Anomalous Topological Phases and Unpaired Dirac Cones in Photonic Floquet Topological Insulators

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by  Daniel Leykam

Anomalous Topological Phases and Unpaired Dirac Cones in Photonic Floquet
Topological Insulators
Daniel Leykam1, M. C. Rechtsman2, and Y. D. Chong1,3
1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371, Singapore
2Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA
3Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore
(Dated: June 8, 2016)
We propose a class of photonic Floquet topological insulators based on staggered helical lattices
andanefficientnumericalmethodforcalculatingtheirFloquetbandstructure. Thelatticessupport
anomalous Floquet topological insulator phases with vanishing Chern number and tunable topo-
logical transitions. At the critical point of the topological transition, the bandstructure hosts a
6
single unpaired Dirac cone, which yields a variety of unusual transport effects: a discrete analogue
1
ofconicaldiffraction,weakantilocalizationnotlimitedbyintervalleyscattering,andsuppressionof
0
Anderson localization. Unlike previous designs, the effective gauge field strength can be controlled
2
via lattice parameters such as the inter-helix distance, significantly reducing radiative losses and

n enabling applications such as switchable topological wave-guiding.
u
J PACSnumbers: 42.82.Et,03.65.Vf,73.43.-f

7

] Photonic topological insulators (PTIs) are an emerg- unpaired Dirac cones, defeating the “fermion-doubling”
s
ing class of photonic devices possessing topologically- principle [29]. It is thus noteworthy that these unusual
c
i nontrivialgappedphotonicbandstructures[1–14],analo- features were not accessed by the experiments of Ref. 6.
t
p gous to single-particle electronic bandstructures of topo- The waveguide array was always observed in a stan-
o logical insulators [15]. They have potential applications dard Chern insulator phase generated by weak periodic
.
s as robust unidirectional or polarization-filtered waveg- driving; transitions to any other topologically nontrivial
c
uides, and as scientific platforms for probing topological phase were unachievable because the strength of the ef-
i
s effects inaccessible in condensed-matter systems. In the fectivegaugefieldwascontrolledbythebendingradiusof
y
technologically important optical frequency regime, only the helical waveguides. Radiative losses, which increase
h
p two PTIs have been demonstrated in experiment: arrays exponentiallywithbending[30],cametodominatebefore
[ of helical optical waveguides [6], and coupled ring res- any“strongfield” topologicaltransitionswerereached[6].

onators [7–9, 11, 13]. These two different designs each
2 This paper describes a class of waveguide arrays over-
possess unique advantages. Waveguide array PTIs, for
v coming the above limitations, allowing for the observa-
4 instance, allow the propagation dynamics of topological
tion of topological transitions between conventional in-
6 edge states to be directly imaged [6]. The design of the
sulator, Chern insulator, and AFI phases, as well as un-
7 waveguide array PTI is based on the “Floquet topologi-
1 paired Dirac cones at the transition points. To the best
calinsulator” concept[16–19],whichoriginallydescribed
0 of our knowledge, AFI phases and unpaired Dirac cones
quantum systems with time-periodic Hamiltonians; the
1. haveneverbeendemonstratedinoptical-frequencyPTIs.
ideaisthattopologicallynontrivialstatescanbeinduced
0 Continuously tuned transitions between trivial and non-
via periodic driving [16–19], rather than via magnetic or
6 trivial topological phases, or into an AFI phase, have
1 spin-orbiteffectsinastaticHamiltonian. InthePTI,the
never been observed in any 2D PTI. Our design is based
: Hamiltonian describes the classical evolution of the op-
v on“staggered” latticesofhelicalwaveguides,witheachof
tical fields in the waveguide array, and its periodic drive
i the two sublattices having a different helix phase. The
X arises from the helical twisting of the waveguides [6, 20].
two-band Floquet bandstructure can be tuned to differ-
r
a ent topological phases by varying the nearest-neighbor
Floquet topological insulators are highly interesting
coupling strength or sublattice asymmetry. We can ac-
because they exhibit topological phenomena that have
cess different topological phases while maintaining small
no counterparts in static Hamiltonians [21–28]. For ex-
bending radii; the bending losses for reaching the con-
ample, there can exist two dimensional (2D) “anomalous
ventional insulator to AFI transition are reduced by
Floquet insulator” (AFI) phases which are topologically
around two orders of magnitude compared to the topo-
nontrivial—including hosting protected edge states—
logical transition discussed in Ref. 6. This design thus
despite all bands having zero Chern number [11, 13, 22–
shows promise for low-loss topological waveguides that
26]. When disorder is introduced, the anomalous topo-
are switchable (e.g. via optical nonlinearity).
logicaledgestatesbecometheonlyextendedstates,with
all other states localized [27]. At critical points between The critical Floquet bandstructure hosts an unpaired
topological phases, Floquet bandstructures can exhibit Dirac cone. This is unlike any other previously ob-
2
separation between waveguides, so that each waveguide
approaches its four nearest neighbors in turn at each
quarter-cycle. Similar schemes can be implemented in
otherlatticegeometries,suchasahoneycomblattice[23].
