Snowflake Topological Insulator for Sound Waves

4.5 MB·

by  Christian Brendel

Snowflake Topological Insulator for Sound Waves
Christian Brendel
Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
Vittorio Peano
Department of Physics, University of Malta, Msida MSD 2080, Malta
Oskar Painter
Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, USA
7
1
Florian Marquardt
0
2 Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany and
Institute for Theoretical Physics, University of Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
n
a
We show how the snowflake phononic crystal structure, which has been realized experimentally
J
recently, can be turned into a topological insulator for sound waves. This idea, based purely on
3 simple geometrical modifications, could be readily implemented on the nanoscale.
2

]
Introduction. – First examples of topologically pro- nanoscale platform.
l
al tected sound wave transport have just emerged during The envisaged system consists of snowflake-shaped
h the past three years. So far, experimental implementa- holes of alternating sizes, in a periodic arrangement on
- tions exist on the centimeter-scale, both for the case of a triangular lattice. This snowflake topological insulator
s
e time-reversal symmetry broken by external driving [1], can be viewed as a metamaterial that supports topolog-
m such as in coupled gyroscopes, as well as for the case ically protected sound waves whose typical wavelength
. withoutdriving[2–5],suchasincoupledpendula. More- is larger than the underlying lattice scale. Such elastic
at over, a multitude of different implementations have been waves propagate along arbitrarily shaped domain walls
m envisioned theoretically [6–19]. However, it is highly de- engineered by appropriately varying the snowflake size.
sirabletocomeupwithalternativedesignideasthatmay We will show that the topological protection is guaran-
-
d berealizedonthenanoscale, eventuallypushingtowards teed if locally (at the lattice scale) the point group sym-
n applications in integrated phononics. The first theoreti- metry of the snowflake design is mantained.
o
calproposalinthisdirection[20]suggestedtoexploitthe Platform. – We assume a planar quasi-two-
c
[ optomechanical interaction to imprint the optical vor- dimensional phononic crystal slab exhibiting a six-fold
ticity of a suitably shaped laser beam to generate chi- rotational symmetry ( 6) as well as a discrete transla-
1 ral sound wave transport in a phononic-photonic crys- tional symmetry (T ) oCn a triangular lattice, with a lat-
v a
tal. On the other hand, if one wants to avoid the strong tice constant a. The most straightforward implemen-
0
driving by an external field, purely geometrical designs tation consists in the snowflake phononic crystal. This
3
3 are called for. One remarkable idea of Mousavi et al. crystalhasbeenexploredbeforeinthecontextofoptome-
6 [21] posited creating a sound wave topological insulator chanics [22], since it is both a photonic and a phononic
0 by designing a phononic crystal structure made from a crystal, although we will only make use of its phononic
1. material that would be carefully engineered by a pat- properties. Its structure is shown in Fig. 1a.
0 tern of small holes to achieve degeneracy between vi- Symmetries and folding. – Due to the -symmetry,
6
7 brations that are symmetric and antisymmetric to the theacousticbandstructureisforcedtohavCeDiraccones
1 plane of the sample. The appearance of a fine-grained at the two high-symmetry points, K(cid:126) = π/a(4/3,0) and
v: length-scalemuchsmallerthanthewavelength,however, K(cid:126) = π/a(2/3,2/√3). Now consider a single snowflake-
(cid:48)
i makes it impossible to use this idea all the way down shaped hole surrounded by six other such holes. Our
X
to wavelengths comparable to the smallest feature sizes
