Direct Geometric Probe of Singularities in Band Structure

Charles D. Brown1,2, Shao-Wen Chang1,2, Malte N. Schwarz1,2, Tsz-Him Leung1,2,

Vladyslav Kozii1,3, Alexander Avdoshkin1, Joel E. Moore1,3, Dan Stamper-Kurn1,2,3

1Department of Physics, University of California, Berkeley CA 947202 Challenge Institute for Quantum Computation, University of California, Berkeley CA 947203Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

(Dated: September 9, 2021)

The band structure of a crystal may have points where two or more bands are degenerate in energyand where the geometry of the Bloch state manifold is singular, with consequences for material andtransport properties. Ultracold atoms in optical lattices have been used to characterize such pointsonly indirectly, e.g., by detection of an Abelian Berry phase, and only at singularities with lineardispersion (Dirac points). Here, we probe band-structure singularities through the non-Abeliantransformation produced by transport directly through the singular points. We prepare atoms inone Bloch band, accelerate them along a quasi-momentum trajectory that enters, turns, and thenexits the singularities at linear and quadratic touching points of a honeycomb lattice. Measurementsof the band populations after transport identify the winding numbers of these singularities to be 1and 2, respectively. Our work opens the study of quadratic band touching points in ultracold-atomquantum simulators, and also provides a novel method for probing other band geometry singularities.

Band structure describes the energy and state of singleparticles within a crystal as a function of their quasi-momentum. While the energy spectrum itself is impor-tant, so too are the local geometry and global topologyof the Bloch-state manifolds for explaining material prop-erties, including those pertaining to quantum Hall effects,electric polarization, and orbital magnetism [1, 2].

The geometry of an eigenstate manifold can be revealedthrough transport of a quantum state along a smoothpath of parameters that define the system’s Hamiltonian.This transport is generally nonholonomic, meaning thatthe state generated by transport from an initial to a finalpoint depends on the path along which the system wastransported. Such transport has been explored mainlyin the two limiting cases that the energy spectrum of asystem is either largely gapped [3, 4] or entirely gapless [5]along a closed loop in parameter space. In the formerlimit, the state-space geometry generates a Berry phase;in the latter, the nonholonomy generalizes to a Wilsonloop operator describing a path-dependent rotation withinthe degenerate subspace. Both dynamics derive from theBerry connection matrix, Anm

q ≡ i〈unq|∂q|umq 〉, whichexpresses the local geometry of state space. Here, n andm are band indices, and q is the quasi-momentum. In thegapped limit, the Berry phase is determined solely by one(Abelian) diagonal element of this matrix; in the gaplesslimit, off-diagonal elements enter, leading to non-Abelianstate rotations [6].

In this work, we explore the nonholonomy of transportthrough a state space containing singular points of de-generacy. One example of such points, which we probeexperimentally, is the Dirac points of degeneracy betweenthe n = 1 and n = 2 bands of the two-dimensional honey-comb lattice, lying at the K and K′ points of the Brillouinzone (Fig. 1). Away from these points, the energy gapbetween the touching bands grows linearly with quasi-momentum. The singular state geometry around eachlinear band touching point (LBTP) has profound implica-

tions for the material properties of graphene, e.g. relatedto Klein tunneling of electrons through potential barriers[7] and the appearance of a half-integer quantum Halleffect [8]. The Dirac point of the honeycomb lattice hasbeen explored also in ultracold-atom experiments [9], in-cluding interferometric measurements of the Berry phaseproduced along trajectories that circle the Dirac point [10]and direct mapping of the Bloch-state structure acrossthe Brillouin zone [11, 12].

Crystalline materials may also host a singular quadraticband-touching point (QBTP), about which the energygap between two bands grows quadratically with quasi-momentum. As before, the QBTP can profoundly affectmaterial properties. For example, the singular QBTP ispredicted to produce an anomalous Landau level spectrum[13]. Interactions can destabilize a QBTP, leading totopologically-protected edge states, nematic phases andboth quantum anomalous Hall and spin phases [14–17].The role of QBTPs is being investigated intensely in bothuntwisted and twisted bilayer graphene [18–22]. Despitetheir importance, QBTPs have remained unexplored inultracold-atom systems.

Here, using ultracold atoms within an optical lattice, wedemonstrate that transport of a quantum system througha singular band touching point leads to a non-Abelian,coherent state rotation between bands, with the rotationdepending on the relative orientation of path tangentsentering and exiting the singular point. Further, we showthat this dependence characterizes and distinguishes theBloch-state geometry surrounding linear and quadraticband-touching singularities.

Let us exemplify our approach by considering the s-orbital LBTP of a two-dimensional honeycomb opticallattice (Fig. 1). To probe this Dirac point, we preparean optically trapped 87Rb Bose-Einstein condensate andthen slowly ramp up an overlain static honeycomb lattice,placing the atoms initially at the Γ-point of the n = 1band. Next, we apply a fictitious force to the gas by

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accelerating the optical lattice potential to a velocityvlat(t). While the atoms remain at zero quasi-momentumin the laboratory frame, they evolve to non-zero velocityv = ~q/m = −vlat in the lattice frame, with q being thelattice-frame quasi-momentum and m being the atomicmass.

To demonstrate the nonholonomy generated by theLBTP, we accelerate the atoms on a trajectory that pro-ceeds at constant acceleration from Γ to K (at quasi-momentum qK), and thence at a different constant ac-celeration to 24 equally-spaced points on a circle that lieat distance of 0.4 ‖qK‖ from the K-point. The turningangle between the rays entering and leaving the K-point,as defined in Fig. 2a, is varied over θ ∈ [0, 2π]. We thenperform “band mapping” by smoothly ramping off thelattice potential at the fixed final quasi-momentum, i.e.with the lattice at a constant final lab-frame velocity. Thisramp maps the population in each band onto a distinctmomentum state. Measuring this momentum distributionquantifies band populations in the moving lattice.

Transport along paths passing through the singularLBTP leads to interband transitions that vary with theturning angle (Figs. 2c-d). For trajectories that enter thesingularity and then reverse onto themselves (θ = 0), thepopulation remains nearly entirely in the initial n = 1band. For trajectories that continue with constant tangentthrough the singularity (θ = π), the atoms undergo a nearcomplete transition to the upper n = 2 band (seen alsoin Ref. [23]). Over the full range of θ, each populationundergoes one cycle of oscillation.

