Accumulation of chiral hinge modes and its interplay with Weyl physics in a three-dimensional periodically driven lattice system
Biao Huang, Viktor Novi?enko, André Eckardt, Gediminas Juzeliūnas
AAccumulation of chiral hinge modes and its interplay with Weyl physics in a three-dimensionalperiodically driven lattice system
Biao Huang, ∗ Viktor Noviˇcenko, † Andr´e Eckardt,
1, 3, ‡ and Gediminas Juzeli¯unas § Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Straße 38, 01069 Dresden, Germany Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio 3, LT-10257 Vilnius, Lithuania Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany (Dated: January 25, 2021)We demonstrate that a three dimensional time-periodically driven lattice system can exhibit a second-orderchiral skin e ff ect and describe its interplay with Weyl physics. This Floquet skin-e ff ect manifests itself, whenconsidering open rather than periodic boundary conditions for the system. Then an extensive number of bulkmodes is transformed into chiral modes that are bound to the hinges (being second-order boundaries) of oursystem, while other bulk modes form Fermi arc surface states connecting a pair of Weyl points. At a fine tunedpoint, eventually all boundary states become hinge modes and the Weyl points disappear. The accumulationof an extensive number of modes at the hinges of the system resembles the non-Hermitian skin e ff ect, withone noticeable di ff erence being the localization of the Floquet hinge modes at increasing distances from thehinges in our system. We intuitively explain the emergence of hinge modes in terms of repeated backreflectionsbetween two hinge-sharing faces and relate their chiral transport properties to chiral Goos-H¨anchen-like shiftsassociated with these reflections. Moreover, we formulate a topological theory of the second-order Floquetskin e ff ect based on the quasi-energy winding around the Floquet-Brillouin zone for the family of hinge states.The implementation of a model featuring both the second-order Floquet skin e ff ect and the Weyl physics isstraightforward with ultracold atoms in optical superlattices. I. INTRODUCTION
In recent years, researches have demonstrated that time-periodically driven systems can show intriguing and uniquee ff ects that find no counterparts in non-driven systems. Exam-ples include anomalous Floquet topological insulators featur-ing robust chiral edge modes for vanishing Chern numbers [1–11] and discrete time crystals [12–23]. The periodic drivingshifts the fundamental theoretical framework from focusingon Hamiltonian eigen problems to the unitary evolution oper-ators genuinely depending on time, resulting in a plethora ofnew concepts and methods such as spacetime winding num-bers [2] and spectral pairing [24]. In this context, it appearsnatural and tantalizing to explore possible new classes of pe-riodically driven systems that go beyond descriptions by tra-ditional theories.In this paper, we show that time-periodic driving of a threedimensional (3D) lattice can give rise to the coexistence ofWeyl physics [25–33] with a new type of hinge (i.e. second-order boundary) states. These states are chiral in the sense thatthey transport particles in a unidirectional fashion along thehinge. As an intriguing e ff ect, we find a macroscopic accumu-lation of these chiral Floquet hinge states, which is associatedwith a complete reorganization of the system’s quasienergyspectrum in response to shifting from periodic to open bound-ary conditions. At a fine-tuned point, even all states of thesystem become hinge states. Tuning away from that pointa pair of Weyl points is created at quasienergy π , leading to ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Fermi-arc surface (i.e. first-order boundary) states that coex-ist with the hinge modes. The localization of an extensivenumber of hinge modes at the boundaries of the system re-sembles the non-Hermitian skin e ff ect [34–37], with a notabledi ff erence being the localization of the modes at increasingdistances from hinges (higher-order boundaries) in such a waythat the hinge modes cover the whole lattice. Di ff erent fromthe case of higher-order topological phases [38–40], the hingestates are buried deeply inside the bulk spectrum. Their exis-tence and robustness is, therefore, not captured by the theoryof higher-order topological insulators / semimetals which relyon the existence of bulk energy gaps. Instead, the chiral hingemodes can be understood as resulting from the repeated back-reflection from two hinge-sharing surface planes and their chi-ral motion as the result of chiral Goos-H¨anchen-like shift as-sociated with the reflection at a boundary face. Furthermore,di ff erent from one-dimensional (1D) periodically driven lat-tices [41], the modes with opposite chirality residing at oppo-site hinges are well spatially separated, so the scattering dueto a local perturbation, such as a local disorder, does not a ff ectthe transport chirality at individual hinges.The model system proposed and studied here consists ofa simple, stepwise modulation of tunnelling matrix elementsinvolving six steps. It generalizes to three dimensions a two-dimensional lattice model introduced by Rudner et. al. [2]for studying anomalous Floquet topolgical insulators (see alsoRef. [1]). The latter 2D tunnel modulation has been recentlyapplied to ultracold atoms for the realization of anomaloustopological band structures [11]. Our 3D model can equallybe implemented with ultracold atoms using such a stepwisetunnel modulation, now in 3D optical superlattices. Further-more, besides giving rise to a new phenomenon, the uncon-ventional chiral second-order Floquet skin e ff ect, the modelproposed here also provides a simple recipe for the robust im-plementation of Weyl physics by means of time-periodic driv- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n ing, which should be easier to realize compared to previousproposals [30].This paper is structured as follows. In the next Section IIa 3D periodically driven lattice is defined. Subsequentlythe characteristic features of the bulk and hinge physics areconsidered in Secs. III and IV. In particular, in Sec IV C atopological theory based on the quasi-energy winding aroundthe frequency-Brillouin zone is formulated for the familyof localized hinge states enforced by reflections from open-boundaries. The experimental implementation of our model,using ultracold atoms in modulated superlattices, is discussedin Sec. V, before the Concluding Section VI. Some technicaldetails are presented in three Appendices A, B and C. (a) Driving (b) Bulk dynamics(c) Reflection by open boundary surfaces at x = y = B → A and A → B , respectively. (d) HingedynamicsFIG. 1. (a) Bonds connected during the driving steps 1 to 6. (b)