aim will be to break the original translational symmetry
r allowed by nanofabrication. In the present manuscript,
a by changing the central snowflake in this configuration,
we propose a very simple modification to an already ex-
thereby enlarging the real-space unit cell by a factor of
isting structure, the so-called snowflake phononic crys-
√3 (Fig. 1a). Conversely, this will reduce the size of the
tal. The snowflake crystal has already proven to be a
first Brillouin zone (BZ) by the same factor. To antici-
reliable platform for nanoscale optomechanics [22], and
pate this reduction, we imagine what happens when the
couldalsosupportpseudomagneticfieldsforsoundwaves
original band structure, obtained for the as-yet unper-
[23]. With the proposed modification, which is inspired
turbed structure, gets folded back into the new BZ (see
byanideafirstanalyzedbyWuandHuforphotonicsys-
Fig.1a-b). ThiswillmaptheDiracconesfromK(cid:126) andK(cid:126)
tems[24](seealso[16,17,25]),onewillbeabletocreate (cid:48)
oftheoldBZtotheΓ-pointofthenewBZ,formingade-
atopologicalinsulatorforsoundwavesbasedonaproven
generatepairofdoubleDiracconesat(cid:126)Γ=(0,0)(Fig.1c-
Name:TI_Paper_Fig1
2
a ra rw+�r 1.7#1c0B9lu�e rUn=itce0ll Redd U�nitrce=ll r0c R= erd U�neitrce=ll r20c 0= 2n0m00nm tHaakminigltaonrioauntevatlhidatfcolratrhifieevsitchineictyononfetchteioΓn-tpooitnhte.oWrigeianrael
11..66 valleydegreeoffreedom. Theresultscanalternativelybe
b xy equency in GHzFreuqncy oin Hz1111....5454 ufittwonrnsodtiecdraocsduotvonbootlceedaxDittneiidrntahcf[eo2cr4for]n.Cam6eWs-seoeywfmsottrmhakreetotrfwreigtcihtuhelsatstrryhu(mec∆temurigre=eetsnry0sit)naasrtntgehousewmoflfpeathnhkotees-
fr11..33 array. Theyarelabeledby ψ ,withσ = 1beingthe
σ,τ
| (cid:105) ±
~� L~ N~ K~ 11..22 qtautaiosin-angular momentum with respect to the 3-fold ro-
K~0 M~
ky kx 1.1!~� M~M K~KN~Nq~�!uNNa~si -L~Lmom~!�enNt~uNNmN~ L~L ~�! N~N Rˆ2π/3|ψσ,τ(cid:105)=e−iσ23π |ψσ,τ(cid:105), (2)
and τ denoting the valley degree of freedom (τ =+1 for
Figure 1. (a) Snowflake crystal slab defined by the parame-
ters (a,r,w,∆r) and the slab’s thickness d. ∆r indicates the K vs. τ = 1 for K(cid:48) ). Under time reversal Tˆ (complex
−
Picture prordaucdeidu sbyd:eviation of every thDierfdau(ltr Feodn)t-Ssinzeo:wflake. For the spe- conjugation) and inversion Rˆ they obey
π
3d-snowflake imacgeisa:lcaseof∆r=0weobtainareguHlaelrvesticnao-Fwonfltsaizek: e16ppxatternwitLahTex-Fonstize:16px
Snowflake/TopoInstuhlatoer/Zedichinsucngrene/*t.meph translational symmHeetlvreyticTa-aFo(nbtsliuzee: 2u0npixt cell)L.aT(ecx)-Fonstize: 20px ψ =Tˆ ψ =Rˆ ψ . (3)
bSannodw flsatrkuec/TtoupreosIn:ssuhlatoor/wFEMs/20t1h6_1e1_1c5o_UrnirtceellsFopldiongn/Pldot1i.nmgbaHndelvsetrtiuccat-uFroentasliozen:g 2a4ppxathLapTaesxs-iFnognstize: 24px | σ,τ(cid:105) | −σ,−τ(cid:105) π| σ,−τ(cid:105)
the high symmetry points (blue BZ in b) that features Dirac The exact mode shapes of ψ for the particular case
σ,τ
cones at K(cid:126) and K(cid:126)(cid:48). Describing the same system with an en- oftheoriginalsnowflakecry|stala(cid:105)reexplainedindetailin
larged unit cell (red) results in a reduced BZ (red), folds the [23]. In order to derive the Hamiltonian, we use two sets
band structure, and maps both Dirac valleys to the (cid:126)Γ-point ofPaulimatricestospanthe4-dimensionalHilbertspace.
(d). ∆r(cid:54)=0breaksT butmaintainsT√ (requiringthered
unit cell and red BZ)aand C6. This gap3saboth Dirac double Waneddaenfionteheorneonseetfoforrtthheeqvuaallseiyadneggurleaerodfefgrreeeedoomf frτˆe{exd,yo,zm}
cones (e). In the band structures (c-e) modes symmetric to
σˆ , such that τˆ ψ = τ ψ and σˆ ψ =
the x-y-plane are displayed in brighter colors. [Here we used σ{xψ,y,z}, and the usuazl|seσt,τo(cid:105)f Paul|i mσ,aτt(cid:105)rices hozld|sσin,τ(cid:105)this
(a,r,w,d)=(5000,1800,750,220)nm and assumed the crys- σ,τ
| (cid:105)
basis.