The unit-cell wavefunction of the n = 1 and n = 2Bloch states near the Dirac point can be represented asa pseudo-spin-1/2 vector, with s-orbital Wannier statesat the lattice sites A and B representing the up- anddown-spin basis states. In this basis, the Bloch statesare eigenstates of the Hamiltonian HLBTP = −B(q) · σwhere B(q) is a pseudo-magnetic field that lies in thetransverse pseudo-spin plane and σ is the vector of Paulimatrices. B(q) has a magnitude B = ~vg|q − qK| thatvaries linearly with distance from the singularity, and asshown in Fig. 2b, it has an orientation (in the propergauge) that is radially outward from K. Here, vg is thegroup velocity near the Dirac point. The 2π rotation ofB(q) about the Dirac point is responsible for the π-valuedBerry phase of trajectories that encircle the Dirac point[10].

This pseudo-spin model explains our observations. Theatomic pseudo-spin entering the Dirac point along a rayexperiences a pseudo-magnetic field whose orientation nremains constant and whose magnitude smoothly tunesto zero. Under this field, the initial-state pseudo-spinremains aligned along n. Departing the Dirac point, thepseudo-spin experiences a magnetic field along a neworientation m, with n · m = cos θ, and a magnitudeincreasing linearly with time. The pseudo-spin is thusplaced in a superposition of eigenstates, with popula-tion cos2(θ/2) in the m-oriented pseudo-spin eigenstate(n = 1 band) and sin2(θ/2) in the orthogonal state (n = 2

band). This simple prediction matches well to our data,with residual differences accounted by numerical simu-lations (see Supplemental Material) of the dynamics ofnon-interacting atoms over the finite duration of our ex-perimental stages [24].

We find that passage through the Dirac point producesa phase-coherent superposition of band states. Such co-herence is demonstrated by allowing the atoms to evolveat the final point of the trajectory for a variable timebefore measuring populations in a basis different from thelocal energy eigenbasis. Temporal oscillations in thesemeasurements, with a frequency matching the calculatedgap between the n = 1 and n = 2 bands, demonstratethe coherence of the atomic state following transport; seeSupplemental Material.

The energy-time uncertainty relation places a boundon how finely the singular point can be located by ourmethod. We consider a trajectory where the accelerationhas magnitude a near the singularity. The system spendsa time δt ∼ (~/ma)δq within δq of the singularity, whilethe energy gap has magnitude δE ∼ ~vgδq in that vicinity.Setting (δt)(δE) ∼ ~ establishes that the band structureis effectively gapless within a quasi-momentum distanceof δq = R ∼

√ma~/vg of the singularity. That is, the

nonholonomy generated by the singular point should beobserved also for finite time trajectories that pass withinthe effective radius, R, of the singularity.

We measure R by driving the atoms along a family oftrajectories, shown in Fig. 3a, that connect between theinitial Γ-point to a final Γ-point that is one reciprocallattice vector away, and performing band mapping mea-surements at the final point. These trajectories cross theboundary between the first and second Brillouin zones atnine equally-spaced points along the K −M −K′ line.As shown in Fig. 3b for various traversal times, τ , weobserve that trajectories that pass directly through eitherDirac point yield a band population distribution that isindependent of τ with ∼ 3/4 of the atoms transferringto the upper band. In contrast, for traversal times thatare longer and for paths that veer farther from the Diracpoints, the transition between bands is increasingly sup-pressed, demonstrating that R decreases with decreasinga (increasing τ).

The singularity at an LBTP can be characterized by twodifferent experimental methods: either by Berry phasemeasurements along trajectories that encircle the singu-larity [10], or, as shown here, through state rotationsproduced along trajectories that pass through the singu-larity. These two methods are related, but nonequivalent.Berry phase measurements measure the integrated Berryflux, which is determined from a diagonal element of theBerry connection matrix Ann

q . A π-valued flux is foundpinned to the singular point. In contrast, the non-Abelianstate rotations detected in our method derive directly fromthe off-diagonal elements Anm

q with n = 1 and m = 2being the two crossing bands; see Supplemental Material.Further, different from Berry phase measurements, ourmethod can be regarded as measuring the Hilbert-Schmidt

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a b

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M K

E(q

)/h (k

Hz)

Quasi-momentum

LBTP

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6

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K

qy

qx

M

K’

ω+δω1(t)

ω

ω+δω2(t)

y

x

2l√27

A

B

FIG. 1. Experimental scheme. a, Illustration of an opticalhoneycomb lattice with a two sites (A and B) in the unitcell, formed by overlapping three λ = 1064 nm wavelengthlight beams (red arrows). Offsetting the optical frequenciesof two lattice beams by δω1,2(t) accelerates the lattice anddrives lattice-trapped atoms through a trajectory in quasi-momentum. b, The n = {1, 2, 3, 4} Brillouin zones of thehoneycomb lattice are shown in green, blue, red and purple,respectively. c, The band structure of the honeycomb lattice(plotted with potential depth of 20 kHz×h) exhibits an LBTPin the s−orbital band manifold at q = K, and a QBTP in thep−orbital band manifold at q = Γ.

quantum distance d2(q,q′) = 1−|〈u1q′ |u1

q〉|2 [25, 26] withq identified as a point along the input path into – andq′ as a point along the exit path from – the singularity.The oscillation of the n = 1 band population as a func-tion of the turning angle reveals the quantum distance toundergo one complete oscillation between zero and unityon a contour encircling the LBTP.

The distinction between these two methods is dramaticin the case of a QBTP. Like the LBTP, a singular QBTPalso carries concentrated Berry flux that is restricted,assuming time-reversal and C6 symmetry, to be 0 or±2π [15]. However, these values of Berry phase are unde-tectable via interference measurements. In contrast, ourmethod uncovers the characteristic nonholonomy of thesingular QBTP and the concomitant modulation of thequantum distance around the singular point.

For a singular QBTP, the geometric structure of Blochstates of the two intersecting bands at the vicinity ofthe singularity can again be described as those of apseudo-spin in a pseudo-magnetic field. Different fromthe LBTP, here, the pseudo-magnetic field, lying in thetransverse pseudo-spin plane, has a magnitude that in-creases quadratically with distance to the singularity, and

a c

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θ = 0 θ = π/2 θ = π θ = 3π/2

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d

θ

θK

K

0.4qK

b

FIG. 2. Non-Abelian state rotations around a Dirac point. a,Zoomed view of Brillouin zone map. Atoms are transported(trajectory marked by red arrows) at constant accelerationsfrom Γ → K and then from K to a final point lying on acircle of radius 0.4‖qK‖ centered at the Dirac point. b, Blochstates near the Dirac point are described as pseudo-spin-1/2states in an pseudo-magnetic field (black arrows) that pointsradially outward from, and wraps once around, the Diracpoint. c, Band mapping images at the final quasi-momentum,with overlain Brillouin zone maps, show the band populationvary with turning angle θ. The spatial scale is indicated bythe black scale bar of length 0.1 mm. d, Fractional bandpopulations Nn/Ntotal (n = 1: green, n = 2: blue, sum ofother bands: gray) vs. θ. Means and standard mean errorsdetermined from 7 repeated measurements. The green andblue dashed lines show cos2(θ/2) and sin2(θ/2), respectively.

has an orientation that wraps by an angle of 4π along apath encircling the singularity; see Fig. 4b.

