Macroscopic Floquet topological crystalline steel pump
MMacroscopic Floquet topological crystalline steel pump
Anna M. E. B. Rossi a , Jonas Bugase a and Thomas M. Fischer a ∗ a Experimental Physics, Institute of Physics and Mathematics,Universit¨at Bayreuth, 95440 Bayreuth (Germany). (Dated: October 10, 2018)The transport of a steel sphere on top of two dimensional periodic magnetic patterns is studiedexperimentally. Transport of the sphere is achieved by moving an external permanent magnet on aclosed loop around the two dimensional crystal. The transport is topological i.e. the steel sphere istransported by a primitive unit vector of the lattice when the external magnet loop winds aroundspecific directions. We experimentally determine the set of directions the loops must enclose fornontrivial transport of the steel sphere into various directions.
I. INTRODUCTION
Topological non trivial matter is a class of material,where the response of the material to external pertur-bations only depends on global properties not on localproperties of the material. Such properties are calledtopological invariants and they change in a discrete way,i.e. a continuous change of the perturbation results ina discrete response of the material. Topological prop-erties of matter play a fundamental role in electronictransport behavior of quantum solid state matter [1, 2],in mesoscopic systems [3–8] and in macroscopic matter[9–12]. One important class of topological material areFloquet topological systems, where the material is sub-ject to a time periodic external perturbation that causesthe pumping of excitations or quasi particles through thematerial. The topological pump effect [13] is usually pro-tected by certain symmetries of the problem. One of suchsymmetries are the point group symmetries of the lattice.The current work presents three macroscopic exam-ples of topological magnetic crystals, with magneticpoint symmetry protected Floquet transport propertiesof paramagnetic (soft magnetic) spheres placed above thecrystal. We experimentally determine regions of orienta-tion we have to wind around an external magnetic fieldto pump the spheres into certain directions.
II. STEEL PUMP SETUP
Three topological two dimensional magnetic crystalsare built from an arrangement of NbB-magnets. Thefirst crystal consists of magnetic cubes of side length d = 2 mm and remanence µ M = 1 . T arranged in afour fold symmetric C checkerboard square lattice of al-ternating up and down magnetized cubes (Fig. 1a). Thesecond, a hexagonal lattice (Fig. 1b) consists of cylin-drical magnets of diameters d = 3 mm and d = 2 mm ,height h = 2 mm and remanence µ M = 1 . T and µ M = 1 . T (respectively). The larger size d = ∗ Electronic address: thomas.fi[email protected] mm magnets are magnetized upwards and they are sur-rounded by six smaller size d = 2 mm magnets that aremagnetized downwards and that touch the larger magnet.The primitive unit cell of the lattice is a six fold symmet-ric C hexagon with corners centered within the smallermagnets. Each unit cell thus contains one large magnetand two smaller magnets. The third lattice (Fig. 1c)is built from a hexagonally closed packed arrangementof d = 3 mm diameter cylinders. The central cylinder(blue) is non magnetic brass, and the surrounding cylin-ders are NbB magnets of alternating up and downwardmagnetization creating an improper six fold S symme-try. All two dimensional lattices have two primitive lat-tice vectors of the same length a = a = a = 2 . mm (square lattice), a = 2 . mm ( C lattice), and a = 5 . mm ( S lattice) and are metastable (the ground state con-figuration of the magnet ensemble is a magnetic rod ofmagnets aligned along one axis) in zero external mag-netic field. We fix the arrangement with an epoxy resinplaced in the voids and the two dimensional surround-ings of the pattern. The pattern then is stable also inthe presence of an external field. The crystals are put ona support and covered with a transparent PMMA spacerof thickness z = 1 − . mm (Fig. 1d). We place a steelsphere of diameter 2 r = 1 mm on top of the spacer andcreate a closed but transparent compartment around thesteel sphere. The topological magnetic crystal with thesteel sphere on top is placed in the center of a goniome-ter set up at an angle of 45 degrees to ensure that rele-vant motion is not effected by the restrictions of motionof the goniometer (Fig. 1e and f) caused by the sup-port. The goniometer holds two NbB-magnets of diame-ter d ext = 60 mm , thickness t ext = 10 mm and remanence µ M ext = 1 . T aligned parallel to each other at a dis-tance 2 R = 120 mm and creating an external magneticfield H ext = 3600 A/m penetrating the two dimensionalcrystal and the steel sphere. The magnetic field gradi-ents ∇ H ext ≈ M ext t ext d ext /R of the external field atthe position of the steel sphere is at least two orders ofmagnitude smaller than the field gradients of the mag-netic field of the crystal ∇ H int ≈ M/a . The two externalmagnets can be oriented to produce an arbitrary direc-tion of the external