Characterizing topology by dynamics: Chern number from linking number

Matthias Tarnowski,1, 2 F. Nur Ünal,3 Nick Fläschner,1, 2 Benno S. Rem,1, 2

André Eckardt,3 Klaus Sengstock,1, 2, 4, ∗ and Christof Weitenberg1, 2

1Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany2The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany

3Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany4Zentrum für Optische Quantentechnologien, Universität Hamburg, 22761 Hamburg, Germany

(Dated: May 1, 2018)

Topology plays an important role in modern solid state physics describing intriguing quantum states suchas topological insulators. It is an intrinsically non-local property and therefore challenging to access, oftenstudied only via the resulting edge states. Here, we measure the topological index directly from the far-from-equilibrium dynamics of the bulk. We use the mapping of the Chern number to the linking number of dynamicalvortex trajectories appearing after a quench to the Hamiltonian of interest. We thereby map out the topologicalphase diagram of quantum gases in optical lattices via a purely dynamical response. Such relations betweentwo topological indices in static and dynamical properties could be also an important approach for exploringtopology in the case of interactions.

PACS numbers: 67.85.-d, 67.85.Lm

Topological quantum matter has recently received much at-tention, because it opens an entirely new class of quantumphases and has potential applications ranging from precisionmeasurements to quantum information and spintronics [1]. Aparadigmatic role is played by the Chern number, which char-acterizes the topology of filled bands in two-dimensional lat-tice systems and also describes the integer Quantum Hall ef-fect. Even richer is the interplay between topological bandstructures and interactions giving rise to topologically orderedstates of matter, such as fractional quantum Hall states, withintriguing emergent properties like abelian or nonabelian any-onic excitations. A widely unexplored field are the fundamen-tal connections between the dynamical behavior of (highly)excited states of a system and the underlying ground-stateproperties.

Ultracold quantum gases are a promising experimental plat-form to explore these questions. On the one hand they allowfor the realization of topologically non-trivial band structuresand artificial gauge fields [2–9] and on the other hand typicaltime scales for dynamical studies are experimentally well ac-cessible. Moreover, they offer the perspective of combiningthese effects with strong interactions (see, e.g., refs. [10–12]).

Here we establish a new approach by connecting a centralquantity of the ground state topology – the Chern number –with the dynamical evolution of highly excited states of thesystem via the measurement of a linking number. That is westudy in detail the contour of dynamically created vortex pairsin momentum space following a sudden quench of the system.We thereby map out the trivial and non-trivial Chern numberareas of the phase diagram. As shown by Wang et al. (ref.[13]), the Chern number of the post quench Hamiltonian mapsonto the linking number between this contour and the positionof the static vortices [Fig. 1(a)]. We thus demonstrate that thedirect mapping between two topological indices – a static anda dynamical one – allows for an unambiguous measurementof the Chern number.

This is a first step in the more general direction of relation-ships between static non-local topological properties and dy-namical properties in complex quantum systems. Besides thisit also circumvents difficulties of other measurement schemesto detect Chern numbers, e.g. drift measurements [5, 6], forwhich the signal cannot unambiguously distinguish betweentrivial and non-trivial Chern numbers when inversion symme-try is broken [5]. It also goes far beyond the characterizationof topology, e.g. by the complete measurement of the Berrycurvature [8], as in these studies non-trivial Chern numberscannot be reached in an adiabatic preparation in the thermo-dynamic limit [14].

Our system is described by a Haldane-like Hamiltonianrealized via Floquet engineering of lattices [2–8, 15–21].We start with a hexagonal optical lattice [22] with nearest-neighbor (NN) tunneling element JAB and sublattice offset ∆AB[see Fig. 1(b)] described by the bare Hamiltonian

H0 =−∑〈l′l〉

JABa†l′ al + ∑

l∈B∆ABnl (1)

(see supplementary [23] for definitions). By circular latticeshaking with a near-resonant angular frequency ω = ∆AB/h−δ with detuning δ and driving strength α , we arrive at a Flo-quet system described by the effective Hamiltonian

HF =−∑〈l′l〉

JeffABa†

l′ al+ ∑〈〈l′l〉〉A

JeffAAa†

l′ al+ ∑〈〈l′l〉〉B

JeffBBa†

l′ al+∑l∈B

∆effnl .

(2)In the limit of low driving strength, the expressions for the ef-fective tunnel elements read Jeff

AB ' ±α

2 JABe∓iφl′l with Peierlsphases φl′l for the NN tunneling and Jeff

AA =−JeffBB' J2

AB/hω forthe next-nearest neighbor (NNN) tunneling, which arises asa super-exchange process. The effective sublattice offset be-comes ∆eff = hδ +3J2

AB/hω [see Fig. 1(b)]. Note that in con-trast to the case without initial sublattice offset [5, 24, 25], werealize the Hamiltonian in a gauge, where the Peierls phasesappear at the NN tunneling, which gives rise to a shifted band

arX

iv:1

709.

0104

6v3

[co

nd-m

at.q

uant

-gas

] 3

0 A

pr 2

018

2

(a)

(c)

non-trivial

ky

t

kx

Shaking phase φ (rad)

Shak

ing

detu

ning

δ/2π

(Hz)

0 π-π π/2-π/2

-500

0

500

-1000

C = 0

C = -1

trivial

C = 1

vortex trajectory

static vortex

A

B

A

B

A

B

A

B

A

B

A

B

2 2 /

− 2 /

offset: ∆ offset: ħ + 3 ∙ 2 /

= ∆ −

2+ / 3

2−

2

2 2−

2

/ 32+ / 32

/ 32(b)

FIG. 1. Mapping between Chern number and linking number. (a)The topology of a lattice system (illustrated here only by the car-toon of 2D surfaces) can be mapped onto the linking number ofdynamical vortex trajectories of the wave function in momentumspace. (b) Calculated Haldane-like phase diagram of the drivenhexagonal lattice with regions of different Chern number (parame-ters: ω = 2π ·6410 Hz, α = 1.28, the detuning δ is varied by chang-ing the initial offset ∆AB via the lattice depth). The experiments areperformed for circular shaking (along the grey line and also for thegrey point in the C = −1 area). (c) Illustration of the tight-bindingmodel of the bare lattice (left) and the effective Hamiltonian (right).

structure with one Dirac point at the Γ point [8]. The Hamil-tonian describes topological phase transitions between C = 0and C = ±1 at hδ = −15J2

AB/hω and hδ = 3J2AB/hω . By

going away from circular shaking to a general shaking phaseφ between x and y direction, one obtains the phase diagramshown in Fig. 1(c) resembling that of the Haldane model [26].