Bulktrajectories within a Floquet cycle at the fine tunned point φ = π/ d = (1 , − , (c) Trajectories atthe same φ = π/ A (or B ) near the y = x = (d) The hinge formed by twointersecting terminating surfaces renders uni-directional modes. Thefigure shows two of the modes closest to the hinge starting at a siteof B (lower plot) or A (upper plot) sublattices directly at the hinge.Each color for the arrow denotes one Floquet cycle. II. MODEL
We consider a bipartite cubic lattice with alternating A-Bsublattices in all three Cartesian directions. The lattice is de-scribed by a time-periodic Hamiltonian H ( t + T ) = H ( t ), with the driving period T divided into 6 steps. In each step tun-nelling matrix elements − J between sites r A of sublattice A and neighboring sites r A ± a e µ of sublattice B are switchedon, with µ = x , y , z . During the driving period T the tunnel-ing steps appear in a sequence µ ± = x + , y + , z + , x − , y − , z − ,as illustrated in Fig. 1(a) [42]. Within each step the evolu-tion is determined by a coordinate-space Hamiltonian H ± µ = − J (cid:80) r A ( | r A (cid:105)(cid:104) r A ± a e µ | + | r A ± a e µ (cid:105)(cid:104) r A | ), where J is the tun-neling matrix element, r A specifies the location of sublattices A , and a is the lattice spacing such that r A ± a e µ denotes thelocations of sites in sublattice B neighboring to the sublattice A site r A . The tunnelling processes occurring in each of thedriving steps are characterized by a single dimensionless pa-rameter, the phase φ = − JT (cid:126) . (1)The one-cycle evolution operator (or Floquet operator), U F = T e − ( i / (cid:126) ) (cid:82) T dtH ( t ) , (2)whose repeated application describes the time-evolution instroboscopic steps of the driving period T , can be decomposedinto terms corresponding to the six driving stages, U F = U z − U y − U x − U z + U y + U x + . (3)When dealing with the bulk dynamics we impose periodicboundary conditions in all three spatial directions. The evo-lution operators for the individual driving steps can then berepresented as: U µ ± = U (cid:16) ± k µ (cid:17) = e − i H µ ± = e − i φ ( τ cos k µ ± τ sin k µ ) , (4)where τ , , are Pauli matrices associated with the sublatticestates A and B and where k µ with µ = x , y , z denotes the Carte-sian components of the quasimomentum vector k . Here and inthe following, we will use a dimensionless description, wheretime, energy, length and quasimomentum are given in units of T , (cid:126) / T , a , and (cid:126) / a , respectively. The quasienergies E n , k andthe Floquet modes | n , k (cid:105) are defined via the eigenvalue equa-tion U F | u n , k (cid:105) = exp( − iE n , k ) | u n , k (cid:105) . (5)We first note that the only global symmetry satisfied bythe Floquet operator (3)-(4) is a particle-hole flip Γ = CK ,where Ki = − iK is complex conjugation and C = τ the thirdPauli matrix. Thus, the system belongs to class D in Altland-Zirnbauer notation [43, 44]. The Floquet operator satisfies CU F ( k ) C − = U ∗ F ( − k ) [4, 45], and therefore the quasiener-gies must appear in pairs E , k = − E , − k . Meanwhile, thesystem obeys the inversion symmetry PU F ( k ) P − = U F ( − k ),with P = τ , enforcing that for each band one has E n , k = E n , − k . Together, we see that the Floquet spectrum has pairsof states with quasi-energies E , k = − E , k . This means possi-ble gaps or nodal points / lines can, modulo 2 π , appear only atquasienergy 0 or π .At k x = k y = k z = ± π (mod π ) Eqs. (3) and (4) yield U F =
1, so that E n , k =
0. Therefore the quasienergy spec-trum is always gapless at quasienergy 0 (modulo 2 π ), regard-less of the driving strength φ . Thus a single band spectrumcould only possibly open up a gap at quasienergy equal to π (modulo 2 π ). To draw a complete phase diagram, we first notethat flipping the sign of φ amounts to a particle-hole transfor-mation U F | − φ = CU F | φ C − , and from the previous analysiswe see that such a flip does not change the spectrum. Fur-thermore, from e − i φ ˆ n · τ = cos φ − i ˆ n · τ sin φ , the periodicity ofthe Floquet operator with respect to the parameter φ is clearlyseen: U F | φ = U F | π + φ . In this way, the irreducible parameterrange is φ ∈ [0 , π/
2] as illustrated in Fig. 2.
III. PHASE DIAGRAM FOR PERIODIC BOUNDARYCONDITIONS
Before investigating the system with open boundary con-ditions and the emergence of chiral hinge modes, let us firstdiscuss the phase diagram considering the case of a transla-tion invariant system with periodic boundary conditions. Letus begin with the topologically trivial high-frequency (weakdriving) limit corresponding to φ (cid:28)
1. In that case one canretain only the lowest order terms in φ when expanding thestroboscopic evolution operator U F of Eq. (3), resulting in U F | φ → (cid:39) cos φτ − i sin φ cos φ (cos k x + cos k y + cos k z ) τ + O (sin φ ) (cid:39) e − i φ k x + cos k y + cos k z ) τ , where τ is the identitymatrix. The spectrum ± φ (cid:80) µ = x , y , z cos k µ corresponds to thatof a static simple cubic lattice artificially described by 2 sub-lattices, where the bands are folded as the Brillouin zone sizeis halved. While the system remains gapless at quasienergy 0for arbitrary φ , a characteristic feature of the high-frequency(weak driving) regime is a finite energy gap at quasienergy π , resulting from the fact that the band width is proportionalto φ and thus is small compared to the dimensionless drivingenergy (cid:126) ω = π . This behaviour can be observed in the spec-trum for φ = π/ φ , the band width growsrelative to 2 π . When φ = π/ π and quasimomentum k = φ ∈ ( π/ , π/ π with topologi-cal charges ±
1, as shown in Fig. 3 (a). They are located at thequasimomenta k = ± k d along the diagonal vector d = (1 , − , , (6)with k = (1 /
2) arccos (cid:104)(cid:16) / − sin φ (cid:17) / sin φ (cid:105) modulo π (seeAppendix A), so that k → π/ φ → π/
2. We ob-serve the emergence of surface Fermi arc states connectingthe Weyl points, when comparing the spectrum with full pe-riodic boundary conditions to that with open boundary con-ditions along z-direction, as illustrated in Fig. 3 (b) and (c)respectively. As the driving strength approaches the fine tuned point, φ = π/ − ε with ε (cid:28)