talslabmadeofsilicon(Si)withYoung’smodulusof170GPa,
mass density 2329kg/m3 and Poisson’s ratio 0.28] WenowwritedowntheHamiltonianasaTaylorseries
up to linear order in(cid:126)k by using the above matrices. We
keep only terms that are invariant under Tˆ and Rˆ ,
π/3
d). Only in the next step we actually do break Ta, by which are the symmetries of our snowflake crystal (even
carrying out the afore-mentioned change of the central for∆r =0). Thisleavesuswithonlythefollowingterms:
snowflake, increasing the lattice constant a a˜ = √3a (cid:54)
andestablishingthenewsymmetry T ,allth→ewhilepre- Hˆ =gτˆ +vτˆ (k σˆ +k σˆ ). (4)
a˜ k x z x x y y
serving around the unit-cell centers. This opens a
6
completeCgapattheΓ-point(Fig.1e),whichcanbetopo- UptoaunitarytransformationthisHamiltonianisequiv-
logical in nature, as will be explained below. alent to the large-wavelength limit of the Bernevig-
The easiest way to implement this modification of the Hughes-Zhangmodelforatopologicalinsulator[26]. The
geometryistochangetheradiusofthecentralsnowflake first term in Eq. (4) is induced by the breaking of the
in each (enlarged) unit cell by ∆r, with ∆r =0 referring Tˆ symmetry and is responsible for gapping the degen-
a
to the original snowflake crystal (Fig. 1a). erate Dirac cones. In other words, g can be interpreted
The band structures shown in Fig. 1 have been ob- as a mass, which can change sign. The unitary sym-
tained from full finite-element simulations of the equa- metry that allows to cast Eq. (4) in a block-diagonal
tions of linear elasticity, solving the eigenvalue problem form is the spin degree of freedom Sˆ=τˆ σˆ . Combined
x z
with the time-reversal operator, it gives rise to a pseudo
div(cid:104)E:(cid:2)gradψ+(gradψ)T(cid:3)(cid:105)= 2(cid:37)ω2ψ, (1) time-reversal symmetry (TˆSˆ), which has the peculiarity
−
that it squares to minus the identity, directly leading to
with ψ being the complex three-dimensional wave func- Kramer’sdegeneracy. AttheΓ-point,thecommoneigen-
tion related to the mechanical displacement field u = states of Hˆ and Sˆ are the states p and d which
Re(cid:2)ψ eiωt(cid:3) of the crystal, E the elasticity tensor, and obey τˆ d k==0 d , τˆ p = p| ±,(cid:105)Sˆ p |=±(cid:105) p ,
· x ± ± x ± ± ± ±
(cid:37) the mass density. Here : is a short-hand for the ten- and Sˆ d| =(cid:105) |d (cid:105). On|eca(cid:105)n sh−ow| th(cid:105)att|hes(cid:105)e sta±te|s ar(cid:105)e
sor product, [E : gradψ] = E ∂ ψ . For concrete- | ±(cid:105) ±| ±(cid:105)
ij ijkl l k actually of p- and d-type with respect to their behavior
ness, we envisioned a phononic crystal slab of thickness
under 60-degree rotations:
220nm, but much smaller slabs have been fabricated al-
ready, down to unit cells of only a few hundred nanome- Rˆπ/3(cid:12)(cid:12)p±(cid:11)=e±iπ/3(cid:12)(cid:12)p±(cid:11), Rˆπ/3(cid:12)(cid:12)d±(cid:11)=e±i2π/3(cid:12)(cid:12)d±(cid:11) .
