the pres-ence of the applied drive. For Fd = 0 the dimers returnto the pinned herringbone (PHB) state. In the jammed(J) state, the critical current is up to three times largerthan in the state where the dimers are aligned in the y-direction. We also note that at incommensurate fields for

0 0.5 1 1.5 2 2.5F

p

0

0.2

0.4

0.6

F

cx,F

cy

0 1 2F

p

0

1

2

3

Fc

y/F

cx

PHBPPP

Fc

y

Fc

x

J

FIG. 27: The critical depinning force in the x-direction, Fxc(open squares), and in the y-direction, Fyc (filled circles), vs Fpfor B/B = 2.0, Rp = 0.35, and np = 0.3125/

2. PP: par-tially pinned phase; P: pinned dimer phase; PHB: pinned her-

ringbone phase; J: jammed state. In the PP phase, Fxc = Fyc ,while in the P phase, Fxc > F

yc . A large enhancement of F

yc

occurs in the PHB phase when dimer jamming occurs. Inset:the ratio Fyc /F

xc vs Fp. The dashed line indicates F

yc /F

xc = 1,

where the depinning thresholds are equal.

Fp > 1.75, the net vortex flow is reduced since some ofthe dimers align in the x-direction and effectively blockthe motion of other vortices along the y-direction.

In order to better characterize the enhancement of Fcin the jammed state, in Fig. 27 we plot the critical depin-ning force in the y-direction, Fyc , and in the x-direction,Fxc

, versus FP

. In the inset of Fig. 27 we show the ratioFyc /F

xc versus Fp. In the partially pinned (PP) phase,

Fyc = Fxc , while in the pinned aligned dimer (P) phase,

Fxc is slightly higher than Fyc since the vortices can depin

more readily into the 1DMI phase in the y-direction. Inthe jammed state that forms from the PHB phase, Fyc is3.1 times higher than Fxc for the same value of Fp.

B. Effects of Changing Rp and B

In Fig. 28 we plot the dynamical phase diagram FDversus Rp for driving in the y-direction with B/B = 2.0,Fp = 0.85, and np = 0.3125/2. For Rp 0.2, the sys-tem depins into the 1DMI phase and makes a transitionto the ML phase at higher drives. For Rp < 0.2, thesystem forms the partially pinned (PP) phase where onlyhalf of the pinning sites are occupied. The PP phase de-pins into a moving interstitial phase (MI2Y) that resem-bles the MI2 state observed for driving in the x-directionin Fig. 17(b), where half the vortices depin while theother half remain pinned. The MI2Y phase is oriented90 from the MI2 phase. At FD = 0.4 for Rp < 0.2,the vortices at the pinning sites begin to depin and re-


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0 0.2 0.4R

p

0

0.2

0.4

0.6

0.8

FD

PP

J

P

MI2Y

1DMI

R

ML

FIG. 28: The dynamic phase diagram for FD vs Rp withFD = FDy, B/B = 2.0, Fp = 0.85, and np = 0.3125/

2.PP: partially pinned phase; J: jammed state; P: pinned phase;MI2Y: y-direction moving interstitial phase 2; 1DMI: one-

dimensional moving interstitial phase; R: random phase; ML:moving locked phase. For Rp < 0.2 the system forms the PPphase. This depins into the MI2Y state for driving in the y-direction, which is similar to the MI2 state shown in Fig. 17(a)for driving in the x-direction. For 0.2 < Rp < 0.3, thesystem forms the jammed J state shown in Fig. 26.

pin, giving a regime of the random (R) phase until FDbecomes large enough for all the vortices to depin into

the ML phase. For 0.2 Rp 0.3, the jammed (J)state discussed in Fig. 26 occurs due to the formationof dimers aligned in the x-direction, which is associatedwith a marked increase in Fc. As discussed earlier, thepinned herringbone (PHB) phase and jammed (J) stateoccur when Fp becomes high enough that the vortices inthe pinning sites cannot shift to allow for dimer order-ing to occur. Similarly, as Rp is reduced, the vorticesin the pinning sites have less room to adjust for dimerordering, so the PHB state forms. The jamming alsoproduces the counterintuitive effect that as Rp increasesabove Rp = 0.35, the depinning threshold decreases.

In Fig. 29 we show the dynamical phase diagram for FDversus np, which determines the value ofB, for B/B =2.0, Rp = 0.35, and Fp = 0.85. As np increases, thecritical depinning force into the 1DMI phase increasessince the repulsion from the pinned vortices experiencedby the interstitial vortices increases as the average vortex-vortex spacing decreases. The transition from the 1DMIphase to the ML phase shifts to higher values of FD asnp decreases since the distance between the pinning siteswhich stabilize the ML flow increases.

0 0.2 0.4 0.6 0.8n

p

0

0.2

0.4

0.6

0.8

1

FD

P

ML

1DMI

FIG. 29: The dynamical phase diagram for FD vs np, whichdetermines B, for driving in the y-direction with B/B =2.0, Rp = 0.85, and Fp = 0.85. P: pinned ordered dimerphase; 1DMI: one-dimensional moving interstitial phase; ML:

moving locked phase.