1, the Weyl dispersion acquires a highlyanisotropic form shown in Fig. 4 (b). The dispersion remainssteep along the diagonal coordinate k · d = k x − k y + k z , butbecomes increasingly flat in other two directions. Exactly atfine tuning, φ = π/
2, the constituent evolution operators (4)reduce to U µ ± ( k ) = − i (cos k µ τ ± sin k µ τ ), and the Floquetstroboscopic operator takes the form U F = − e − i τ k · d . Thisprovides the quasi-energies E k , ± = ± k · d + (2 m + π , where m ∈ Z labels the Floquet bands, and where the upper andlower branches labeled by ± now directly correspond to sub-lattices A and B, i.e. ± → τ . In that case the Weyl pointsdisappear and the dispersion E k , ± is completely flat for themomentum plane normal to d , as one can see in Fig. 4 (a).Hence a particle can only propagate along the diagonal d witha dimensionless velocity v ± = ± d , depending on whether theparticle occupies a site on the sublattice A or B at the begin-ning of a driving cycle. It is noteworthy that for fine tuneddriving, the e ff ective Floquet Hamiltonian, H F ≡ − i ln U F = τ d · k + π for φ = π/ , (7)is periodic in the momentum space only by taking into accountthe periodicity in quasi-energies. Such an e ff ective Hamilto-nian does not have a static counterpart, and can only be pro-duced in periodically driven systems.The fine-tuned dispersion can be understood by consideringthe dynamics in real space. For φ = π/ µ ± theparticle is fully transferred from a sublattice A site positionedat r A to a neighboring site B situated at r B = r A ± e µ or viceversa. During the six steps composing the driving period, theparticle follows the trajectory shown in Fig. 1 (b). Thus, aftercompleting each period the particle located on a site of sublat-tice A ( B ) is transferred by + d ( − d ) to an equivalent site ofthe same sublattice, giving rise to stroboscopic motion alongthe diagonal directions ± d at the velocity v ± . IV. ACCUMULATION OF CHIRAL HINGE MODES FOROPEN BOUNDARY CONDITIONS
Let us now consider the case of open boundary conditionswith the faces of the boundary planes oriented along the direc-tions ± µ , either with µ = x , y , z (fully open boundary condi-tions) or with µ = x , y , keeping periodic boundary conditionsalong the z direction in the latter case. As an intriguing e ff ect,we find that, when the system is subjected to open boundaryconditions with at least two properly chosen boundary planes,an extensive number of chiral Floquet hinge modes is formed.This phenomenon is best understood by considering the real-space propagation of the particle at the fine-tuned parameter φ = π/ r = ( x , y , z ), with coordinates x , y , z taking integervalues between 1 and L x , y , z . The lattice sites are considered tobelong to sublattice A (or B ) if s = ( − x + y + z equals + − s = = A and s = − = B to label thesublattice. φ (1) PBC x , y , z (2) PBC z , OBC x , y (3) OBC x , y , z π π π φ = π/ , π/ , π/
2, corresponding to the metallic phase, the Weyl semimetal / hinge phase, and the fine-tunedpoint, respectively. The columns represent periodic / open boundary conditions (PBC / OBC) along the specified directions. The system sizesare L x = L y =
40 in column (1), 20 in (2), and 16 in (3). Columns (1) and (2) contain energy spectra projected onto k z . Although the spectrafor fully periodic and open boundary conditions in x and z direction are almost identical in the metallic phase ( φ = π/ ff erentin the WSM / hinge phase ( φ = π/
3) and at fine tuning ( φ = π/ ff erence is a result of the formation of chiral hinge modes for openboundary conditions in x and z directions (column 2). This can be inferred also from the dot size reflecting the inverse participation ratiowith respect to the site basis as a measure for localization, as well as from the color code indicating the mean distance to two opposite hingesand interpolating from blue for one hinge over black near the center of the system to red for the other hinge. The green dots mark the Weylpoints. We also plot the real-space densities of various Floquet modes for open boundary conditions along x and y (column 2) or all (column3) directions. A. Fine-tuned driving
Let us consider a hinge along the z axis confined by the − x and − y surface planes. In that case the particle is restricted tothe lattice sites with x ≥ y ≥
1. Starting from a site of,say, sublattice B , a particle will propagate on this sublatticeat constant velocity in steps − d in diagonal direction, untilit reaches the − x boundary face positioned at x =
1. At theboundary, tunnelling between the lattice sites B and A cannotoccur during one of the six steps of the driving cycle. As aresult, the particle changes the sublattice and starts to prop- agate on the A sublattice in opposite direction. The micro-scopic processes leading to such a reflection are illustrated inFig. 1 (c). The left plot shows the two possible ways of howa change of sublattice can occur at the boundary face orientedin the − x direction. The two processes are distinguished bywhether they start on a B lattice site directly at the bound-ary with x = x = − x surface, the particle φ Weyl points π π π π (a) Positions of Weyl points. (b) φ = π/
3, PBC x , y , z (c) φ = π/
3, PBC x , y (d) φ = π/
6, PBC x , y , z FIG. 3. (a) The di ff erent phases can be distinguished by the pres-ence or absence of Weyl points at quasienergy π . For φ = π/
8, inthe metallic phase ( φ < π/ φ = π/
6, the band touching point appears at k = (greencircle). For π/ < φ < π/ ± sign) that separatealong the diagonal k x = − k y = k z , as shown for φ = π/
3. At finetuning φ = π/ φ > π/ φ = π/ − k x , k y for periodic boundary conditionsin x and y and either periodic (b,d) or open (c) boundary conditionsin z direction. A surface Fermi arc can be observed for φ = π/ z direction. The orange surface denotes the contour formedby quasi-energies closest to E = π at each ( k x , k y ). For φ = π/ E = π . travels in reversed direction in steps of 2 d until it eventuallyreaches the − y surface and is again backreflected, this timewith the sublattice change A → B . The microscopic detailsof such a reflection at the − y surface are depicted on the righthand side of Fig. 1 (c). In this way, the particle will moveback and forth between the − x and the − y surfaces sharingthe hinge. Such a dynamics is illustrated in Figs. 5 and 6, (a) Fine tuned point (b) Slight deviationFIG. 4. (a) Anisotropic 1D-like dispersion along the corrdinate k x − k y + k z at the fine-tuned point φ = π/
2. (b) For a small detuning, φ = π/ − .