We probe this geometric structure at the QBTP thatoccurs at Γ between the n = 3 and n = 4 bands of the hon-eycomb lattice. For this, we first load the Bose-Einsteincondensate into the n = 3 band of the lattice by “inverseband mapping.” In previous work [27], such loading intoexcited bands was realized with moving atoms in a staticlattice; here, we realize similar state-preparation withstatic atoms loaded into a moving lattice. Specifically,we gradually increase the depth of a honeycomb latticemoving with velocity vQ that, as shown in Fig. 4a, islocated within the third Brillouin zone in the extendedzone scheme. Thence, we accelerate atoms (in the latticeframe) at constant acceleration along the path Q → Γ,into the QBTP, and then, turning by an angle θ, accel-erate the atoms at a different constant acceleration outof the QBTP and to the edge of the Brillouin zone. Fi-nal points are chosen as ones where populations in then = 3 and n = 4 bands are easily distinguished by bandmapping.

We observe a nonholonomy at the QBTP that is distinctfrom that observed at the LBTP. Specifically, we observetwo cycles of oscillation in the final band populations overthe interval θ ∈ [0, 2π]. This behavior is explained wellby the pseudo-spin representation of the singular QBTP.

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a

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K K’MTrajectory midpoint

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N 1 / N

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τ (ms)

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0 1 20

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FIG. 3. Effective size of a Dirac singularity. a, Illustration ofquasi-momentum trajectories over which the atoms are trans-ported for the measurements in b. Each trajectory connectstwo different Γ-points, but traverses one of nine equally spacedpoints along the K−M−K′ line. b, Fractional n = 1 bandpopulation plotted, for different trajectory traversal times τ ,against the trajectory midpoints from a. Means and standardmean errors are generated from 3-5 repeated measurements.c, An effective radius R (plotted normalized by ~k = h/λ)is defined for each τ by the distance along the K−M−K′

line for which the threshold N1/Ntotal = 0.5 is fulfilled. Rdiminishes with longer τ .

Different from the LBTP, here, pseudo-magnetic-fieldorientations along the incoming (n) and outgoing (m)paths are related as n ·m = cos 2θ. The nonholonomyof the QBTP now produces populations of cos2(θ) andsin2(θ) in the n = 3 (initial) and n = 4 bands, respectively.

The amplitude of the observed oscillation is lower thansuggested by this simple theory; again, we ascribe thisdifference to dynamical effects of our finite-duration ac-celeration and band-mapping stages. Nevertheless, theperiodicity of the oscillations, combined with the knowntime-reversal and C6 symmetry of our lattice, unambigu-ously determines the topological winding number aroundthe QBTP to be well defined and equal to 2; see Supple-mental Material.

In conclusion, we have demonstrated transport of aquantum system through singular band-touching pointswith different topological winding numbers. We observenon-Abelian, coherent state rotation between bands. Thedependence of this rotation on the relative orientationof path tangents entering and exiting the singular pointunambiguously measures the winding number.

Our method of probing band structure could be ap-plied to gain insight on other band structure singularitiesand on interaction effects. It would be interesting tostudy higher-order singular band-touching points, i.e., be-

a b

c

θ

Turning angle, θπ 3π/2π/2 2π0

Nn / N

tot

0

0.5

1.0

Q

θ

FIG. 4. Non-Abelian state rotations around a QBTP. a, Atomsare prepared in the n = 3 band at Q, transported (along redarrows) to the QBTP at Γ, and then transported to a finalquasi-momentum for band mapping. b, The pseudo-magneticfield (black arrows) describing the Bloch state geometry wrapstwice in orientation for one revolution around the QBTP.c, A plot of normalized band population as a function of θ,collected by analyzing band mapping data; see SupplementalMaterial. Red circles: n = 3 band; purple circles: n = 4band; gray circles: bands with n 6= 3, 4. Means and standardmean errors are determined from 7 repeated measurements.Dashed red (purple) lines are predictions based on a simplepseudo-spin model for n = 3 (n = 4) populations. Ournumerical simulations suggest that the data does not reachunity oscillation amplitude due to nonadiabaticity in the bandmapping procedure; see Supplemental Material.

tween more than two bands. We deliberately minimizeinteraction effects in these experiments, but in futurework it will be interesting to observe potential interaction-induced instabilities of Dirac points, QBTPs or otherband touching points. The path-dependent non-Abeliannonholonomy observed here may also pertain to chemicalsystems, where potential energy surfaces are endowedsimilarly with conical intersections [28–31], suggesting anew route for quantum state control in optically drivenmolecules.

I. METHODS

We prepare Bose-Einstein condensates of 1− 2× 104

87Rb atoms in an optical dipole trap with trap frequenciesωx,y,z ≈ 2π × {23, 41, 46} Hz. The chemical potentialof the harmonically trapped gas is roughly 0.4 kHz × h.We load a condensate into a honeycomb lattice withlattice depth ≈ 10− 20 kHz× h, which is formed by themutual interference of three 1064-nm-wavelength lightbeams propagating in the horizontal plane, intersecting at

5

equal angles of 120◦, and polarized in-plane. We operateour lattice in a “lattice of tubes” configuration, withonly weak confinement in the vertical direction. Ourcalculations suggest that, with the lattice turned on, thereare approximately 90 atoms per lattice site at the center ofthe trap. One lattice beam has a fixed angular frequencyω, while the other two beams have time-dependent angularfrequencies ω1(t) = ω + δω1(t) and ω2(t) = ω + δω2(t).Under a closed feedback loop that dynamically varies thephase difference between the intersecting beams that formthe lattice, time-dependent relative detuning betweenlattice beams allows us to accelerate the lattice potentialreproducibly in two dimensions. The lattice accelerationdrives the atoms through a two-dimensional trajectory ofquasi-momentum in the lattice frame. After transportingthe atoms along this trajectory, we adiabatically rampdown the lattice potential to map Bloch states onto aplane wave basis (band mapping) and then measure theatomic velocity distribution using velocity-space focusingand resonant imaging. This distribution reveals the latticeband populations prior to band mapping.