The hexagonal lattice possesses two types of sites forming

the A and B sublattices. For each quasimomentum k, the twosublattice states form a pseudo-spin 1/2 degree of freedom thatis described by the vector of Pauli matrices σ . Each Hamilto-nian can be written in the form

h(k) = h0(k)+σ ·h(k). (3)

Here h(k) plays the role of a "magnetic field" that inducesa k-dependent precession dynamics with angular velocity2|h(k)|/h around itself. Its direction h(k) = h(k)/|h(k)| de-termines the two eigenstates |k,±〉 forming both Bloch bandsand, therefore, completely characterizes the topology of thesystem. In fact, in such a two-band system, the Chern numberhas a simple geometric interpretation: it counts the number oftimes h(k) wraps around the Bloch sphere when k runs overthe first Brillouin zone [1].

The central idea is thus to measure this wrapping by ob-serving whether h(k) visits both poles as will be explained indetail in the following (see Fig. 2). We start with a filled low-est band |k,±〉 of the initial Hamiltonian describing the barelattice and quench into the final Hamiltonian describing theshaken lattice, i.e. between the two "magnetic fields" hi(k)and hf(k). After a variable evolution time, we perform statetomography in the basis of the initial lattice [8, 27], i.e. weswitch back to the initial Hamiltonian and observe the preces-sion around hi(k). It translates into an oscillation of the mea-sured momentum distribution after time-of-flight expansion,whose phase yields the momentum-dependent phase profileof the time-evolved state. The topology of hf(k) is entirelyencoded in the vortices of this phase profile. Namely, the link-ing number associated with the trajectories of vortices directlycorresponds to the Chern number [13, 28]. While static vor-tices appear at the Dirac points, where hi(k) points to one ofthe poles of the Bloch sphere, the contours of dynamical vor-tices correspond to those k where hi(k) points to the equator.A topologically nontrivial Hamiltonian visiting both poles re-quires this contour to encircle a static vortex so that it has tobe crossed once (or an odd number of times) when going fromone static vortex to the other. The absence of a dynamical vor-tex contour can therefore be identified with a Chern numberzero.

Note that the argument can be formulated in a more generalframework by considering the inverse images of any two or-thogonal vectors on the Bloch sphere [13]. The Chern numberthen maps onto the linking number of the two trajectories inthe space spanned by kx, ky and time and can be related to aHopf invariant. Such a linking number characterizing a Hopfinsulator was recently observed in a quantum simulation usinga nitrogen vacancy center [29].

Our scheme has the advantage that it can be performed inany band structure (flat or non-flat), which offers a topologi-cally trivial basis for the tomography. Our method of analyz-ing only the vortices in momentum space does therefore notrequire flat bands as measurement basis in the state tomogra-phy. In contrast to previous work [30], in which the non-trivialregion was not resolved, we start here with dispersive bands

3

(a)

timePreparation in lowest bare band

Dynamics in Floquet bands

0 Tomography in bare bands

(i) (ii) (iii)

K

ΓK‘

(b)

K

ΓK‘

(c)

FIG. 2. Illustration of the measurement scheme. (a) The states ofthe two-band model can be visualized on a Bloch sphere with theeigenstates of the two bare bands on the poles. (i) We initialize thestate of the system in the lower bare band of hi(k) (south pole). (ii)We quench into the final Floquet system hf(k) by suddenly switch-ing on the lattice shaking. The states (black arrow) evolve on theBloch sphere according to the Floquet Hamiltonian. (iii) We mea-sure the time-evolved state by projecting back onto the bare bandsof hi(k) and following the dynamics. When the time evolved state|k, t〉 was at one of the poles, this leads to the absence of dynamicsin the tomography and to a vortex in the azimuthal phase profile. (b)The inverse images of the poles on the Bloch sphere form contoursin the Brillouin zone. At the Dirac points, where the Hamiltonianpoints to one of the poles, there is no dynamics and the state will stayat the south pole and give rise to a static vortex (green and orangedots). Where the Hamiltonian lies on the equator, the time-evolvedstate will reach the north pole dynamically and will give rise to a dy-namic vortex. These dynamic vortices move on a contour, which isthe inverse image of the equator of the Bloch sphere (grey line). TheChern number of the Hamiltonian can be inferred from the linkingnumber of the contour: if the dynamic vortex contour encloses oneof the static vortices, then both static vortices correspond to oppositepoles so that the Hamiltonian is topologically nontrivial. This canbe seen by following the dynamics along a path connecting the twoDirac points (dashed line). In the depicted case, the Chern number is1. (c) Same as (b), but for the case of a Chern number 0.

(JAB/∆AB ' 0.08), which yield a much broader non-trivial re-gion (18J2

AB/hω ' h ·500 Hz). State tomography in dispersivebands in principle yields distorted phase profiles, but the link-ing number of vortex trajectories remains a robust measure forthe Chern number [23], such that the above arguments hold forany trivial basis.

In order to get a better resolution of the vortex dynamics, wemeasure the dynamics in steps of a quarter of the driving pe-riod T = 2π/ω = 156 µs. We thus sample the micromotion ofthe Floquet system [18–20, 31]. Because the micromotion ofthe vortex positions is small compared to the contours of theirtrajectories in our case [23], it has no influence on the mea-surement of the Chern number. However, our method wouldbe suitable to detect the non-trivial topological effects thatcan arise from the large micromotion at driving frequenciescomparable to the single-particle band width of the system.A prominent example is the appearance of anomalous edgestates protected by a winding number, which is constructedfrom the evolution operator for all times within a single driv-ing period [32, 33].

We use the relation between the linking number and theChern number to map out the topological phase transition ofthe effective Hamiltonian. Fig. 3 shows as a key result data ofthe time-integrated vorticity for different quenches into Chern0 and Chern 1 areas of the phase diagram (different detuningsof the lattice shaking). While the static vortices at the Γ andK points are visible in all data sets, one clearly recognizesadditional vortex contours in the data sets for near-resonantshaking. We easily count the linking number of these contoursand thereby obtain the Chern number of the final Hamiltonian.As a further central result, the region with non-trivial Chernnumber 1 agrees well with the prediction from a full numericalcalculation of the system [23].

The sign of the Chern number can be obtained from an anal-ysis of the chiralities of the observed vortices and their direc-tion of motion (see Fig. 4) (compare ref. [13]). We can definea chirality χd of the dynamical vortex contour from the di-rection in which the dynamical vortices of positive chiralitymove. The sign of the linking number can then be defined asthe product −χdχs with χs denoting the chirality of the en-closed static vortex. From this sign, one directly obtains thesign of the Chern number of the lower Bloch band [23]. Fig. 4shows time-resolved vortex data for two different directionsof the circular lattice shaking, which leads to Chern numbersof opposite sign. While the chirality of the vortex contour isthe same in both cases, the chirality of the enclosed vortexchanges with the driving direction, directly indicating the op-posite sign of the Chern number.

Our measurements establish a one-to-one relation betweentwo different topological indices and demonstrate a direct sig-nature of topology in highly-excited states. This approachsheds new light on the identification of topology and mightbe fruitful also for developing protocols for the detection ofstrongly correlated topologically ordered states of interact-ing matter, such as fractional quantum Hall states, e.g. viaa quench of the interaction strength.