ters lateral extension (acoustic frequencies would scale
inversely with the linear dimensions, as usual). Note that away from the Γ point only states of the same
Effective Hamiltonian. – We now derive an effective helicity (s = 1) will get mixed to form the finite k
± −
Name:TI_Paper_Fig1
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3
�-mode frequencies a 1.50 b upper domain only f g max
p± -type + 1.48
5 MHz | i
band gap |d±i-typbeand gap 0 p3a·kx y in GHz 1111....45448064 c lower domain only rms displacement
nc 1.46
ue 1.44
-200 -100 0 +100 +200 - q 0
e
� r in nm fr d e
1.50
Figure2. Bandinversion: Frequenciesofthe|p±(cid:105),|d±(cid:105)modes y 1.48 (f)
at the Γ-point (cid:126)k = 0 evolving for a sweep of the central 1.46 (e)
x 1.44
snowflake radius r . A snapshot of the corresponding dis- (g)
c
placement fields for rc = 1600nm and rc = 2000nm is also z -⇡ 0 -⇡ 0
shown. The in-plane displacement field is directly visualized unt quasi momentum p3a·kx
e
bythedeformation,whereastheout-of-planedisplacementis m
�r=-200nm encoded in the colorscale. d(p)-�orrb=it+a2ls00anrme (anti-z)sy-mcmoetmricpoFnigueren3t. (acea) Snowflake strip configuration comprising two
under rotation by 180 degrees. different dodisplmains. The lower domain has smaller central
snowflakes (blue, ∆r=−200nm) whereas the corresponding
ones in the upper domain are enlarged (red ∆r=+200nm).
Picture produced by: Default Font-Size:
The depicted strip hosts n = 11 resized snowflakes (n = 6
l
Snowflake/TopoInsulator/FEM/2017_e0i1g_1e4n_psdtmaodteess/.The Hamiltonian teHrmelvsetiwcah-Ficohntsiazree: 16lipnxear in bLlauTeexa-Fnodnstnize:=16p5x red). (b) Acoustic band structure of the
PSrnoodwuflcaekDe/aTtoapTohInirsdu.mlator/FEM/2017_t0h1_e14_qpudmaosdeims/Plootm.mentum (cid:126)k willHeinlvdeuticcea-tFroanntssiitzieo:n 2s0opnxly be-LaTebxlu-Feodnostmizaei:un2o0nplxy, with a system size n = 17, i.e. a strip
SPnlootwBflaankdeG/TaoppDoaIntsau.mlator/FEM/2017_t0w1_e14e_npdmstodaets/eswhose60-dHeegrlveeetqicuaas-iF-aonngtusliazrem: 2o4mpexntaLdaifT-ex-Fcoomnpstriiszien:g2ju4sptxone domain (∆r=−200nm). The blue area
ferbyonequantum: p+tod+andp toPdict.uOren plyrofudrutcheerd by:shows the band structure of tDheefcaourlrte Fspoonntd-iSnigzein:finite crys-
− −
away from the Γ-point, higher-order terms (e.g. k2) tal,therebyindicatingthestrip’sbulkbanHdes,lvwethicear-eFaosntthseizeso: l1e6px LaTex-Fonstize: 16px
∼ remainingbandsaredegeneratepairsofedgestateslocalized
can eventually couple the + and states, i.e. mix dif-
− folder/subfolder/suabtstuhbefoulpdpeerr/c(oud)ea.nmd lower (l) bouHnedlavreyticoaf-tFhoensttsriizpe.: (2c0)px LaTex-Fonstize: 20px
ferent helicities. An indirect signature of this coupling
Counterpart to (b) for the upper domain (∆r = +200nm),
is the lifting of the degeneracy of the two helicities. Re- Helvetica-Fontsize: 24px LaTex-Fonstize: 24px
essentiallythestripversionofFig.1e. Astripcomprisingtwo
markably, for our specific design, the splitting remains
domains (∆r = ∓200nm,n = 18,n = 17) has the band
l/u l u
smaller than 1.5% of the band gap even for a quasimo- structuredepictedin(d,e). Inadditiontothebulkmodesand
mentum as large as 1/4 of the distance to the boundary the edge modes at the physical boundaries (g), it reveals the
of the Brillouin zone, (cid:126)k π/(6a). two topologically protected counterpropagating modes local-
| |≤ izedatthedomainwallwherethegapcloses[wavefunctionin
In a topological insulator, the helical edge states are
(f)]. Forclarity, inallthesebandstructureswejustdepicted
confined along domain walls that separate regions of op-
the modes symmetric to the sample plane.
posite mass g. Here, we can simply tune the mass g by
changing the radius of the central snowflake, see Fig. 2.