II. ACKNOWLEDGEMENTS

We thank E. Altman, M. Zaletel, N. Read and J. Harrisfor early insights into this work. We acknowledge supportfrom the NSF QLCI program through grant number OMA-2016245 and also NSF grant PHY-1806362, and fromthe ARO through the MURI program (grant numberW911NF-17-1-0323). C. D. B. acknowledges supportfrom the National Academies of Science, Engineering,and Medicine Ford Postdoctoral Fellowship program. V.K. and J. E. M. were supported by the Quantum Materialsprogram at LBNL, funded by the U.S. Department ofEnergy under contract number DE-AC02-05CH11231. A.A. and J. E. M. acknowledge support from the NSF undergrant number DMR-1918065 and from a Kavli ENSIfellowship. J. E. M. acknowledges support from a SimonsInvestigatorship.

Supplemental Material for “DirectGeometric Probe of Singularities in Band

Structure”

SI. ARBITRARY LATTICE TRANSLATION

The optical honeycomb lattice potential is created byinterfering three laser beams, with wavelength λ = 1, 064nm, that intersect at relative angles of 120◦. Written

explicitly, the potential is given by

V =

∣∣∣∣∣∣√

2Vlat

9

∑i=1,2,3

εieiki·r

∣∣∣∣∣∣2

=2

9Vlat

3−∑

i=1,2,3

cos (Gi · r)

,

(S1)

where kn = k(cos θi, sin θi, 0) [32] are the wavevectorsof lattice beams with k = 2π/λ, θi = ( 1

2 −2i3 )π, εi =

(sin θi,− cos θi, 0) are the (in-plane) polarization vectors,Vlat is the depth of the lattice, and Gi =

∑j,k εijkkj are

the reciprocal lattice vectors (εijk is the antisymmetricLevi-Civita tensor). Here, the script index i is not to beconfused with the imaginary number i. When one of thebeams with angular frequency ω1 = ω + δω1, is detunedby δω1 relative to the beam with angular frequency ω,

the lattice will translate at a velocity v1 = a∆ω1k1 in thelab frame, where a = 2λ

3 is the lattice constant. Here, weignore the small change in the wavevector k1 due to thedetuning. Similarly, detuning beam 2 by ∆ω2 translates

the lattice at a velocity of v2 = a∆ω2k2. Therefore, bychoosing an appropriate linear combination of v1 and v2,we can translate the lattice at any velocity in the latticeplane. Acceleration along a linear trajectory in quasi-momentum space corresponds to varying the detuningsof the two lattice beams linearly in time.

In the experimental setup, each lattice beam is con-trolled by an acousto-optic modulator (AOM), which weuse to control the detuning between lattice beams. Wechange the drive frequency of AOMs by tuning the setvoltage of the voltage-controlled oscillators (VCOs) thatcontrol the AOMs, as shown in Fig. S1b. For each linesegment in momentum space, we change the detuning lin-early, which creates a uniform acceleration of the lattice.In the reference frame where the lattice is stationary, theatoms are thus accelerated in the opposite direction.

SII. GENERAL EXPERIMENTAL SEQUENCE

The ramp shapes of beam intensities are given inFig. S1a. After creating a Bose-Einstein condensate, wefirst adiabatically ramp up the lattice intensity, such thatall atoms are loaded into the ground band at Γ; see “slowload” in Fig. S1. This also allows the atoms to follow theminimum of the combined trap potential created by theoptical dipole trap (ODT) beams and the lattice beams,which varies as the lattice intensity is ramped up, due topossible misalignment between ODT beams and latticebeams.

For the QBTP experiment we prepare atoms in then = 3 band, which can be done by rapidly loading atomsinto a running lattice. In reality, to make sure that theatoms are at the minimum of the trap potential at the startof the experiment, we first load atoms into the groundband with the slow loading procedure. Next, we perform

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a

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Time (ms; not to scale)

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State preparation Lattice acceleration Band mappingi

i

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FIG. S1. Experimental sequence. a, Typical ramp shape oflattice beam intensity. The same ramp function is shared forall three lattice beams. b, Typical ramp shape of lattice beamdetuning. Generally, the detunings between two beam pairsare different, depending on the trajectory in momentum space.The shaded bars on top and bottom of both a and b indicatethe time intervals (in units of milliseconds) used for each stageof the experiment: top (gray) = slow load, bottom (red) =fast load.

150

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iv

FIG. S2. Band mapping data. Shown in the figure are individ-ual measurements from our characterization of the honeycomb-lattice QBTP, as summarized in Fig. 4 of the main text. Imagesi through iv (in a and c) show the results of band mappingafter trajectories with four different turning angles θ, at thefinal quasi-momenta marked by black points on the Brillouinzone map in b. a, Band mapping images obtained in the labframe. These are absorption images of the atomic spatial dis-tribution taken after (1) adiabatically ramping off the latticebeams (and also optical traps) at the final lattice velocity, and(2) allowing the atoms to expand in a weak harmonic magneticconfinement so as to map velocity onto position. The spatialscale of the images in a and c is indicated by the black scalebar of length 0.1 mm. The scale of the images in terms ofvelocity distributions is indicated by the overlain Brillouinzone maps. b, Here the Brillouin zone is rotated to matchthe actual orientation of the lattice created in the experimentand enlarged for clarity. c, The band mapping images in thelattice frame. Each of them are obtained by translating thelab-frame image in a by the corresponding quasi-momentum(one of the red arrows) in b.

band mapping to recover a stationary BEC, before rapidlyramping up the lattice beam intensities again but with arunning lattice. This is shown in Fig. S1 as “fast load.”With this technique, we can load into the n = 3 bandwith > 92% fidelity on average.

Band population detection (or “band mapping”) isachieved by ramping down the lattice beams slowlyenough, such that atoms in different bands are adiabat-ically mapped to plane waves with momenta separatedby reciprocal lattice vectors. In the resulting absorptionimages, the positions of atom peaks then correspond totheir momentum in the lab frame. In the main text,we show our band mapping measurement results in thelattice frame, which is obtained by translating the labframe image by a distance determined by the final quasi-momentum in a lattice acceleration sequence. The posi-tions of atom peaks in the images now correspond to theirmomenta in the lattice frame, and the association betweenband label n and these peaks can then be determinedby overlapping the Brillouin zone of the lattice with theimage. Fig. S2 provides an example of images obtainedfrom band mapping measurements.

In the measurement of Fig. 2c of the main text, ifthe angle of the outgoing trajectory would lie close to aBrillouin zone edge, we accelerate the atoms further beforeband mapping to a point far away from any Brillouin zoneedges. This further acceleration is slow and adiabatic,such that the measured population in each band is notaffected by the additional translation.