For simplicity, this paper focuses on the square lattice.
First, we model the lattice in a tight-binding approxi-
mation similar to a 2D discrete-time quantum walk [39].
Since the inter-waveguide couplings are evanescent, we
assume each waveguide couples to one neighbor at a
time. The Floquet evolution operator, Uˆ, is defined by
ψ(z + Z) = Uˆψ(z), where Z is the helix period and
y ψ = (ψ ,ψ ) are the tight-binding amplitudes on each
g A B
er sublattice. Uˆ factorizes into a series of independent two-
n
e waveguide couplings, separated by free evolution:
i
s
a
u Uˆ =Sˆ(−k )Sˆ(−k )Sˆ(k )Sˆ(k ), (1)
Q − + − +
√
with the notation k ≡ (k ± k )/ 2, where k are
± x y x,y
the crystal momenta in units of the inverse waveguide
separation in the absence of modulation; and
FIG. 1. (Color online) A square staggered lattice of helical
(cid:18) ei∆cosθ −iei(∆+κ)sinθ (cid:19)
waveguides. (a) Schematic of two neighboring waveguides, Sˆ(κ)= c c , (2)
twisting clockwise along the propagation axis z with relative −ie−i(∆+κ)sinθc e−i∆cosθc
phaseshiftπ. (b)Crosssectionofthelatticepotentialateach
helixquarter-cycle,withthecirculartrajectoriesoverlaid. (c) where∆isasmalldetuningbetweenthesublatticeprop-
Phase diagram of the tight-binding model, in terms of the agation constants (which can be implemented by having
sublattice asymmetry ∆ and coupling strength θc. Red dots different waveguide refractive indices), and θc is the cou-
indicatetheparametersforthebanddiagramsinFig.2. The pling strength. Since Uˆ is unitary, its eigenvalues have
nontrivial phase forms an AFI along the ∆ = 0 line and a theformeiβ(k)whereβ(k)isthe“quasienergy” spectrum.
Chern insulator for ∆(cid:54)=0. (d) Tight-binding bulk spectrum
Notethatthismodelresemblesthe2Dquantumwalkde-
at the black square in (c), along the phase boundary.
scribedinRef.[22],withtimeevolutionreplacedbyprop-
agationinz,andthatSˆ(κ)isthemostgeneralscattering
matrix permitted by the lattice symmetries [40].
served photonic band-crossing points, which involve ei-
Fig. 1(c) shows the phase diagram of the quasienergy
ther paired Dirac points [33], quadratic dispersion [34],
bandstructure, as a function of ∆ and θ . The system is
or an attached flat band [35, 36]. The unpaired Dirac c
a trivial insulator at weak couplings, and a topological
coneisreminiscentofthechiralbandstructureoftheHal-
insulator above a critical coupling strength [40]. At the
dane model with broken parity and time-reversal sym-
transition,thebandstructurehasanunpairedDiraccone
metries [31], or surface states of 3D topological insula-
attheΓpoint,asshowninFig.1(d). Increasing∆pushes
tors [15, 32]. Here, it arises from a Floquet process, and
the two bands away from quasienergy β = 0 and closer
specifically the fact that the Floquet bandstructure is a
towards reconnecting at β = ±π, reducing the critical
“quasienergy” spectrum (see below). Wave propagation
coupling strength. At θ = π/2, the bands merge into a
inthecriticalPTIisimmunetotheinter-valleyscattering c
topological flat band [22].
thatoccurswithpairsofDiraccones[37]. Basedonthis,
The ∆ = 0 case is particularly interesting. Here, the
we demonstrate a novel “discrete” conical diffraction ef-
sublattice symmetry enforces a line degeneracy at the
fect,generatedbyexcitingasingleunitcell,aswellasre-
Brillouin zone edge, so there is a single band gap. For
sistancetolocalizationinthepresenceofshort-rangedis-
smallθ ,thespectrumresemblesthatofanunmodulated
order [38]. We note that although similar Floquet band- c
square lattice with a single Bloch band folded back onto
structuresthatcanhostunpairedDiracconeshaveprevi-
itself. At the critical point θ = π/4, the formation of
ously been studied theoretically [9, 17, 29], those studies c
the Dirac cone leads to a completely gapless spectrum.
lacked information about the propagation dynamics of
A long-wavelength expansion of Uˆ ≈ exp[−i(Hˆ −π)]
theDiracconestates,whichwecaninvestigateusingour D
abouttheΓpointyieldsaneffectiveDiracHamiltonian,
experimentally realistic waveguide array models.
An example ofthe staggered helix design is the square Hˆ (k)=−k σˆ +k σˆ −4(θ −π/4)σˆ , (3)
D x z y y c x
latticeshowninFig.1(a)–(b). Therearetwosublattices,
forming a checkerboard pattern; the helices on each sub- where σˆ are the Pauli matrices. For θ > π/4, the
x,y,z c
lattice are shifted relative to each other in the z direc- system is an AFI [22–26] with unidirectional topological
tion, by half a helix cycle. This produces a z-dependent edge states.