4

Tim

e (

T)

En

erg

y/h

(k

Hz)

(a)

Ch

ern

nu

mb

er

Detuning ħδ (J /ħω)

-25 -20 15 -10

0

1

Detuning δ/2π (Hz)-1000 -500 0 500

151050-5 20

AB2

6

11

6

i)

ii) iii)

iv) v)

i) iv)iii)ii) v)

0 0.5

1

1.5

(b)

(c)

FIG. 3. Mapping out the topological phase diagram using the linking number. (a) Original data of the observed vortices summed over all timesteps (red dot: positive chirality, blue dot: negative chirality; the hue indicates the time step where the vortex was present). The hexagon marksthe first Brillouin zone. The dynamical vortex contours are highlighted by a guide-to-the-eye (grey line). (b) The Chern number is obtainedfrom the linking number of these dynamical vortex contours (or the absence of a contour) and plotted for various shaking detunings (cutthrough the phase diagram corresponding to the grey line in Fig. 1). The region with non-trivial Chern number agrees well with the predictionfrom a full numerical calculation (solid line). (c) Calculated Floquet bands for various detunings illustrating the closing of the Dirac points atthe topological phase transitions.

(a)

(b)

= +1 = −1 = +1

t3 t2 t1

= +1 = +1 = −1

=2

=−2

t3 t2 t1

6

1

6

Tim

e (T

)

1

6

1

6

Tim

e (T

)

1

FIG. 4. Sign of the linking number. (a) Vortex data in the non-trivial regime (shaking phase of π/2 and shaking detuning of δ/2π =−372 Hz).The first subfigure shows the time-integrated data, while the other subfigures show successive stroboscopic time steps t1 = 13 · T/4, t2 =17 ·T/4, t3 = 21 ·T/4 after the quench. The vortex contour has a positive chirality, while the enclosed static vortex has a negative chirality,revealing the Chern number +1 (see text). (b) Reverse shaking (grey point in Fig. 1) for δ/2π = −359 Hz and for time steps t1 = 14 ·T/4,t2 = 18 ·T/4, t3 = 22 ·T/4 after the quench. The chirality of the enclosed vortex is now inverted and the Chern number is −1.

5

We acknowledge financial support from the DeutscheForschungsgemeinschaft via the Research Unit FOR 2414and the excellence cluster “The Hamburg Centre for Ultra-fast Imaging - Structure, Dynamics and Control of Matterat the Atomic Scale”. BSR acknowledges financial supportfrom the European Commission (Marie Curie Fellowship).We acknowledge fruitful discussions with Ramanjit Sohal andChristoph Sträter.

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[5] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger,D. Greif, and T. Esslinger, Nature 515, 237 (2014).

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[8] N. Fläschner, B. Rem, M. Tarnowski, D. Vogel, D.-S. Lühmann,K. Sengstock, and C. Weitenberg, Science 352, 1091 (2016).

[9] M. Tarnowski, M. Nuske, N. Fläschner, B. Rem, D. Vogel,L. Freystatzky, K. Sengstock, L. Mathey, and C. Weitenberg,Phys. Rev. Lett. 118, 240403 (2017).

[10] N. R. Cooper and J. Dalibard, Phys. Rev. Lett. 110, 185301(2013).

[11] A. G. Grushin, Á. Gómez-León, and T. Neupert, Phys. Rev.Lett. 112, 156801 (2014).

[12] E. Anisimovas, G. Žlabys, B. M. Anderson, G. Juzeliunas, andA. Eckardt, Phys. Rev. B 91, 245135 (2015).

[13] C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, Physical Re-view Letters 118, 185701 (2017).

[14] L. D’Alessio and M. Rigol, Nature Comm. 6, 8336 (2015).[15] A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. 95,

260404 (2005).[16] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini,

O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403(2007).

[17] C. V. Parker, L. C. Ha, and C. Chin, Nature Phys. 9, 769 (2013).[18] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014).[19] M. Bukov, L. D’Alessio, and A. Polkovnikov, Advances in

Physics 64, 139 (2015).[20] A. Eckardt and E. Anisimovas, New J. Phys. 17, 93039 (2015).[21] A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).[22] P. Soltan-Panahi, D.-S. Lühmann, J. Struck, P. Windpassinger,

and K. Sengstock, Nature Phys. 8, 71 (2012).[23] see supplementary material.[24] T. Oka and H. Aoki, Phys. Rev. B 79, 081406(R) (2009).[25] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podol-

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[26] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).

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[28] J. Yu, Phys. Rev. A 96, 023601 (2017).[29] X. X. Yuan, L. He, S. T. Wang, D.-L. Deng, F. Wang, W. Q.

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6

SUPPLEMENTAL MATERIAL

Tight-binding description

We consider a system of spinless fermions in a hexago-nal lattice with sublattice offset ∆ = ν hω + hδ that is near-resonantly driven by a circular force F (t) =−F [cos(ωt)ex +sin(ωt)ey] [See Fig. 1(b)]. Here ν is an integer and hδ � hω

the detuning. (Our experiment is described by ν = 1, whereasthe case ν = 0 captures the Floquet topological insulator pro-posed in Ref. [S1], which was realized both with optical waveguides [S2] and in an optical lattice experiment [S3]). In gen-eral, the system is described by the Hubbard Hamiltonian

H(t) =− J ∑〈`′`〉

a†`′ a`

+∑`

[−r` ·F (t)+(ν hω + hδ )δ`∈B

]n`,

(S1)

where a†` , a`, and n` = a†

` a` denote the creation, annhilation,and number operator for fermions on lattice site ` at positionr`, respectively, where J describes tunneling between nearestneighbor pairs 〈`′`〉, and where δ`∈B is one if ` lies in sublat-tice B and zero otherwise. Since we do not consider any barenext-nearest-neighbor hopping, here we have dropped the sub-script indices used in the main text (J ≡ JAB and ∆ ≡ ∆AB) inorder to make the notation simpler. The force F (t) is an iner-tial force created by moving the lattice along a circular orbitin space, so that the Hamiltonian describes the system in thereference frame co-moving with the lattice.

Let |ψ(t)〉 denote the state of the system in the lattice frame.It is convenient to perform a gauge transformation |ψ ′(t)〉 =U†(t)|ψ(t)〉 and H ′(t) = U†(t)H(t)U(t)− ihU†(t) ˙U(t), withthe unitary operator

U(t) = Ushift(t)Urot(t)

= exp(

i∑`

[χ

shift` (t)+χ

rot` (t)

]n`

),

(S2)

where

χshift` (t) =

Fhωr` · [−sin(ωt)ex + cos(ωt)ey],

χrot` (t) =−νωtδ`∈B.