As discussed above when all snowflakes have the same
radius (corresponding to ∆r = 0) the p and d bands mass term g(y) varying along the transverse (finite) y-
are degenerate at the Γ point (g = 0).−For a d−ecreased direction from g0 to +g0. We employ the continuum
−
(increased) radius r of the central snowflakes, the d limit based on Eq. (4), using the envelope function ap-
c
orbitals have larger (smaller) energy, corresponding to−a proximation. Following this standard procedure [27–29],
positive (negative) mass g, cf. Fig. 2. In order to under- we obtain a right (left) moving state, with a linear dis-
standthisbehavior, itisusefultoobservethatthep-or- persion E = vkx (E = vkx) for s = 1 (s = +1), that
− −
bitalshaveextranodesattheexternallinksleadingoutof decays exponentially away from the domain wall, with a
the(enlarged)unitcell,enforcedbyaphase-differenceof penetrationdepthξ =v/g0. Thisbehaviour,obtainedin
π across those links. When all snowflakes have equal ra- thecontinuumlimit, isconfirmedindirectfinite-element
dius,theadditionalenergycostassociatedwiththelarger simulations of the microscopic equations of elasticity for
phasegradient(comparedtoad-orbital)acrosstheseex- a strip geometry.
ternallinksexactlyoffsetsthebenefitofareducedphase Helical edge channels in finite-element simulations. –
gradientonapathencirclingthecentralsnowflake. Obvi- Wewillnowverifytheabovestatementsforthesnowflake
ously, stronger (weaker) internal links [corresponding to crystal,usingthefullmicroscopicacousticequations. For
adecreased(increased)centralsnowflakeradiusrc]favor that purpose, we consider a strip with a finite extent
energetically the p- (d-) states, eventually leading to the alongy. Beforeweinvestigatetheeffectsofdomainwalls,
behavior displayed in Fig. 2. we first briefly discuss the strip with a spatially homoge-
Strip in the continuum model. – To ascertain the ap- neousmassterm,i.e. composedofthehexagonalbuilding
pearance of edge states at an interface with a band in- blocks comprising three snowflakes (Fig. 1a, red shaded
version, we first consider a strip configuration with a area), with the central snowflake’s radius deviating by
Name:PMF_Geomentry.pdf 4
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tained by the COM−SOL finite20element solver. The Dirac 20 |p�i-excitation 330535
cones are replaced by a complete bulk band gap. There 15 e2530
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are states that appear in ad1d5ition to the bulk-derived 10 on20
bands and that are localized at the boundaries (Figure p1525
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15 15 400 40
erate; one state is localized at the upper and the other 10 10 300 30 5
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cally a superposition of the band structures of the bare -10-40 -30 -20 -10 0 10 20 30 40 0
strips with ∆r = 200nm. However, in addition to the Figure 4. Finite size sample with arbitrarily shaped sharp
±
bulkbandsandtheafore-mentionedstatesatthesPaicmtuprel eproduc2d50eodChma nbgeMyaaxA:...inwall,simulatedusinDgefaautlti gFhotn-tb-Siinzde:ingmodel. Onepo-
boundaries, two states appear that traverse the gap en- larization (p−) is injected and propHaelgveatitcea-sFotnotsizteh: 1e6prxight. (a)LaTex-Fonstize:16px
tirely, with a linear dispersion of opposite slope (gfroolduepr/subfoDldeert/asuilbesdubrfeopldreers/ceondtea.tmion, showinHgelvaeltlic6a-Ftorniatsnizgele: s20ipnxside eaLcahTex-Fonstize: 20px
velocity). Moreover,thereisnodiscernibleavoidedcross- hexagonal unit cell. ColorHinedlivceatticeas-tFhoenstsqiuzaer: e24ofpxtheLsaoTuenxd-Fonstize: 24px
wave amplitude, i.e. the energy. (b) Extracting the compo-
ing between these two states, underlining the absence of
nentofthep− modeinsideeachunitcell. (c)Scatteringinto
back-scattering expected for topological insulators due 35 ChangeCircleSize
theotherhelicity(p+)ispresent,butstillstronglysuppressed
to the symmetry-protection. Figure 3e shows the quasi-
evenatsharpcorners. Notethatinpanel(c)weenhancedthe
momentum resolved wave function of the right-moving
depictedenergybyafactorof10tomaketheweakscattered
state(redenergydispersioninpaneld). Forsmallquasi-
component visible. (d) Weak component of opposite polar-
momenta it is highly confined around the domain wall, ization (p+) appearing at a corner [color scale different from
withatypicalpenetrationdepthinverselyproportionalto before]. (e)Domainwallgeometry(transversetocorner). (f)
thesizeofthebulkbandgap(asexpectedfromξ =v/g Thetotalfractionofp+ polarizationdecreasesforasmoother
0
derived in the continuum model). domain wall.