3
Toapplytheseideastoarealisticphotoniclattice,such
asfemtosecondlaser-writtenwaveguidesinfusedsilica[6,
25, 41], we now go beyond the tight-binding description.
Aphotoniclatticeisdescribedbyaparaxialfieldψ(r,z)
governed by the Schrödinger equation
1 k δn(x,y,z)
i∂ ψ =− ∇2ψ− 0 ψ, (4)
z 2k ⊥ n
0 0
where ∇2 = ∂2 +∂2, k = 2πn /λ, and the refractive
⊥ x y 0 0
index is n =1.45 at wavelength λ=633nm, with mod-
0
ulation δn∼7.5×10−4. Similar to real experiments, we
give the waveguides elliptical cross sections with axis di-
ameters 11µm and 4µm [6], as shown in Fig. 1(b). They
form a square lattice with mean waveguide separation
a, helix radius R , and pitch Z. We can increase the
0
FIG.2. (Coloronline)Bandstructuresforasemi-infinitestrip
effective coupling, θ , by increasing 1/a, R , or Z.
c 0
10 unit cells wide. Blue points are obtained from the contin-
Direct calculation of the Floquet bandstructure for a
uummodel,andredcurvesfromthetight-bindingmodel. (a)
continuum model (as opposed to a tight-binding model)
Trivialinsulator(a=25µm,θ ≈0.17π,∆=0). (b)Anoma-
c
is a nontrivial task, because the quasienergies βn,k are lous Floquet insulator (a = 20µm, θc ≈ 0.4π,∆ = 0). (c)
defined modulo 2π/Z, so there is no ground state for Critical phase (a = 23µm, θ ≈ π/4,∆ = 0). (d) Chern
c
numerical eigensolvers to converge on, and continuum insulator (a=23µm, θc ≈0.15π,∆≈π/4).
(unguided) modes enter in an uncontrolled way. We de-
vised an efficient method for performing this calculation
bytruncatingtheevolutionoperatorUˆ toabasisformed
by the static Bloch waves at z = 0. This amounts to
a quasi-static approximation neglecting coupling to un-
bound (continuum) modes. Bending losses can be es-
timated via the norm of the Floquet evolution operator
eigenvalues,byfindingsitsdeviationfromunitarity. Fur-
ther details are given in the Supplemental Material [40].
We now fix R = 3µm and Z = 2cm. This yields a
0
loss of <∼ 0.02dB/cm, independent of topological phase,
which we tune by varying a and/or ∆. By constrast, the
strength of the effective gauge field in the unstaggered
lattice of Ref. 6 was tuned by increasing R , which also
0
increasedthebendinglossesexponentially[30]. Thatlim-
FIG.3. (Coloronline)Outputintensityprofileafterprogation
ited the system to the “weak field” perturbative regime;
through 5Z, with one edge site initially excited (red arrow).
losses exceeded 3dB/cm before reaching a predicted
Reducingthelatticeperiodcausesatransitionintoananoma-
strong field topological transition (between two Chern
lous Floquet insulator phase with topological edge states.
insulators), making that transition unobservable.
Fig. 2 shows the band structure for a strip geome-
try. For comparison, results from the truncated-Bloch trivial gap and a nontrivial gap, and the two bands have
methodareplottedtogetherwiththeresultsfromafitted Chern numbers 1 and -1, as in the Haldane model [42].
tight-binding model [40]. The two methods agree well, The topological transitions can be probed via beam
particularly in the weak-coupling regime. In Fig. 2(a), propagation experiments. Fig. 3 shows beam propaga-
weseethatthesystemisatrivialinsulator,withasingle tion simulations with a single waveguide initially excited
band and a single gap (note that the spectrum is peri- along the edge. For large a, with the lattice in the triv-
odic along the β axis), whereas in Fig. 2(b), the gap has ial phase, the excitation simply spreads into the bulk.
closedandreopened,inducingchiraledgestatescentered Upondecreasinga,weobserveastrongly-localizedmode
at β ∼ π. This is the AFI phase; the Chern number of that propagates unidirectionally along the edge, includ-
the single band is necessarily zero, despite the presence ing around corners. This is a clear signature of a topo-
of chiral edge states. The transition point is shown in logical transition to the AFI, which has never been ex-
Fig. 2(c), which features an unpaired Dirac point at the perimentally demonstrated in a 2D photonic lattice. We
center of the Brillouin zone. Fig. 2(d) shows a ∆ (cid:54)= 0 stress that varying a is just one of many possible tuning
case, corresponding to a Chern insulator; there is both a methods. Due to the strong sensitivity of the evanescent
4
FIG. 5. (Color online) Disorder-insensitivity at the critical
point. (a) Fourier intensity of a broad (width 5a) probe
beam,afterpropagating60Zthroughaweakly-disordered[40]
100×100 lattice, averaging over 20 disorder realizations. A
FIG.4. (Coloronline)Conicaldiffractionarisingfromtheun- weak antilocalization dip occurs in the backscattering direc-
pairedDiracpoint. Intheupperpanels,theinitialexcitation tion (white arrow); the color scale saturates in the forward
isGaussian; inthelowerpanels, onlyoneunitcellisexcited. direction. (b)ParticipationnumberP normalizedbynumber
(a,d)Inputand(b,e)outputbeamintensity. (c,f)Phasepro- of waveguides 2N2 =1800, for the tight-binding eigenmodes
fileofoutputcross-polarizedpseudospincomponent,showing of a strongly-disordered lattice. Localization is absent for
clockwisephasecirculation(blackarrows). Thelatticesizeis ∆=0, but Lifshitz localization tails appear for ∆=π/8.