(S3)

While Ushift(t) integrates out the time-periodic shift in quasi-momentum induced by the circular force, Urot(t) captures arotation of the pseudospin defined by the sublattice degree offreedom and integrates out the resonant part ν hω of the sub-lattice imbalance ∆. The resulting transformed Hamiltonianreads

H ′(t) =−∑〈`′`〉

Jeiθ`′`(t)a†`′ a`+∑

`

δδ`∈Bn`, (S4)

with time-periodic Peierls phases θ`′`(t) = α sin(ωt−ϕ`′`)−σ`νωt. Here, we have introduced the dimensionless driving

strength α = Fa/hω , with a = 1√3

23 λL denoting the distance

between adjacent lattice sites, ϕ`′` denotes the azimuthal angleof the vector r`′ −r`, and σ` = 1 (σ` =−1) for ` ∈ A (` ∈ B).

The transformation preserves the periodic time dependenceof the Hamiltonian and removes large energy offsets of orderhω between neighboring sites. With that it provides a goodstarting point for computing the effective time-independentHamiltonian HF and the periodic micromotion operator UF(t)in a high frequency approximation [S4, S5], in terms of whichthe time-evolution operator for the dynamics induced by H ′(t)takes the transparent form

U ′(t, t0) = UF(t)exp(− i

h(t− t0)HF

)U†

F(0). (S5)

Note that the transformation U(t) restores also the transla-tional symmetry of the lattice, which was broken by the on-site potential−r` ·F (t), so that the Floquet states of H ′(t) andthe eigenstates of HF are Bloch states.

Effective Hamiltonian

In order to compute the effective Hamiltonian, we will keepthe two leading terms of the high-frequency expansion [S4],

HF ≈ H(1)F + H(2)

F with H(1)F = H0,

H(2)F =

∞

∑m=1

[Hm, H−m

]mhω

.(S6)

Here,

Hm =1T

∫ T

0dt H ′(t)e−imωt

=−∑〈`′`〉

J(m)`′` a†

`′ a`+δm,0 ∑`

hδδ`∈Bn`(S7)

denote the Fourier components of the Hamiltonian, with tun-neling parameters J(m)

`′` = JJm+σ`ν(α)e−i(m+σ`ν)ϕ`′` , where

Jn(x) is an ordinary Bessel function of the first kind.Evaluating these terms, we find

HF ≈− ∑〈`′`〉

Jeff〈`′`〉a

†`′ a`− ∑

〈〈`′`〉〉Jeff〈〈`′`〉〉a

†`′ a`

+∑`

∆effδ`∈Bn`,(S8)

where 〈〈`′`〉〉 denote pairs of next-nearest neighbors. The ef-fective nearest-neighbor tunneling matrix elements,

Jeff〈`′`〉 = JJσ`ν

(α)e−iσ`νϕ`′` , (S9)

originate from the first-order term H(1)F . In turn, the effective

next-nearest-neighbor tunneling matrix elements

Jeff〈〈`′`〉〉 =−

∞

∑m=1

J2

mhω

[J 2

m−σ`ν(α)ei(m−σ`ν)(π+σ〈〈`′`〉〉π/3)

−J 2m+σ`ν

(α)ei(m+σ`ν)(π+σ〈〈`′`〉〉π/3)],

(S10)

7

stem from the first-order term and can be understood as a su-perexchange process. Here, σ〈〈`′`〉〉 = 1 (σ〈〈`′`〉〉 =−1) for tun-neling clockwise (counterclockwise) around a hexagonal pla-quette of the lattice. The effective sublattice offset

∆eff = hδ +∞

∑m=1

zJ2

mhω

[J 2

m−σ`ν(α)−J 2

m+σ`ν(α)], (S11)

with coordination number z = 3, possesses contributions fromboth orders.

Comparison of models with and without initial sublattice offset

There is a fundamental difference between the case ν 6= 0;corresponding to our experiment with ν = 1, and the experi-ments with the case ν = 0 described in Refs. [S2, S3]. Forν = 0, nearest-neighbor tunneling is present already in theundriven system and second-order next-nearest neighbor tun-neling is a driving induced process. Conversely, for ν 6= 0nearest-neighbor tunneling has to be induced by the driving(since it is off-resonant in the undriven lattice), while second-order next-nearest-neighbor tunneling occurs already in theundriven system. This fact is reflected in the behavior of theeffective tunneling matrix elements in the limit of small driv-ing strength α , where we have

Jeff〈`′`〉 ' J+O(α2),

Jeff〈〈`′`〉〉 '

√3α2

4J2

hωeiσ〈〈`′`〉〉π/2

+O(α4) for ν = 0,(S12)

whereas

Jeff〈`′`〉 '

α

2Jσ`e−iσ`ϕ`′` +O(α3),

Jeff〈〈`′`〉〉 '−σ`

J2

hω+O(α2) for ν = 1.

(S13)

Here, we have used that Jn(x) = 1|n|![sgn(n)x/2

]|n|+

O(x|n|+2). The opposite sign of the effective next-nearest-neighbor tunneling on the two sublattices arises from the op-posite sign of the offset to the intermediate state in the su-perexchange process. This difference between the cases ν = 0and ν = 1 has two major consequences. The first one is re-lated to the fact that the Peierls phases appear at the drivinginduced tunneling matrix elements. For ν = 0, the effectivenext-nearest neighbor tunneling matrix elements are complex,corresponding to the configuration of the Haldane model [S6].In our case, for ν = 1, instead the nearest-neighbor tunnelingmatrix elements acquire a phase. While the model can still bemapped to the Haldane model via a gauge transformation, thisimplies that one of the Dirac cones is shifted from one of theK points at the corner of the first Brillouin zone to the Γ pointat its center. The second consequence is more important: Thetopologically non-trivial properties of the effective Hamilto-nian emerge from the interplay between nearest-neighbor tun-neling processes on the one hand and next-nearest-neighbors

tunneling processes on the other. If the energy scale of oneof these processes is much smaller than that of the other one,the topological band gap will be of the order of this smallerenergy scale. For ν = 0 the next-nearest neighbor tunnelingmatrix elements, which are suppressed already by a factor ofJ/(hω) with respect to nearest-neighbor tunneling, scale onlyquadratically with the driving amplitude α , so that for not toostrong driving the band gap scales like α2J2/(hω). In con-trast, for ν = 1 the gap should roughly scale like αJ as longas α <∼ J/(hω) and like J2/(hω) for larger driving strength(as long as α ≤ 1). This suggests that the case ν = 1 is favor-able for the realization of robust topological band structures.Indeed, the width of the region with non-trivial Chern numberis 100 Hz in Ref. [S3], but 500 Hz in this work. However, inan implementation with an inhomogeneous lattice, where theresonance condition of the global shaking varies across thesample, the ratio of the width of the non-trivial region to thedriving frequency is also relevant.