1, the dispersion is no longer completely flat in other twodirections, and a pair of non-equivalent Weyl points is formed alongthe diagonal at k = ± k (1 , − , showing the path of a particle projected onto the xy -plane. In-terestingly, within this plane, the particle returns to the sametransverse position ( x , y ) only after having travelled twice be-tween both faces, as one can see in Figs. 5 and 6.It is noteworthy that the change in the sublattice B → A or A → B during the backward reflection from the correspondinghinge planes is accompanied by a lateral Goos-H¨anchen-like(GH) shift of particle’s trajectory. This is similar to changinga track for a train before sending it backwards. Importantlythe back reflected particle propagates in a trajectory situatedcloser the hinge or further away from it for the B → A or A → B reflections, respectively. Because of such chiral GHshifts, the particle visits a larger number of B sites than A sites when travelling between the two surface planes. Sincethe ballistic motion along the B ( A ) sites is the accompaniedby a spatial shift in − z ( z ) direction, one arrives at an overallsteady advance in − z direction, i.e. along the hinge, duringthe forward and backward motion of the particle between thehinge-sharing surfaces oriented in − x and − y direction. Moreprecisely, as demonstrated in Appendix B, the particle tracesout two slightly mismatched “loops” shown in Figs. 5 and6 that involve four reflections by the hinge surfaces beforeit comes back to exactly the same point in the xy plane, butshifted by − z -direction at an equivalent (i.e. B-type) lat-tice site. In this way the particle’s trajectory roughly coversa two-dimensional strip along the z-direction, whose width isabout twice its distance from the hinge. Although such a pic-ture applies to a particle situated further away from the hinge,the advance in the − z direction takes place also for trajectoriessituated very close to the hinge where the particle is reflectedsimultaneously from both hinge planes x = y =
1, asillustrated in Fig. 1(d).Suppose the particle intially occupies a site of the sublattice B with transverse coordinates y = x = M +
1, so the par-ticle is situated at the hinge surface oriented in − y direction,and is M ≥ − x hinge surface. Inthat case, it takes (2 M +
1) driving periods for the particle tocome back to the initial position ( M + ,
1) in the xy plane, BA1 2 3 4 5 6123456 x y FIG. 5. An example of a fine-tunned stroboscopic motion of a par-ticle at the lower-left hinge. The picture shows the projection of theparticle’s trajectory in the xy plane. The sites of the B and A sub-lattices are marked in blue and red, respectively. The particle is ini-tially at the site of the sublattice B characterized by the coordinates x = M + y =
1, with even M =
4. This corresponds to the lowerrow ( y =
1) and the fifth column ( x = x = y =
1. Bulk ballistic trajectories overone / two driving periods are indicated by thin / thick solid arrows. Theparticle returns to its initial site after 2 M + = while having shifted by − z direction, i.e.( U F ) M + | B , M + , , z (cid:105) = − | B , M + , , z − (cid:105) . (8)After averaging over such a full reflection cycle, the parti-cle travels with an M -dependent mean velocity of v M − = − / (2 M +
1) along the z (hinge) axis (see Appendix B formore details). Here | s , x , y , z (cid:105) describes a particle located atsite ( x , y , z ) belonging to sublattice s = B = − s = A = s = ( − x + y + z .Let us now consider periodic boundary conditions in z di-rection with L z = N z , i.e. | s , x , y , z + N z (cid:105) = | s , x , y , z (cid:105) , whilekeeping open boundary conditions in the x and y . It is thenconvenient to introduce mixed basis states | s , x , y , k z (cid:105) (cid:48) = √ N z (cid:88) z e ik z z | s , x , y , z (cid:105) . (9)They are characterized by quasimomenta k z , which are de-fined modulo π corresponding to the lattice period of 2 whenmoving in z direction, and obey( U F ) M + | B , M + , , k z (cid:105) (cid:48) = − | B , M + , , k z (cid:105) (cid:48) e ik z . (10)In a similar manner, the system returns to any state of the stro-boscopic sequence | M , k z , p (cid:105) = ( U F ) p | B , M + , , k z (cid:105) (cid:48) (11) BA1 2 3 4 5 6123456 y x FIG. 6. Like in Fig. 5 the particle is initially at the site of thesublattice B with x = M + y =
1, but now with odd M =
5. Theparticle returns to the initial site after 2 M + =
11 periods. after 2 M + U F ) M + | M , k z , p (cid:105) = − | M , k z , p (cid:105) e ik z , (12)with p = , , . . . M . By superimposing the subharmonichinge states | M , k z , p (cid:105) , we can now construct Floquet hingestates: | M , k z , q (cid:105) = √ M + M (cid:88) p = | M , k z , p (cid:105) exp (cid:32) i 2 π q − k z + π M + s (cid:33) , (13)where the index q = , , . . . , M labels these modes. Thecorresponding quasienergies are given by E M , k z , q = (cid:32) π q − k z + π M + (cid:33) mod 2 π . (14)An analogous dispersion but with an opposite slope (op-posite sign) is obtained for the states formed at the oppositehinge confined by the planes at x = y = L facing the + x and + y directions. The dispersion at both hinges reproduces the spec-trum for the beam geometry shown in the row 1 and column(2) of Fig. 2. Such a spectrum of the system with open bound-ary conditions in x and y directions looks completely di ff erentfrom the one for full periodic boundary conditions shown incolumn (1) of Fig. 2 or Fig. 4 (a), where all the modes havethe same positive or negative dispersion slope (group veloc-ity) v z = ±
2. In contrast, for the beam geometry [column (1)of Fig. 2] the spectrum due to the chiral hinge modes is noworganized in linear branches given by Eq. (14) and the anal-ogous dispersion with inverted sign for the opposite hinge.Each branch is characterized by a di ff erent group velocity v M ± = ± M + M , where thelower and upper sign in ± correspond to the states locatedaround opposite hinges x = y = x = y = L . The red / blue colors in Fig. 2 indicate the mean distance of each modefrom the two relevant hinges. The dark red (blue) mode asso-ciated with M = x = y = x = y = L ) and propagates at the largest velocity in nega-tive (positive) z direction. Modes with a smaller slope havelarger M and thus are located further away from the particu-lar hinge, as indicated by the color. The real-space density offour di ff erent hinge modes at φ = π/ | ψ (cid:105) is the inverse participa-tion ratio IPR = (cid:80) j |(cid:104) j | ψ (cid:105)| , with real-space site states | j (cid:105) andthe Floquet eigenstates | ψ (cid:105) . It is shown in the spectra of Fig. 2via the dot size roughly indicating the inverse of the numberof sites a mode is spread over. Thus 1D-like modes localizedat the hinges have larger IPR than those that are spread over a2D-like ribbon further away from the hinges.All in all, for the given geometry, all modes are hingemodes at fine tuning. This e ff ect resembles an extensive ac-cumulation of boundary modes featured in the non-Hermitianskin e ff ect [34–37]. It is noteworthy that the modes of thepresent periodically driven lattice are localized at the second-order boundaries, viz. at the hinges rather than directly atthe boundary faces. Therefore, the formation of an exten-sive number of hinge modes might be called chiral second-order Floquet skin e ff ect , in analogy to the terminology usedfor non-Hermitian systems [37]. An important di ff erence isthat in non-Hermitian systems the skin modes are localizeddirectly at the boundaries [34–37], whereas in the present pe-riodically driven system the hinge modes are localized at var-ious distances from the hinges to which they are bound to insuch a way that the hinge modes cover the whole lattice. Thisis because the eigenstates of a unitary Floquet evolution op-erator are orthogonal to each other (like those of a Hermi-tian operator) implying that there can be at most one Floqueteigenstate per lattice site on average, so an accumulation ofboundary states is not possible directly at the boundary. Toput in another way, the non-Hermitian skin e ff ect is associatedwith the exceptional points of the non-Hermitian Hamiltonianwhen the boundaries are introduced [35, 36], whereas no ex-ceptional points are formed for periodically driven systemsdescribed by the unitary Floquet evolution operators. Moredetails on these issues are available in Appendix C. B. Beyond fine-tuned driving
Although the above discussion is based on ballistic trajecto-ries at the fine-tuned driving parameter φ = π/
2, we expect thechirality of the hinge modes to be robust also against perturab-tions and tuning away from φ = π/
2. This applies especiallythe hinge states with larger chiral velocities, which are situ-ated closer to the hinge than those with smaller chiral veloci- (a) without defect (b) with defect(c1) t = t = t = x , y , z ) = (1 , ,
16) for a system of 16 × ×
16 sites with full openboundary conditions and fine tuned φ = π/
2. The squared wave func-tion at di ff erent times is reflected in the opacity of the plotted dots.Di ff erent colors indicate time. (a) Without defect. (b) In the presenceof a potential defect of energy ∆ = π at the two sites marked by thegreen tube. Additionally, we plot the squared wave function of onehinge mode. (c) Snapshots of the time evolution in the presence ofthe defect at di ff erent times.(a) without defect (b) with defect(c1) t = t = t = φ = . × π/
2, away from fine tuning. ties, and are spatially well separated from counter-propagatingmodes at the opposite hinge. The chiral hinge modes per-sist for a rather wide range of φ beyond φ = π/
