Probing disorder-driven topological phase transitions via topological edge modes with ultracold atoms in Floquet-engineered honeycomb lattices

TL;DR

This paper demonstrates how disorder can induce phase transitions between different Floquet topological phases in ultracold atom systems, using edge mode properties in driven honeycomb lattices, advancing the study of topology under disorder.

Contribution

It introduces a method to probe disorder-driven topological phase transitions with ultracold atoms, overcoming previous limitations related to lattice translational invariance.

Findings

Disorder drives phase transitions between Floquet topological phases.

Disorder enhances the robustness of the anomalous Floquet topological regime.

Confirmation that disorder favors anomalous Floquet topological phases.

Abstract

One of the most fascinating properties of topological phases of matter is their robustness to disorder and imperfections. Although several experimental techniques have been developed to probe the geometric properties of engineered topological Bloch bands with cold atoms, they almost exclusively rely on the translational invariance of the underlying lattice. This prevents direct studies of topology in the presence of disorder, further hindering an extension to disordered interacting topological phases. Here, we identify disorder-driven phase transitions between two distinct Floquet topological phases using the characteristic properties of topological edge modes with ultracold atoms in periodically-driven two-dimensional (2D) optical lattices. Our results constitute an important step towards studying the rich interplay between topology and disorder with cold atoms. Moreover, our measurements confirm that disorder indeed favors the anomalous Floquet topological regime over conventional Hall systems, indicating an enhanced robustness and paving the way towards observing exotic out-of-equilibrium phases such as the anomalous Floquet Anderson insulator.

Equations

$${\sigma }_r=\frac{\lambda }{4\cdot {\mathrm{NA}}_{\mathrm{eff}}}.$$

$${\sigma }_z\approx 0.89\frac{\lambda }{{\mathrm{NA}}_{\mathrm{eff}}^2},$$

$$⟨\vec{k}|V|\overrightarrow{k^{\prime }}⟩=\int d\vec{x}V(\vec{x})e^{-i(\vec{k}-\overrightarrow{k^{\prime }})\vec{x}}=\tilde{V}(\Delta \vec{k}).$$

$$|\psi (\vec{k},\tau ){|}^2=A|\psi (\vec{k},0){|}^2+B{e}^{-k^2/2{\sigma }_k^2},$$

$$\overline{y}=\frac{{\sum }_i{y}_i{w}_i}{{\sum }_i{w}_i},$$

$$J_i(t)=A{e}^{B\cos (\omega t+{\varphi }_i)}+C,$$

$$\hat{U}(T)=\prod _{j=0}^{N-1}{e}^{-\frac{i}{\hslash }\left(\hat{H}(t_{N-j-1})+{\hat{V}}_D\right)\frac{T}{N}},$$

$${\hat{H}}_{\text{eff}}=\frac{i\hslash }{T}\ln (\hat{U}(T)).$$

$${\hat{G}}_{\alpha }=\hat{P}{e}^{\frac{2\pi i}{L_{\alpha }}\hat{\alpha }}\hat{P},\alpha =x,y,$$

$$\mathcal{B}=\frac{1}{2\pi }\mathrm{Im}\left(\text{Tr}(\ln ({\hat{\mathcal{G}}}_y{\hat{\mathcal{G}}}_x{\hat{\mathcal{G}}}_y^{†}{\hat{\mathcal{G}}}_x^{†}))\right),$$

$$|\phi (t=nT)⟩=\hat{U}(nT)|{\phi }_0⟩={\left(\hat{U}(T)\right)}^n|{\phi }_0⟩.$$

Full text

The discovery of topological phases of matter has revolutionized condensed matter physics [1, 2]. Remarkably, disorder plays a crucial role in revealing topological robustness. A prominent example is the quantization of Hall plateaus in the integer [3] and fractional quantum Hall effects [4]. While strong disorder eventually induces a transition to a trivial regime when its amplitude approaches the spectral gap, weak disorder is essential for pinning the Fermi energy within the gap, thereby stabilizing the plateaus with extraordinary precision.

