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5

Orientational Ordering, Buckling, and Dynamic Transitions for Vortices Interacting

with a Periodic Quasi-One Dimensional Substrate

Q. Le Thien1,2, D. McDermott1,2, C. J. Olson Reichhardt1, and C. Reichhardt11Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA

2Department of Physics, Wabash College, Crawfordsville, Indiana 47933 USA

(Dated: July 23, 2018)

We examine the statics and dynamics of vortices in the presence of a periodic quasi-one dimen-sional substrate, focusing on the limit where the vortex lattice constant is smaller than the substratelattice period. As a function of the substrate strength and filling factor, within the pinned statewe observe a series of order-disorder transitions associated with buckling phenomena in which thenumber of vortex rows that fit between neighboring substrate maxima increases. These transi-tions coincide with steps in the depinning threshold, jumps in the density of topological defects, andchanges in the structure factor. At the buckling transition the vortices are disordered, while betweenthe buckling transitions the vortices form a variety of crystalline and partially ordered states. In theweak substrate limit, the buckling transitions are absent and the vortices form an ordered hexagonallattice that undergoes changes in its orientation with respect to the substrate as a function of vortexdensity. At intermediate substrate strengths, certain ordered states appear that are correlated withpeaks in the depinning force. Under an applied drive the system exhibits a rich variety of distinctdynamical phases, including plastic flow, a density-modulated moving crystal, and moving floatingsolid phases. We also find a dynamic smectic-to-smectic transition in which the smectic orderingchanges from being aligned with the substrate to being aligned with the external drive. The differentdynamical phases can be characterized using velocity histograms and the structure factor. We dis-cuss how these results are related to recent experiments on vortex ordering on quasi-one-dimensionalperiodic modulated substrates. Our results should also be relevant for other types of systems suchas ions, colloids, or Wigner crystals interacting with periodic quasi-one-dimensional substrates.

PACS numbers: 74.25.Wx,74.25.Uv,74.25.Ha

I. INTRODUCTION

Commensurate-incommensurate transitions are rele-vant to a number of condensed matter systems that canbe effectively described as a lattice of particles inter-acting with an underlying periodic substrate. A com-mensurate state occurs when certain length scales ofthe particle lattice match the periodicity of the under-lying substrate, such as when the number of particlesis equal to the number of substrate minima1,2. Typi-cally when commensurate conditions are met, the sys-tem forms an ordered state free of topological defects,while at incommensurate fillings there are several pos-sibilities depending on the strength of the substrate. Ifthe substrate potential is weak, the particles maintaintheir intrinsic lattice structure which floats on top of thesubstrate, while for strong substrates a portion of theparticles lock into a configuration that is commensuratewith the substrate while the remaining particles formexcitations such as kinks, vacancies, or domain walls.At intermediate substrate strengths, the lattice order-ing can be preserved but there can be periodic distor-tions or rotations of the particle lattice with respect tothe substrate lattice3–8. These different cases are as-sociated with differing dynamical responses of the par-ticles under the application of an external drive2,7–11.When kinks or domain walls are present, multi-step de-pinning process can occur when the kinks become mo-bile at a lower drive than the commensurate portionsof the sample7,8. Examples of systems that exhibit

commensurate-incommensurate phases include atoms ad-sorbed on atomic surfaces1,3,4, vortices in type-II super-conductors interacting with artificial pinning arrays12–21,vortex states in Josephson-junction arrays22,23, super-fluid vortices in Bose-Einstein condensates in the pres-ence of co-rotating optical trap arrays24–26, cold atomsand ions on ordered substrates27–30, and colloidal par-ticles on periodic6–8,31–34 and quasi-periodic opticalsubstrates35,36.

In the superconducting vortex system, commensura-bility occurs when the number of vortices is an inte-ger multiple of the number of pinning sites, and varioustypes of commensurate vortex crystalline states can oc-cur with different symmetries12,13,16,19. At fillings wherethere are more vortices than pinning sites, it is pos-sible to have multi-quantized vortices occupy the pin-ning sites, and a composite vortex lattice can form thatis comprised of individual or multiple flux-quanta vor-tices localized on pinning sites coexisting with vorticeslocated in the interstitial regions between the pinningsites12,19,20. Ordered commensurate vortex states havebeen directly imaged with Lorentz microscopy13 andother imaging techniques37,38, and the existence of com-mensuration can also be deduced from changes in thedepinning force needed to move the vortices when peaksor steps appear in the critical current as a function ofvortex density12,14–16,18,19. It is also possible for orderedvortex structures such as checkerboard states to form atrational fractional commensuration ratios of n/m withinteger m and n, where n is the number of vortices and

2

m is the number of pinning sites37–40. Experiments7 andsimulations8,41 of colloidal assemblies on optical trap ar-rays examined the depinning transitions and subsequentsliding of the colloids and show that the depinning thresh-old is maximum for one-to-one matching of colloids andtraps, while it drops at incommensurate fillings due tothe presence of highly mobile kinks, anti-kinks, and do-main walls.

Commensurate-incommensurate transitions can alsooccur for particles interacting with a periodic quasi-one-dimensional (q1D) or washboard potential, where theparticles can slide freely along one direction of the sub-strate but not the other. An example of this type ofsystem is shown in Fig. 1 for a two-dimensional systemof vortices interacting with a quasi-one dimensional si-nusoidal substrate. The potential maxima are indicatedby the darker shadings, and the vortices are attractedto the light colored regions. This type of system hasbeen studied previously for colloids interacting with q1Dperiodic substrate arrays, where it was shown that var-ious melting and structural transitions between hexago-nal, smectic, and disordered colloidal arrangements canoccur42–48. In general, the colloidal studies focused onthe case where the particle lattice constant a is largerthan the substrate lattice constant w. Martinoli et al.

investigated vortex pinning in samples with a 1D peri-odic thickness modulation49–51 and observed broad com-mensuration peaks in the depinning threshold that wereargued to be correlated with the formation of orderedvortex arrangements that could align with the substrateperiodicity. Other vortex studies in similar samplesalso revealed peaks in the critical depinning force as-sociated with commensuration effects52,53, while studiesof vortices interacting with 1D magnetic strips showedthat commensurate conditions were marked by depinningsteps rather than peaks54. Under an applied dc drive, de-pinning transitions occur into a sliding state, and whenan additional ac drive is added to the dc drive, a series ofShapiro steps in the voltage-current curves appears whenthe frequency of the oscillatory motion of the vortex lat-tice over the periodic substrate locks with the ac drivingfrequency50. Similar commensuration effects and Shapirostep phenomena were also studied for vortices interact-ing with periodic washboard potentials or q1D periodicsawtooth substrates55. Vortices interacting with periodicq1D planar defects have also been studied in layered su-perconductors when the field is aligned parallel to thelayer directions. Here, different vortex lattice structures,smectic states, and oscillations in the critical current oc-cur as a function of applied magnetic field56–62.

For higher vortex densities in the presence of a q1Dsubstrate where the vortex lattice constant a is smaller

than the substrate lattice constant, a < w, there are sev-eral possibilities for how the vortices can order. In theweak substrate limit, they can form a hexagonal latticecontaining only small distortions, while in the strong sub-strate limit the vortices can be strongly confined in eachpotential minimum to form 1D rows, so that the over-

x(a)

y

x(c)

(b)

y

x(e)

(d)

y

x(g)

(f)

y

(h)

FIG. 1: The real space images (left column), with the sub-strate minima indicated by lighter regions and the vortex posi-tions marked with circles, and the structure factor S(k) (rightcolumn), for a system with a periodic quasi-one-dimensionalsubstrate with Fp = 1.5. (a,b) At w/a = 1.85, each substrateminimum contains a single row of vortices (r1) and the struc-ture factor shows smectic order. (c,d) w/a = 2.054, at theonset of a buckling transition. (e,f) At w/a = 2.651 there isan ordered zig-zag r2 vortex lattice. (g,h) At w/a = 3.0 thereis a mixture of r2 and r3 lattices.

all two-dimensional (2D) vortex structure is anisotropic.Between these limits, the vortices can exhibit bucklingtransitions by forming zig-zag patterns within individualpotential troughs, so that for increasing vortex densitythere could be a series of transitions at which increas-ing numbers of rows of vortices appear in the potential

3

troughs. Transitions from 1D rows of particles to zig-zag states or multiple rows have been studied for par-ticles in single q1D trough potentials in the context ofvortices63–65, Wigner crystals66–68, colloids70,70–72, q1Ddusty plasmas73,74, ions in q1D traps75–77, and othersystems78,79 where numerous structural transitions, dif-fusion behavior and dynamics can occur. In the case of aperiodic array of channels such as shown in Fig. 1, muchless is known about what buckling transitions would oc-cur and what the dynamics would be under an applieddriving force. Recently Guillamon et al. studied vortexlattices in samples with a periodic q1D array of grooves.As a function of the commensuration ratio p = w/a, theyfound that for p < 6, the vortex lattice remains triangularbut undergoes a series of transitions that are marked byrotations of the angle θ made by the vortex lattice withrespect to the substrate symmetry direction80. They alsoobserved that at much higher fields, the system transi-tions into a disordered state with large vortex densityfluctuations. Open questions include what happens tothese reorientation transitions as the substrate strengthis increased, and what the vortex dynamics are when adriving force is applied. Dynamical phases and structuraltransitions between different kinds of nonequilibrium vor-tex flow states have been extensively studied for drivenvortex systems interacting with random81–86 and 2D pe-riodic pinning arrays87–89; however, there is very littlework examining the dynamic vortex phases for vorticesmoving over q1D periodic substrates. It is not knownwhether the vortices would undergo dynamical structuraltransitions or exhibit the same types of dynamic phasesfound for vortices driven over random disorder, such as adisordered plastic flow state that transitions to a movingsmectic or anisotropic crystal as a function of increasingdrive.