16×16 unit cells, and the propagation distance is L=6Z.
plane. WhilethatisthecaseforstaticHamiltonians,here
coupling strength to waveguide mode localization, there diffractionispreservedbytheuniquefeaturesoftheFlo-
areotherinterestingwaystoachievecontrollableswitch- quetbandstructure: thespectrumisentirelygapless,and
ing between topological phases, such as the Kerr effect hasnolocalbandmaximaorminima. Consequently, the
or thermal tuning. band velocity is nonzero almost everywhere, and the ini-
It is also interesting to study the behavior of the lat- tial excitation evolves into a discrete conical-like diffrac-
tice at the critical point of the topological transition, tionpatternwithadarkcentralspotandnonzerovortex
where the quasienergy bandstructure contains an un- charge. As shown in Fig. 4(d)–(f), this holds true even
paired Dirac cone at the Brillouin zone center. A direct whentheinitialexcitationisreducedtoasingleunitcell.
methodforrevealingtheexistenceofaDiracconeis“con- Wave propagation at the critical point should be in-
ical diffraction”, which involves constructing an initial trinsically robust against disorder, due to the enforced
wavepacket from Dirac cone states, which then evolves chirality and absence of band edges. To show this, we
(underlinearrelativisticdispersion)intoaringwithcon- introduce random site-to-site fluctuations in the waveg-
stant thickness and nonzero phase winding. In honey- uide detunings of the tight-binding model (1). For weak
comb lattices with two Dirac cones, conical diffraction disorder,Diracmodesexperiencesuppressedbackscatter-
requires selectively exciting one cone, e.g. using a tilted ing,aphenomenonknownas“weakantilocalization” [38].
spatially-structured input beam [33]. With an unpaired Usually, weak antilocalization disappears when the dis-
Dirac cone, however, we can generate conical diffraction order is short-ranged, due to inter-valley scattering [37].
using simple unstructured Gaussian beams at normal in- However, Fig. 5(a) shows that weak antilocalization per-
cidence, as shown in Fig. 4(a)–(b). This exclusively ex- sists in our system even for completely short-range (site-
cites “pseudospin-up” Dirac modes governed by Eq. (3), specific) disorder. Furthermore, Anderson localization
with chirality determined by the chirality of the modu- normallysetsinatlargedisorderstrengths, commencing
lation δn(r,z). This intrinsic chirality is revealed by the atthebandedges. InFig.5(b),weprobethelocalization
phaseofthediffractedfield. Pseudospinangularmomen- ofthetight-bindingeigenmodesbytheirmodeparticipa-
tum generates an optical vortex in the “cross-polarized” tion numbers, and find that localization is defeated in
pseudospin-down component of the diffracted field, with the critical ∆=0 system due to the lack of band edges.
vortex charge sensitive to the chirality of the Dirac dis- For∆(cid:54)=0, theFloquetbandstructureshavewell-defined
persion [43]. Here, “pseudospin-down” corresponds to bandedges,andwecorrespondinglyobserveLifshitztails
light scattered into the second Brillouin zone, readily of strongly-localized modes [40].
measured via Fourier filtering [40]. Fig. 4(c) shows the In summary, we have shown how to realize Floquet
phaseprofile,exhibitingthepredictedtopologicalcharge. PTIs in staggered helical waveguide arrays. Novel topo-
What happens as we reduce the width of the initial logical transitions, beyond those characterized by Chern
Gaussian excitation? One might expect conical diffrac- numbers, can be accessed by tuning lattice parame-
tiontobedestroyed,sinceEq.(3)isbasedonaneffective- ters other than the bending radius; this allows for low-
mass (carrier-envelope) approximation in the transverse loss operation and raises the prospect of nonlinear or
5
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7
Supplemental Material We can now extend this treatment to the full 2D lattice.
for Each unit cell consists of two sublattices, with wave am-
Anomalous Topological Phases and Unpaired plitudes a and b . Here n = (n,m) indexes the unit
n n
Dirac Cones in Photonic Floquet Topological cells, and we consider a lattice in the “diamond” config-
Insulators uration illustrated in Fig. 1 of the main text. The two
DanielLeykam,M.C.Rechtsman,andY.D.Chong sublattices are displaced by a= √a (1,1) and the lattice
√ √2
vectors are d =a 2(1,0), d =a 2(0,1).