2. This canbe observed from the example of φ = π/ .
3% detuning,half-way across the Weyl phase transition) displayed in Fig. 2column (2) around k z = π/ E =
0. We can see thatthe hinge modes at smaller distances M from the hinge stillpreserve their chirality. In turn, hinge modes with larger M that are closer to the sample center, are gradually mixed withmodes of opposite chirality and become bulk modes when φ deviates away from φ = π/ x , y and z presented in column (3) of Fig. 2showing that for φ = π/ φ = π/ ff erent hinges are joined to form a closed loop respectinginversion symmetry of the system [see also Fig. 7 (a)]. Thesix hinges not participating in this closed loop do not carryhinge modes, since their two boundary planes are not con-nected along the diagonal direction d . Meanwhile, non-hingemodes, representing the bulk dynamics all center along thecubic diagonal [column (3) of Fig. 2].To further confirm the robustness of the hinge states, inFigs. 7 and 8 we simulate the dynamics of a particle in thepresence of a defect for a system with open boundary condi-tions in all three directions, corresponding to column (3) ofFig. 2. Figure 7 illustrates the dynamics of a particle initiallylocated at a corner ( x , y , z ) = (1 , ,
16) of the system, wheretwo transporting hinges intersect each other, (a) for the finetuned situation without defect and (b) in the presence of astrong potential o ff set of ∆ = π on two neighboring hingesites (indicated by a green tube) at ( x , y , z ) = (1 , , , (1 , , φ = . π/
2) are presented in Fig. 8. We find that de-spite this strong defect the chiral nature of the hinge modes en-sures that no backscattering occurs at the defect and the major-ity of the wave-packets continues to follow chiral trajectoriesalong the hinges. In Figs. 7 (b) and 8 (b) we also plot repre-sentative eigenstates of the system with the defect. The eigen-states remain delocalized along the hinge, with only a smalldistortion compared to the situation without defect shown incolumn (3) of Fig. 2. The fact that the defect does not inducescattering away from the hinge (modes) can be seen also fromFig. 9. It shows the time-evolved state after 100 driving cyclesfor the non-fine-tuned system ( φ = . × π/
2) both without de-fect (a) and with defect (b). Very similar distributions are alsofound after even longer evolution, e.g. over 1000 driving cy-cles; the densities are, thus, representative for late-time statesin the limit t → ∞ . C. Topological origin
It is an interesting question, whether the robust chiral hingestates are a consequence of topological properties of thedriven system. However, as we see from column (2) of Fig. 2,the quasi-energies of hinge states are fully mixed with the bulkspectrum, and therefore no traditional topological band theo-ries for gapped or semimetallic systems apply. Here, it is thecollaboration of boundary geometry and Floquet driving thatgenerates such topologically protected states.In section IV A we have obtained equation (12) describingthe evolution over 2 M + M th hingestate | M , k z , p (cid:105) at the fine tunned point. Using this equation (a) ∆ = ∆ = π FIG. 9. Density distribution of a particle initially localized at thecorner site ( x , y , z ) = (1 , ,
16) after an evolution over 100 drivingcycles for the non-fine-tuned parameter φ = . × π/
2, without defect(a) and with a defect (b) [corresponding to the parameters of Fig. 8(b)]. The densities are representative for late-time states in the limit t → ∞ ; similar distributions are found also after 1000 driving cycles. one can define the the quasienergy winding number [1] forthe M th hinge state via the Floquet evolution operator overthe 2 M + W M = π i (cid:90) π dk z U − k z , M + ∂ k z U k z , M + = , (16)where U k z , M + = (cid:104) M , k z , p | ( U F ) M + | M , k z , p (cid:105) = − e ik z (17)and p = , , . . . M . In Eq. (16) the integration over k z ex-tends over one Brillouin zone of width π , as the distance be-tween two non-equivalent lattice sites equals to 2 in z direc-tion. A similar procedure can be applied to the opposite hingeat ( x , y ) = ( L , L ), where the hinge modes shown in blue incolumn (2) of Fig. 2 are characterzed by the opposite groupvelocity and thus the opposite winding number W M = − W M is associated with fine tuning, φ = π/
2. However, the spatialseparation between hinge modes of opposite chirality allowsto preserve their chiral character also away from fine tuningpoint φ = π/
2, as one can see in column (2) of Fig. 2. Thus,the formation of bulk states via the mixing of hinge modes ofopposite chirality happens mostly in the center of the system,where hinge modes corresponding to large M and small chiralvelocities lie close by to their counter propagating partnersassociated with the opposite hinge. In turn, the states with thelargest chiral velocity, which are situated close to the hingeand far away from counter-propagating modes of the oppositehinge, are much less a ff ected by a small detuning.In this way, tuning away from φ = π/ φ = π/ k z = E = π , as can be seen from the spectrum shown for φ = π/ steps 1 2 3 4 5 6 α α α α α ab according to the protocol given in the table, gives riseto di ff erent dimerizations of the cubic lattice in each driving step, thatenables tunneling along the desired bonds. V. EXPERIMENTAL REALIZATION WITH ULTRACOLDATOMS IN OPTICAL LATTICESA. Engineering of the driven lattice
Above, we have shown that the proposed modulation oftunnelling gives rise to a variety of phenomena, including therobust creation of a pair of Weyl points, unidirectional bulktransport, chiral Goos-H¨anchen-like shifts, and the macro-scopic accumulation of chiral hinge modes for open bound-ary conditions corresponding to a chiral second-order Floquetskin e ff ect. The model itself is, nevertheless, rather simpleand its implementation with ultracold atoms in optical latticescan be accomplished using standard experimental techniques.All what is needed is a static cubic host lattice potential ofequal depth V in each Cartesian direction and a superlatticepotential, whose amplitudes along various diagonal lattice di-rections are modulated in a stepwise fashion in time in order tosuppress / allow tunneling along the six di ff erent bonds speci-fied by our protocol. This can be achieved using the followingoptical lattice potential: V ( r ) = V (cid:88) µ = x , y , z cos (2 k L r µ ) + V (cid:88) a , b = , α ab ( t ) cos k L ( x + ( − a y + ( − b z ) , (18)where only two of the four modulating lasers α ab with a , b = , B. Detection of hinge dynamics