In this work we consider ordering and dynamics of vor-tices interacting with a periodic q1D sinusoidal poten-tial for fillings 0 < w/a < 5.5. In the strong substrateregime, the system undergoes a series of structural tran-sitions that are related to the number of rows rn of vor-tices that fit within each substrate trough. These tran-sitions include transformations from 1D vortex rows tozig-zag patterns that gradually increase rn. The vortexstructure contains numerous dislocations at the bucklingtransitions and is ordered between the buckling transi-tions. For strong substrates, the onset of the bucklingand ordered phases produces a series of steps in the crit-ical depinning threshold as a function of vortex density,while for weaker substrate strengths, some of the statesin which the vortices order produce peaks in the criticaldepinning force. For the weakest substrates, the vorticesform a triangular lattice that undergoes rotations with re-spect to the underlying substrate symmetry direction as afunction of applied magnetic field, similar to the behaviorobserved by Guillamon et al.

80. Under an applied drivewe observe plastic flow states, moving density-modulatedcrystals, and dynamic floating solids. For certain fillingswe also find smectic-to-smectic transitions where the two

smectic states have different orientations. These differentflowing phases produce distinct features in the velocityhistograms and the structure factor. The commensura-bility ratio w/a also strongly affects the driving force atwhich the transition to a moving floating solid occurs.Our results should be general to other types of systemsthat can be represented as a collection of repulsive par-ticles interacting with a periodic q1D substrate, such ascolloids on optical line traps, ions in coupled traps, andWigner crystals on corrugated substrates.

II. SIMULATION

We model a two-dimensional system of vortices in-teracting with a periodic q1D potential with period w,where there are periodic boundary conditions in the xand y-directions. The vortices are modeled as point par-ticles and the dynamics of an individual vortex i obeysthe following equation of motion:

ηdRi

dt= F

ivv + F

si + F

id + F

iT . (1)

Here η is the damping constant which we setequal to unity. The vortex-vortex forces F

ivv =

∑Nv

j=1 F0K1(Rij/λ)Rij , where F0 = φ20/2πµ0λ

3, φ0 is theelementary flux quantum, µ is the permittivity, K1 is themodified Bessel function, Ri is the location of vortex i,Rij = |Ri − Rj|, Rij = (Ri − Rj)/Rij , and λ is thepenetration depth. The vortices have repulsive interac-tions and form a triangular lattice in the absence of asubstrate. The vortex interaction with the substrate isgiven by F

si = −∇V (xi)x where the substrate has the

sinusoidal form

V (x) = V0 sin(2πx/w). (2)

We define the pinning strength of the substrate to beFp = 2πV0/w. The dc driving force F

id arises from the

Lorentz force induced by a current applied along the easydirection (y-axis) of the substrate which produces a per-pendicular force on the vortices and causes them to movein the x-direction in our system. We measure the vortexvelocity 〈Vx〉 along the driving direction as we increasethe external drive in increments of δFd, and average thevortex velocities over a fixed time in order to avoid anytransient effects. The thermal forces FT are modeledas random Langevin kicks with the properties 〈FT 〉 = 0

and 〈FiT (t)F

jT (t

′)〉 = 2ηkBTδijδ(t − t′), where kB is theBoltzmann constant. The initial vortex positions are ob-tained by annealing from a high temperature state andcooling down to T = 0. The dc drive is applied onlyafter the annealing procedure is completed. We considera range of vortex densities, which we report in terms ofthe ratio w/a of the periodicity of the substrate to thevortex lattice constant that would appear in the absenceof a substrate. We denote a state containing n rows ofvortices in each potential minimum as rn.

4

III. PINNED PHASES

In Fig. 1(a,c,e,g), we plot the real space locations ofthe vortices on the potential substrate after annealingfor a system with Fp = 1.5 at fillings of w/a = 1.85,2.054, 2.651, and 3.0, while in Fig. 1(b,d,f,h) we showthe corresponding structure factors S(k). At w/a = 1.58in Fig. 1(a), the vortices form single 1D rows in each po-tential minimum, corresponding to an r1 state, and theoverall vortex structure is highly anisotropic with latticeconstants ax = 4.5 in the x−direction and ay = 1.31in the y−direction. Additionally, each potential troughcaptures a slightly different number of vortices, introduc-ing disorder in the alignment of rows in adjacent minima,and leaving the system with periodic ordering only alongthe x-direction. The corresponding structure factor inFig. 1(b) exhibits a series of spots at ky = 0 and finitekx, indicative of the 1D ordering associated with a smec-tic phase. As the magnetic field increases, the vortexordering must become increasingly anisotropic in orderto maintain single rows of particles in each minimum.This is energetically unfavorable, so instead a transitionoccurs to a zig-zag or buckled state in which there aretwo partial rows of vortices in every substrate minimum.Figure 1(c) illustrates the real space vortex positions forw/a = 2.054 at the beginning of the zig-zag transition,where some of the troughs contain a buckled vortex pat-tern. In the corresponding S(k) plot in Fig. 1(d), thesmectic ordering develops additional features at large kassociated with the shorter range structure that arises onthe length scale associated with the zig-zag pattern. Asthe magnetic field is further increased, the zig-zag patternappears in all the substrate minima and the system formsan ordered anisotropic 2D r2 lattice as shown in Fig. 1(e)for w/a = 2.651, where there are two rows of vortices ineach potential minimum that form a zig-zag structurewhich is aligned with zig-zag structures in neighboringminima. The corresponding S(k) in Fig. 1(f) has a se-ries of peaks at small and large k indicating the presenceof a more ordered vortex structure. For higher fields,the zig-zag lattice becomes increasingly anisotropic un-til another buckling transition occurs to produce r3 withthree vortex rows per substrate minimum. Figure 1(g)shows the transition point at w/a = 3.0 where certainpotential troughs contain three vortex rows while otherscontain two vortex rows or mixtures of two and threevortex rows. In Fig. 1(h), S(k) for this case shows thatthe system is considerably more disordered than at thecommensurate case illustrated in Fig. 1(e,f).

In Fig. 2 we show the continuation of the evolution ofthe vortex lattice from Fig. 1 in both real space and k-space. At w/a = 3.3 in Fig. 2(a), there is an ordered r3structure with three vortex rows in each potential mini-mum, producing the ordered S(k) shown in Fig. 2(b). Asthe vortex density is further increased, the row structuredisorders as shown in Fig. 2(c) for w/a = 3.644, corre-sponding to a ring like structure in S(k) as indicated inFig. 2(d). There are still peaks along the ky = 0.0 line

x(a)

y

x(c)

(b)

y

x(e)

(d)

y

x(g)

(f)

y

(h)

FIG. 2: The continuation of the real space images (left col-umn) and S(k) (right column) from the system in Fig. 1 withFp = 1.5. (a,b) At w/a = 3.3, there is an ordered structurewith three vortex rows per potential minimum (r3). (c,d)At w/a = 3.644, there is a partially disordered state withroughly three vortex rows per potential minimum. (e,f) Atw/a = 4.1455 there is a partially ordered state with four vor-tex rows per minimum (r4). (g,h) The disordered state atw/a = 5.15 showing ring structures in S(k).

due to the anisotropy induced by the substrate. For thisvalue of Fp, further increasing the vortex density doesnot produce a more ordered configuration; however, cer-tain partially ordered structures can occur as illustratedin Fig. 2(e) for w/a = 4.1455, where there are four vortexrows per trough (r4) with mixed peaks and smearing in

5

0.4

0.6

0.8

1P

6

0.2

0.4

0.6

0.81

P6

0.4

0.6

0.8

P6

1 2 3 4 5w/a

0.4

0.6

0.8

1

P6

ac e

g

A C EG

(a)

(b)

(c)

(d)

FIG. 3: The fraction of six-fold coordinated vortices P6 vsw/a for (a) Fp = 0.5, (b) Fp = 1.0, (c) Fp = 1.5 and (d)Fp = 2.0. In panel (c), the labels a, c, e, g indicate the valuesof w/a at which the images in Fig. 1 were obtained, while thelabels A, C, E, and G indicate the values of w/a at which theimages in Fig. 2 were obtained. The dips in P6 coincide withtransitions in the number of vortex rows contained withineach potential minimum.

the corresponding structure factor shown in Fig. 2(f). Athigher fields, the vortex structures become disordered asshown in Fig. 2(g) at w/a = 5.15, where S(k) in Fig. 2(h)has pronounced ring structures. There are still two peaksat ky = 0 and finite kx due to the smectic ordering im-posed by the q1D substrate.

