Axial-field-induced chiral channels in an acoustic Weyl system
Valerio Peri, Marc Serra-Garcia, Roni Ilan, Sebastian D. Huber
AAxial field induced chiral channels in an acoustic Weyl system
Valerio Peri, Marc Serra-Garcia, Roni Ilan, and Sebastian D. Huber Institute for Theoretical Physics, ETH Zurich, 8093 Z¨urich, Switzerland Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel (Dated: June 27, 2018)Condensed-matter and other engineered systems, such as cold atoms, photonic, or phononic meta-materials, have proven to be versatile platforms for the observation of low-energy counterparts ofelementary particles from relativistic field theories. These include the celebrated Majorana modes,as well as Dirac and Weyl fermions. An intriguing feature of the Weyl equation is the chiral symme-try, where the two chiral sectors have an independent gauge freedom. While this freedom leads to aquantum anomaly, there is no corresponding axial background field coupling differently to oppositechiralities in quantum electrodynamics. Here, we provide the experimental characterization of theeffect of such an axial field in an acoustic metamaterial. We implement the axial field through aninhomogeneous potential and observe the induced chiral Landau levels. From the metamaterialsperspective these chiral channels open the possibility for the observation of non-local Weyl orbitsand might enable unidirectional bulk transport in a time-reversal invariant system.
Three-dimensional Weyl semimetals are characterizedby a touching of two non-degenerate Bloch bands. Aroundthe touching point, the low-energy physics can be de-scribed by an equation akin to Weyl’s equation for mass-less relativistic particles H = (cid:88) α,β = x,y,z v αβ ( δk α + k WP α ) σ β , (1)where k WP is the location of the touching point and v αβ denotes the velocity tensor. The Pauli-matrices σ β encodesome pseudo-spin degree of freedom reflecting the two in-volved bands. A key property of Weyl systems is their chi-rality s = sign[det( v αβ )] = ± Much of the recent interest in Weyl systems arisesfrom their magneto-transport properties.
The appli-cation of a magnetic field B leads to Landau levels thatare dispersing along the field direction and have zero groupvelocity perpendicular to it. In particular, the dispersionof the Landau levels ω ( k ) = sign( B (cid:107) ) sv (cid:107) k (cid:107) (2)depends on the chirality s of the WPs and on the projec-tion of the WP velocity onto the field direction v (cid:107) . Thismakes Weyl systems the momentum-space bulk analogof the quantum Hall effect: Unidirectional channels areseparated in momentum space rather than in real space.Moreover, these chiral channels live in the three dimen-sional bulk as opposed to on the edge of a two-dimensionalsample. Such unidirectional channels might harbor inter-esting physical effects or technological promises also forclassical systems such as acoustic or electromagneticmetamaterials. However, as neither phonons nor photonscarry an electromagnetic charge e , it remained unclear ifsuch chiral Landau levels can be observed in neutral meta-materials. Looking at Eq. (1), we observe that the effect of a mag-netic field can be viewed as a space-dependent shift of theWP locations k WP ( x ). This is most easily seen in the Lan-dau gauge. For a magnetic field B = B ˆ e y , the momentum k z → k z + eBx (3)is shifted in space as a function of x as a result of mini-mally coupling the corresponding gauge field. Taking thisviewpoint of inhomogeneous WP positions, it has recentlybeen realized that the application of other space depen-dent perturbations can lead to effects alike the ones in-duced by a magnetic field both for graphene as well asfor Weyl semimetals For example, in reaction to aninhomogeneous uniaxial strain, the locations of the WPsare moved in space.