V. DISCUSSION

Our results are for honeycomb pinning arrays whereit was shown in previous work that n-merization of theinterstitial vortices into vortex molecular crystal statesoccurs for B/B > 1.5. Many of the dynamical effectspresented in this work are due to the n-merization ef-fect. In kagome pinning arrays, similar types of vortexmolecular crystal states appear, so we expect that manyof the same types of dynamic phases described here willalso occur for kagome pinning arrays, although we do ex-pect that there will be certain differences as well. Inthe kagome pinning array, the vortex dimer state ap-pears at B/B = 1.5 and has a herringbone orderingeven for large, weak pins. There are no easy flow chan-nels along 30 to the x-axis, so the symmetry breakingflows should be absent. Additionally, since there is noeasy flow channel in the y-direction, the anisotropic de-pinning dynamics may be different as well.

We have only considered B/B < 2.5 in this work.At higher fields, a wide array of vortex molecular crystalstates occur that should also have interesting dynamicalphases. Since the low matching fields are more robust,

observing the dynamics near these low fields experimen-tally is more feasible. Although our results are specifi-cally for pinning sites with single vortex occupation, sim-ilar dynamics should occur if the first few matching fieldshave multiple vortices at the pinning sites. In this case,the effective dimerization of the interstitial vortices wouldbe shifted to higher magnetic fields.

Although true phase transitions are associated onlywith equilibrium phenomena, the nonequilibrium phasesconsidered here have many analogies to equilibrium


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18

phases. For example, several of the transitions betweenthe nonequilibrium phases have a continuous type be-havior, while in other cases the transitions are sharp,indicative of a first order nature. Future studies couldexplore the possible emergence of a growing correlationlength near the transitions to see whether they exhibitthe true power law behavior associated with continuousphase transitions or whether they show crossover behav-

ior. For transitions that exhibit first order characteris-tics, it would be interesting to prepare a small patch ofpinning sites with different characteristics that could actas a nucleation site for one of the phases in order to un-derstand whether there is a length scale analogous to acritical nucleus size.

We also note that the dynamics we observe shouldbe general to systems with similar geometries and re-pulsively interacting particles. For example, in colloidalsystems, square pinning arrays with flat regions betweenthe pinning sites (muffin-tin potentials) have been fab-ricated, and in these systems the interstitial colloids aremuch more mobile than in washboard-type pinning po-tentials. Honeycomb pinning arrays could be created us-ing similar techniques for this type of system.

VI. SUMMARY

We have shown that vortices in honeycomb pinning ar-rays exhibit a rich variety of dynamical phases that aredistinct from those found in triangular and square pin-ning arrays. The honeycomb pinning arrays allow for theappearance of n-mer type states that have orientationaldegrees of freedom. We specifically focused on the casewhere dimer states appear. At B/B = 2.0, the dimerscan have a ferromagnetic type of ordering which is three-

fold degenerate. At depinning, the dimers can flow inthe direction in which they are aligned. For the caseof driving along the x-axis, the dimers flow at 30 tothe applied drive, giving a transverse velocity response.At incommensurate fields where dimers are present, eventhough the orientational order is lost, the moving statescan dynamically order into a broken symmetry statewhere the vortices flow with equal probability at either+30 or 30 to the x-axis. As the driving in the x-direction increases, there is a depinning transition for the

vortices in the pinning sites, and the transverse responseis lost when the vortices either flow in a random phase orchannel along the pinning sites. As a function of pinningforce, we find other types of vortex lattice ordering atzero driving, including a partially pinned lattice and aherringbone ordering of the dimers. These other order-ings lead to new types of dynamical phases, including anelastic depinning for weak pinning where all the vortices

depin simultaneously into a moving crystal phase, andan ordered interstitial flow in which the moving dimersbreak apart. The transitions between these flow phasesappear as clear steps in the velocity force curves, and wehave mapped the dynamical phase diagrams for varioussystem parameters. We also showed that the differentphases have distinct fluctuations and noise characteris-tics. When the temperature is high enough, the dimerstates lose their orientational ordering and begin to ro-tate within the interstitial sites. This destroys the sym-metry breaking flow; however, the moving locked phasecan still occur at high drives.

The transition in the vortex ground state ordering as

a function of pinning force causes the critical depinningforce for driving in the x and y-directions to differ. Whendriving along the y-direction, the initial depinning oc-curs in the form of one-dimensional interstitial channels,and at high drives the vortices can form an anisotropicmoving locked phase. We find a large enhancement ofthe depinning force in the y-direction associated with thepinned herringbone phase when the dimers align in thex-direction and creates a jamming effect. The jammedstate can enhance the critical depinning force by a factorof three, and can also arise for decreasing pinning size.We expect that many of the general features we observewill carry over to the higher matching fields in the honey-comb pinning arrays and in kagome arrays since ordered

n-mers states occur for the kagome lattice as well.

VII. ACKNOWLEDGMENTS

This work was carried out under the auspices of theNational Nuclear Security Administration of the U.S. De-partment of Energy at Los Alamos National Laboratoryunder Contract No. DE-AC52-06NA25396.

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