We can also characterize the system using the fraction

of six-fold coordinated vortices P6 = N−1∑N

i=1 δ(6−zi),where zi is the coordination number of vortex i obtainedfrom a Voronoi construction. In general, we find that P6

drops at the buckling transitions due to the formationof dislocations that are associated with the splitting ofa single row of vortices into two rows, creating a kinkat the intersection of the two rows. In Fig. 3(a) we plotP6 versus w/a for a system with Fp = 0.5. Over therange 1.0 < w/a < 1.7, each pinning trough containsan r1 state, while the dip in P6 at w/a = 1.77 corre-sponds to the middle of the buckling transition whenthere is roughly a 50:50 mixture of r1 and r2. For1.85 < w/a < 2.35, the system forms an ordered r2 statesimilar to that shown in Fig. 1(e), but less anisotropicsince the weaker substrate compresses the zig-zag struc-ture less and permits it to be wider. Near w/a = 2.4,there is another buckling transition from r2 to r3 and thesystem forms a disordered state similar to that shown inFig. 1(g). As w/a is further increased, there is a partiallyordered state near w/a = 3 that is similar to the state inFig. 2(a); however, due to the weaker substrate strengtha fully ordered r3 state does not form. For w/a > 3.2 the

system adopts a polycrystalline configuration that be-comes more ordered at high vortex densities. In Fig. 3(b),we show that a similar set of features associated withbuckling transitions occurs for a stronger substrate withFp = 1.0; however, in this case the transition from r2 tor3 is sharper and a fully ordered three row state appearsnear w/a = 3.0.

In Fig. 3(c) we plot P6 versus w/a for samples withFp = 1.5, the same pinning strength at which the im-ages in Figs. 1 and 2 were obtained. Here the dips in P6

associated with the r1 to r2, r2 to r3, and r3 to r4 tran-sitions are sharper. We also observe the development ofa small dip near w/a = 4.4 corresponding to a partialtransition from r4 to r5. The values of w/a at which rowtransitions occur shift upward with increasing Fp. Forexample, the r1 to r2 transition occurs at w/a = 1.768for Fp = 0.5 but at w/a = 2.05 for Fp = 1.5, since thehigher Fp stabilizes the r1 state up to higher lattice con-stant anisotropies. Figure 3(d) shows P6 versus w/a forsamples with Fp = 2.0. Here the dip in P6 at the r1 to r2transition broadens, while a pronounced jump emergesat w/a = 4.7 corresponding to the r4 to r5 transition.We expect that for higher values of Fp, additional dipsin P6 for transitions from rn to rn+1 states for n ≥ 5 willappear at w/a values higher than those we consider here.

It is difficult to determine if the buckling transitionsare first or second order in nature. For particles in anisolated trough, the transition from a single row to azig-zag pattern is second order, and there have been sev-eral studies in cold ion systems of quenches through thistransition in which the density of kinks was calculated fordifferent quench rates and compared to predictions fromnonequilibrium physics on quenches through continuousphase transitions75–77. We expect that the buckling tran-sitions we observe are second order; however, it may bepossible that the additional coupling to particles in neigh-boring potential minima could change the nature of thetransition, and we have observed a coexistence of chainstates which is suggestive of phase separation. For vor-tex systems it could be difficult to change the substratestrength as a function of time, but for colloidal systemsit is possible to create q1D periodic optical substrates ofadjustable depth and use them to study time dependenttransitions by counting the number of kinks that form asa function of the rate at which the substrate strength ischanged.

In Fig. 4 we plot the depinning force Fc versus w/afor Fp = 0.1, 0.25, 0.5, 1.0, 1.5, and 2.0 to show thatthe buckling transitions are associated with changes inthe slope of the depinning force, which decreases withincreasing w/a in a series of steps. The first drop in Fc

near w/a = 2.0 corresponds to the r1 to r2 transition.In the r1 state, the particle-particle interactions roughlycancel in the x-direction, so the depinning force is approx-imately equal to Fp, while close to the buckling transitionthe vortices on the right side of a zig-zag experience anadditional repulsive force in the driving direction fromthe vortices on the left side of the zig-zag, decreasing the

6

1 2 3 4w/a

0

0.5

1

1.5

2F

c

1 1.5 2 2.5w/a0

0.05

0.1

0.15

Fc

a

c

e

gA

C E

FIG. 4: The depinning force Fc vs w/a for Fp = 0.1 (redcircles), 0.25 (green squares), 0.5 (blue diamonds), 1.0 (orangeup triangles), 1.5 (purple left triangles), and 2.0 (pink downtriangles) showing that for Fp > 0.25 the buckling transitionscorrespond to step features in Fc. The labels a, c, e, g indicatethe values of w/a at which the images in Fig. 1 were obtained,while the labels A, C, and E indicate the values of w/a atwhich the images in Fig. 2 were obtained. Inset: a highlightof the main panel illustrates that for weaker pinning, peaks inFc occur, as shown for Fp = 0.1 (red circles) and 0.25 (greensquares). The peak is associated with the formation of anordered zig-zag lattice similar to that shown in Fig. 1(e).

x(a)

y

(b)

FIG. 5: (a) Real space image of the vortex configuration atthe peak in Fc at Fp = 0.25 and w/a = 1.94 for the systemshown in the inset of Fig. 4 where an ordered zig-zag structureoccurs. (b) The corresponding S(k) contains various peaksreflecting the ordered nature of the state.

driving force needed to depin the vortices. To estimatethe magnitude of this reduction in Fc, we note that theaverage x-direction spacing between vortices in a givenzig-zag is approximately dx = 2.0. For a zig-zag with a30◦ angle between the two closest neighbors on the otherside of the chain from each vortex, the vortex-vortex in-teraction force of K1(2) produces an additional repulsiveforce of fr = 0.55, giving a value of Fc = Fp − fr that isclose to the value of Fc = 0.78 observed after the r1 to r2step for the Fp = 1.5 system. Similar arguments can bemade for the magnitudes of the drops in Fc at the higher

order transitions as well. At w/a = 1.85 in the Fp = 1.5sample, Fc already begins to drop below Fp even thoughthe pinned configuration shown in Fig. 1(a) is an r1 state.This occurs because in this range of w/a, application ofa finite Fd < Fc induces a slight buckling of the vortices,while for w/a < 1.5 the r1 rows remain in a 1D pinnedstate up to Fd = Fc. The inset of Fig. 4 shows a blowupof Fc versus w/a for the weaker pinning cases Fp = 0.25and Fp = 0.1. At w/a = 1.94 there is a peak in Fc forthe Fp = 0.25 sample coinciding with the formation ofthe long range ordered zig-zag state shown in Fig. 5(a).The corresponding S(k) in Fig. 5(b) contains a series ofpeaks that reflect the ordered nature of the state, whichresembles the zig-zag state in Fig. 1(e,f) except that thesystem is more ordered and the zig-zag structure is wider.For Fp = 0.1 the zig-zag state transitions into a hexago-nal lattice and the peak in Fc begins to disappear. Someexperiments examining vortices in q1D periodic pinningstructures show that peaks in the critical current occurat certain fillings49,51–53 in regimes where the pinning isweak, whereas other experiments performed in the strongpinning limit reveal more step-like features in the criticalcurrent. This suggests that the experiments in the strongpinning limit are producing buckling transitions54.In Fig. 6 we plot representative real space images with

the matching S(k) for some other substrate strengths tohighlight other types of ordering we observe. Figure 6(a)shows the real space ordering of the vortices at Fp = 0.5and w/a = 4.825, where the vortex lattice is polycrys-talline and contains regions of triangular ordering withdifferent orientations. The corresponding structure fac-tor in Fig. 6(b) has ring features with some remnant ofthe smectic ordering appearing at smaller values of k.In Fig. 6(c), at Fp = 2.0 and w/a = 4.33 an r4 stateappears, while S(k) in Fig. 6(d) has smectic orderingfeatures along with additional crystalline ordering sig-natures due to the ordered arrangement of the particleswithin the troughs. At Fp = 2.0 and w/a = 4.67 inFig. 6(e), a new type of ordered structure appears inwhich the vortices can pack more closely by forming al-ternate regions of r3 and r4 states, producing a consider-able amount of triangular ordering as seen in the plot ofS(k) in Fig. 6(f), where there are sixfold peaks at largek and smectic peaks at smaller k. In Figs. 6(g,h), forFp = 2.0 and w/a = 5.15, a more disordered structureappears, with some regions of the sample containing r4or r5 states.