The main difference to a realmagnetic field is the chirality dependence of the shift k z → k z + sB x, (4)where B is the y-component of an axial magnetic field and moves WPs of opposite chiralities s in opposite direc-tions, cf. Fig. 1a. Interestingly, the axial nature of thefield, together with the chirality-dependence of Eq. (2)leads to co-propagating chiral channels in the presence ofan axial magnetic field.The axial field B is deeply connected to the chiralanomaly discussed in high-energy physics. Here, weset out to measure the effect of an axial field in an acous-tic system, where a B is not a formal consequence of thechiral symmetry, but manifests itself in a set of chiral Lan-dau levels. Moreover, the chosen “gauge” for B will haveobservable effects in the surface physics of our phononiccrystal.In a series of recent papers a tight-binding modelfor an acoustic Weyl system has been proposed andimplemented. The starting point of the model of Ref. 15are honeycomb layers with an in-plane hopping t n shownin gray in Fig. 1b. The stacked honeycomb layers are cou-pled via small direct hoppings t d (red) and large chiral hop-pings t c (yellow) which strongly break inversion symmetry. a r X i v : . [ c ond - m a t . o t h e r] J un K K’H H’ Γ x K K’H H’ π / π / xy z R Lhdr mny-momentum f r equen cy ab cd e FIG. 1:
Inhomogeneous Weyl point separation and ax-ial magnetic fields. a,
Hexagonal Brillouin zone with thehigh-symmetry points Γ, K , K (cid:48) , H , and H (cid:48) . The blue (red) dotsindicate Weyl points of chirality s = +1 ( −
1) whose location inmomentum space is shifted as a function of the spatial coordi-nate x due to the presence of an axial magnetic field. b, Tightbinding model and the local acoustic orbitals in the structureshown in c . The density plot indicates the pressure field in themid-plane of the the cavity at h/
2. The colors mark the phasesof the local modes. c, Unit-cell of the hexagonal acoustic crys-tal. The parameters are explained in the main text. d, ChiralLandau levels emerging at the high-symmetry points due tothe axial magnetic field. The color again indicates the chiralityof the Weyl points. e, Photo of the acoustic crystal.
The resulting spectrum is characterized by the minimalnumber of four WPs for time reversal invariant systems. Moreover, the strong inversion symmetry breaking leads toa maximal separation of the WPs which lie in the vicin-ity of the high symmetry points K , K (cid:48) , H , H (cid:48) shown inFig. 1a (see App. A).We can now induce an axial field via a space-dependentsub-lattice potential ∝ B xσ z . As the WPs with differ-ent chirality s differ by the sign of v zz in Eq. (1), such aglobal term creates an axial field ∝ sB xσ z in the low-energy theory (App. A). In other words, the WPs shift asshown in Fig. 1a.To take the step from the tight-binding model to a con-crete acoustic structure we employ the following strategy: The starting point is the unit-cell shown in Fig. 1c, wherewe are interested in the pressure waves in air surroundingthe shown structure. The pillars of radius R and height h create a hexagonal layer. The modes at the in-plane K and K (cid:48) -points are shown in Fig. 1b. We see that thelow-frequency physics around these points is governed bymodes localized at the corners of the unit cell, i.e., on ahoneycomb lattice with nearest neighbor distance d . Thewidth r of the “ventilator holes” controls the strength ofthe interlayer coupling, while the turning angle ϑ of theventilators determines the ratio t d /t c . Finally, the radius n and depth m of the holes below one sub-lattice detunethe local mode and amount to the sub-lattice potentialinducing the axial field.We now optimize the parameters of this unit cellvia finite-element simulations to find degenerate WPs atpre-determined locations δk x = δk y = 0 and δk z ∈ [ − π/ a z , π/ a z ] in the vicinity of the high-symmetrypoints K and H . The location of the WPs around K (cid:48) and H (cid:48) are fixed by time-reversal symmetry. The finalsample is then constructed by