Effects of disorder. – The engineered symmetry Sˆ will
not protect against competely arbitrary (generic) disor-
der. This behaviour is in fact common to all bosonic effectsofdisorder. Tokeepthecomputationaleffortman-
topological insulators [30]. However, Sˆ is valid near the ageable,weapproximatethesnowflakecrystalbyatight-
Γ-point. Therefore,genericallyspeaking,onemayexpect bindingmodel,wherethesitesofthismodeldirectlycor-
that this protection is preserved near smooth defects, respond to the physical triangles that are arranged in a
interfaces, and sample boundaries, which admix only honeycomb lattice. Note that in order to obtain most
wavevectorsclosetoΓ. Ontopofthis,evenforsharpdo- of the insights we are aiming for here, it would be more
mainboundaries,wefindinthefullfinite-elementsimula- generally sufficient to use any tight-binding model that
tions that there is no discernible back-scattering for the exhibits the same symmetries that underly the topolog-
parameters we have explored. Such a scattering would ical protection in our system. We use a hexagonal unit
showupintheformofaminigap,i.e. anavoidedcrossing cell comprising six sites with equal eigenfrequencies. To
between the counterpropagating edge states. This unex- mimicthemassterm,hoppingrateswithinaunitcell,J˜,
pected robustness in the presence of sharp interfaces has deviate from hopping rates between different unit cells,
been observed as well in works analyzing photonic struc- J. Again there is a direct interpretation: By changing
tures based on C symmetry [24, 25]. Moreover, one theradiusofthecentralsnowflakeintherealmicroscopic
6
can show that even short-range defects of a certain kind, structure, the links connecting the triangles also change,
that are compatible with the symmetry Sˆ, do not in- consequently leading to different coupling between two
duce scattering between counterpropagating edge states. triangles.
For example, both of the following perturbations have As shown above, the unidirectional edge states are su-
zero matrix element between sectors of opposite helicity: perpositions of the states with the same helicity s (e.g.
changing the masses of all 6 triangles in a given unit cell p+ andd+). Figure4showstheenergydistributionfora
by the same amount, or leaving out a single snowflake mechanical wave that is propagating at the domain wall
hole. between two domains with different mass terms. We ex-
Arbitrary boundaries. – We now consider a finite sys- cite a whole unit cell (indicated by the yellow arrow)
tem with anarbitrarily shaped boundary to observe uni- witha p -typemodeshape, therebylaunchingasound
−
| (cid:105)
directional transport of mechanical excitations and the wave that just propagates to the right. By calculating
5
the linear response of each lattice site to this particular ideas presented in [23]). The simplicity of the nanoscale
excitation, we obtain the propagation probability (mod- design (and the small dimensions of the unit cell) will
ulus squared of the Green’s function) of the mechanical turn such a modified snowflake crystal into a versatile
excitation. As mentioned in the general discussion near platform for generating arbitrary phononic circuits and
Eq. (4), the symmetry Sˆ is obeyed to a very good ap- networks[31,32]onthechip,whichmaycoupletohybrid
proximationwithinaratherlargefractionoftheBrillouin quantumsystemsofvariouskindsandcouldalsocontain
zone. As a consequence, any remaining admixture of the optically tuneable non-reciprocal elements [33].
opposite helicity is quickly suppressed when a sharp do- Acknowledgments. –WeacknowledgefundingbyERC
mainwall(withasuddenjumpinhoppingamplitudes)is StG OPTOMECH and the EU HOT network, as well
replaced by even only a slightly smoothened wall. This as the Max Planck Society. O.P. acknowledges sup-
is confirmed by the numerical simulations displayed in port by the AFOSR-MURI Quantum Photonic Mat-
Fig. 4f. ter, the ARO-MURI Quantum Opto-Mechanics with
Conclusions. –Thesnowflaketopologicalinsulatorfor Atoms and Nanostructured Diamond (grant N00014-15-
soundwavesproposedhereisstraightforwardtofabricate 1-2761), and the Institute for Quantum Information and
atanyscale,downtothenanoscale. Itcanbeexcitedand Matter, an NSF Physics Frontiers Center (grant PHY-
read-outusingavarietyofdifferentapproaches,including 1125565) with support of the Gordon and Betty Moore
electrical,mechanical,andoptomechanical(adaptingthe Foundation (grant GBMF-2644).
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