Effective Hamiltonian in quasimomentum representation

It is instructive to express the effective Hamiltonian givenin Eq.(S8) in quasimomentum representation,

H = ∑k

(a†Ak, a

†Bk) [h0(k) · I +h(k) ·σ ]

(aAk

aBk

). (S14)

Here, I is 2× 2 identity matrix, σ denotes the Pauli matricesacting on the pseudospin space defined by the two sublatticestates σ = A,B, and aσk = 1√

M ∑`∈σ e−ik·r` a` the annihilationoperator for a fermion with quasimomentum k on sublatticeσ , where M is the number of lattice cells. The components ofthe Hamiltonian on the Bloch sphere follow as

h fx (k) =−JJν(α)

3

∑j=1

cos(k ·a j−νϕ j), (S15)

h fy (k) = JJν(α)

3

∑j=1

sin(k ·a j−νϕ j), (S16)

h fz (k) =−

J2

hωcz(α)

3

∑j=1

2cos(k ·b j)−∆eff/2, (S17)

h f0(k) =−

J2

hωc0(α)

3

∑j=1

2sin(k ·b j)+∆eff/2, (S18)

where a j is the vector that connects the nearest-neighborsites, with j labeling the three possible directions for mov-ing from an A site to a B site, a j = a[cos(ϕ j)ex + sin(ϕ j)ey]with their corresponding angles ϕ j defined from the positivex-axis. b j denotes the lattice vectors b1 = a(

√3,0),b2 =

a(−√

32 , 3

2 ),b3 =−b1−b2 which connect next-nearest neigh-bors, and c0,z(α) are some constants coming from taking thesum in Eq. (S10).

8

FIG. S1. Micromotion of the static Dirac points. Position of the static vortices for the observed time steps (multiples of 39 µs, driving periodis four time steps). (a) Static vortex at the Γ point. (b) Static vortex at the K point. The kx position is shifted by +0.2 in (a) and +0.7 in (b)for better visibility. Both vortices move in a circular fashion: the kx direction (red points) is out of phase with the ky direction (blue points).The vortex at the Γ point moves with the driving frequency. The vortex at the K point moves at twice the driving frequency and with oppositechirality. Both amplitudes are very small (few percent of the lattice vector length |b|). The detuning is δ/2π =−478 Hz.

State tomography with dispersive bands

In the following, we discuss the effect of the dispersivebands on the state tomography and show that it does not af-fect the mapping between the linking number and the Chernnumber. While the phase profile is, in general distorted for to-mography in non-flat bands, we will show here that the topo-logical information encoded in the vortex dynamics is not al-tered. Here, we are not after full state tomography, but ratherwe are interested in whether two vortex-trajectories in mo-mentum space link or not. Since this linking number can onlyhave discrete quantized values, it is topologically protectedand cannot be altered by the distorted phase profile measuredin the tomography in dispersive bands. This robustness oftopological defects is a general feature and was also used inthe related work of Ref. [S7]. While the effect of dispersivebands of the initial Hamiltonian was discussed in [S8, S9], theeffect of dispersive bands of the tomography Hamiltonian wasnot discussed previously.

The momentum distribution after a time-of-flight mea-surement on a state given by |ψ(k)〉 = cos(θk/2)|A〉 +sin(θk/2)eiφk |B〉, can be expressed as

n(k) = f (k)|〈A|ψ(k)〉+ 〈B|ψ(k)〉|2

= f (k){1+ sin(θk)cos(φk)},(S19)

where(|A〉, |B〉) are the poles of the Bloch sphere and f (k) isthe Fourier transform of the Wannier function. This measure-ment is nothing but a projection onto the x-axis of the Blochsphere, |x〉 = (|A〉+ |B〉)/2. One can see this easily by ex-pressing the Bloch vector ψ(k) representing the wave function|ψ(k)〉 as

ψ(k) =

(sin(θ(k))cos(φ(k)), sin(θ(k))sin(φ(k)), cos(θ(k))) .(S20)

We can immediately see that the x-component is given bysin(θ(k))cos(φ(k)).

We now consider the double quench protocol between theinitial Hamiltonian hi(k) and the final Floquet Hamiltonianh f (k). We start with an initial state |ψ i(k)〉 and quench to theHamiltonian h f (k), letting the system evolve for some time t;|ψ f (k, t)〉= e−ih f (k)·σt |ψ i(k〉). We then quench to the tomog-raphy Hamiltonian ht(k) = hi(k) and let it evolve for a timet ′; so that |ψ t(k, t + t ′)〉= e−iht (k)·σt ′ |ψ f (k, t)〉. The quenchedstate precesses around ht(k), giving rise to the oscillatory sig-nal in the momentum distribution (S19). As long as the initialstate is trivial, the linking numbers observed after this double-quench protocol reflects the Chern number of the ground stateof the final Hamiltonian h f (k). In the case of dispersive bands,the initial and the tomography Hamiltonians do not define amap from the Brillouin zone to the Bloch sphere that wrapsaround the whole sphere. Thus, we can continuously deformht(k) so that it points to the north pole for every k-point whichis again a topologically trivial state. We do this by the k-dependent rotation defined by

ht(k) = R(n,θ t(k))ht(k)

= R(sinφt(k)x− cosφ

t(k)y, θt(k))ht(k),

(S21)

where the transformation is captured by the vector θ t(k)(pointing along n(k)) and having length of θ t(k). Sinceht(k) is smooth in k, fully gapped and topologically trivial,R(n,θ t(k)) is continuous in k as well. When this rotation isapplied to the Hamiltonians in the remaining stages of the ex-periment, h f (k) and hi(k), it does not change the topology oftheir band structures. Namely, it does not change the num-ber of times h f (k) wraps around the Bloch sphere. One canconsider a patch in the k-space and its image under the maph f (k). The continuous rotation R(n,θ t(k)) can stretch, com-press, rotate, or shift this patch on the sphere but can not cutit open. Once we perform this rotation on the Hamiltonians in

9

all three stages, hi(k),h f (k) and ht(k), the rest of the discus-sion follows as described by Wang et al. in Ref. [S13].

In this rotated frame, the tomography Hamiltonian ht(k) isparallel to the z-axis and again we have a precession aroundthe z-axis. but now the signal that we measure is the projectionon the rotated x-axis. As a result, the phase of the measuredoscillatory dynamics is not the azimuthal angle of |ψ f (k, t)〉.Nevertheless, the phase distribution possesses vortices when-ever |ψ f (k, t)〉 ‖ z. At k-values for which |ψ i(k)〉 ‖ h f

(k), theinitial state can not precess and when projected onto ht

(k), weobserve a static singularity in the tomography. On the otherhand, at some k-value, if the rotated quench Hamiltonian isperpendicular to the initial state |ψ i(k)〉 ⊥ h f

(k), after someprecession time t, the state reaches −hi

(k) direction (effectivenorth pole) and gives rise to a dynamic vortex in the tomogra-phy.