magnetic field with respect to thecrystal. A laser pointer pointing along H ext is mountedon the goniometer creating a stereographic projection of a r X i v : . [ c ond - m a t . o t h e r] J u l FIG. 1: a) Top view of the magnetic pattern of symmetry C , sample left and scheme right b) symmetry C and c) symmetry S . Silver areas in the sample (red areas in the scheme) are magnetized up and black (green) areas are magnetized downrespectively. Blue areas are non magnetic brass cylinders inserted for mechanical stability. One unit cell is emphasized in fullcolors. The vector Q is one of the primitive reciprocal lattice vectors. d) Sideview of the pattern and the compartment holdingthe steel sphere. e) Goniometer and external magnets surrounding the sample. f) A photo of the setup. the instantaneous external magnetic field direction on arecording plane. III. TOPOLOGICALLY NONTRIVIALTRANSPORT LOOPS
We reorient the external magnets by moving along aclosed reorientation loop that starts and ends at the sameinitial orientation. The steel sphere responds to the re-orientation loop with a motion that starts at one positionof the lattice and ends at a final position. A topologicaltrivial motion of the steel sphere is a motion where thesteel sphere responds to a closed reorientation loop witha closed loop on the lattice. Not every closed reorien-tation loop causes a trivial response of the steel sphere.There are topologically non trivial trajectories, where thesteel sphere trajectory ends at a position differing fromthe initial position by one vector of the lattice.We choose a collection of different non self intersectingreorientation loops and measured the corresponding dis-placement of the steel sphere. Each non self intersectingreorientation loop cuts the sphere of orientations that wecall the control space into two areas. One of the areasis circulated by the loop in the positive sense the otherin the negative sense. We define the intersection of allpositive areas of loops causing the same net transportof the steel sphere as the positive common area. Simi-larly we can define the negative common area of the sametransport directions.In Fig. 2a we show the common area (blue) determinedin this way for the transport into the n × Q -direction forthe fourfold symmetric pattern. The common area is a rectangle centered around the primitive reciprocal vector − Q of the lattice. Whenever we wind the modulationloop around the common area in a way that does nottouch the area the result is the same non-trivial trans-port. Entering the common area leads to a statisticaltrivial or non-trivial response transport direction of thesteel sphere. The sphere passes from the up-magnetizedregion toward the down-magnetized regions or vice versawhen the loop crosses the gates (red circles). In the ex-periments we observe a hysteresis, i.e. the gate in controlspace is positioned at the red circles of the southern hemi-sphere when the external field moves from north towardthe south and the northern hemisphere for the oppositedirection. The region of the hysteresis is shown as thered area in control space. Note that similar common ar-eas repeat every 2 π/ C -symmetry. In previous work [8] we have computedthe theoretical position of the common area as well asthe position of the gates. Theoretically the common areais just one point, the − Q -direction, and the gate is a(red) line on the equator showing no hysteresis.In Fig. 2b we show the control space of the S -symmetric patterns. Non-trivial transport into the n × Q -direction occurs if we wind the loop around theyellow common area. We call the borders of the yellowarea the fence. A new feature of the S -symmetric pat-tern is that the transport is still predictable if we enterand exit the yellow area with a loop through fence seg-ments marked in red. A loop exiting the common area inthe north (south) has the same result as a common areaavoiding loop with the same winding number around the B n ( B s ) point. Although the transport direction of thosecommon area passing loops are the same as the common FIG. 2: Control spaces of a) the C -symmetric, b) the S -symmetric, and c) the C -symmetric lattice. The − Q -direction ofcontrol space is opposite to the reciprocal lattice vector direction Q in d). a) theoretical fence points are shown in blue, gatesas red lines. The fence points that one must wind around to achieve non-trivial transport enlarges to the fence area (blue) inthe experiment. Gates are shown as red circles and show a hystereses (red area) when winding around the fence area in differentdirections. We depict a purple loop encircling the blue area as an example loop inducing non-trivial colloidal transport into the n × Q -direction. b) theoretical fences for paramagnets are shown in green and for diamagnets in blue. The experimentallydetermined fence for the steel sphere lies further outside with two separate regions of instability when leaving the yellow areatoward the north or south. We depict a palindrome modulation loop in purple that cycles through the