IV. LATTICE ROTATIONS FOR WEAK

SUBSTRATES

We have also studied systems with a pinning strengthof Fp = 0.02. Here, for w/a > 1.77 the vortices forma triangular lattice and the features associated with thebuckling transitions observed in Fig. 3 are lost. In thiscase the vortex lattice can orient at various angles withrespect to the underlying substrate. In Fig. 7(a) we show

7

x(a)

y

x(c)

(b)

y

x(e)

(d)

y

x(g)

(f)

y

(h)

FIG. 6: Real space images (left column), with the substrateminima indicated by lighter regions and the vortex positionsmarked with circles, and S(k) (right column). (a,b) At Fp =0.5 and w/a = 4.825 there is a polycrystalline structure. (c,d)At Fp = 2.0 and w/a = 4.33 there is a partially ordered r4structure. (e,f) At Fp = 2.0 and w/a = 4.67, an orderedstructure appears. (g,h) At Fp = 2.0 and w/a = 5.15 thestructure is disordered.

the real space vortex positions at w/a = 2.8, where thevortices form a triangular lattice that is aligned at an an-gle θ = 27.6◦ with respect to the y-axis. In Fig. 7(b) atw/a = 3.272, θ = 24◦, while in Fig. 7(c) at w/a = 3.53,θ = 2.9◦, and in Fig. 7(d) at w/a = 3.75, θ = 13.2◦.In Ref.80, Guillamon et al. observed experimentally thatvortices on a q1D substrate retained triangular order-

x(a)

y

x(b)

y

x(c)

y

x(d)

y

FIG. 7: Real space images of the vortices in a sample withFp = 0.02 showing that the hexagonal vortex lattice can adoptvarious orientations θ with respect to the substrate. (a) Atw/a = 2.8, θ = 27.6◦. (b) At w/a = 3.272, θ = 24◦. (c) Atw/a = 3.53, θ = 2.9◦. (d) At w/a = 3.75, θ = 13.2◦.

ing but that the vortex lattice was oriented at an angleθ ranging from θ = 0 to θ = 30◦ with respect to thesubstrate. In several cases, they found that the systemlocked to specific angles close to θ = 30◦, θ = 24◦, andθ = 0◦. We find a much larger variation in the orien-tation of the lattice with respect to the substrate as afunction of filling than was observed in the experiments,which may be due to differences the pinning strength orthe finite size of our simulations. Our results show thatfor weak pinning, the buckling transitions are lost andare replaced with orientational transitions of the vortexlattice with respect to the substrate. Another feature weobserve when the pinning strength is increased is thatthe vortex lattice becomes disordered or polycrystalline.Guillamon et al. also observe that at higher fillings thevortex lattice becomes disordered; however, in their sys-tem there are strong random vortex density fluctuations,while for our thermally annealed samples the vortex den-sity at higher fields is generally uniform.

V. DYNAMIC PHASES

In Fig. 8 we plot simultaneously P6 and the aver-age nearest neighbor spacing dnn versus Fd for a sam-ple with Fp = 1.5 at varied w/a. Here, we obtain dnnby performing a Voronoi tesselation to identify the zinearest neighbors of particle i, and then take dnn =

8

00.20.40.60.8

1P

6

00.20.40.60.8

P6

00.20.40.60.8

P6

0 2 4 6 8 10Fd

00.20.40.60.8

P6

1.2

1.25

1.3

1.35

d nn

1.31.4

1.5

1.6

d nn

1.61.7

1.8

1.92

d nn

2.7

2.75

2.8

2.85

d nn

(a)

(b)(c)

(d)a c

e

FIG. 8: The fraction of six-fold coordinated vortices P6 (darkblue curves) and the average nearest-neighbor distance dnn

(light red curves) vs Fd for a system with Fp = 1.5. (a)At w/a = 1.767 the system depins from an r1 state. (b)At w/a = 2.5 the system dynamically orders into a movingtriangular lattice with P6 = 0.97. (c) w/a = 3.061. (d)At w/a = 3.535, the onset of the dynamically ordered phasecoincides with a drop in dnn near Fd = 4.0. The labels a, c,and e correspond to the values of Fd used for the images inFig. 9.

(N∑

i zi)−1

∑N

i=1

∑zij=1 rij , where rij is the distance be-

tween particle i and its jth nearest neighbor. For w/a =1.767 in Fig. 8(a), P6 = 0.83 in the pinned r1 state thatoccurs for 0 < Fd < 1.4. There is a dip in P6 over therange 1.4 < Fd < 1.7, corresponding to the plastic flowstate in which some of the vortices remain immobile whileother vortices hop in and out of the potential wells. ForFd > 1.7, P6 increases and reaches a saturated value ofP6 = 0.93 when the vortices form a moving triangularlattice containing a small fraction of dislocations. Thevalue of dnn drops at the depinning transition, and sev-eral additional drops in dnn occur at higher drives. Inthe r1 pinned state, each vortex has two close nearestneighbors that are in the same pinning trough, and fourmore distant nearest neighbors that are in adjacent pin-ning troughs. Once the vortices depin and enter a movingstate, they adopt a more isotropic structure, causing dnnto drop as the distance to the four more distant near-est neighbors decreases. The additional drops in dnn athigher Fd occur whenever the vortex lattice rearrangesto become still more isotropic.In Fig. 8(b) we plot P6 and dnn versus Fd for the same

sample at w/a = 2.5 where an ordered zig-zag state withP6 = 0.89 appears at zero drive, similar to that shownin Fig. 1(e). The depinning threshold is Fc = 0.65, muchlower than the value of Fc for the w/a = 1.767 filling inFig. 8(a), and the depinning transition is marked by adrop in P6 to P6 = 0.2. Over the range 0.65 < Fd . 2.0,the vortices are in a dynamically disordered state, while

x(a)

y

x(c)

(b)

y

x(e)

(d)

y

(f)

FIG. 9: Real space images (left column), with the substrateminima indicated by lighter regions and the vortex positionsmarked with circles, and S(k) (right column) for the dynamicsystem from Fig. 8(d) with Fp = 1.5 and w/a = 3.535 at thevalues of Fd labeled a, c, and e. (a,b) At Fd = 0.5 the samplecontains pinned vortices coexisting with individual vorticesthat hop from trough to trough. (c,d) At Fd = 3.5, all thevortices move together to form a disordered lattice with aperiodic density modulations. (e,f) At Fd = 7.0 the systemforms a moving floating triangular lattice.

the saturation of dnn above Fd ≈ 2.0 indicates that apartially ordered state has formed. The value of P6 doesnot reach a maximum until Fd = 3.9, where P6 ≈ 0.97and an ordered state appears. For 1.5 < Fd < 3.9, weobserve a moving density-modulated solid. The value ofF trc , the drive at which the sample reaches a moving tri-

angular lattice state, is higher for w/a = 2.5 than forw/a = 1.767, even though the depinning threshold Fc issmaller for the w/a = 2.5 system. At w/a = 2.5, dnnis initially small and jumps up at the depinning transi-tion, unlike the decrease in dnn at depinning found inFig 8(a). Since the vortices in Fig. 8(b) form a zig-zag r2structure in the pinned state, each vortex has four close

9

0.25 0.50 0.75 1.00Vx

0.0

0.3

0.6

0.9

1.2

P(Vx)

×10−1

(a) −0.8 0.0 0.8Vy ×10−1

0.0

1.5

3.0

4.5

6.0

P(Vy)

×10−2

(b) 0.25 0.50 0.75 1.00Vx

−0.6

0.0

0.6

Vy

×10−1

(c)

2.4 3.2 4.0 4.8Vx

0.0

0.2

0.4

0.6

0.8

1.0P(Vx)

×10−1

(d) −2 0 2 4Vy ×10−2

0

1

2

3

4

5

P(Vy)

×10−2

(e) 2.4 3.2 4.0 4.8Vx

−2

0

2

4

Vy

×10−2

(f)

5.6 6.4 7.2 8.0Vx

0

2

4

6

8

P(Vx)

×10−2

(g) −2.5 0.0 2.5 5.0Vy ×10−2

0.00

0.25

0.50

0.75

1.00P(Vy)

×10−1

(h) 5.6 6.4 7.2 8.0Vx

−2.5

0.0

2.5

Vy

×10−2

(i)

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

FIG. 10: Vortex velocities for the system in Fig. 9 with Fp = 1.5 and w/a = 3.535. Left column: Histogram P (Vx) ofthe instantaneous vortex velocities in the driving direction Vx. Center column: Histogram P (Vy) of the instantaneous vortexvelocities in the transverse direction Vy. Right column: Heightfield map of Vy versus Vx. (a,b,c) The plastic flow regime atFd = 0.5. (d,e,f) The moving modulated solid regime at Fd = 3.5. (g,h,i) The moving floating solid regime at Fd = 7.0.

nearest neighbors in the same pinning trough, and twomore distant nearest neighbors in an adjacent pinningtrough. This causes dnn to be smaller in the pinned statethan it was for the r1 structure in Fig. 8(a), and whenthe vortex lattice becomes more isotropic in the movingstate, dnn increases rather than decreasing as the twohalves of each zig-zag structure move further apart. AtFd = 4.1 we observe a drop in dnn that coincides witha dip in P6. This feature is associated with a transi-tion from a density-modulated lattice to a more uniformmoving floating lattice.