To observe the dynamics associated with the hinge modes,one can apply the boxed potential achieved in recent exper-iments [46–52]. There, thin sheets of laser beams penetratethrough the quantum gases creating a steep potential barrier.Three pairs of such beams are imposed in a three-dimensionalsystem, creating the sharp “walls” for the box potential whileleaving the central part of the gases homogeneous.Essentially, such a potential combined with our lattice driv-ing scheme immediately leads to the particle dynamics de-scribed in Fig. 7 and Fig. 8. To take into account realisticexperimental situations, two modifications are adopted in ourfollowing simulations. First, we consider the e ff ect of a rela-tively “softer” wall for the box potential with V box ( r ) = V b (cid:88) µ = x , y , z + tanh r (1) µ − r µ ξ + tanh r µ − r (2) µ ξ , (19)where the potential ramps up over a finite distance of roughly4 ξ near the boundaries r (1 , µ , see Fig. 11 for instance. Thesecond modification we adopt is that the initial state is not0taken to be localized on a single lattice site but described by agaussian wave packet of finite width, ψ i = ( x , y , z ) ( t = = e − [( x − x ) + ( y − y ) + ( z − z ) ] / s . (20) (a) | ψ i ( t = | (b) Projection to x - y (left) and x - z (right) x V box ξ = (c1) Ideal boundary(c2) | ψ i ( t = | (c3) | ψ i ( t = | xV box ξ = (d1) Softer boundary(d2) | ψ i ( t = | (d3) | ψ ( t = | FIG. 11. The dynamics of particles in a box potential with di ff er-ent softness of the boundary. The initial density distribution takes aGaussian profile spreading over several lattice sites. The parametersare φ = π/ , V b T / (cid:126) = . π . The initial Gaussian profile has thecenter ( x , y , z ) = (4 , ,
12) and width s = . The results of the dynamics are presented in Fig. 11 (c1)–(c3) and (d1)–(d3), for the ideal sharp boundary (as in Fig. 7)and the realistic softer boundaries in experimental setting re-spectively. We see that the chiral motion snapshots for theideal / softer boundaries exhibit qualitatively the same charac-ters, signalling that a softer boundary does not cause signifi-cant changes. This is in consistent with the previous simula- tion showing the robustness of hinge states and their resultingchiral dynamics against local defects in Fig. 7 and Fig. 8. Themajor di ff erence from previous cases, then, derives from theinitial state that overlaps with more than one set of eigenstatesnear the hinge, each with di ff erent group velocities as givenin Eq. (15). Finally, we mention that some portion of initialparticle distribution would reside within the region with sig-nificant changes in V box . That portion of the particles could bepermanently confined to the initial hinge due to a mechanismsimilar to Wannier-Stark localization. However, the majorityof the particles are still traveling into the connecting hinges,as shown in Fig. 11 (c3) and (d3).In cold atom experiments, the density profiles are usuallydetected by taking a certain projection plane, where the inte-grated (column-averaged) densities are observed. To this end,we point out that the hinge dynamics can be confirmed by ob-serving the density profiles in two perpendicular planes. Aschematic plot is given in Fig. 11 (b), corresponding to thedynamics along the hinge x = y ∼
1. The density profiletaken at x − y plane (i.e. the “top” view) would show a lo-calized distribution at the corner, verifying the particles onlylocate at x = y ∼
1. Meanwhile, the profile at x − z plane(i.e. “side” view) indicates the movement / spreading along z .In a more general situation, i.e. at long time limit with all6 hinges populated as in Fig. 9, additional image projectionplanes could be exploited. We also mention that a simulta-neous implementation of multiple imaging planes have beenapplied in experiments [31, 32]. C. Detection of Floquet Weyl points
Weyl physics has been explored in recent cold atom exper-iments and theoretical proposals [28, 31, 32, 53, 54], and alsoextensively in solid state systems [25]. Here, we discuss ascheme closely related to a recent experiment [11, 55] detect-ing the spacetime singularities in anomalous Floquet insula-tors.First, the band touching at Weyl points can be verified usingthe St¨uckelberg interferometry [11, 56–58]. Such a methodmeasures the smaller gap for the two bands at E ∼ π ,see Ref. [11] for details. Compared with the experiments forinsulators, a di ff erence here is that the two bands are, overall,always gapless at E =
0. That means if a global gap is mea-sured, it will prevent us from gaining information about thegaps or band touching at quasienergy E = π . But fortunately,there exists a finite region neighboring to k = π/ , − , E = φ , seeFig. 12 (a1), (a2) for example. Then a local gap closure canbe measured near k .The specific measurement for our case can be performed inthe following way. An example for k = ( π/ − . , − , k with E ± ( k ) = ± E . Here, weuse the branch cut along π in taking the logarithm of Floqueteigenvalues e − iE ± ( k ) . They have the same magnitudes and op-posite signs due to particle-hole and inversion symmetry asexplained in Sec. II. Then, the local gap around E ∼ (a1) ∆ (0) at φ = π/ (a1) ∆ (0) at φ = π/ Weyl point π ϕ π π Δ ( ) Δ ( π ) Measure (b) Exemplary gap at k = k (1 , − ,
1) with k = π/ − . E =
0, for φ = π/ φ = π/ E ∼ E ∼ π ,and the measured gap which takes the smaller one of the two. ∆ (0) ≡ E , while the other gap around the Floquet Brillouinzone boundary E ∼ π is ∆ ( π ) ≡ π − E . Therefore, ∆ (0) = ∆ ( π ) can only occur at 0 , π mod 2 π . In experiments, one can startfrom the high frequency limit ( φ →
0) where the band width issmall compared to 2 π and therefore the measured gap alwayscorresponds to ∆ (0) . Slowing down the driving, the gap ∆ ( π ) shrinks while the other gap ∆ (0) expands. At some point, thetwo gaps coincide with their magnitudes, as shown in Fig. 12(b). Since it is always the smaller one of ∆ (0) and ∆ ( π ) thatwill show up in experimental measurement, one will observea cusp shape of the measured gap, i.e. near φ ≈ .
41 inFig. 12 (b). One could then imply from the occurrence ofthe cusp that for φ > .
41, the experimental data starts to re-veal ∆ ( π ) , whose vanishing at φ ≈ .