SIII. COHERENT EVOLUTION AFTERPOPULATION TRANSFER AT A SINGULARITY

Here, we show that after the population transfer atK, the atoms in bands n = 1 and n = 2 still interferecoherently. The trajectory used for the coherence test isshown in Fig. S3a. After creating an even superpositionin the lowest two bands by turning at K, the atoms areaccelerated to the center of the second Brillouin zone, atpoint Q in momentum space, and held there for a varyingamount of time t. During the hold, the state acquires arelative phase ϕ = t∆ωQ between the |n = 1〉 and |n = 2〉components, where ~∆ωQ is the band gap between thelowest two bands at Q. Then, we accelerate the atomsquickly back to Γ, with an acceleration large enough suchthat the state is projected onto the basis states formedby the Bloch states at Γ. Finally, we perform a bandmapping measurement to project the phase difference intoa population difference between bands n = 1 and n = 2.In Fig. S3b, we indeed see an oscillation in the measuredpopulation that is consistent (with the empirically addeddecay) with what is expected from theory.

7

Q

ab

0 400 800Hold time (ms)

0.2

0.6

1.0

exp.sim.fit

N1 /

Nto

t

FIG. S3. Detecting coherence of superposition producedby transport through the LBTP singularity. a, Atoms aretransported on a trajectory with three constant accelerations:Γ→ K, K→ Q, and Q→ Γ. The first two acceleration stepsare done slowly, in 0.3 ms each. Atoms are then held at Q fora variable hold time. The last acceleration step is done quicklyin 0.2 ms. This faster acceleration causes the atomic state(a superposition in the n = 1 and n = 2 bands at Q) to beprojected onto the band eigenbasis at Γ prior to band mapping.b, n = 1 band population seen from band mapping at Γ. Thedynamical phase acquired by the superposition state Q leadsto temporal oscillation in the band-mapping n = 1 population.We fit the data to a sinusoid with a decaying exponential enve-lope (red curve), which finds that the frequency is 1.47± 0.53kHz, and the decay constant 319±205 µs. This frequency is inagreement with the band gap between n = 1 and n = 2 bandsat Q, which is calculated as 1.77 kHz at a lattice depth ofVlat = h× 24 kHz for non-interacting atoms. The black curveis the simulation result obtained by a numerical simulation ofthe entire transport, hold time, and projective measurementprocess, in the non-interacting limit, as described in Sec. SVI.The simulation does not account for decay: To make a moredirect comparison between theory and experiment, we applyexponential damping to the simulation result using the decayconstant found from our fit to experimental data.

SIV. COMPLEMENTARY REPRESENTATIONOF DATA IN FIG. 3 OF THE MAIN TEXT

In Fig. 3 of the main text, we show a measurement ofthe effective size of a Dirac point singularity by plottingthe population in the lowest band against the midpointof various trajectories (the trajectories are shown bothin Fig. 3 of the main text, and in Fig. S4a). Instead,here we provide a different visualization of the data. InFig. S4b we plot the n = 1 band population normalizedby the total atom number against the acceleration timeused to traverse the trajectories in Fig. S4a. A projectionof the initial state

∣∣ψ1Γ

⟩onto the n = 1 Bloch state at the

final Γ-point in the two-band model gives 0.25, which isthe expected normalized population in the n = 1 bandunder the sudden approximation. For trajectories faraway from singularities, the adiabatic theorem predictsthat the n = 1 band population should increase withlonger acceleration times, which we indeed observe in ourexperiments. On the other hand, when we cross K in thetrajectory, the band gap between n = 1 and n = 2 bandsvanishes, and the adiabatic theorem no longer applies.

τ (ms)0 0.5 2.01.0 1.5

N1 /

Nto

t

Κ

Μ. . .

0

0.2

0.4

0.6

0.8

1

Γ

Γ

Μ

Κ’Κ

a b

FIG. S4. Complementary representation of the same dataset of Fig. 3 in the main text. a, Trajectories taken in theexperiment copied here for reference. b, normalized n = 1band population as a function of acceleration time τ . Onlydata for trajectories that go through points between K and Mare shown in the plot, for simplicity. Here, the solid lines arethe simulation results obtained by numerical simulations ofthe entire transport, hold time, and projective measurementprocess, in the non-interacting limit, as described in Sec. SVI.For the shortest acceleration time used, we expect the n = 1band population to deviate from 0.25 (dashed line), predictedby the two-level model in Sec. SVII, because of excitation tohigher bands.

SV. MOMENTUM SPACE EXTENT OF THEQBTP

To characterize the singular behavior of the QBTP,we perform a measurement similar to the one shown inFig. 3 of the main text. As seen in Fig. S5b, for thesame acceleration time τ , more atoms stay in the thirdband since the condition for adiabatic evolution is bettersatisfied. At very short acceleration times, transitionsto higher bands are significant, which is also capturedby simulations. At longer acceleration times (τ > 1.4ms), the actual quasi-momentum of atoms in the latticedeviates from the set values due to acceleration in the labframe, which appears in the images as an overall shift ofpeak positions after band mapping (Fig. S5c). There aretwo immediate consequences: first, the actual trajectory isdifferent from expected; second, band mapping becomesunreliable if the actual final quasi-momentum is closeto the boundary between the third and fourth Brillouinzones. Therefore, we only keep data points whose finalmomentum in the lab frame is ≤ 0.1‖qK‖. We attributethis acceleration to the fact that atoms are dragged alongthe direction of lattice translation during the sequence(plus corrections due to group velocity of atoms in thelattice) [33]. The dragging effect grows with the durationof the experiment. The QBTP experimental sequenceis significantly longer than the LBTP experiment (thereare more steps in the procedure and the band mappingtime is necessarily longer), so the dragging effect is morepronounced. Once the atoms are dragged away fromthe potential minimum of the optical trap, this confiningpotential applies an acceleration to the atoms. Finally, we

8

285 290 295 300 305x [a.u.]

236

240

244

248

252

y [a

.u.]

0.2

0.40.6

0.81

1.21.41.6

1.8

2

2.2

2.4

Acceleration time (ms)0 1.0 2.0 3.0

i

ii

iii

n = 3n = 4n 3,4

0

0.5

0

0.5

0

0.5

N n / N to

t

iii

i

ii

a b

c

FIG. S5. Momentum-space extent of the QBTP. a, Trajectorieschosen for the experiments. b, The population in each bandas a function of acceleration time τ . Here, τ is the total timetaken to accelerate the atoms from the initial state to thefinal state. For this choice of band mapping point, only thethird and fourth bands are uniquely distinguished. Error barsare one standard error. c, Acceleration in lab frame. Asdescribed in Sec. SII, the positions of atom peaks in imagescorrespond to their lab frame velocities. Therefore, by tracingthe position of one of the peaks in band mapping images in thelab frame, we can measure the change in lab-frame momentumthat atoms experience during the experiment. The figure isplotted with data from b-ii, and different colors are assignedfor each acceleration time labeled next to the data points. Theradius of the dashed circle is 0.1‖qK‖, centered at the finalquasi-momentum along the trajectory.

indeed observe that, like the LBTP, the region in whichband population rotations occur is confined at the QBTP.