assembling unit-cells withthe pre-determined locations of the WPs to obtain thesought after term ∝ B xσ z . With this design strategy,we matched the low-frequency physics around the WPsof the acoustic sample to the low-frequency expansion ofthe tight-binding model. The details of the structure aregiven in the Appendix C. Note, that for a single WP, thefield we apply corresponds to a magnetic flux equivalentto 1% of a flux quantum or a few hundred Teslas in typicalelectronic Weyl systems.In summary, for our setup [see Fig. 1e] we expect fourWPs, where the pair K and H sees an axial field B andtheir time-reversed partners K (cid:48) and H (cid:48) the same with − B . Together with Eq. (2) this leads to the chiral Landaulevels depicted in Fig. 1d, where the center of the chiralLandau levels are at f = 7 . v ≈
40 m / s. On the surfaces with surface normal ± ˆ x , weclose the sample with hard walls, leading to well-definedzig-zag edges. The other four surfaces are left open, cf.App. C.To characterize the properties of the three-dimensionalsample we measure the spectral response of the acous-tic field. We excite sound waves at different locationson the surface of the system. The response is then mea-sured via a sub-wavelength microphone on the inside of thesystem. Using a lock-in measurement we measure phaseand amplitude information of the acoustic field (App. D).This amounts to the measurement of the Greens function G ( r i , r j , ω ) = (cid:104) ψ ∗ i ( ω ) ψ j ( ω ) (cid:105) , where ψ i ( ω ) is the acousticfield at site r i and frequency ω . By taking the spatialFourier-transform of this signal, we obtain the spectral re-sponse shown in Fig. 2.Owing to the gauge choice for the field B , the momen-tum in x -direction is not well defined. We therefore showthe projection of the spectra onto k y and k z along thehigh-symmetry lines shown in Fig. 2a. The top and bot-tom panels of Fig 2d display the touching points of twobands at the H (cid:48) , H and K (cid:48) and K points, respectively. k y k x k y k z AH’ HL LM M Γ K’ KM/LK’/H’ K/H
K K’ K K’00.15 r e l a t i v e w e i gh t right left L H’ A H L4.06.08.010.0 F r equen cy [ k H z ] L-H’-H-L
K H7.07.58.0 F r equen cy [ k H z ] K-H
M K’ Γ K M4.06.08.010.0 F r equen cy [ k H z ] M-K’-K-M F r equen cy [ k H z ] H’ F r equen cy [ k H z ] H F r equen cy [ k H z ] K’ δ k y = − π /6 K’,K,H’,H δ k y = + π /67.58.0 F r equen cy [ k H z ] K w e i gh t s abc d e top/bottom right left . k H z . k H z . k H z FIG. 2:
Observation of chiral Landau levels. a,
High-symmetry lines in the Brillouin zone and projection of the x -axis onto k y , k z . b, Fourier transform to the first Brillouin zone at k z = 0 ( k z = π/a z shows similar behavior which is not shown.) of theacoustic response at different frequencies for different speaker positions. (top/bottom, right, or left, see text). The concentrationof the response at 7 . K and K (cid:48) points shows that physics at these frequencies is governed by the band-touchingsat the corners of the Brillouin zone. The excitation sensitivity when excited from the left or right motivates the presences ofchiral channels where at K (cid:48) and K only one sign of the group velocity v y is present. At lower (higher) frequencies, the responseis dominated by a spherically symmetric spectrum. c, Quantitative analysis of the above density plots for 7 . . . K and K (cid:48) points. At 7 . K and K (cid:48) as a function of excitation location. d, Measured spectrum alongthe high-symmetry lines indicated in a . Both for the top and bottom panel one can observe the touching of two bands around7 . e . The middle panel shows a gap opening along the K - H lines. e, Dispersion alongthe chiral Landau levels at the four high-symmetry points in the Brillouin zone. The mometum-frequency relation was obtainedvia fitting the phase evolution of the measured frequency response (see text). The horizontal error-bars are smaller than thesymbol size. The vertical error bars indicate our frequency uncertainty given the dissipation in the acoustic field.