Micromotion

In order to describe the influence of the periodic micromo-tion described by UF(t) on the dynamics, let us consider thefirst non-trival term of the high-frequency expansion UF(t) =exp[G1(t)+ G2(t)+ · · ·

]. We approximate [S4]

UF(t)' exp[G1(t)], G1(t) =−∞

∑m 6=0

Hmeimωt

mhω, (S22)

and find

G1(t) = ∑〈`′`〉

g〈`′`〉(t)a†`′ a`,

g〈`′`〉(t) =−∞

∑m6=0

Jeimωt

mhωJm+σ`ν

(α)e−i(m+σ`ν)ϕ`′` .(S23)

This correction is of the same origin as the effective next-nearest neighbor tunneling terms in the effective Hamiltonian.In leading order with respect to the driving amplitude, the co-efficients read g〈`′`〉(t) = −α

Jhω

cos(ωt−ϕ`′`) for ν = 0 andg〈`′`〉(t) =

Jhω

σ`e−iσ`ωt for ν = 1. For ν = 1, this correctionis again present already for infinitely weak driving, i.e. forα → 0.

The operator G1(t) describes a time-periodic micromotionin real space, where a particle at a given site ` explores neigh-boring lattice sites. With respect to quasimomentum, it can beexpressed like

G1(t) = ∑k

(a†Ak, a

†Bk) [gx(k, t)σx +gy(k, t)σy]

(aAk

aBk

).

(S24)Here, gx(k, t) = Re

(g(k, t)

)and gy(k, t) = Im

(g(k, t)

), with

g(k, t) = −i∑m 6=0 ∑3j=1 g j(t)e−a j ·k, where g j(t) describes

g〈`′`〉(t) for processes connecting an A site ` with a neighbor-

ing B site `′ at r`′ = r`+a j. For ν = 1, we find

g(k, t) = ∑m 6=0

Jmhω

Jm+1(α)×

3

∑j=1

exp(

mωt−a j ·k− (m+1)ϕ j +π/2).

(S25)

For small driving amplitudes α , the leading contributionstems from the m = −1 term. Neglecting all other terms,UF(t) describes a rotation in pseudospin, by a k-dependentangle ∼ J/(hω) around an axis in the xy-plane that itselfrotates around the z-axis with angular velocity ω and k-dependent phase. Increasing α , however, for α ' 1 both thethe m = −2 term and the m = 1 term become relevant sothat also higher harmonics of the driving frequency will makethemselves felt in the micromotion described by UF(t).

Apart from the real-space micromotion described by UF(t),another contribution to the micromotion is given by the trans-formation U(t) = Ushift(t)Urot(t) back to the original latticeframe of reference. It describes a phase rotation betweendifferent lattice sites, which corresponds to both a shift inquasimomentum and a rotation around the z-axis of the sub-lattice pseudospin. Moreover, there is another effect. In or-der to predict the dynamics observed in the experiment, wealso have to consider the experimental protocol, where latticeshaking is switched on at time t0 and switched off again atthe measurement time t. The shaking is performed in such away that the relative lattice position x(t + t ′) changes con-tinuously when the shaking is switched on. It is given byx(t + t ′) = 0 for t + t ′ < t0, x(t + t ′) = ξ(t + t ′)− ξ(t0) fort0 < t + t ′ < t, and by x(t + t ′) = ξ(t)− ξ(t0) for t + t ′ > t,where ξ(t + t ′) = −∆x[cos(ω(t + t ′))ex + sin(ω(t + t ′))ex]with ∆x = F/(Mω2) and atomic mass M. Accordingly thelattice velocity x(t + t ′) is discontinuous, featuring jumps byξ(t0) and −ξ(t)) at t + t ′ = t0 and t + t ′ = t, respectively.As a result the inertial force Fini(t + t ′) = −Mx(t + t ′) in-duced in the lattice frame of reference, which is given byF (t + t ′) =−Mξ(t + t ′) between t0 and t and vanishes beforeand after that, possesses also a contribution Fboost(t + t ′) =−Mξ (t0)δ (t + t ′− t0)− ξ (t)δ (t − t). These boosts shift thesystem’s state in quasimomentum by q(t0) and −q(t) withq(t + t ′) = −(M/h)ξ(t + t ′), which is described by the uni-tary operator Uq = exp(i∑`q ·r`n`). Thus, starting from thetrivial insulator state |ψ0〉 at time t + t ′ < t0, for times t ′ > 0the time-evolved state reads

|ψ(t + t ′)〉= e−ih t ′Ht |ψ(t)〉, (S26)

where Ht denotes the static tomography Hamilto-nian describing the system for times t ′ > 0 andwhere the state to be measured is given by |ψ(t)〉 =U−q(t+t ′)Ushift(t)Urot(t)U ′(t, t0)U

†rot(t0)U

†shift(t0)Uq(t0)|ψ0〉.

Employing Eq. (S5) as well as the fact that Ushift(t + t ′) =Uq(t + t ′), we find

|ψ(t)〉= Urot(t)UF(t)e−ih (t−t0)HF U†

F(t0)U†rot(t0)|ψ0〉︸ ︷︷ ︸≡|ψ ′0〉

. (S27)

10

FIG. S2. Sign of the Chern number. (a) In our Floquet system, the sign of the Chern number can change either by inverting the shakingdirection or by increasing the shaking amplitude. Phase diagram calculated for a detuning of δ/2π = −40 Hz. (b) Sense of wrapping theBloch sphere in the four different cases.

Thus, the full micromotion, as it can be observed in the exper-iment is described by

Umicro(t) = Urot(t)UF(t). (S28)

One should note that |ψ ′0〉 = U†micro(t0)|ψ0〉 is not an eigen-

state of the initial Hamiltonian. We can overcome thisby transforming the initial Hamiltonian as well; H i′ =U†

micro(t0)HiUmicro(t0). When we also rotate the tomography

Hamiltonian Ht′ = U†micro(t)H

iUmicro(t), it is now clear thatthese tomography and initial Hamiltonians are equal to eachother only for tomography times t = t0 + nT with integer n.For any other sub-stroboscopic time steps, the tomographyHamiltonian will not be parallel to the initial Hamiltonian onthe Bloch sphere. In Fig. 2(a) of the main text, we omit thesecontributions due to the micromotion and just aim to illustratethe experimental procedure.

The Hamiltonian Ht is represented by a quasimomentum-dependent vector ht(k) playing the role of a magnetic fieldwith respect to the sublattice pseudospin and the state |ψ(t)〉 isrepresented by a quasimomentum-dependent unit vector ψ(t)denoting a point on the Bloch sphere of that pseudospin. Thepositions of the measured vortices correspond to those pointsin k-space, where both vectors are parallel (or antiparallel).Thus, as long as ht(k) points to the south (or north) pole ev-erywhere, the pseudospin rotation Urot(t) at angular velocityω will not make itself felt. However, as soon as ht(k) tiltsaway from the north pole, as it is the case in the present ex-periment, this rotation will cause an oscillatory behavior of thevortex position with respect to the time t. Thus, the interplaybetween the oscillations induced by UF(t) and that by Urot(t)is another source for the generation of higher harmonics in themotion of the vortex position observed in the experiment.