SVI. SIMULATIONS BY FULL HAMILTONIANCONSTRUCTION

The dynamics of the system are governed by the timedependent Schrodinger-equation:

i~∂t |ψ〉 = H(t) |ψ〉 . (S2)

We divide the time sequence into small time steps witheffectively constant Hamiltonian H(t) such that we canuse the time-independent Schrodinger equation. Bloch’stheorem allows us to write the Hamiltonian in the planewave basis

⟨r|G(n,m)

∣∣r|G(n,m)

⟩= exp

{(iG(n,m) · r)

},

where G(n,m) = nG1 + mG2 (n,m ∈ Z) is a recipro-cal lattice vector of the 1064 nm 2D optical honeycomblattice. The Schrodinger equation then takes the form(

1

2m

(p + ~q

)2+ V (r)

)unq(r) = Enqu

nq(r), (S3)

where q is the quasi-momentum vector, n is the bandindex and unq(r) = 〈r|unq〉 is the real space representationof the lattice periodic part |unq〉 of a Bloch state

∣∣ψnq⟩ =

eiq·r|unq〉. By expanding V (r) and unq(r) in the plane wavebasis the Hamiltonian can be expressed explicitly as

H(n,m),(n′,m′)(q) =

(2

3V0 +

~2k2

2m|G(n,m) + q|2

)δn,n′δm,m′ −

V0

9δn,n′+jδm,m′+l with (j, l) ∈ J , (S4)

where J = {(−1, 0), (0,−1), (1, 0), (0, 1), (−1,−1), (1, 1)},the tilde implies that the quantity is normalized by thelattice laser wavenumber k = 2π/λ with λ = 1, 064 nmand V0 is the optical lattice depth. Here, V0 < 0 be-cause the lattice light is red-detuned with respect to theprincipal atomic resonances of 87Rb.

We simulate the system dynamics by truncating theplane wave basis such that it incorporates 81 differentplane wave components and we divide the time evolu-tion into time steps of τ = 5 µs. We ensured thatthe simulation captures all relevant dynamics of thesystem with these settings. The initial state of thesystem |ψ(0)〉 is often taken to be the ground band(n = 1) eigenstate. We apply H(t) to the state such

that |ψ(t)〉 = T∑t′<t e

−iH(t′)τ/~ |ψ(0)〉, where T impliesthat the sum is properly time ordered, by utilizing spec-tral decomposition to speed up our computation. Withthis method we can easily extract the populations of the

state in each of the eigenstates |unq〉 at each time stepby first calculating the eigenstates and subsequently theoverlap with the state |ψ(t)〉 (defined as the fidelity), asshown in Fig. S6. In that manner we can ensure that theeffects we observe originate in the system dynamics andthat we are able to quantify the effect of non-adiabaticpopulation transfer. We use this technique to make goodguesses for initial experimental parameters that minimizenon-adiabatic behavior. Although these simulations donot involve any interaction effects, they nevertheless showno significant discrepancies to the observed data.

9

Time (ms)

Nn /

Nto

t

i ii iii iv v vi

FIG. S6. Decomposition of dynamic system state into Blocheigenfunctions of H(t). The state |ψ(0)〉 = |Γ, 1〉 evolvesover the following steps: i) lattice ramp down, ii) ramp toq = 1.25 K, iii) inverse band mapping, iv) ramp to q = Γ,v) ramp to q = qr=0.9

255° , before vi) band mapping. In v), thesuperscript labels the distance away from Γ, in units of “qK

′′.Due to nonadiabatic evolution, bands other than n = 3, 4 arepopulated.

SVII. POPULATION DYNAMICS ANDTOPOLOGY OF THE BAND TOUCHING

POINTS WITHIN THE TWO-BAND MODELS

In this section, we show that all the main results pre-sented in the main text can be qualitatively understoodwithin an effective two-band model. To do that, we derivethe equations which describe the band population of aparticle moving in a lattice in the presence of an exter-nally applied force within a two-band model. Relevant toour experiment, we only focus on the case of a constantforce. In our derivation, we closely follow that from thesupplementary materials from Ref. [11]. Note that in thefollowing sections i is the imaginary number.

We start with the two-band lattice Hamiltonian of theform

H0 =∑n=1,2

∑q

εnq|ψnq〉〈ψnq |, (S5)

where |ψnq〉 are Bloch eigenstates of the n-th band atquasi-momentum q having energy εnq. The Bloch states

can be parameterized as |ψnq〉 = eiq·r|unq〉, where |unq〉 hasperiodicity of the lattice in real space and r is the positionoperator. The Bloch states satisfy the normalizationconditions 〈ψnq |ψmq′〉 = δ(q− q′)δnm and 〈unq|umq 〉 = δnm.

In the presence of an external constant force F theHamiltonian of the system reads as H = H0 − F · r,leading to the Schrodinger equation

i∂t|Ψ(t)〉 = (H0 − F · r)|Ψ(t)〉, (S6)

where we use the units with ~ = 1.We assume that at t = 0 the particle has quasi-

momentum q0, implying that its wavefunction is givenby

|Ψ(0)〉 =∑n=1,2

ϕn(0)|ψnq0〉, (S7)

and the band population at t = 0 equals |ϕn(0)|2. In thepresence of an external force, the quasi-momentum of theparticle changes according to

q(t) = q0 + Ft, (S8)

meaning that the solution has form

|Ψ(t)〉 =∑n=1,2

ϕn(t)|ψnq(t)〉. (S9)

The Schrodinger equation can then be rewritten as:

i∂t

(ϕ1

ϕ2

)=

(ε1q(t) − ξ

11q(t) −ξ12

q(t)

−ξ21q(t) ε2

q(t) − ξ22q(t)

)(ϕ1

ϕ2

),

(S10)where we have defined

ξnmq(t) ≡ F ·Anmq(t) = i〈unq(t)|∂t|u

mq(t)〉, (S11)

and

Anmq ≡ i〈unq|∂q|umq 〉 (S12)

is the Berry connection. As we see from Eq. (S10), itis precisely the off-diagonal components of the Berryconnection that drives the population change of the twobands.