We further analyze the nature of these touching pointsthrough their associated surface physics below. The mid-dle panel shows the evolution of the band-structure from K to H , indicating a gap opening in k z direction assuringthat we indeed couple different layers via the “ventilators”.Around the frequency of the WPs, most of the acousticresponse is concentrated around the K , K (cid:48) , H , and H (cid:48) points. In Fig. 2b, we present the response for three se-lected frequencies in the k z = 0 plane. The three columnsshow the response when excited from the surfaces withsurface normal ± ˆ x at the same time (bottom/top), fromthe surface ˆ y (right) and − ˆ y (left), respectively. Three ob-servations can be made: (i) Only those modes which havea group-velocity that allows energy to be transported intothe bulk are excited. (ii) Around 7 . K and K (cid:48) are involved. (iii) When excited from the right (left), only the K ( K (cid:48) ) are excited. Thesethree observations together essentially prove that we dealwith a system with chiral channels around this frequency.This sensitivity on the excitation point is further quanti-fied in Fig. 2c, where we show the integrated weight forall x along k y at the k z = 0 plane.The open boundary conditions on the ± ˆ y surfaces al-low for a further analysis of the chiral channels. Radiationinto free space essentially allows for any mode with arbi-trary wave number to be supported inside our sample. Inparticular, no finite size quantization k α = 2 πm/L α , with α = y, z and m ∈ Z occurs. We use the fact that theeigenstates are Bloch waves ψ i ( k ) = e i k · r i (cid:18) u k v k (cid:19) , (5) − π/ − π/ k y − π π k z top surface x C = = − − π/ − π/ k y bottom surface − π y − π π k z − π/ − π/ k y − π k z top surface − π k z bottom surface FIG. 3:
Surface Fermi arcs. a,
The left and right pan-els show the numerically simulated surface dispersion on thetwo surfaces ± ˆ x . Outside the shown momentum range, onlybulk modes exist. The lines indicate equifrequency contours atthe original Weyl point frequency (white) and ±
150 Hz aboveand below. The endpoints of the white Fermi arcs are clearlymoved between the two surfaces. The horizontal lines indicatethe original Weyl points and their designed extremal locations.The middle panel shows the evolution of the k z -layer Chernnumber as a function of x . b, Surface Fourier-transform of themeasured Fermi arc on the +ˆ x surface. The white lines showthe location of the maximal response. c, Evolution of the whitelines in b as a function of frequency f on the two surfaces ± ˆ x .The opposite sign of the group velocity v y = ∂ k y f indicate thechiral nature of the surface modes winding around the sampleaccording to the k z -layer Chern number. where u k and v k are the sub-lattice weights. Using theabove structure, we can fit the phase evolution from unit-cell to unit-cell to obtain k y ( ω ). In particular, we selectthe Fourier components ψ k x ,k z ( y ) of the measured fieldscorresponding to the high-symmetry lines. We then fitarg[ ψ k x ,k z ( y )] ≈ k y ( ω ) y . This analysis is only possibleowing to the fact that with the excitation-point selectivityshown in Fig. 2b/c, we can ensure that in the chiral chan- nel region we fit the phase evolution of only one mode . Theresulting dispersion curves ω ( k y ) are shown in Fig. 2e. Theclearly visible four chiral channels in accordance with theexpectations in Fig. 1d are the main result of this study.A key property of Weyl systems manifests itself onthe two-dimensional boundaries of a finite sample: Openequifrequency contours in the surface Brillouin zone endat the projections of the bulk WPs. These open contours(or Fermi surfaces called Fermi arcs for electronic systems)can be understood via layer Chern numbers, where each k z -layer is seen as a two-dimensional system. When pass-ing a layer with a WP, this Chern number is changing by ± s . The celebrated surface arcs are then nothing but theedge states induced by these layer Chern numbers. For oursetup we expect an x -dependent k z -Chern number as wemove the location of the WPs with the axial field. In otherwords, we can expect the Fermi-arcs to terminate at differ-ent locations on the two ± ˆ x surfaces (called bottom/topbefore). This can be directly observed in the numericalsimulations shown in Fig. 3a.The experimentally measured Fermi arcs are shown inFig. 