common area back andforth in control space but causes an open trajectory with ratchet jumps of the steel sphere above the lattice. For the part ofthe loop moving in the mathematical positive sense the winding number around B n is non-zero and causes nontrivial transport,while for the same loop traveling in the mathematical negative sense the winding number around B s is zero and causes trivialmotion c) theoretical fences as green and blue lines with the experimental fence shown as pink and red crosses. The purpleexample loop causes adiabatic transport in the ( Q − Q )-direction. d-f) Trajectories (blue and red) of the steel sphere subjectto the purple loops shown in a-c. Red segments correspond to faster motion than blue segments. The adiabatic motion d)and f) smoothly changes from fast to slower, while the ratchet motion e) discontinuously changes from slow adiabatic to fastratchet jumps. The yellow arrow corresponds to the primitive unit vector pointing into the transport direction. Movies of themotion are provided in [14]. area avoiding loops, their character is that of a ratchet.A jump of the steel sphere from one point on the lat-tice to a different point occurs when we exit the commonarea. Loops avoiding the common area cause a smoothquasi adiabatic transport of the steel sphere. Entering orexiting the common area in regions where there is no redfence segments yields statistical results for the steel trans-port direction. Two further common areas exist at thelocation turned by ± π/ π/ C -symmetric pattern. The transport is adiabatic if we en-circle an even number of B − points, i.e. when enteringand exiting segments that have the same pink or red(bright or dark green) color, and of ratchet characterotherwise. Non trivial transport into the σ ( Q i − Q j )-direction ( σ = ± i, j = 1 , ,
3) occurs when the modu-lation loop enters the yellow area via a neighbor segmentof the reciprocal unit vector σ Q i and exits via a nearestor next nearest neighbor segment of the reciprocal latticevector σ Q j . The sign σ of the nearest or next nearest exitreciprocal vector σ Q j must be the same as that of near-est reciprocal vector σ Q i of the entry. The experimentalposition of the fence (pink and red) has been determinedfrom the irreversible jumps of the steel sphere when theexternal field exits the yellow area near a reciprocal lat-tice vector having opposite sign to the reciprocal latticevector of the entry. The match between experiment andtheory here is almost perfect.In Fig. 2d we show the trajectory of the steel spheresubject to a purple loop in Fig. 2a encircling the recip-rocal vector − Q in the mathematical positive sense. Amovie of the motion can be found in [14]. In Fig. 2ewe show the trajectory of a palindrome modulation loopfor the S -symmetric lattice shown in Fig. 2b. The loopcrosses the common area by entering the left southernfence segment and exiting at the northern right segmentand returning to the initial orientation left of the com-mon area. Immediately afterwards the palindrome loopretraces the path in the opposite direction. The trajec-tory of the steel sphere is of the ratchet type and doesnot close because of irreversible jumps that happen whenthe modulation loop leaves the common area at differentfence segments during the forward and backward period.Finally in Fig. 2f we depict the adiabatic trajectory ofthe steel sphere above a C -symmetric lattice for a looppassing through the yellow area via the red fence seg-ments in Fig 2c. IV. DISCUSSION AND CONCLUSION
From the measurements we see that the experimentsare in topological agreement with the theory [6–8]. Themost striking difference between experiment and theoryis the existence of a hysteresis visible in the C - and S - symmetric patterns. The theory assumes that particleswill jump from an instable position toward the new equi-librium position at the fence, where the curvature of thepotential minimum vanishes. We explain the shift of theexperimental fence with respect to the theoretical predic-tions as well as the hysteresis by solid friction that letsthe steel particle move only when the magnetic potentialexceeds a certain slope. Slopes for a forward and back-ward jump will have opposite sign explaining the splittingof the closed theoretical fence in the S -pattern into twoseparate forward and backward loop fences.Let us note that the topological protected transporttheory has been developed for colloidal particles not forsteel spheres. The scale invariance of the theory demon-strates the robustness of the topological concept. Pre-sumably it is also possible to down scale the experimentfrom the colloidal toward molecular scales, which wouldprovide a transport mechanism for molecular magnetsabove magnetic nano structures. V. ACKNOWLEDGMENTS
J. B acknowledges financial support by a Ghana MOE- DAAD joined fellowship. We appreciate scientific sup-port by Johannes Loehr and by Daniel de las Heras. [1] M. Z. Hasan, and C. L. Kane; Colloquium: Topologicalinsulators.
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