A. Linear band touching

To study linear band touching, we apply the aboveanalysis to the two lowest bands of the typical bandstructure of the honeycomb lattice arising from the s-orbitals. Under the tight-binding approximation withonly nearest neighbor hopping, the Hamiltonian of the(non-driven) system has the form

H0(q) =

(0 M(q)

M∗(q) 0

), (S13)

with

M(q) = −t

(2eiqya/2 cos

√3qxa

2+ e−iqya

), (S14)

10

0

0.25

0.5

1

Trajectory midpoint

0.25

0.5

1

N1 / N

tot

K K’M

τ

R/ℏk

τ

N1 / N

tot

τ

M

K

FIG. S7. Final band population as a function of the acceleration time τ and different trajectories shown in Fig. S4a calculatedwithin a two-band model, Eq. (S13). a, Population of the n = 1 band as a function of the trajectory midpoint at different fixedτ and the size of the singularity R as a function of τ . The black arrow indicates the increase of τ . The results are in goodqualitative agreement with the measured data shown in Figs. 3b,c of the main text. b, The normalized n = 1 band populationpresented as a function of the acceleration time for different fixed trajectories. Again, the result of the calculation is in goodqualitative agreement with the experimental data shown in Fig. S4b.

where −t is the hopping integral and a is the latticeconstant.

With this model Hamiltonian, we use Eq. (S10) to cal-culate the final band population as the particles movealong the trajectories shown in Fig. S4a (or Fig. 3a ofthe main text), assuming that originally they were pre-pared in the n = 1 state. The result as a function ofthe trajectory midpoint and the acceleration time τ ispresented in Fig. S7 and is in good qualitative agreementwith Figs. 3b,c of the main text and Fig. S4b. The onlydifference appears at short acceleration times, which isquite natural since at such large acceleration the popula-tion of the higher bands becomes substantial and mustbe taken into account.

The Dirac singularities are located at the points qK =(4π/3

√3a, 0

)and qK′ =

(2π/3

√3a, 2π/3a

)in the Bril-

louin zone and those related by the reciprocal latticevectors. Near the K-point, the Hamiltonian can be ex-panded to linear order in quasi-momentum k ≡ q− qK

as

H0(k) ≈ vg(

0 k+

k− 0

), k± = kx ± iky, (S15)

where vg = 3at/2 is the group velocity at the Dirac point.The wave functions of the n = 1 and n = 2 bands aregiven by

|u1k〉 =

1√2

(−k+/k

1

), |u2

k〉 =1√2

(k+/k

1

),

(S16)

with k± ≡ kx ± iky, k ≡ |k| =√k2x + k2

y, and the compo-

nents of the Berry connection equal

A11k = A22

k = −A12k = −A21

k =1

2(k2x + k2

y)

(ky−kx

).

(S17)We emphasize that while the specific form of the wavefunctions and Berry connection depend on the gauge ofthe wave functions, the final answer for the observables,e.g. band population, is gauge-independent.

When a particle moves exactly towards or away fromthe singularity in momentum space, it does not experi-ence any interband transitions away from the singularpoint itself since F ·Amn

k = 0. This is a general featureof any homogeneous Hamiltonian satisfying the relationH0(sk) = sαH0(k). Indeed, if choosing a proper gaugelike the one in Eq. (S16), the eigenstates of the Hamilto-nian depend only on the direction of the quasi-momentum,

k = k/|k|, but not on its absolute value. Then, according

to Eq. (S11), F ·Amnk = 0 because k(t) and, consequently,

|umk(t)〉 do not depend on t. This implies that all the tran-

sitions happen exactly at the singularity. This conclusionis true provided we stay in the region of the Brillouinzone where the Hamiltonian of the system can be approx-imated by a homogeneous function (i.e., we must remainsufficiently close to the singular point). Assuming that

the particle moving along the direction k scatters off the

singularity in the direction k′ at time t0, the transitionprobability between the two bands can then be foundfrom the continuity condition |Ψ(t0 − 0)〉 = |Ψ(t0 + 0)〉,

11

which translates into

ϕ1(t0 − 0)|u1k〉+ ϕ2(t0 − 0)|u2

k〉=ϕ1(t0 + 0)|u1

k′〉+ ϕ2(t0 + 0)|u2k′〉. (S18)

If the particle was originally in the n = 1 band, ϕ2(t0 −0) = 0, then the transition probability is given by

d2(k,k′) = |ϕ2(t0 + 0)|2 = |〈u2k′ |u1

k〉|2 = 1− |〈u1k′ |u1

k〉|2.(S19)

This quantity is exactly the Hilbert-Schmidt quantumdistance between the states |u1

k′〉 and |u1k〉 [25, 26]. For

two infinitesimally close states k and k′ = k + dk, itreveals the gauge-invariant quantum metric tensor gij(k):

d2(k,k + dk) = gij(k)dkidkj , (S20)

which is defined as

gij(k) =1

2[〈∂iuk|∂juk〉 − 〈∂iuk|uk〉〈uk|∂juk〉+ (i↔ j)].

(S21)In the case of the isotropic Dirac dispersion, the quan-

tum distance is expressed through the angle θ betweenthe momenta k and k′:

d2Dirac(k,k′) = sin2 θ

2. (S22)

This result is in perfect agreement with the pseudo-magnetic field picture discussed in the main text andleads to the same prediction.

It is worth noting that such a clear and simple relationbetween the band population change and the quantumdistance (and Hilbert space geometry) is, strictly speak-ing, only possible in the vicinity of the singular point,where the Hamiltonian can be approximated by a homo-geneous function. In the experiment, on the other hand,the start and end points of the trajectory in the Brillouinzone clearly lie outside the singular region. Nevertheless,all the conclusions derived above remain valid providedthe particle’s motion is sufficiently slow. Indeed, out-side the singular region, the system is effectively gapped.Hence, according to the adiabatic theorem, the interbandtransitions are rare if the drive is slow.

B. Quadratic band touching

The entire analysis for the quadratic band touchingpoint is similar to that of the previous section, yet the re-sults show some important differences, which we highlightand discuss in detail below.

The lowest-energy quadratic band touching is formedby the n = 3 and n = 4 bands at the Γ-point. To derivethe effective low-energy theory for the QBTP, we followRefs. [34, 35] and start with the tight-binding model band

structure that arises from the px and py orbitals. We onlyinclude the σ-bonding and nearest-neighbor hopping forsimplicity. The Hamiltonian in this case takes the form

H = tσ

3∑i=1

∑~r∈A

{p†i (~r)pi(~r + aei) + H. c.

}, (S23)

where pi = (pxex + py ey) · ei are the projections of the porbitals parallel to the nearest-neighbor bond directionsei, p

†x,y represent the creation operators of atoms on the

corresponding orbitals, and tσ is the σ-bonding strength.Vectors ei are three unit vectors from one A cite to its

three neighboring B cites and are given by e1,2 = ±√

32 ex+

12 ey and e3 = −ey.