1 2
ToderivetheBlochwavespectrum,wetaketheansatz
TIGHT-BINDING MODEL √
(a ,b ) = (a ,b eik·a)eik·na 2, with crystal momenta
n n 0 0
Here, we present a detailed derivation of the tight- kx,y ∈(−a√π2,a√π2). A complete clockwise cycle consists
of the sequence
binding model of the staggered square lattice PTI, lead-
ing to Eqs. (1)–(2) of the main text. Consider a generic (cid:18) (cid:19) (cid:18) (cid:19)
a a
two-waveguidecoupler,whichisdescribedbythecoupled b ei1k·a =Sˆ0 b ei0k·a , (S4a)
1 0
mode Hamiltonian [1],
(cid:18) (cid:19) (cid:18) (cid:19)
a a
2 =Sˆ 1 , (S4b)
Hˆ =(cid:18) ω C (cid:19), (S1) b2eik·(a−d1) 0 b1eik·(a−d2)
coupler C −ω (cid:18) a (cid:19) (cid:18) a (cid:19)
3 =Sˆ 2 , (S4c)
where C is the coupling strength and ω is the mode b3eik·(a−d2) 0 b2eik·(a−d1)
(cid:18) (cid:19) (cid:18) (cid:19)
detuning. Reciprocity requires that the coupling C be a0 =Sˆ a3 , (S4d)
purely real (one can see this by rotating the coupler b0eik·(a+d1) 0 b3eik·(a+d2)
by π, which reverses the apparent waveguide velocities
whilerepresentingthesamephysicalsystem,soC =C∗). whichdescribesthesequentialcouplingbetweenawaveg-
uide and its four neighbors. It is convenient to combine
Hencetherelativevelocitybetweenthetwohelicalwaveg-
the Bloch phase factors exp(ik ·a) = exp(iκ) with Sˆ
uides can at most renormalize the magnitude of C. Note 0
into the scattering matrix,
thatthisreasoningonlyappliestonearestneighborinter-
actions. Nextnearestneighborcoupling(betweenwaveg- (cid:18) ei∆cosθ −iei(∆+κ)sinθ (cid:19)
uides belonging to the same sublattice) can be analyzed Sˆ(κ)= c c , (S5)
−ie−i(∆+κ)sinθ e−i∆cosθ
using the frame transformation method of Ref. [6] and c c
displays a time-dependent complex phase. where κ is the relative phase. S(κ+π) = σˆ S(κ)σˆ is
z z
The evolution operator for propagation by a distance periodic (σˆ are the Pauli matrices). The evolution
x,y,z
L is, to first order in ω, operator Uˆ(k) can be compactly written as
Uˆcoupler =e−iHˆcouplerL Uˆ(k)=Sˆ(−k−)Sˆ(−k+)Sˆ(k−)Sˆ(k+), (S6)
(cid:18)cos(LC)− iω sin(LC) −isin(LC) (cid:19)
= C . where we have defined
−isin(LC) cos(LC)+ iω sin(LC)
C √
(S2) k =a(k ±k )/ 2. (S7)
± x y
As a simplifying assumption (which does not affect the From Eq. (S6), we can find the quasienergy spectrum by
topological properties of the model [2]), we consider a
vanishing interaction length L → 0 with C → ∞ such
(cid:18) (cid:19) (cid:18) (cid:19)
a (k) a (k)
that the coupling strength is encoded by the dimension- Uˆ(k) 0 =e−iβ(k) 0 , (S8)
b (k) b (k)
less parameter LC → constant = θ , i.e. the coupler’s 0 0
c
principal Euler angle. Then the detuning (ω/C)≈0 has
where β(k) is the quasienergy.