In Fig. 8(c), we show P6 and dnn versus Fd at w/a =3.061 where there is an r3 pinned state, similar to thatillustrated in Fig. 2(a). Here the depinning thresholdFc = 0.3, and the system transitions into a moving tri-angular lattice at F tr

c = 1.6, which is somewhat lowerthan the value of F tr

c for w/a = 2.5 in Fig. 8(b). Thebehavior of dnn in Fig. 8(c) follows a similar pattern asin Fig. 8(b), with dnn increasing with increasing Fd. Weplot the same quantities for w/a = 3.535 in Fig. 8(d),where the depinning threshold Fc ≈ 0.087 and the sys-tem dynamically orders for Fd > 4.0. There is a small dipin dnn at Fd = 4.15 along with a saturation in P6 which iscorrelated with a structural change to a dynamic floatinglattice.

In order to characterize the nature of the dynamic vor-

tex structures in the moving states, in Fig. 9(a,b) we plotthe real space images and S(k) for the system in Fig. 8(d)at w/a = 3.535 and Fd = 0.5. Here, individual vorticesjump from one pinning well to the next while a portionof the vortices remain immobile in the substrate minima.As shown in the plot of S(k), the vortex configurationis fairly ordered and takes the form of a distorted non-triangular structure which causes P6 to be low for thisvalue of Fd. For Fd > Fp, the vortices move together sothere is no plastic motion, and form a distorted latticecontaining pronounced density modulations as shown inFig. 9(c,d) for Fd = 3.5. For Fd > 4.0 we find a tran-sition from the density modulated lattice to a movinghomogeneous floating triangular lattice which coincideswith the drop in dnn in Fig. 8(d) and the maximum inP6. Fig. 9(e) shows the floating lattice at Fd = 7.0, whereas indicated in Fig. 9(f) S(k) contains sixfold peaks thatare indicative of triangular ordering. The smectic order-ing induced by the substrate is substantially weaker oralmost absent at this drive, as shown by the weaknessof the spots in S(k) at ky = 0, indicating that the sys-tem has formed a floating solid. We find similar types oftransitions in the dynamics at other fillings as well.

We can also characterize the different dynamic statesin Figs. 8 and 9 by examining histograms of the vortexvelocities. In Fig. 10(a) we plot the distribution P (Vx) of

10

x(a)

y

x(c)

(b)

y

x(e)

(d)

y

(f)

FIG. 11: Real space images (left column), with the substrateminima indicated by lighter regions and the vortex positionsmarked with circles, and S(k) (right column) for a systemwith Fp = 0.5 and w/a = 1.767. (a,b) The plastic flow phaseat Fd = 0.3 where there is individual vortex hopping fromwell to well. The S(k) peaks indicate a smectic phase withperiodic ordering along the x−direction. (c,d) At Fd = 0.6,all the vortices are flowing and form chains that are alignedin the x-direction. S(k) shows that a new smectic order hasappeared with periodic ordering along the y-direction. (e,f)At Fd = 3.6 there is a moving floating triangular crystal.

Vx in the driving direction at Fd = 0.5 for the system inFig. 9(a,b) with Fp = 1.5 and w/a = 3.535. Figure 10(b)shows the transverse velocities P (Vy), while in Fig. 10(c)we plot Vy versus Vx as a heightfield map. At this drive,the motion is plastic and occurs by individual vortex hop-ping, so there is a sharp peak in P (Vx) at Vx = 0.15 whichreflects the fact that most of the vortices are slowly mov-ing within an individual pinning trough. When a singlevortex jumps into an adjacent pinning trough, it createsa pulse of motion through the trapped vortices that trig-gers the jump of another single vortex to the next pinningtrough, where the process repeats. This depinning cyclecreates two peaks in P (Vx). The peak at low Vx cor-

responds to the motion of a velocity pulse through thedense assembly of vortices at the bottom of the pinningtrough, while the peak at high Vx is produced by in-dividual vortices escaping over the potential maximum.This peak falls near Vx = 1.1, which is larger than Fd,reflecting the fact that after an individual vortex passesthe crest of the substrate maximum, the substrate con-tributes an additional force term in the driving directionas the vortex moves toward the next substrate minimum.In this case the maximum force exerted by the pinningsite is Fp = 1.5 while the driving force is Fd = 0.5, sothat the maximum possible instantaneous vortex velocitywould be Vx = 2.0; however, vortex-vortex interactionsprevent individual vortices from moving this rapidly. InFig. 10(b), P (Vy) is centered at Vy = 0 since there isno driving force in the transverse direction; however, weobserve some asymmetry in P (Vy) as well as peaks atfinite Vy due to the fact that the vortex lattice segmentsinside the pinning troughs are oriented at an angle withrespect to the substrate symmetry direction, as shown inFig. 9(a). This asymmetry also appears in the Vy versusVx plot in Fig. 10(c), which has two prominent features.The first is a wide band of Vy values at low Vx that areassociated with soliton-like pulses moving through thedense regions of the vortex clusters, which push vorticesin both the positive and negative y-direction. The secondis the loop shape at larger Vx values which correspondsto motion in which the vortices are accelerated or de-celerated as they pass over the substrate maxima andminima.

In Fig. 10(d,e,f) we show instantaneous velocity plotsfor the system in Fig. 9(c,d) with Fd = 3.5 where thevortices are moving elastically in the density-modulatedsolid phase. Here P (Vx) in Fig. 10(d) has peaks atVx = 2.01 and Vx = 4.75 which are smoothly connectedby finite P (Vx) values. The shape of this histogram showsthe velocity imposed by the driving force of Fd = 3.5when the substrate forces alternately act with or againstthe driving force. When the substrate force is againstthe drive the velocity is Vx = Fd − Fp = 2.0, while whenthe substrate and driving forces are in the same direc-tion, Vx = Fd + Fp = 5.0, close to the observed valuesof the peaks in P (Vx). In Fig. 10(e), P (Vy) has twopeaks close to Vy = 0.1 and Vy = −0.1, indicating thatthere is an oscillatory motion in the y-direction. Thiseffect can be seen more clearly in the Vy versus Vx plotin Fig. 10(f) which has two symmetric lobes. Since thevortices are in a density-modulated lattice, shearing inthe y-direction occurs between adjacent density modula-tions, with one density modulation moving in the pos-itive y-direction while the other moves in the negativey-direction.

The velocity plots for the system in Fig. 9(e,f) atFd = 7.0 appear in Fig. 10(g,h,i). In Fig. 10(g), P (Vx)has a two-peak feature similar to that in Fig. 10(d), butwith peak values at Vx = 5.52 and Vx = 8.4. Figure10(h) shows that P (Vy) has a single peak centered atVy = 0, while in Fig. 10(i), there is a single lobe in the

11

0 2 4 6Vx ×10−1

0.0

0.4

0.8

1.2

1.6

2.0

P(Vx)

×10−1

(a) −0.8 0.0 0.8Vy ×10−1

0.0

0.2

0.4

0.6

0.8

P(Vy)

×10−1

(b) 0 2 4 6Vx ×10−1

−0.8

0.0

0.8

Vy

×10−1

(c)

0.25 0.50 0.75 1.00Vx

0.00

0.25

0.50

0.75

1.00

P(Vx)

×10−1

(d) −1.2 −0.6 0.0 0.6Vy ×10−1

0.0

0.2

0.4

0.6

0.8

P(Vy)

×10−1

(e) 0.25 0.50 0.75 1.00Vx

−1.0

−0.5

0.0

0.5

Vy

×10−1

(f)

3.25 3.50 3.75 4.00Vx

0.0

0.2

0.4

0.6

0.8

P(Vx)

×10−1

(g) −1 0 1 2Vy ×10−3

0.0

0.6

1.2

1.8

2.4

3.0

P(Vy)

×10−2

(h) 3.25 3.50 3.75 4.00Vx

−1

0

1

2

Vy

×10−3

(i)

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

FIG. 12: Vortex velocities for the system in Fig. 11 with Fp = 0.5 and w/a = 1.767. Left column: P (Vx). Center column:P (Vy). Right column: Heightfield map of Vy versus Vx. (a,b,c) The plastic flow phase at Fd = 0.3 where there is a large peakin P (Vx) at Vx = 0 due to the pinned vortices. (d,e,f) The moving smectic phase from Fig. 11(c,d) at Fd = 0.6. (g,h,i) Themoving triangular solid phase at Fd = 3.6.

Vy versus Vx plot. Here the vortices have formed a float-ing triangular solid, and their motion is close to one-dimensional along the driving direction. As Fd is furtherincreased, the width of the lobe feature in the Vy direc-tion gradually decreases. We find similar histograms forthe other fillings in the strong pinning limit for the plasticflow, moving modulated solid, and moving floating solidregimes.