In Fig. S1 we evaluate the micromotion of the static Diracpoints in the experimental data. We find indeed the circularmotion with the driving frequency or multiples of it predicted

by the above derivation. As expected from the scaling of themicromotion with J/(hω) which is in the order of ∼ 0.1, theamplitude of the micromotion is very small (few percent ofthe lattice vector length |b|) and does not affect the topologyof the system.

Sign of the linking number

We determine the sign of the linking number by comparingthe relative chirality of the static (χs) and dynamic vortices(χd), i.e. the total sign is set by χsχd . The chirality of the dy-namic vortex contour is given by the multiplication of the chi-rality of a vortex (or an antivortex) χv and the chirality of thepath that it travels χp. Since the dynamic vortex contour is theinverse image of the equator of the Bloch sphere, the directionof the motion is set by the gradient of the Hamiltonian |h(kv)|at the equator. This direction can be reversed by modifyingthe magnitude of the gap parameter |h(kv)|, without closingthe gap at the Dirac point itself, i.e. without changing the chi-rality of the static vortex at the Dirac point. In the following,we show that changing the gradient of the Hamiltonian at theequator also converts the vortices into antivortices, hence, pre-serves the chirality χd and with that also the sign of the linkingnumber. This means that the chirality of the dynamic vortexcontour χd reflects indeed the topology of the Hamiltonianand cannot be changed by topologically-trivial deformationsof the energy band. Our definitions are inspired by a relatedargument in ref. [S13].

The state of the system ψ(k, t) is given by Eq. (S20) forθ(k, t) and φ(k, t). The initial state points to the south polefor all k, ψ(k,0) = −ez (in the case of dispersive bands, af-ter performing the rotation given in Eq. (S21)). Quenchingto the Floquet Hamiltonian induces a rotation by the angleα(k, t) ≡ ω(k)t = 2h f (k)t around the direction of h

f(k) =

h f (k)/|h f (k)|. The time evolved state thus reads ψ(k, t) =

11

FIG. S3. Phase profiles. Phase profiles corresponding to the time steps shown in Fig. 4. The vorticity data is obtained from such phase profiles.The identified vortices are marked by red and blue circles as a guide to the eye. While the phase profile itself is distorted for state tomographyin dispersive bands, the vortices can be clearly identified and their interpretation is not compromised.

R(α(k, t), hf(k))ψ(k,0) where R(α(k, t), h

f(k)) denotes the

rotation matrix. For the given initial conditions, this gives

ψ(k, t) =

hxhz[1− cos(α(k, t))]+ hy sin(α(k, t))hyhz[1− cos(α(k, t))]− hx sin(α(k, t))

h2z [1− cos(α(k, t))]+ cos(α(k, t))

. (S29)

In the tomography, we observe a static vortex wheneverh

f(k) ‖ ez. Dynamic vortices occur when h

f(k) ⊥ ez, i.e.

when hf(k) lies on the equator, so that for α(k, tn) = nπ and

ψ(k, tn) points to the north (south) pole for odd (even) integersn, where tn = nπ/2|h f (k)|. In the following, we will focuson the case n = 1, where a dynamic vortex is found at timet1(k). The condition α(k, tn) = π defines the trajectories ofthe dynamic vortices in quasimomentum k, corresponding tothe inverse image, P, of the equator of the Bloch sphere withrespect to the map h(k) : k→ h. Note that in Ref. [S13], thiscorresponds to the inverse image of the north pole with respectto the map [k, t]→ h.

Direction of vortex motion

Let kv ε ` ⊂ P be a point on the line ` which lies in theinverse image of the equator and e‖(kv) denote a tangentialunit vector of ` at kv which defines a direction on this line.Then, the vortex which passes kv at time t(kv) moves withvelocity kv = k‖e‖ where k‖ = −ω(kv)/g‖(kv) with g‖(k) =t(k)∇kω(k) · e‖(kv). Thus, as long as the gap does not close[ω(k) > 0], the direction of motion χp is determined by thegradient of the gap along the line `,

χp =−sgn[g‖(kv)]. (S30)

Here, e‖ is the unit vector obtained from ∇khz(k) by an az-imuthal rotation by π/2; e‖ = R(ez,π/2)∇khz(k).

Vortex chirality

In order to obtain the chirality of a dynamic vortex at pointkv, we expand the wave function ψ(kv, t(kv)) in the vicinity ofkv,

ψ(k) =

00−1

+δk

hx(kv) f (kv)− hy(kv)g(kv)

hy(kv) f (kv)− hx(kv)g(kv)0

, (S31)

where f (kv) = 2∇khz(k) and g(kv) = π/h(k)∇k|h(k)| as be-fore. This can be also expressed as δψ(k) = f (kv)δkh(k)+g(kv)δkh

′(k), where h

′(k) = (−hy(kv), hx(kv)) is a unit vector

orthogonal to h(kv) and that, like h(kv), lies on the equator.These two unit vectors (h, h

′) span a coordinate system that is

rotated by φ(kv) with respect to the one spanned by (ex, ey).The chirality χv of a dynamical vortex is now determined

by whether the azimuthal phase φ(k) winds in positive ornegative direction while δk is taken around a closed loop;δk = δk[cos(γ)ex + sin(γ)ey] for γ : 0→ 2π . The chiralityreads,

χv = sgn[g(kv)× f (kv)] = sgn[g(kv) · e‖]. (S32)

Therefore, both the direction a vortex travels and its chi-rality depend on the gradient g(kv) of the gap at the vortexposition kv. Inverting the direction of the motion requires toinvert the gradient of the gap g‖(kv) along the line `. On theother hand, inverting the chirality of the vortex via a change

12

FIG. S4. Dynamical vortex contours. Countors of the dynamical vor-tices for different shaking detunings according to the simple modeldescribed in the text.

of g(kv) requires to invert g(kv) · f⊥(kv)≡ g f⊥(kv). Note thatthe unit vector e‖, which is defined to point along the directionwhere hz(kv) keeps the constant value zero, stands perpendic-ular to the gradient of hz(kv). Thus, changing the direction ofmotion of the vortex without closing the gap implies that thevortex changes its chirality, which preserves the overall signof the dynamic vortex counter χvχp. Hence, any deformationin the Hamiltonian that does not change the topology cannotchange the observed sign of the linking number.

Sign of the Chern number in our system

The sign of Chern number is given by the sense in which theBloch sphere is covered. This is fixed by (i) which one of thetwo Dirac points is at the north pole of the Bloch sphere and(ii) in which sense the states wrap around the Bloch sphereazimuthally. These questions can be simply related to the chi-ralities of the observed static and dynamical vortices: the chi-rality of the dynamical vortex contour χd determines, whichDirac point is at the north pole, while the chirality of the en-closed static vortex χs determines the azimuthal winding ofthe states. For our choices of sign conventions, the sign of theChern number of the lowest band is given by sgn[C] =−χd χs.