a negligible effect during the interaction length. A phase
difference between the waveguides ∆ ≈ ωZ/4 only ac-
cumulates during the “free evolution” parts of the cycle,
Symmetric lattice (∆=0)
whenthewaveguidesaredecoupled. Undertheseapprox-
imations, the scattering matrix describing the evolution
When ∆ = 0, the lattice is invariant under the in-
of the field over one quarter cycle Z/4 is
terchange of the two sublattices (described by the σˆ
x
(cid:18)ei∆ 0 (cid:19) operation). The two bands are degenerate along the
Sˆ0 = 0 e−i∆ Uˆcoupler Brillouin zone edge, and for small θc the band structure
(S3) β ≈±2θ (cosk +cosk )isequivalenttothedispersion
(cid:18) ei∆cosθ −iei∆sinθ (cid:19) c + −
= c c . of an ordinary square lattice with its single band folded
−ie−i∆sinθ e−i∆cosθ
c c back onto itself. Intuitively this is because in the weak
8
(a) (b) and Uˆ factorizes into
2π 5π/2
y y
enereg enereg Uˆ ==σ(cid:104)ˆσxˆSˆSˆ(k(k−)σˆ)Sxˆσˆ(kxSˆ)((cid:105)k2+)σˆxSˆ(k−)Sˆ(k+)
asi asi x − +
Qu 0 Qu ≡Uˆ12/2. (S13)
π/2
kx ky k+ k- IfoneconsiderstheunfoldedBrillouinzonek± ∈[−π,π],
thespectrumofUˆ showninFig.S1isthesame(upto
1/2
a global phase shift) as the Floquet operators appearing
FIG. S1. (a) Gapless Floquet spectrum at θ =π/4 of Uˆ(k)
c
and (b) unfolded spectrum of Uˆ (k ), revealing the same inthenetworkmodelofRef.[5]. Themaindifferencebe-
1/2 ±
tween the two is that in our helix lattice the degeneracy
structureasthemodelofRefs.[4,5]withanadditionalDirac
cone appearing at the corner of the extended Brillouin zone betweenthetwosublatticesisliftedateachpointintime,
k ∈[−π,π]. and it is only restored when one considers the evolution
±
operator for a complete cycle. This explains the line de-
generacy in Fig. S1(a): the eigenvalues “see” a square
coupling limit, the difference between sequential and si- lattice with period a, while the symmetry-breaking in-
multaneous coupling becomes negligible, so the stagger- stantaneous potential increases the actual lattice period
√
ing of the helices is irrelevant. to a 2, folding the Brillouin zone back onto itself. This
As θc is increased to π/4, band edges at the Γ point zone folding is what allows the “single” Bloch band of
approach β = ±π, reconnecting to form the “critical” the square lattice to reconnect with itself to form the
gapless band structure shown in Fig. S1(a). To reveal anomalous Floquet topological insulator.
the long wavelength dynamics near this critical point,
we expand Uˆ about the Γ point to first order in k and
δθ =θ −π/4, Asymmetric lattice (∆(cid:54)=0)
c
(cid:110) (cid:111)
UˆΓ ≈exp −i[HˆD(ak)−π]+... , (S9) For ∆ (cid:54)= 0, the sublattice symmetry is broken so Uˆ
no longer factorizes into Uˆ and there is no longer any
1/2
where directcorrespondencewiththenetworkmodelofRefs.[4,
5]. The broken symmetry lifts the line degeneracy at the
Hˆ (k)=−k σˆ +k σˆ −4δθσˆ , (S10)
D x z y y x Brillouin zone edge, opening a topologically trivial gap.
Because the two bands are pushed apart, the formation
is an effective Dirac Hamiltonian with mass 4δθ, group
oftheΓpointDiracconeisshiftedtotheweakercoupling
velocity v = 1, and associated “pseudospin” σˆ . The
F √ x
strength (cf. Fig. 1 in main text),
eigenstates of σˆ , which are (a,b) = (1,±1)/ 2, involve
x
excitations of both sublattices with equal intensity, in √
θ =cos−1[sec(∆)/ 2]≤π/4. (S14)
contrasttothepseudospineigenstatesingraphenewhich c
excite only a single sublattice. In the eigenbasis of σˆ
x This critical coupling now marks the transition to a
and using polar coordinates (k ,k ) = (kcosϕ,ksinϕ),
x y Chern insulator phase because there is a second, trivial
the Dirac Hamiltonian takes the canonical form
bandgap. Neverthelessthelongwavelengthdynamicsat
(cid:18)−4δθ ke−iϕ (cid:19) this critical point remain almost the same as the ∆ = 0
HˆD(k)= keiϕ 4δθ . (S11) case: performing a similar series expansion to the one
above, we obtain an effective Dirac Hamiltonian with a
(cid:112)
ThiseffectiveDiracHamiltonianhasthechiralsymmetry ∆-dependent group velocity v = sec(∆) cos(2∆) and
F
Γˆ = 2Sˆ . The evolution operator Uˆ shares this symme- a rotated pseudospin,
z
try [3],
1 (cid:112) 1
Sˆ (∆)= (σˆ cos∆−σˆ sin∆) 1−tan2∆− σˆ tan∆,
ΓˆUˆΓˆ =Uˆ−1, (S12) z 2 x y 2 z
(S15)
tofirstorderink. Thus,eventhoughthefullsystemdoes continuously interpolating between Sˆz(0) = 12σˆx and
not have any obvious chiral symmetry, the symmetry is Sˆ (π) = −1σˆ , resembling the pseudospin in graphene.
z 4 2 z
locally restored at the Dirac point. This pseudospin is derived by introducing the total an-
BoththefullbandstructureandeffectiveDiracHamil- gularmomentumoperatorJˆ =Lˆ +Sˆ ,whereLˆ =i∂
z z z z ϕ
tonian of the symmetric lattice resemble the ring res- is the usual orbital angular momentum operator, and
onatormodelofRefs.[4,5]. Infact,thereisanexactcor- enforcing its conservation via [Jˆ,Uˆ] = 0, which has a
z
respondence between the two: When ∆=0, the scatter- nontrivial solution because the spectrum has rotational
ing matrix Sˆ(κ) has the symmetry σˆ Sˆ(κ)σˆ = Sˆ(−κ), symmetry.