3b. From the shown Green’s function at a given fre-quency on the left, we can extract ω ( k y , k z ). Owing toour resolution in k z and the dissipation broadened fea-tures, we cannot determine the end-points of this Fermiarc. However, we can observe the evolution of ω ( k y , k z )and hence determine the group velocity on the two op-posing surfaces. In Fig 3c the +ˆ x and − ˆ x surfaces showindeed opposite group velocities, in line with the expec-tation of edges states induced by a Chern number. Theupper half of the Brillouin zone is not shown and deter-mined by time-reversal symmetry.By directly observing chiral Landau levels in a Weylsystem we have shown that axial fields rooted in the the-ory of high-energy physics can be implemented and ob-served in condensed matter systems. Many of the the-oretically predicted phenomena, such as the chiral mag-netic effect, the chiral vortical effect, strain inducedquantum oscillations and many more seem now to bereachable in systems of classical metamaterials, in cold-atoms, or in low-temperature electronic systems.We acknowledge insightful discussions with DmitryPikulin and Ady Stern. We are greateful for the finan-cial support from the Swiss National Science Foundation,the NCCR QSIT, and the ERC project TopMechMat. Appendix A: Tight-binding model
The dynamics governed by the tight-binding model ofFig. 1b can be written as ∂ t (cid:18) u k v k (cid:19) = (cid:18) γ ( k ) + α ( k ) β ( k x , k y ) β ∗ ( k x , k y ) γ ( k ) − α ( k ) (cid:19) (cid:18) u k v k (cid:19) , (A1)with γ ( k ) = 2 cos( k z a z ) (cid:20) t d + t c cos( k y a n )+ 2 t c cos( √ k x a n /
2) cos( k y a n / (cid:21) ,β ( k x , k y ) = t n (cid:20) e − i √ k x a n / + 2 cos( k y a n / e i √ k x a n / (cid:21) ,α ( k ) = 2 t c sin( k z a z ) (cid:20) sin( k y a n ) − √ k x a n /
2) sin( k y a n / (cid:21) . Here, u k and v k describe the amplitudes of the Blochwaves of Eq. (5). The lattice constants are related to thesample geometry through a n = √ d and a z = L + h . Note,however, that we only match the low-frequency physicsaround the WPs of model (A1) to the corresponding low-frequency physics of the acoustic structure. In particular,we do not intend to match the full lattice model at alllattice momenta.The model (A1) features band touchings whenever α ( k ) = β ( k x , k y ) = 0. The function β ( k x , k y ) describesa simple honeycomb layer and hence has gap closings atthe K = (0 , π/ a n ) and K (cid:48) = (2 π/ √ a n , π/ a n )points. A straight-forward analysis of α ( k ) shows thatthe full model has WPs at the K = (0 , π/ a n , K (cid:48) = (2 π/ √ a n , π/ a n , H = (0 , π/ a n , π/a z ), and H (cid:48) = (2 π/ √ a n , π/ a n , π/a z ) points, respectively, cf.Fig. 1a.The low-frequency physics around the Weyl points isfully described by the velocity tensors v K/K (cid:48) αβ = t n / ∓ t n / ∓ √ t c ,v H/H (cid:48) αβ = t n / ∓ t n / ± √ t c . Going from K ( H ) to K (cid:48) ( H (cid:48) ) two rows change sign. Whencomparing K and H , on the other hand, only one rowdiffers in sign. This explains the distribution of chiralities(red and blue) in Fig. 1a.From the above velocity tensors we see that only k z couples to the σ z matrix. If we now want to shift the WPin k z direction as k z → k z − sB x , we need to couple aspace dependent sub-lattice potential of the form V ( x ) = 3 √ t c B xσ z . (A2)It is crucial to note that the axial nature of the field arisesfrom the chirality dependent pre-factor of v zz . In otherwords, the above potential V ( x ) acquires the axial natureonly in the low-energy theory.So far, γ ( k ) in Eq. (A1) has not been addressed. Inorder to avoid any frequency shift or tilt of the conical dispersion, we need γ ( k ) = 0 for k = K, K (cid:48) , H, H (cid:48) . Astraight-forward analysis shows how this happens when-ever 2 t d = 3 t c . Appendix B: Chirality and Berry curvature