Synthetic quantum systems offer unique opportunities to study the properties and exotic excitations of topological phases of matter [5, 6] with microscopic resolution and control [7]. With neutral atoms in optical lattices, topological phases are typically engineered via periodic driving (Floquet engineering) [8, 9, 10], or synthetic dimensions [11, 12, 13, 14]. This has enabled the realization of Hofstadter [15, 16, 12, 13, 17] and Haldane-type models [18, 19, 20], with recent advances in studying strongly-interacting regimes [21, 22, 23]. Moreover, numerous techniques have been developed to characterize the engineered geometric properties of Floquet-Bloch bands [6] with high precision [24, 25, 18, 26, 27, 19, 28, 29]. While this provides an excellent starting point for studying the rich interplay between topology and disorder [30, 31, 32, 33], experimental studies remain scarce.

Topological pumps [34] – 1D dynamical versions of 2D topological systems – offer a simpler experimental and theoretical route for investigating topology and disorder [35]. Indeed, disorder-induced phase transitions have been studied in the context of 1D topological Thouless pumps in photonic waveguide arrays [36], with neutral atoms [37], superconducting qubits [38] and in metamaterials [39]. Disorder can further drive a transition from a topologically trivial to a non-trivial phase known as topological Anderson insulators and has been studied experimentally in 1D topological pumps [37] and 1D momentum-space lattices [40], while 2D realizations have so far only been reported in photonic systems [41, 42, 43].

Chiral edge modes serve as clear signatures of phase transitions between topologically trivial and non-trivial regimes [44, 45]. In this work, we extend this idea by demonstrating that mode matching between a localized Bose-Einstein condensate (BEC) – precisely controlled via an optical tweezer [29] – and topologically protected chiral edge modes can be exploited to selectively prepare edge modes in distinct Floquet topological regimes. This enables the study of topological phase transitions in the presence of disorder, which we introduce using an optical speckle potential [46, 47]. Specifically, we investigate disorder-driven transitions between the anomalous Floquet (AF) and Haldane (H) topological regimes in a 2D periodically-modulated honeycomb lattice (Fig. 1a). Our experimental observations are in agreement with numerical simulations, underscoring the reliability of our approach. Central to this is a newly developed method for calibrating the average disorder strength, based on Kapitza-Dirac diffraction of a BEC [48]. In contrast, earlier methods relied on optical calibrations in a conjugate imaging plane [49] or state-dependent disorder potentials [50, 51] limiting their applicability to more general settings.

The technique builds on our recent work on the observation of topological chiral edge modes [29]: A localized BEC is confined in a tightly-focused optical tweezer trap near a topological interface generated via a programmable repulsive potential step (Fig. 1b). After release, the in-situ evolution of the atoms reveals the presence or absence of chiral motion. Distinct Floquet topological phases are realized via chiral modulation of the three tunnel couplings (Fig. 1a), as demonstrated in Ref. [20]. Floquet topological phases are characterized by so-called winding numbers, which determine the number and chirality of topological edge modes in a specific quasienergy gap. For a generic two-band model there are two different energy gaps, one around zero energy (0-gap) and one at the Floquet Brillouin zone edge ($\pi$-gap) with winding numbers ($W^{0}$,$W^{\pi}$). The generalized bulk-boundary correspondence connects the Chern number of each band to the winding number according to: $\mathcal{C}^{\mp}=\pm(W^{0}-W^{\pi})$ [52, 53, 54], where $\mathcal{C}^{-}$ ($\mathcal{C}^{+}$) denotes the Chern number of the lower (upper) band. Here, we focus on two distinct topological regimes: the anomalous Floquet (Fig. 1c) with ($W^{0}$, $W^{\pi}$)=(1,1) and the Haldane regime (Fig. 1d) with (1,0), both supporting topological edge modes with the same chirality.

The overlap between the BEC and the edge mode can be tuned by changing the strength of the harmonic confinement of the tweezer allowing us to distinguish the two regimes with high precision [48, 29]. Here, we make use of the unique property of anomalous Floquet topological systems which support an edge mode at each quasienergy $\epsilon$, making it extremely robust to the initial-state parameters of the wave packet (Fig. 1c). In contrast, the observation of edge modes in the Haldane regime (0-gap) sensitively depends on the initial state (Fig. 1d) since it is located at non-zero quasimomentum ($K$ and $K^{\prime}$ points). By adjusting the tweezer depth we find that a shallow confinement $\omega_{s}$ suppresses the overlap with the edge mode in the Haldane regime, while maintaining a large overlap with the edge mode in the $\pi$-gap in the anomalous regime. Intuitively, this can be understood by looking at the spatial wavefunction of the edge mode in both regimes. A shallow tweezer generates a BEC at zero quasimomentum, which has good overlap with the $\pi$-gap edge mode in the anomalous regime, but essentially no overlap with the [math]-gap edge mode at finite quasimomentum. In order to further enhance the differential signal we choose a large potential step $V_{s}$ for generating the interface in the anomalous regime, as it increases the edge-mode velocity, while a shallow potential step $V_{d}$ is beneficial for large edge-mode velocities in the Haldane regime due to the smaller characteristic energy gap (Fig. 1c,d). For experiments where we study edge-mode propagation in the Haldane regime, we find that a larger radial confinement $\omega_{d}$ results in good overlap with the [math]-gap edge mode in both regimes.