A. Smectic to Smectic Transitions

For Fp < 1.0, we find that a dynamically induced smec-tic to smectic transition can occur. In Fig. 11(a,b) weshow the real space and S(k) images for a system withFp = 0.5 and w/a = 1.767 in the plastic flow regimewhere there is a combination of vortices that are trappedin the pinning troughs and a smaller amount of vorticesthat hop by jumping from one trough to the next andthen triggering a jump of another vortex from one troughto the next. Here, S(k) indicates that the overall systemhas smectic ordering due to the chain-like structure ofthe vortices within the pinning troughs. In Fig. 12(a) weplot P (Vx) at Fd = 0.3, where there is a peak at Vx = 0due to the pinned vortices along with a small bump atVx = 0.6 due to the vortex hopping. Figure 12(b) shows

that P (Vy) has a maximum at Vy = 0, while in Fig. 12(c),the Vy versus Vx plot is asymmetric in Vy, with a peak atVx = Vy = 0.0 and a second peak at higher Vx producedby the moving vortices.

In Fig. 11(c) we show the real space vortex configu-ration at Fd = 0.6, which is higher than the maximumpinning force of Fp = 0.5. All the vortices are in mo-tion, but instead of retaining their alignment along they-direction induced by the substrate, they form a chain-like structure aligned in the x-direction with a slight tiltin the positive y-direction. This alignment in the drivedirection is more clearly seen in the corresponding S(k)in Fig. 11(d), where the peaks fall along kx = 0, indicat-ing a smectic phase with ordering along the y-direction.There are some very weak peaks on the ky = 0 axis due tothe substrate, but overall the vortex structure is a smec-tic state rotated 90◦ from the y-axis. The peaks do notfall exactly at kx = 0 but are at a slight angle, due to thechannels in Fig. 11(c) being slightly tilted in the positivey-direction. In Fig. 12(d), P (Vx) for this case shows apeak at Vx = 0.19, while there is an absence of weightin P (Vx) at Vx = 0.0, indicating that the vortices are al-ways in motion. Figure 12(e) shows that P (Vy) peaks atVy = 0.0 and has an overall asymmetry, which also ap-pears in the Vy versus Vx plot in Fig. 12(f). The vortexchanneling occurs when the vortices form effective pairs

12

1 2 3 4 5w/a

0

1

2

3

4

Fd

01234567

For

c

P Plastic

ML

MFS

1 2 3

(b)

(a)

FIG. 13: (a) F orc , the drive at which the system transitions

from a density modulated moving crystal to an ordered mov-ing floating solid, vs w/a at Fp = 0.5 (red circles), 1.0 (greensquares), 1.5 (orange diamonds), and 2.0 (black triangles).(b) Dynamic phase diagram as a function of Fd and w/a for asystem with Fp = 1.5. P: pinned phase. Plastic: plastic flowregime. ML: moving modulated lattice state. MFS: movingflowing solid state. Dashed lines are guides to the eye thatindicate the transition from r1 to r2, r2 to r3, and r3 to adisordered pinned state.

aligned along the x-direction. In each pair, one vortexis slowed by the backward-sloping side of the potentialtrough, while the other vortex is sped up by the forward-sloping side of the potential. The faster vortex pushesthe slower vortex, giving the pair an increased net mo-tion along the x direction. This pairing effect is visiblein the real space image in Fig. 11(c).

As Fd further increases, there is a transition to aflowing solid phase as shown in Fig. 11(e,f) for Fd =3.6, where S(k) has sixfold ordering. The correspond-ing P (Vx) in Fig. 12(g) has two peaks, while P (Vy)in Fig. 12(h) has a symmetrical distribution with threepeaks indicating that there is an oscillation in the vor-tex orbits in the y-direction. The plot of Vy versus Vx inFig. 12(h) contains a single lobe similar to that found forthe moving floating solid in Fig. 10(i). As Fd is increasedstill further, the width of this lobe in the Vy direction de-creases. The smectic-to-smectic transition is limited tothe range 1.5 < w/a < 2, in which two vortices can fitbetween adjacent potential maxima in the dynamicallymoving regime.

VI. DYNAMICAL PHASE DIAGRAM

For Fp > 0.25, the drive F orc at which the system tran-

sitions from a density modulated moving crystal to anordered moving floating solid shows considerable varia-tion with w/a, particularly for the larger values of Fp. InFig. 13(a) we plot F or

c , determined from the location of afeature in P6, versus w/a for Fp = 0.5, 1.0, 1.5, and 2.0.For Fp = 0.5, F or

c has a local maximum near w/a = 2.5,and then drops for w/a > 3.0. In the pinned phase forw/a > 3.0, the system forms a polycrystalline state, andin the moving state the grains realign to form a mov-ing crystalline state. For Fp = 1.0, 1.5, and 2.0, whenw/a < 1.75 the system depins from a single chain of vor-tices and can partially form a moving crystal state. Whenw/a is large enough that a pinned zig-zag state forms, themoving density-modulated state can persist up to muchhigher drives. For Fp = 1.0 the system forms a pinnedpolycrystalline state for w/a > 4.5 which coincides withthe drop in F or

c for w/a > 4.5. For Fp = 1.5 and 2.0,there are local peaks in F or

c that correlate with the mov-ing buckled phases which occur when groups of vorticescan fit between adjacent pinning maxima as the vorticesmove. This effect is most pronounced for Fp = 2.0.In Fig. 13(b) we plot a dynamic phase diagram as a

function of Fd and w/a for a system with Fp = 1.5.Above depinning in the regime where Fd < Fp, the sys-tem is in a plastic flow state in which there is a coexis-tence of moving vortices and immobile vortices. In thisregime the structure factor generally shows disorderedfeatures. For Fd > Fp, all the vortices are moving andthe system is either in a modulated lattice (ML) stateor a moving flowing solid (MFS) state. We find similardynamical phase diagrams for other values of Fp > 1.0,while for the weaker substrates, the size of the plasticflow region is reduced and the ML phase is replacedwith a smectic moving state similar to that shown inFig. 11(c,d).

VII. DISCUSSION

The dynamic phases we observe have certain similar-ities to the dynamic states observed for vortices movingover random pinning arrays in that there can be pinned,plastic, and dynamically ordered phases as a functionof external drive81–86. There are some differences, in-cluding the fact that the moving modulated lattice weobserve does not form a smectic state that is alignedin the drive direction, as found for vortices moving overrandom pinning arrays83–86. Future studies might con-sider combinations of random disorder with periodic dis-order, which would introduce a competition in the mov-ing phase between the smectic ordering imposed by thesubstrate and the smectic ordering induced by the drive.Additionally, for random pinning arrays simulations in-dicate that for increasing vortex density, the drive atwhich the transition to the ordered state occurs decreases

13

due to the increase in the strength of the vortex-vortexinteractions86. For the q1D substrate, the location ofthe ordering transition shows strong fluctuations due tothe ability of the moving lattice to become dynamicallycommensurate with the periodicity of the substrate.

VIII. SUMMARY

We examine the statics and dynamics of vortices inter-acting with a periodic quasi-one-dimensional substratein the limit where the vortex lattice spacing is smallerthan the spacing of the periodic lattice. For weak sub-strate strengths, we find that the vortices retain hexago-nal ordering but exhibit numerous rotations with respectto the substrate, similar to recent experimental observa-tions. For stronger substrates there are a series of buck-ling transitions where the vortices can form anisotropic1D chains, zig-zag patterns, and higher order numbers ofchains within each substrate minimum. At some fillingsthe overall lattice has long range order and becomes par-tially distorted at the transitions between these states.For higher fillings the buckling transitions are lost andthe system forms a polycrystalline state. We also findthat the depinning shows a series of step like features

when the system transitions from a state with n chainsto a state with n+ 1 chains in each substrate minimum,and that for weaker pinning there are some cases wherethere is a peak in the depinning force as a function of fill-ing. For weak substrates, under an applied drive the vor-tices depin elastically and retain their triangular order-ing, while for strong substrates the buckled states tran-sition to a partially disordered flowing state followed byvarious other transitions into moving modulated crystalor homogeneous floating moving crystal states. Our re-sults should also be applicable to other systems of par-ticles with repulsive interactions in the presence of a pe-riodic quasi-one dimensional substrate, such as electroncrystals, colloids, and ions in optical traps.

Acknowledgments

This work was carried out under the auspices of theNNSA of the U.S. DoE at LANL under Contract No.DE-AC52-06NA25396. The work of DM was supportedin part by the U.S. Department of Energy, Office of Sci-ence, Office of Workforce Development for Teachers andScientists (WDTS) under the Visiting Faculty Program(VFP).

1 P. Bak, Commensurate phases, incommensurate phasesand the devil’s staircase, Rep. Prog. Phys. 45, 587 (1982).

2 O.M. Braun and Y.S. Kivshar, The Frenkel-Kontorova

Model: Concepts, Methods, and Applications (Springer-Verlag, Berlin, 2004).

3 A.D. Novaco and J.P. McTague, Orientational epitaxy theorientational ordering of incommensurate structures, Phys.Rev. Lett. 38, 1286 (1977).

4 S.N. Coppersmith, D.S. Fisher, B.I. Halperin, P.A. Lee,and W.F. Brinkman, Dislocations and the commensurate-incommensurate transition in two dimensions, Phys. Rev.B 25, 349 (1982).

5 M. Peyrard and S. Aubry, Critical behaviour at the tran-sition by breaking of analyticity in the discrete Frenkel–Kontorova model, J. Phys. C: Solid State Phys. 16, 1593(1983).