87 shows the existence ofthe Weyl point around E ∼ π . Similar measurements can beperformed for k slightly deviating from k , which will showthat at φ = . ∆ ( π ) remains finite, proving that the band clo-sure around E ∼ π is a point contact. When the designated φ is slowly approached, one can perform a measurement of thegap at a certain k . A shortcut for our system is that focusingon momenta along the diagonal k = k (1 , − ,
1) is su ffi cientto determine the Weyl point, as discussed in Sec.III.With the Weyl points determined, one could further ap-ply band tomography [59, 60] method for momentum statessurrounding a certain Weyl point in order to determine itscharge. Note that one does not need the eigenstate informationthroughout the whole Brillouin zone as the two bands are gap-less in certain regions, except for just an arbitrarily small sur-face wrapping a Weyl point k (Weyl) determined previously. Asshown before, near the Weyl points in our model, there existsa finite region where the two bands are fully gapped in both E ∼ π , which allows for populating eigenstates withbosons at a certain band [11, 60]. As an example, in Fig. 13(a) we illustrate the surfaces formed by 6 faces q x , y , z = ± . q = k − k (Weyl) , with k (Weyl) ≈ . × (1 , − , φ = π/
3, as in Fig. 2. From the full information of theFloquet eigenstates | u n , k (cid:105) given by Eq. (5), the Berry curva-ture penetrating out of a plane normal to the unit vector e µ can be computed as Ω µ ( k ) = ± i (cid:80) νρ ε µνρ (cid:16) (cid:104) ∂ k ν u n , k | ∂ k ρ u n , k (cid:105) (cid:17) ,where ε µνρ is the Levi-Civita symbol, and ± sign denotes thatthe unit vector penetrating out of the cube is along ± e µ direc-tions. Figure 13 shows momentum resolved Berry curvaturesin each wrapping surface and their net fluxes (cid:82) surf d k Ω µ ( k ) inthat plane. (a) The surfaces wrapping a Weyl point(b1) Ω z + , net 1.284 (b2) Ω z − , net 0.866 (b3) Ω x + , net 1.284(b4) Ω x − , net 0.866 (b5) Ω y + , net 1.469 (b6) Ω y − , net 0.513FIG. 13. The Berry curvatures for the surfaces wrapping a Weylpoint. Adding the net Berry curvatures up we have 2 π . VI. CONCLUSION
In this paper, we have shown that three-dimensional peri-odically driven lattice systems can show a macroscopic ac-cumulation of chiral hinge modes, when subjected to openboundary conditions. This corresponds to a chiral second-order Floquet skin e ff ect. An intuitive understanding of thise ff ect was given by considering the system at a fine-tunedpoint of the periodic driving, where the bulk motion can onlytake place forwards or backwards along a single diagonal di-2rection. As a consequence, for open boundary conditions, par-ticles are reflected back and forth between hinge-sharing sur-face planes, with a drift along the direction of the hinge beingaccomplished by chiral Goos-H¨anchen-like shifts associatedwith these reflections. This e ff ect is di ff erent from higher-order Floquet topological insulators (HOFTI) not only regard-ing the underlying mechanism, but also because no macro-scopic accumulation of hinge modes takes place in HOFTI.The e ff ect resembles, however, the accumulation boundarymodes in non-Hermitian systems, even though here notice-able di ff erences are also found. Namely in the non-Hermitiancase the accumulation of boundary modes occurs close to thehinge. This is not possible in our system, since di ff erent fromthe eigenmodes of a non-Hermitian Hamiltonian, the Floquethinge modes are orthogonal to each other, as they are eigen-states of the unitary Floquet evolution operator U F . Thuswe find hinge bound modes also at larger distances from thehinge. Another interesting aspect is the competition or inter-play between the hinge modes and the emergence of robustWely points in our system, so the hinge states can co-existwith the Fermi arc surface states. The implementation of themodel featuring both the second-order Floquet skin e ff ect andthe Weyl physics is straightforward with ultracold atoms inoptical superlattices. VII. ACKNOWLEDGMENT
The authors thank E. Anisimovas and F. Nur ¨Unal for help-ful discussions. We acknowledge funding by the EuropeanSocial Fund under grant No. 09.3.3-LMT-K-712-01-0051and the Deutsche Forschungsgemeinschaft (DFG) via the Re-search Unit FOR 2414 under Project No. 277974659.
Appendix A: Evolution operator and quasienergies along thediagonal1. Evolution operator
In the bulk the stroboscopic evolution operator U F is gen-erally given by Eqs. (3)-(4) in the main text. Let us considerthe operator U F for wave-vectors k along the cubic diagonaldirection d = (1 , − , k = ± k d and, thus, | k | = √ k . (A1)with k >
0. In that case Eqs. (3)-(4) simplify to U F = [ U ( ∓ k ) U ( ± k )] , (A2)where U ( ∓ k ) U ( ± k ) = (cid:0) cos φ − i τ ∓ k sin φ (cid:1) (cid:0) cos φ − i τ ± k sin φ (cid:1) , (A3)with τ ± k = τ cos k ± τ sin k and Pauli matrices τ , , forthe sublattice freedom. Explicitly one, thus, has U ( ∓ k ) U ( ± k ) = (cid:104) cos φ − sin φ cos (2 k ) (cid:105) − i d , (A4) with d = sin φ cos (2 k ) τ −
2i cos φ sin φ cos k τ . (A5)
2. Quasi-energies
The evolution operator U F = e − i H F defines the quasiener-gies representing the eigenvalues of the of tthe Floquet Hamil-tonian H F , which describes the stroboscopic time evolution inmultiples of the driving period T =
1. Using Eqs. (A2) and(A4) for the evolution operator, one arrives at the followingequation for the quasi-energies E k cos ( E k / = cos φ − sin φ cos (2 k ) . (A6)This provides the dispersion (modulo 2 π ) along the diagonal k x = − k y = k z = k E k d ,γ = γ arccos (cid:104) cos φ − sin φ cos (2 k ) (cid:105) , with γ = ± . (A7)In particular, quasienergies E k d ,γ = π (modulo 2 π ) corre-spond to cos φ − sin φ cos (2 k ) = / , (A8)and thus cos (2 k ) = / − sin φ sin φ , (A9)giving (modulo π ) k = (1 /
2) arccos (cid:104)(cid:16) / − sin φ (cid:17) / sin φ (cid:105) . (A10)At the fine tunned point ( φ = π/
2) the condition Eq.(A8) re-duces tocos (2 k ) = − / , giving k = π/ . (A11)On the other hand, at φ = π/ φ = /
4, so thatcos (2 k ) = , giving k = . (A12)In this way, two band touching points are formed atquasienergy π for π/ < φ < π/
2, as well as for π/ < φ < π/ φ = π/ φ < π/ φ > π/
6, Eq.(A8) can no longer be fulfilled, so aband gap is formed at quasienergy π . Appendix B: Stroboscopic hinge motion at fine tuning
In this Appendix we give a detailed description of the stro-boscopic real-space dynamics of the system at fine tuning, φ = π/