The experimental sequence starts by generating a BEC of 39K atoms in a crossed optical dipole trap. By adding a tightly-focused optical tweezer trap at $\lambda_{T}=$1064\text{,}\mathrm{nm} a tightly-localized BEC of about 200 atoms is formed, whose position is controlled using a 2D acousto-optical deflector (AOD) that allows us to precisely align the position of the BEC relative to the topological interface. Residual atoms not loaded into the tweezer trap are expelled by lowering the large-volume crossed optical dipole trap. While holding the atoms in the tweezer trap, a programmable repulsive optical potential realizing the topological interface is ramped up. The potential is generated using a digital micromirror device (DMD), which is illuminated by a spatially and spectrally incoherent light source with a wavelength centered around $\lambda_{S}=$638\text{\,}\mathrm{nm} [29]. Subsequently, a hexagonal optical lattice, generated by three free-running laser beams ($\lambda_{L}=$745\text{,}\mathrm{nm}), interfering under an angle of $120\text{\,}\mathrm{\SIUnitSymbolDegree}$, is ramped up to a depth of 5.9$E_{R}$ (Fig. [1](https://arxiv.org/html/2508.20154v1#S0.F1)a), where $E_{R}/h=h/(2\lambda_{L}^{2}m_{\mathrm{K}})=$9.23\text{\,}\mathrm{kHz} is the recoil energy, $h$ Planck’s constant and $m_{\mathrm{K}}$ the mass of 39K. The periodic modulation is realized by varying the intensities of the three laser beams out of phase according to $I_{i}(t)=I_{0}(1-m+m\cos(\omega t+\phi_{i}))$, with $\omega$ the modulation frequency, $m$ the modulation amplitude, $\phi_{i}=\kappa\frac{2\pi}{3}i$ the phase for the $i$th lattice beam ($i=\{1,2,3\}$), and $\kappa=\pm 1$ the chirality of the modulation. After turning on the lattice, the modulation is linearly ramped to its final value $m=0.25$ within $5T$, where $T$ is one modulation cycle. At the end of the ramp, the atoms are suddenly released from the tweezer trap and we monitor the in-situ time evolution of the cloud after a variable time $t$. The vertical overlap of the tweezer focus and dipole trap is optimized for each experimental run to avoid excitations perpendicular to the 2D lattice, which, together with the dipole trap, has a combined trap frequency $\omega_{z}/(2\pi)=$0.33(3)\text{,}\mathrm{kHz}$$, and interaction effects are suppressed by setting the scattering length to $6a_{0}$, where $a_{0}$ is the Bohr radius.

To introduce controlled disorder, we generate a repulsive optical speckle potential at a wavenlength of $532\text{\,}\mathrm{nm}$. Optical speckle potentials are well-established tools in quantum simulation experiments [55, 47], particularly in studies of Anderson localization [56, 46, 57, 58, 59, 60], offering disorder with well-characterized statistics and tunable strength over several orders of magnitude. In our setup, a lens transforms a random phase pattern, generated by a holographic diffuser, into a speckle field in the Fourier plane of the lens. This pattern is then projected into the atomic plane using a high-resolution imaging objective with $\mathrm{NA}=0.5$ (Fig. 2a). Assuming diffraction-limited performance of the objective, we estimate a speckle correlation length of $\sigma_{r}=$0.3\text{,}\mathrm{\SIUnitSymbolMicro m} in the horizontal plane and $\sigma_{z}\simeq$2.4\text{\,}\mathrm{\SIUnitSymbolMicro m} in the vertical direction, much larger than the harmonic oscillator length in the $z$-direction. The intensity distribution of the resulting disorder potential closely follows the expected probability density function $p_{\bar{I}}(I)=\bar{I}^{-1}\exp[-I/\bar{I}]$, with $\bar{I}$ being the average intensity [47]. Its strength has been calibrated using diffraction measurements [48], providing the proportionality constant $\alpha_{I}$ for the potential energy shift seen by the atoms, $\overline{V}_{D}=\alpha_{I}\bar{I}$. In the optical setup we further displace the diffuser from the optical axis, such that rotating it results in different potential disorder realizations with similar statistics, as different patches on the diffuser will be illuminated. We use this to average over different disorder realizations.