6 S. Bleil, H.H. von Grunberg, J. Dobnikar, R. Castaneda-Priego, and C. Bechinger, Strain-induced domain forma-tion in two-dimensional colloidal systems, Europhys. Lett.73, 450 (2006).

7 T. Bohlein, J. Mikhael, and C. Bechinger, Observation ofkinks and antikinks in colloidal monolayers driven acrossordered surfaces, Nature Mater. 11, 126 (2012).

8 A. Vanossi, N. Manini, and E. Tosatti, Static and dynamicfriction in sliding colloidal monolayers, Proc. Natl. Acad.Sci. (USA) 109, 16429 (2012).

9 J. Tekic, O.M. Braun, and B.B. Hu, Dynamic phases in thetwo-dimensional underdamped driven Frenkel-Kontorovamodel, Phys. Rev. E 71, 026104 (2005).

10 A. Vanossi, N. Manini, M. Urbakh, S. Zapperi, and E.Tosatti, Modeling friction: From nanoscale to mesoscale,Rev. Mod. Phys. 85, 529 (2013).

11 D. Mandelli, A. Vanossi, N. Manini, and E. Tosatti, Fric-tion boosted by equilibrium misalignment of incommensu-rate two-dimensional colloid monolayers, Phys. Rev. Lett.114, 108302 (2015).

12 M. Baert, V.V. Metlushko, R. Jonckheere, V.V.Moshchalkov, and Y. Bruynseraede, Composite flux-linelattices stabilized in superconducting films by a regular ar-ray of artificial defects, Phys. Rev. Lett. 74, 3269 (1995).

13 K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tono-mura, and V.V. Moshchalkov, Direct observation of vortexdynamics in superconducting films with regular arrays ofdefects, Science 274, 1167 (1996).

14 J.I. Martın, M. Velez, J. Nogues, and I.K. Schuller, Fluxpinning in a superconductor by an array of submicrometermagnetic dots, Phys. Rev. Lett. 79, 1929 (1997).

15 D.J. Morgan and J.B. Ketterson, Asymmetric flux pinningin a regular array of magnetic dipoles, Phys. Rev. Lett. 80,3614 (1998).

16 C. Reichhardt, C. J. Olson, and F. Nori, Commensurateand incommensurate vortex states in superconductors withperiodic pinning arrays, Phys. Rev. B 57, 7937 (1998).

17 C. Reichhardt and C. J. Olson Reichhardt, Vortex molec-ular crystal and vortex plastic crystal states in honey-comb and kagome pinning arrays, Phys. Rev. B 58, 064523(2007).

18 U. Welp, Z.L. Xiao, V. Novosad, and V.K. Vlasko-Vlasov,Commensurability and strong vortex pinning in nanopat-terned Nb films, Phys. Rev. B 71, 014505 (2005).

19 G.R. Berdiyorov, M.V. Milosevic, and F.M. Peeters, Novelcommensurability effects in superconducting films with an-tidot arrays, Phys. Rev. Lett. 96, 207001 (2006).

20 S. Goldberg, Y. Segev, Y. Myasoedov, I. Gutman, N.

14

Avraham, M. Rappaport, E. Zeldov, T. Tamegai, C.W.Hicks, and K.A. Moler, Mott insulator phases and first-order melting in Bi2Sr2CaCu2O8+δ crystals with periodicsurface holes, Phys. Rev. B 79, 064523 (2009).

21 I. Swiecicki, C. Ulysse, T. Wolf, R. Bernard, N. Bergeal, J.Briatico, G. Faini, J. Lesueur, and J.E. Villegas, Strongfield-matching effects in superconducting YBa2Cu3O7δ

films with vortex energy landscapes engineered via maskedion irradiation, Phys. Rev. B 85, 224502 (2012).

22 T.C. Halsey, Josephson-junction arrays in transverse mag-netic fields: Ground states and critical currents, Phys. Rev.B 31, 5728 (1985).

23 H.D. Hallen, R. Seshadri, A.M. Chang, R.E. Miller, L.N.Pfeiffer, K.W. West, C.A. Murray, and H.F. Hess, Directspatial imaging of vortices in a superconducting wire net-work, Phys. Rev. Lett. 71, 3007 (1993).

24 J.W. Reijnders and R.A. Duine, Pinning of vortices in aBose-Einstein condensate by an optical lattice, Phys. Rev.Lett. 93, 060401 (2004).

25 H. Pu, L.O. Baksmaty, S. Yi, and N.P. Bigelow, Structuralphase transitions of vortex matter in an optical lattice,Phys. Rev. Lett. 94, 190401 (2005).

26 S. Tung, V. Schweikhard, and E. A. Cornell, Observationof vortex pinning in Bose-Einstein condensates, Phys. Rev.Lett. 97, 240402 (2006).

27 H.P. Buchler, G. Blatter, and W. Zwerger, Commensurate-incommensurate transition of cold atoms in an optical lat-tice, Phys. Rev. Lett. 90, 130401 (2003).

28 B. Granz, S.E. Korshunov, V.B. Geshkenbein, and G.Blatter, Competing structures in two-dimensional trappeddipolar gases, Phys. Rev. B 90, 060101 (2014).

29 A. Benassi, A. Vanossi, and E. Tosatti, Nanofriction incold ion traps, Nature Commun. 2, 236 (2011).

30 A. Bylinskii, D. Gangloff, and V. Vuletic, Tuning frictionatom-by-atom in an ion-crystal simulator, Science 348,6239 (2015).

31 K. Mangold, P. Leiderer, and C. Bechinger, Phase tran-sitions of colloidal monolayers in periodic pinning arrays,Phys. Rev. Lett. 90, 158302 (2003).

32 S. El Shawish, J. Dobnikar, and E. Trizac, Colloidal ioniccomplexes on periodic substrates: Ground-state configu-rations and pattern switching, Phys. Rev. E 83, 041403(2011).

33 D. McDermott, J. Amelang, L.M. Lopatina, C.J. Olson Re-ichhardt, and C. Reichhardt, Domain and stripe formationbetween hexagonal and square ordered fillings of colloidson periodic pinning substrates, Soft Matter 9, 4607 (2013).

34 T. Neuhaus, M. Marechal, M. Schmiedeberg, and H.Lowen, Rhombic preordering on a square substrate, Phys.Rev. Lett. 110, 118301 (2013).

35 J. Mikhael, J. Roth, L. Helden, and C. Bechinger,Archimedean-like tiling on decagonal quasicrystalline sur-faces, Nature 454, 501 (2008).

36 J. Mikhael, G. Gera, T. Bohlein, and C. Bechinger, Phasebehavior of colloidal monolayers on quasiperiodic sub-strates, Soft Matter 7, 1352 (2011).

37 S. Field, S. James, J. Barentine, V. Metlushko, G. Crab-tree, H. Shtrikman, B. Ilic, and S. Brueck, Vortex config-urations, matching, and domain structure in large arraysof artificial pinning centers, Phys. Rev. Lett. 88, 067003(2002).

38 A.N. Grigorenko, S.J. Bending, M.J. Van Bael, M. Lange,V.V. Moshchalkov, H. Fangohr, and P.A.J. de Groot, Sym-metry locking and commensurate vortex domain formation

in periodic pinning arrays, Phys. Rev. Lett. 90, 237001(2003).

39 M. Baert, V.V. Metlushko, R. Jonckheere, V.V.Moshchalkov, and Y. Bruynseraede, Flux phases and quan-tized pinning force in superconductor with a periodic lat-tice of pinning centres, Europhys. Lett. 29, 157 (1995).

40 C. Reichhardt and N. Grønbech-Jensen, Critical currentsand vortex states at fractional matching fields in supercon-ductors with periodic pinning, Phys. Rev. B 63, 054510(2001).

41 D. McDermott, J. Amelang, C.J. Olson Reichhardt, and C.Reichhardt, Dynamic regimes for driven colloidal particleson a periodic substrate at commensurate and incommes-nurate fillings, Phys. Rev. E 88, 062301 (2013).

42 A. Chowdhury, B.J. Ackerson, and N.A. Clark, Laser-induced freezing, Phys. Rev. Lett. 55, 833 (1985).

43 J. Chakrabarti, H.R. Krishnamurthy, A.K. Sood, and S.Sengupta, Reentrant melting in laser field modulated col-loidal suspensions, Phys. Rev. Lett. 75, 2232 (1995).

44 J. Hu and R.M. Westervelt, Commensurate-incommensurate transitions in magnetic bubble arrayswith periodic line pinning, Phys. Rev. B 55, 771 (1997).

45 Q.-H. Wei, C. Bechinger, D. Rudhardt, and P. Lei-derer, Experimental study of laser-induced melting in two-dimensional colloids, Phys. Rev. Lett. 81, 2606 (1998).

46 E. Frey, D. Nelson, and L. Radzihovsky, Light-inducedmelting of colloidal crystals in two dimensions, Phys. Rev.Lett. 83, 2977 (1999).