For completeness, we present the derivation of theBerry-monopole represented by a WP. A WP is a coni-cal touching of two bands, hence its general Hamiltoniancan be written as H = (cid:88) α d α ( k ) σ α , (B1)where d ( k ) is a vector linear in k . in polar coordinates d = | d | (sin ϑ cos ϕ , sin ϑ sin ϕ , cos ϑ ). The eigenvalues of theHamiltonian (B1) are (cid:15) ( k ) = ±| d ( k ) | . The eigenvectorsare |−(cid:105) = (cid:18) sin (cid:0) ϑ (cid:1) e iϕ − cos (cid:0) ϑ (cid:1) (cid:19) | + (cid:105) = (cid:18) cos (cid:0) ϑ (cid:1) e iϕ sin (cid:0) ϑ (cid:1) (cid:19) . (B2)To characterize the WP the lower band is relevant. Ne-glecting A | d | = i (cid:104)−| ∂ | d | |−(cid:105) , the remaining components of A are A ϑ = i (cid:104)−| ∂ ϑ |−(cid:105) = 0 , (B3) A ϕ = i (cid:104)−| ∂ ϕ |−(cid:105) = sin (cid:18) ϑ (cid:19) . (B4)From this the Berry curvature F = ∇ ∧ A follows: F | d | = sin ϑ . (B5)To get back to the original coordinates, this result needsto be multiplied by the Jacobian of the coordinate trans-formation. If d ( k ) = k , the Jacobian is the one of a spher-ical coordinate transformation (1 / sin ϑ | k | ) and the Berrycurvature is F = k | k | . (B6)Integrated over a shell around WP, this amounts to a fluxof 2 π , i.e., the WP corresponds to a monopole charge.However, so far we have considered the case of isotropicWPs.In the most general case the Weyl Hamiltonian is givenby Eq. (1). In this case d ( k ) (cid:54) = k . However, there is stilla linear relation between d and k . If the tensor v αβ wasdiagonal it would amount to a simple rescaling. In themost general case, this transformation amounts to a per-mutation of the rows of the Jacobian and a rescaling. Theparity of this permutation is captured by the determinantof the velocity tensor. Therefore the Berry curvature forthe generic Weyl point described by Eq. (1), is F = sign[det( v αβ )] k | k | = s k | k | . (B7)In other words, the chirality of the WP defines the chargeof the Berry monopole. Appendix C: Sample details
The sample shown in Fig. 1e is printed on a StratasysConnex Objet500 with PolyJet technology by Stratasys.The printed material is VeroWhitePlus. The 300 µ m res-olution in the xy -plane and a resolution of 30 µ m in the z -direction assure a fine enough surface finish for goodhard-wall boundary condition for the acoustic field. Atleast in the frequency/wavelength regime we are interestedin. Each layer was printed in four parts and later assem-bled and stacked.The full sample consists of L x × L y × L z = 20 × × ϑ = 2 . d = 13 mm, a z = 18 . R + r = 9 mm. Thedepth m = L/ − . ≤ n ≤ . . ≤ R ≤ . . ≤ h ≤ . ± ˆ z (ventilators) and ± ˆ y (armchair) edges. These open bound-ary conditions allow for the phase fitting mentioned in themain text. Along the ± ˆ x surface we close the sample withhard walls. In their absence the outermost mode localizedon the corner of the unit cell in Fig. 1b is largely detuned.This leads effectively to a bearded edge, where spurioussurface states appear around the Γ and A point. To avoidthese states, we close the sample on the ± ˆ x surfaces. Appendix D: Measurement and signal analysis
The acoustic signals are generated with speakers SR-32453-000 from Knowles. The pressure fields are mea- sured via a sub-wavelength microphone FG-23629-P16from Knowles with a diameter of 2 . i x , i y , i z )running from i x , i y = 0 , . . . ,
20 and i z = 0 , . . . ,
12 weexcited at (0 , ,
6) [called bottom in the main text],(20 , ,
6) [top], (10 , ,
6) [left], and (10 , ,
6) [right]. Thecrystal is scanned in a grid of 19 × × × ×
20 honeycomb unit-cells.In fact, the zig-zag terminations along the surfaces ± ˆ x limit the number of points that can be measured at thesurfaces. The data displayed in Fig. 2 are based on purebulk measurement. The first two unit cells closest to the ± ˆ x and the closest unit cells to ± ˆ z and ± ˆ y surfaces havenot been take into consideration in these analysis to avoidspurious surface effects. On the other hand, the data ofFig 3 are based on surface measurements only, i.e., datataken on the first two unit-cells. Finally, for the densityplots, the discrete spatial Fourier transforms are displayedwith the Lanczos interpolation method for visual clarity,except in Fig 3b, where the discrete nature of the mea-surements is relevant. Note, that the chiral channel phasefitting in Fig 2e is not based on an interpolation of theFourier-transform data. H. Weyl,
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