x x
9
Increasing θ further, the bands become increasingly Theminimumfeasiblecenter-to-centerwaveguidesepara-
c
flat before the spectrum becomes completely degener- tionisapproximately12µm[7](otherwisethewaveguides
ate at θ = π/2, corresponding to Uˆ = ˆ1. Similar to would overlap and coalesce), leading to the constraint
c
Ref. [2], this second gap closing marks the formation of a≥12µm+2R =18µm. Thehelixradiusandpitchare
0
an“anomalous” Floquetinsulatorphase,withedgemodes chosen to minimize the waveguide acceleration and asso-
traversing the remaining band gap. Since the bulk evo- ciated bending losses. Array lengths of L = 10−15cm
lutionoperatoristrivial(simplytheidentity),newtopo- are feasible, corresponding to 5−7 complete modulation
logicalinvariantsspecifictoFloquetsystemsarerequired cycles used in the main text. The number of cycles can
to describe this phase. befurtherincreasedbyreducingthepitch,attheexpense
of stronger losses.
Westressthatthemechanismforthetopologicalphase
CONTINUUM MODEL transition in these staggered helical lattices is distinctly
different from the Floquet photonic topological insula-
Our continuum model uses parameters similar to tor realized in the unstaggered helical lattice of Ref. [6],
Refs. [6, 7], describing femtosecond laser-written waveg- which relied on the effective gauge field induced by the
uide arrays in fused silica [8]. Propagation is governed helical modulation. Indeed, the dimensionless effective
by the paraxial (Schrödinger) equation, gauge field strength A = ak R Ω ∼ 0.3 is always small
0 0 0
compared to the dimensionless quasienergy bandwidth
i∂ ψ =− 1 ∇2ψ− k0∆n(x,y,z)ψ, (S16) 2π. The reason why this small effective field is sufficient
z 2k n toinduceanontrivialphaseinRef.[6]isthattheunper-
0 0
turbed lattice already hosts Dirac points, so a nontrivial
where k = 2πn /λ, the background refractive index is
0 0 phase is induced by an infinitesimal perturbation. In
n =1.45atwavelengthλ=633nm,andtheindexmod-
0 contrast, phase transition displayed by the square helix
ulation forms a square helix lattice:
latticeisintrinsicallynon-perturbative,akintothetopo-
logical transition at large field strengths from C = 1 to
(cid:88)(cid:110)
∆n(x,y,z)=∆n V (X−,Y−) C =−2 predicted (but not observed) in Ref. [6].
1 0 n m
nm The only way to close the quasienergy gap at φ=±π
(cid:111)
+αV (X+ ,Y+ ) , (S17) and reach the anomalous Floquet topological insulator
0 n+1/2 m+1/2 phaseusingthissmalleffectivegaugefieldiswithamod-
ulation frequency Ω near-resonant with the unperturbed
where
bandwidth ∼ t, where t ≤ 2.7cm−1 is the nearest neigh-
X±(z)=x±x (z)−nd (S18) bor coupling strength, set by a. In this resonant cou-
n 0
plingregime,perturbationtheoryyieldsagapsize∼A .
Y±(z)=y±y (z)−md. (S19) 0
m 0 The only way to tune the gap size is using the sole re-
maining free parameter, the helix radius R , which in-
∆n = 7.5 × 10−4 is the modulation depth, α is the 0
1 √ creases the bending loss exponentially [9]. For example,
asymmetry between the sublattice depths, and d=a 2
the maximum practical value of A ∼ 2.2 from Ref. [7]
is the lattice period (recall that a is the separation be- 0
requiresR =16µmleadingtoprohibitivebendinglosses
tween the sublattices). Putting lattice in the “diamond” 0
of3dB/cm. Forcomparison,inthemaintextweachieved
configurationminimizestheeffectofthex−y anisotropy
a topological gap size ∼π with negligible bending losses
of the individual waveguide profiles V , described by the
0 and conservative waveguide parameters.
hypergaussian function
The main difference between the staggered and un-
V (x,y)=exp(cid:0)−[(x/σ )2+(y/σ )2]3(cid:1), (S20) staggeredhelicallatticesisintheformofthemodulation
0 x y
potential. Themodulationinthelatterisequivalenttoa
withwidthsσ =2µmandσ =5.5µm. Usingthesepa- spatially uniform effective gauge field modulated in time
x y
rameters, representative values of the coupling constant atasinglefrequencyΩ. Thisharmonicdrivingconserves
between neighboring waveguides range from C =2.7/cm the Bloch momentum k and can be easily understood in
(for separation 12µm) to C = 0.1/cm (for separation terms of local coupling in the frequency domain using a
27µm). truncated repeated zone scheme [2]. The phase transi-
The waveguide centers move in helices according to tionisinducedbycouplingbetweenneighboringzonesin
the frequency domain. In contrast, the step-like, on/off
x (z)=R cos(Ωz), y (z)=R sin(Ωz), (S21) nature of the evanescent coupling terms in our staggered
0 0 0 0
lattice is more naturally understood as a local modu-
where R = 3µm is the helix radius, and Ω = 2π/Z lation in the time domain: it involves many harmonics
0
with pitch Z = 2cm. The helices of the two sublattices nΩinthefrequencydomain,andtheperiod-doublingin-
are staggered and π out of phase, as given by Eq. S17. ducesscatteringbetween Blochmomentak. Thus,acom-
10
parativelyweaktimemodulationcanbecompensatedby 1
strongtransversescatteringtoinduceaphasetransition.