47 C. Richter, M. Schmiedeberg, and H. Stark, A colloidalmodel system with tunable disorder: Solid-fluid transitionand discontinuities in the limit of zero disorder, Eur. Phys.J. E 34, 107 (2011).

48 L. Zaidouny, T. Bohlein, R. Roth, and C. Bechinger, Light-induced phase transitions of colloidal monolayers withcrystalline order, Soft Matter 9, 9230 (2013).

49 O. Daldini, P. Martinoli, J. L. Olsen, and G. Berner,Vortex-line pinning by thickness modulation of supercon-ducting films, Phys. Rev. Lett. 32, 218 (1974).

50 P. Martinoli, O. Daldini, C. Leemann, and E. Stocker, A.C.quantum interference in superconducting films with peri-odically modulated thickness, Solid State Commun. 17,205 (1975).

51 P. Martinoli, Static and dynamic interaction of supercon-ducting vortices with a periodic pinning potential, Phys.Rev. B 17, 1175 (1978).

52 O.V. Dobrovolskiy, E. Begun, M. Huth, and V.A.Shklovskij, Electrical transport and pinning properties ofNb thin films patterned with focused ion beam-milledwashboard nanostructures, New J. Phys. 14, 113027(2012).

53 O.V. Dobrovolskiy and M. Huth, Dual cut-off directcurrent-tunable microwave low-pass filter on superconduct-ing Nb microstrips with asymmetric nanogrooves, Appl.Phys. Lett. 106, 142601 (2015).

54 D. Jaque1, E.M. Gonzalez, J.I. Martin, J.V. Anguita, andJ. L. Vicent, Anisotropic pinning enhancement in Nb filmswith arrays of submicrometric Ni lines, Appl. Phys. Lett.81, 2851 (2002).

55 O.V. Dobrovolskiy, AC quantum interference effects innanopatterned Nb microstrips, J. Supercond. Novel Mag.28, 469 (2015).

56 J. Guimpel, L. Civale, F. de la Cruz, J. M. Murduck, andI.K. Schuller, Dimensional phase transition in supercon-ductors with short coherence length, Phys. Rev. B 38, 2342

15

(1988).57 S.H. Brongersma, E. Verweij, N.J. Koeman, D.G. de

Groot, R. Griessen, and B.I. Ivlev, Phys. Rev. Lett. 71,2319 (1993).

58 L.S. Levitov, Phyllotaxis of flux lattices in layered super-conductors, Phys. Rev. Lett. 66, 224 (1991).

59 L. Balents and D.R. Nelson, Quantum smectic and super-solid order in helium films and vortex arrays, Phys. Rev.B 52, 12951 (1995).

60 N. Fogel, M. Mikhailov, Y. Bomze, and O. Yuzephovich,Intrinsic pinning, commensurability, and reentrant behav-ior on superconducting Mo/Si multilayers, Phys. Rev. B59, 3365 (1999).

61 S. Baily, B. Maiorov, H. Zhou, F. Balakirev, M. Jaime,S. Foltyn, and L. Civale, Smectic vortex phase in opti-mally doped YBa2Cu3O7 thin films, Phys. Rev. Lett. 100,027004 (2008).

62 P.J.W. Moll, L. Balicas, X. Zhu, H.-H. Wen, N.D. Zhi-gadlo, J. Karpinski, and B. Batlogg, Critical current oscil-lations in the intrinsic hybrid vortex state of SmFeAs(O,F),Phys. Rev. Lett. 113, 186402 (2014).

63 G. Carneiro, Equilibrium vortex-line configurations andcritical currents in thin films under a parallel field, Phys.Rev. B 57, 6077 (1998).

64 J.J. Barba and J. Albino Aguiar, Bi-dimensional chain-likevortex structure in a mesoscopic superconductor, J. Phys.:Conf. Ser. 150, 052015 (2009).

65 G. Karapetrov, M. Milosevic, M. Iavarone, J. Fedor, A.Belkin, V. Novosad, and F. Peeters, Transverse insta-bilities of multiple vortex chains in magnetically cou-pled NbSe2/permalloy superconductor/ferromagnet bilay-ers, Phys. Rev. B 80, 180506 (2009).

66 G. Piacente, I.V. Schweigert, J.J. Betouras, and F.M.Peeters, Generic properties of a quasi-one-dimensionalclassical Wigner crystal, Phys. Rev. B 69, 045324 (2004).

67 J.E. Galvan-Moya and F.M. Peeters, Ginzburg-Landautheory of the zigzag transition in quasi-one-dimensionalclassical Wigner crystals, Phys. Rev. B 84, 134106 (2011).

68 J.E. Galvan-Moya, V.R. Misko, and F.M. Peeters, Genericordering of structural transitions in quasi-one-dimensionalWigner crystals, Phys. Rev. B 90, 094111 (2014).

69 W. Ferreira, J. Carvalho, P. Oliveira, G. Farias, and F.Peeters, Structural and dynamical properties of a quasi-one-dimensional classical binary system, Phys. Rev. B 77,014112 (2008).

70 D. Lucena, W.P. Ferreira, F.F. Munarin, G.A. Farias, andF.M. Peeters, Tunable diffusion of magnetic particles in aquasi-one-dimensional channel, Phys. Rev. E 87, 012307(2013).

71 A.V. Straube, R.P.A. Dullens, L. Schimansky-Geier, andA.A. Louis, Zigzag transitions and nonequilibrium patternformation in colloidal chains, J. Chem. Phys. 139, 134908(2013).

72 J.E. Galvan-Moya, D. Lucena, W.P. Ferreira, and F.M.Peeters, Magnetic particles confined in a modulated chan-nel: Structural transitions tunable by tilting a magneticfield, Phys. Rev. E 89, 032309 (2014).

73 B. Liu and J. Goree, Phonons in a one-dimensional yukawachain: Dusty plasma experiment and model, Phys. Rev. E71, 046410 (2005).

74 T.E. Sheridan and A.L. Magyar, Power law behavior ofthe zigzag transition in Yukawa clusters, Phys. Plasmas17, 113703 (2010).

75 S. Fishman, G. De Chiara, T. Calarco, and G. Morigi,Structural phase transitions in low-dimensional ion crys-tals, Phys. Rev. B 77, 064111 (2008).

76 A. del Campo, G. De Chiara, G. Morigi, M.B. Plenio, andA. Retzker, Structural defects in ion chains by quenchingthe external potential: the inhomogeneous Kibble-Zurekmechanism, Phys. Rev. Lett. 105, 075701 (2010).

77 M. Mielenz, J. Brox, S. Kahra, G. Leschhorn, M. Albert, T.Schaetz, H. Landa, and B. Reznik, Trapping of topological-structural defects in Coulomb crystals, Phys. Rev. Lett.110, 133004 (2013).

78 J.-B. Delfau, C. Coste, and M. Saint Jean, Transversesingle-file diffusion near the zigzag transition, Phys. Rev.E 87, 032163 (2013).

79 T. Dessup, T. Maimbourg, C. Coste, and M. Saint Jean,Linear instability of a zigzag pattern, Phys. Rev. E 91,022908 (2015)

80 I. Guillamon, R. Cordoba, J. Sese, J.M. De Teresa, M.R.Ibarra, S. Vieira, and H. Suderow, Enhancement of long-range correlations in a 2D vortex lattice by an incommen-surate 1D disorder potential, Nature Phys. 10, 851 (2014).

81 S. Bhattacharya and M.J. Higgins, Dynamics of a disor-dered flux line lattice, Phys. Rev. Lett. 70, 2617 (1993).

82 A.E. Koshelev and V.M. Vinokur, Dynamic melting of thevortex lattice, Phys. Rev. Lett. 73, 3580 (1994).

83 L. Balents, M.C. Marchetti, and L. Radzihovsky, Nonequi-librium steady states of driven periodic media, Phys. Rev.B 57, 7705 (1998).

84 P. Le Doussal and T. Giamarchi, Moving glass theoryof driven lattices with disorder, Phys. Rev. B 57, 11356(1998).

85 F. Pardo, F. de la Cruz, P.L. Gammel, E. Bucher, andD.J. Bishop, Observation of smectic and moving-Bragg-glass phases in flowing vortex lattices, Nature (London)396, 348 (1998).

86 C.J. Olson, C. Reichhardt, and F. Nori, Nonequilibriumdynamic phase diagram for vortex lattices, Phys. Rev.Lett. 81, 3757 (1998).

87 C. Reichhardt, C.J. Olson, and F. Nori, Dynamic phasesof vortices in superconductors with periodic pinning, Phys.Rev. Lett. 78, 2648 (1997).

88 C. Reichhardt, C.J. Olson, and F. Nori, Nonequilibriumdynamic phases and plastic flow of driven vortex latticesin superconductors with periodic arrays of pinning sites,Phys. Rev. B 58, 6534 (1998).

89 J. Gutierrez, A.V. Silhanek, J. Van de Vondel, W. Gillijns,and V. Moshchalkov, Transition from turbulent to nearlylaminar vortex flow in superconductors with periodic pin-ning, Phys. Rev. B 80, 140514 (2009).