The four different possible combinations of χd = ±1 andχs = ±1 can be realized in our system by changing the shak-ing parameters: (i) the occupation of the poles by the Diracpoints inverts for large shaking amplitudes and (ii) the senseof wrapping around azimuthally inverts with the direction ofshaking. Data for the two shaking directions at small shakingamplitude is presented in Fig. 4 of the main text. Fig. S2 il-lustrates the four different possibilities in the phase diagramspanned by shaking phase and shaking amplitude.

Data analysis and full numerics

The experiments start with an ultracold cloud of about3 · 105 spin-polarized 40K atoms in the F = 9/2, mF = 9/2state. We linearly ramp up the hexagonal optical lattice in 10ms and hold for another 5 ms before switching on the latticeshaking. In the direction orthogonal to the hexagonal lattice,the sample is harmonically confined, i.e. realizing a lattice oftubes. The lattice is formed by the interference of three laserbeams of wavelength λL = 1064 nm and we introduce an AB-offset by polarization control of the beams [S10]. We imagethe sample on a CCD camera after 21 ms of time-of-flightexpansion, which leads to a magnification where one latticevector length |b| corresponds to 56 pixel). The state tomogra-phy uses 32 time steps of 8 µs and a sinusoidal fit including anexponential damping. Fig. S3 shows exemplary phase profilesas obtained from the tomography in the dispersive bands.

To obtain the tight-binding description of our lattice, westart from the known lattice geometry fixed by the polarizationof the lattice beams (linear polarization tilted 9o out of thelattice plane, with a relative phase of the in-plane and out-of-plane polarization of 0, 2π/3 and 4π/3 for the three beams).We calculate the exact band structure for this geometry anddifferent lattice depths V0. To determine a precise value of thelattice depth, we use the band distance data of the bare latticefrom the state tomography and fit the exact band structure toit (see Fig. S5). In this way, we compensate for small driftsof the lattice depth. We then fit a tight binding model to theexact band structure and obtain ∆AB, JAB, JAA and JBB. Thevalues of hδ = ∆AB− hω and JAB are given in Fig. S5. Thevalues of JAA/h are in the range of 80 to 115 Hz and JBB/h inthe range of -2 to -6 Hz. For the comparison with the effectiveHamiltonian Eq. S8, the small JAA and JBB are neglected.

We compare our data to exact numerics of the driven tight-binding model (Fig. 3 and Fig. S5). In this calculation, thetime evolution operator U(t,0) is calculated via time slicingas a product of time evolution operators for constant Hamilto-nians (compare Ref. [S10]). This method works for any evo-lution time t including sub-stroboscopic time steps, where themicromotion is automatically taken into account. It dependson the initial phase of the shaking, which we set to zero as inthe experiment. To obtain a prediction for the dynamical vor-tex contours, we calculate the overlap of the time evolved state|ψ(t)〉=U(t,0)|ψ(0)〉 with the initial state |ψ(0)〉 and countmomenta where this overlap is below a threshold of 0.02. Inthe last column of Fig. S5, these numerical data are summedup for all time steps (using a resolution of eight time steps perdriving period, in order to better resolve the contours). Thephase diagrams in Fig. 1(c) and Fig. S2 are obtained from thisexact numerics and the Chern number is calculated as the in-tegral of the Berry curvature. The non-integer values from thecalculation on a finite grid in momentum space are removedby setting the Chern number to zero or one based on a thresh-old of 0.5.

13

Experimental data for different detunings

The experimental vortex data for different detunings lead-ing to the phase diagram of Fig. 3 is presented in Fig. S5. Thedifferent rows correspond to different detunings ranging from-938 Hz to 515 Hz. The detuning is varied by changing thelattice depth at fixed driving frequency of 6.410 kHz and driv-ing amplitude of 1 kHz. The first column shows the band gapbetween the two lowest bare bands as obtained from the os-cillation frequency of the tomography along three equivalenthigh symmetry paths (red, blue and green. The data is aver-aged over the six first time steps after the quench into the Flo-quet system). The collapse of the curves indicates the goodbalance of the three lattice beam intensities. The lattice pa-rameters are obtained from a fit of the exact band structure tothe band gap (black curve, regions around the static vorticesare excluded from the fit). The extracted detuning and next-neighbor tunneling are states in the respective subfigures. Thedashed horizontal line indicates the shaking frequency and in-dicates the near-resonant nature of the driving.

The second column shows experimental data with the time-integrated static and dynamic vortices after the quench intothe respective Floquet system. The hue indicates the time afterthe quench, at which the vortex appeared (lighter color meanslater time). The static vortices at the Γ and K points are presentin all images. For non-trivial Chern number and in a regimeof larger detunings, closed contours of dynamical vortices ap-pear. The contour calculated from the effective Hamiltonian(green line) gives a reasonable approximation for the contourshape in the non-trivial regime.

The third column shows the expected dynamical vorticesfrom a full numerical calculation including the initial state andthe micromotion. Each black dot indicated a zero scalar prod-uct between the initial state and the time-evolved state.

Dynamical vortex contours from the effective Hamiltonian

The effective Hamiltonian allows deriving a simple es-timate for the dynamical vortex contours (green lines inFig. S5). We neglect here the dispersion of the initial bandsand the micromotion. In this approximation only non-trivialcontours can be described, because the trivial contours arisefrom the finite dispersion of the initial bands. The vortex con-tour corresponds to the momenta, where the final Hamilto-nian lies on the equator, i.e. where the z-component vanisheshf

z(k) = 0. Using hfz(k) = ∆eff/2+∑

3j=1(J

effAA−Jeff

BB)cos(k ·b j)(compare Eq. S15) and the effective tunneling elements in thelow driving limit, this corresponds to

S(k) =3

∑j=1

cos(k ·b j) =−∆eff

2(JeffAA− Jeff

BB)

=−hδ +3J2

AB/hω

4J2AB/hω

=− δ +34

(S33)

with δ = hδ/(J2AB/hω). The sum of the three cosines S(k)

can obtain values between +3 and −3/2. This means thatcontours only exist for detunings between δ = −15 and δ =−+3, which defines the non-trivial region.

Fig. S4 shows the value of the sum of the three cosines S(k),which correspond to the vortex contours for different detun-ings δ . The contour closes around the Γ point for δ = −15(where S(k = Γ) = 3) and around the K and K’ points forδ = +3 (where S(k = K) = −3/2). In Fig. S5 the contoursare plotted together with the data for the respective detunings(green lines). While we don’t expect quantitative agreementon this level of approximation, the predictions qualitativelyexplain the behavior of the data in the non-trivial regime. Fur-thermore, this discussion gives an intuitive picture for the de-tunings, where the topological phase transitions occur.

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FIG. S5. Experimental data for different detunings and comparison to calculations. See text for details.