0.995
m
FLOQUET BAND STRUCTURE CALCULATION
r
o
n
Here we explain our method for calculating the Flo- e
u 0.99
quetbandstructureofthecontinuummodel. Waveguides al
v
fabricated using the femtosecond laser writing technique n
e
canbechosentobesinglemode,withgoodconfinement. g
i
Therefore, to a good approximation we can project the E0.985
time-dependent field onto the bound waveguide modes,
neglectingradiationlosses(whicharesmall,shownbelow
self-consistently). This reduces the difficult problem of 0.98
continuum Floquet band structure calculation to a low- 16 18 20 22 24 26 28
dimensionaldiscretemodel, resemblingthetightbinding Waveguide spacing (µm)
limit.
WeobtaintheboundmodesbysolvingthestaticBloch FIG. S2. Norm of the eigenvalues of the truncated evolution
operator Uˆ(k) averaged over the Brillouin zone as a function
mode eigenvalue problem at z =0,
of waveguide separation (symmetric lattice; α = 1). Error
E (k)|u (k)(cid:105)=Hˆ(k,0)|u (k)(cid:105), (S22) barsdenoteonestandarddeviationovertheensembleofBloch
n n n
eigenmodes.
where Hˆ(k,0)=(∇+ik)2+V(r,0) is the Bloch Hamil-
tonian. For single mode waveguides, we can limit the
number of bands N to the number of waveguides per
unit cell (N = 2). This defines a truncated basis
the high symmetry points k =(0,0),(π,0),(0,π),(π,π),
{| u (k,0)(cid:105)}, n = 1,...,N. We numerically propagate
n
consistent with our symmetry argument above that the
each basis element for one modulation period, assuming
modulation-induced effective gauge field does not intro-
twisted boundary conditions corresponding to the Bloch
duceaphasetothewaveguidecoupling-itmerelyrenor-
momentum k,
malizes the coupling strength.
|un(k,Z)(cid:105)=e−i(cid:82)0ZHˆ(k,z)dz |un(k,0)(cid:105),
To fit the tight binding band structure to the contin-
=Uˆ(k,Z)|u (k,0)(cid:105), (S23)
n uum model’s in Fig. 2 in the main text, we set ∆ and
where Uˆ(k,Z) is the Bloch mode evolution operator. θc to reproduce the band gap at ky = π/a and ky = 0
respectively. This is already sufficient to demonstrate a
Since | u (k,0)(cid:105) is in general not an eigenstate of
Hˆ(k,z (cid:54)= 0n), this evolution generically mixes the basis reasonable agreement between the two models, although
it obvious from Fig. 2(b) in the main text that for short
elements. We obtain a truncated evolution operator by
lattice periods the functional form of the dispersion be-
projecting the final state back onto the basis states,
comes qualitatively different (band edges are displaced
U (k,Z)=(cid:104)u (k,0)|Uˆ(k,Z)|u (k,0)(cid:105). (S24) from the Γ point), which we attribute to corrections be-
mn m n
yond the tight binding approximation (namely the cou-
ThismethodassumesthattheevolutionUˆ doesnotcou- pling no longer occurs purely sequentially).
ple modes beyond the Nth band (including unbound
radiative modes). If there is coupling to these modes, There is an intuitive and efficient way to obtain a cor-
then the matrix Umn will be non-unitary. For our real- respondence between the continuum and tight binding
istic choice of potential parameters, this quasistatic ap- modelparameters: onesimplyhastosimulateinthecon-
proximation is good and we calculate the mean norm of tinuum model a single two-waveguide coupler to obtain
the eigenvalues of Umn to exceed 0.99 in all cases, see itsscatteringmatrixSˆ0. ∆andθc canthenbeextracted
Fig. S2. This corresponds to an amplitude decay rate of using Eq. (S3). Therefore the optimization of the de-
∼0.02dB/cm. As R0/Z →0 the quasistatic approxima- signparametersfortheanomalousFloquetinsulator(eg.
tion gets better and the norms approach unity. waveguide depths, separation, size) merely requires the
Given Uˆ(k), we can compute its eigenvectors and optimizationofatwo-waveguidecoupler,arelativelysim-
hence evaluate the Chern number numerically [10], and ple task. For example, for a symmetric coupler (∆ = 0)
we can also determine the effective Hamiltonian via with a given coupling constant C, one can obtain the
Hˆeff = τi log(Uˆ). Neglecting (non-Hermitian) loss terms, required coupling angle θc by choosing an interaction
in the symmetric (α = 0) lattice Hˆ is purely real at length L=θ /C.
eff c