The Fourier transform of the Hamiltonian can be writ-ten as

H(k) =

(0 h(k)

h†(k) 0

),

h(k) =tσ2

3∑i=1

eiak·ei (I + σx cos 2θi + σy sin 2θi) , (S24)

where σx,y are Pauli matrices in the chiral orbital basis

(p−/√

2, p+/√

2), p∓ = px ∓ ipy, I is the identity matrix,and we have defined θ1 = π/6, θ2 = 5π/6, and θ3 = 3π/2.Importantly, this Hamiltonian is symmetric under time-reversal, inversion, and six-fold rotations.

The Hamiltonian in Eq. (S24) exhibits a quadratic bandtouching at k = 0. Expanding to O(k2) we find

h(k) =3tσ2

(1 00 1

)+

3tσa

4

(0 k+

−k− 0

)−

− 3tσa2

16

(2k2 k2

−k2

+ 2k2

)+O(k3). (S25)

After performing a unitary rotation in the sublattice space(two different sublattices originate from non-equivalentcites A and B in the primitive cell of the honeycomblattice), we obtain

H(k)→ U†H(k)U ≈ (S26)

≈ 3tσ2

1− (ak)2

4 − (ak−)2

8 0 ak+2

− (ak+)2

8 1− (ak)2

4 −ak−2 0

0 −ak+2 −1 + (ak)2

4(ak−)2

8ak−

2 0 (ak+)2

8 −1 + (ak)2

4

,

where U = (I − iτy)/√

2 and τy is the Pauli matrix in thesublattice space. Rewritten in this form, the Hamiltonianis block-diagonal: the lower right (upper left) block de-scribes quadratic band touching at energy −3tσ/2 (3tσ/2),while the off-diagonal blocks describe virtual hopping

12

between the low- and high-energy bands. Accountingfor such hopping within second-order perturbation the-ory [18, 26], we end up with the effective low-energyquadratic Hamiltonian describing just two lower bands(which correspond to n = 3 and n = 4) that exhibit aQBTP:

Heff(k) ≈ 1

4m

(k2 k2

−k2

+ k2

), (S27)

where the effective mass is given by m−1 = 3tσa2/4 and

the constant term −(3tσ/2)I has been dropped. Theeigenenergies of this Hamiltonian are given by ε3 = 0and ε4 = k2/2m, i.e., we have a band touching betweena quadratically-dispersing band and a flat band. Theeigenvectors equal

|u3k〉 =

1√2

(−k2−/k

2

1

), |u4

k〉 =1√2

(k2−/k

2

1

),

(S28)resulting in the Berry connection of the form

A33k = A44

k = −A34k = −A43

k =1

k2x + k2

y

(−kykx

).

(S29)The Berry phase around the QBTP is twice that aroundthe LBTP, i.e., equals ±2π. Since the Berry phase isdefined only mod 2π, such a characteristic is not par-ticularly useful in distinguishing a topologically-trivialQBTP from a non-trivial one. Instead, as we discussbelow, one can use a gauge-independent integer windingnumber, provided certain symmetries are present.

Repeating the same derivation as for the LBTP, we findthat the population of the n = 4 band after scattering offthe singularity (assuming the particles are originally inthe n = 3 band) and the quantum distance are given by

d2quad(k,k′) = |〈u3

k′ |u4k〉|2 = sin2 θ. (S30)

It is worth discussing now certain important differencesbetween the geometries of the linear and quadratic bandcrossings. For a comprehensive analysis of this and relatedquestions we refer to Ref. [26], while we only focus on themost important points.

In general, linear band touchings are always charac-terized by a Berry phase of π (mod 2π) around them,and the maximal quantum distance between the statesencircling the point always equals dmax

Dirac = 1, meaningthat there are always orthogonal states no matter howclose we are to the singularity. The linear singularitiesare always topological in this sense and no symmetriesplay any role in this conclusion.

Quadratic band touching points are quite different inmany ways. Despite the common belief, the Berry phasearound the QBTP, in general, does not have to be quan-tized to integers of 2π. Related to this fact, the maximalquantum distance between the states near the band touch-ing point, in general, can take an arbitrary value between0 and 1, and so can the amplitude of the transition prob-ability oscillations |〈u3

k′ |u4k〉|2.

However, this arbitrariness is removed when certainsymmetries are present in the system. In particular, whentime-reversal and rotational C6 symmetries are present,which is the case in our system, the Berry phase becomesquantized in units of 2π, and the quantum distance canonly take values 0 or 1. We emphasize that the quantumdistance is a more useful quantity for characterizing thenon-trivial topology of the band touching. Indeed, whilethe Berry phase is defined only mod 2π, its values of ±2πcan be easily changed to 0 by a simple gauge transfor-mation. It implies that the Berry phase cannot reliablydistinguish between a trivial QBTP with dmax = 0 and anon-trivial one with dmax = 1.

Equivalently, when time-reversal and rotational C6 sym-metries are present, the non-trivial topology of the QBTPcan be characterized by a well-defined gauge-independentwinding number [26]. In general, the effective two-bandHamiltonian can always be rewritten as a pseudo-spin in apseudo-magnetic field B(k), Heff(k) = B0(k) ·I+B(k) ·σ,where σ is the vector of Pauli matrices. If the aforemen-tioned symmetries are present, one can choose a basissuch that {B(k) · σ, σz} = 0, i.e., Bz(k) = 0. In this case,the pseudo-magnetic field lies entirely in the transversepseudo-spin plane, where it traces out a great circle onthe unit Bloch sphere as k goes around the band touchingpoint. In this sense, the winding of B(k) around the ori-gin (singularity) is a topological property since it cannotbe changed smoothly. The winding number counts howmany times B(k) encircles the origin and is well definedas long as B(k) 6= 0 on a trajectory that encompasses thesingularity. It can be explicitly defined as [36–38]

w =i

2π

∮dk∇k lnB−(k), (S31)

where the contour of integration encircles the singularityand B− = Bx − iBy. Near the QBTP in our system,B0(k) = k2/4m, B(k) = (1/4m)(k2

x− k2y, 2kxky, 0)T , and

B−(k) = k2−/4m, and so the winding number for the

QBTP wquad = 2. For comparison, the winding numberfor Dirac Hamiltonian (S15) equals wDirac = −1.

The amplitude of the oscillations measured experimen-tally and shown in Fig. 4c of the main text differs from1 due to nonadiabaticity in the lattice acceleration stepsand, particularly, in the band mapping procedure. Never-theless, according to the discussion above, the non-zeroamplitude and two cycles of oscillations combined withour lattice symmetries unambiguously indicate that thetopological winding number around the QBTP we studyis well-defined and equals 2.

13

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