2, giving rise to chiral hinge-bound Floquet modes.We will consider the hinge that is shared by the two surfaceplanes oriented in the − x and − y direction, which is parallelto the z -axis. The projection of the particle’s trajectory in the xy plane is illustrated in Figs. 5 and 6. A particle of sublattice3 s = + , − ≡ A , B is translated by 2 s d during each drivingcycle, provided x + s ≥ y − s ≥ | s , x , y , z (cid:105) transforms according to the following rule after asingle driving period: U F | s , x , y , z (cid:105) = − | s , x + s , y − s , z + s (cid:105) . (B1)The particle thus propagates with a stroboscopic velocity v = s (1 , − ,
1) in opposite directions s = ± ff erent sublat-tices A and B .Suppose initially the particle occupies a site of the sublat-tice B at the boundary y = M sites away from thehinge ( x = M +
1) with odd M + z , so that s = B = − | s , M + , , z (cid:105) ≡| B , M + , , z (cid:105) . The subsequent stroboscopic trajectory pro-jected to the xy plane is shown in Fig. 5 for M = M =
5. Generally it takes (2 M +
1) driving periodsfor the system to return to its initial state | M + , , z (cid:105) . To seethis, consider the stroboscopic evolution of the particle withan even M > z . The stroboscopic motion of theparticle then splits into four bulk and four boundary segmentsillustrated in Fig. 5 for M = M / B sub-lattice, and the state vector transforms as | B , M + , , z (cid:105) →| B , , M + , z − M (cid:105) . Subsequently the particle is reflectedfrom the plane x = A situated closerto the hinge, | B , , M + , z − M (cid:105) → | A , , M − , z + − M (cid:105) ,as shown in Fig. 1(c) of the main text. During the next M / − A sublattices, giving | A , , M − , z + − M (cid:105) →| A , M , , z (cid:105) . The subsequent reflection from the plane y = B sublattice situated furtheraway to from the hinge, | A , M , , z (cid:105) → | B , M , , z − (cid:105) . Theevolution takes place in the similar way during final four seg-ments. Explicitly the full stroboscopic dynamics is given by:( U F ) M / | B , M + , , z (cid:105) = ( − M / | B , , M + , z − M (cid:105) , (B2) U F | B , , M + , z − M (cid:105) = − i | A , , M − , z + − M (cid:105) , (B3)( U F ) M / − ( − i) | A , , M − , z + − M (cid:105) = i ( − M / | A , M , , z (cid:105) , (B4) U F i | A , M , , z (cid:105) = | B , M , , z − (cid:105) , (B5)( U F ) M / − | B , M , , z − (cid:105) = − ( − M / | B , , M , z − M (cid:105) , (B6) U F ( − | B , , M , z − M (cid:105) = i | A , , M , z − M (cid:105) , (B7)( U F ) M / − i | A , , M , z − M (cid:105) = ( − i) ( − M / | A , M − , , z − (cid:105) , (B8) U F ( − i) | A , M − , , z − (cid:105) = − | B , M + , , z − (cid:105) , (B9)In this way, after (2 M +
1) driving periods the particle returnsback to the intial position ( M + ,
1) in the xy plane and isshifted by 2 lattice units to an equivalent point of the sublattice B in the direction opposite to the z axis. The same holds forthe initial state vector | B , M + , , z (cid:105) characterized by an odd M and even z (see Fig. 6 for M = ≤ M ≤
3) where the reflectionscan take place simultaneously from both planes x = y =
1, as illustrated in Fig. 1(d) in the main text. Thus one canwrite for any distance M ≥ U F ) M + | B , M + , , z (cid:105) = − | B , M + , , z − (cid:105) , (B10)This means the particle propagates along the hinge inthe − z direction with the stroboscopic velocity equal to − / (2 M + M + B is reflected to a siteof the A sublattice situated closer to the hinge, whereas theparticle in the sublattice A is reflected to a site of the sublat-tice B situated further away from the hinge, as one can seein Figs. 5 and 6, as well as in Eqs. (B3), (B5), (B7), (B9).Consequently the number of B sites visited over all 2 M + M +
1) exceeds the corresponding number of A sites ( M ). The four reflections do not yield any total shift ofthe particle in the z direction. On the other hand, the ballisticmotion between sites the same sublattice B ( A ) is accompa-nied by a shift by 2 lattice sites in the z ( − z ) direction for eachdriving period. This leads to the overall shift of the particle toan equivalent site in the − z direction is due to the di ff erence inthe number of the visited B and A sites after 2 M + Appendix C: Non-Hermitian Hamiltonian corresponding tostroboscopic operator
Recently it was suggested [61] to associate a non-HermitianHamiltonian H NH ( k ) to the momentum space stroboscopicevolution operator i U F ( k ). Let us consider such a non-Hermitian Hamiltonian for our 3D periodically driven lattice H NH = i U F . (C1)For the fine tunned driving ( φ = π/
2) the bulk stroboscopicevolution operator corresponds to a non-Hermitian Hamilto-nian describing a unidirectional transfer between the latticesites along the diagonal d = (1 , − ,
1) and in the opposite di-rection − d for the sublattices A and B , respectively: H bulkNH = − i (cid:88) r A | A , r A + d (cid:105) (cid:104) A , r A | − i (cid:88) r B | B , r B − d , (cid:105) (cid:104) B , r B | . (C2)The open boundary conditions for the hinge correspondingto x ≥ y ≥ | s , r s (cid:105) = r s · e x , y ≤ , with s = A , B . (C3)The bulk non-Hermitian Hamiltonian (C2) supplied with theopen boundary conditions (C3) describes a unidirectional cou-pling between unconnected linear chain of the A or B sites4terminating at the hinge planes. The eigenstates of each lin-ear chain represent non-Hermitian skin modes which are lo-calized at one end of the chain depending on the direction ofasymmetric hopping [35, 36]. In the present situation suchskin modes would be trivially localised on di ff erent planes ofthe hinge for the chains comprising di ff erent sublattice sites A or B , and no chiral motion is obtained along the hinge.Yet the open boundary conditions (C3) are not su ffi cientto properly represent the boundary behavior of a particle inour periodically driven lattice. In fact, bulk non-HermitianHamiltonian (C2) supplied with the boundary conditions (C3)is no longer a unitary operator. Thus one can not associate such an non-Hermitian operator with the evolution opera-tor, in contradiction with Eq. (C1). The unitarity is restoredby adding to H bulkNH extra terms [in addition to the conditions(C3)] to include e ff ects of the chiral backward reflection atthe hinge planes to a neighboring linear chain composed ofthe sites of another sublattice. These terms are described byEqs.(B3), (B5), (B7) and (B9), and correspond to the dashedlines in Figs. 5-6. 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