We start by investigating the propagation of chiral edge modes in the presence of disorder. For weak disorder, it is expected that disorder induces couplings between different quasimomenta within the edge-mode dispersion, resulting in a renormalization of the edge-mode velocity. This is of particular relevance in the context of topological photonics [61, 62], where disorder and imperfections can hinder practical applications. We focus on the Haldane regime, where the system hosts a single chiral edge mode within the 0-gap. The modulation parameters are chosen to lie well within the topological phase, sufficiently far from any phase transition. To probe the effect of disorder, we measure the difference in the center-of-mass (COM) position $\Delta\bar{y}=\bar{y}_{+1}-\bar{y}_{-1}$ of the wavepacket for both chiralities along the topological interface (aligned along the $y$-direction) at various evolution times, where $\bar{y}_{\kappa}$ denotes the COM position evaluated from in-situ absorption images, as shown in Fig. 2c,d. The in-situ images further reveal that disorder not only slows down edge transport but also induces propagation into the bulk. This is further reflected in a reduced overall transport velocity, evident from the shorter COM distance traveled by the atoms. The velocity observed in the experiment is further suppressed compared to numerical simulations. We attribute this to the finite width of the topological interface and the more finely tuned overlap with the initial state used in the numerics [63, 64, 65, 29]. We find qualitatively similar behavior in the anomalous regime.

To study the disorder induced shift of the transition between the Haldane and Anomalous Floquet regimes we utilize the preparation scheme introduced in Fig. 1c,d. While the specifics of edge-mode propagation depend on the disorder potential, the presence or absence of chiral motion at the topological interface provides a robust observable for distinguishing between topological regimes. To enhance the contrast between the anomalous Floquet and the Haldane regime, we choose a large potential step $V_{s}$ in combination with a shallow tweezer $\omega_{s}$ and measure the COM displacement between the two chiralities after a fixed evolution time, as a function of modulation frequency (Fig. 3a). In the anomalous regime at lower modulation frequencies, we observe clear chiral motion, which gradually diminishes near the topological phase transition. Beyond a critical frequency, the signal vanishes, marking the transition to the Haldane regime. This behavior is in agreement with numerical simulations. We determine the critical point by fitting a kink function to the experimental and numerical data. Note that the absolute value of the COM distances differs between the numerical simulations and the experiment, as described above. The robustness of this method for identifying the phase transition is further validated by benchmarking against calculations of the Bott index $\mathcal{B}$ [66, 67] – a topological invariant equivalent to the Chern number that is particularly suited to study disordered systems. Here, we compute the Bott index for the lowest band of a simplified two-band model, as described in detail in Ref. [48]. We identify the region in the phase diagram with a disorder-averaged Bott index $\mathcal{B}=1$ with the Haldane regime, where $\mathcal{C}^{\mp}=\pm 1$ and the region with $\mathcal{B}=0$ with the anomalous regime, where $\mathcal{C}^{\mp}=0$.

We further benchmark our method without disorder ($\overline{V}_{D}=0$) via comparison to standard gap-closing measurements based on Stückelberg interferometry (red data point in Fig. 3b) [68, 69, 20]. This measurement determines the location of the phase transition at $11.13(8)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The systematic shift between this gap-closing measurement and the numerical simulations based on a simple two-band model (Bott index in Fig. 3b) is due to mixing with higher bands in the experiment. Extending the simulations to a 6-band model, we identify the phase transition at $10.92\text{\,}\mathrm{kHz}$ in agreement with the experimental results. Tracking the phase transition as a function of the applied disorder potential, we observe that the transition shifts to higher modulation frequencies with increasing disorder (Fig. 3b), suggesting that disorder favors the anomalous Floquet topological regime, in agreement with theoretical predictions [32]. We attribute the systematic shift between the experiment and theory to the role of higher bands. Numerical studies further suggest that the observed transition shift is quite generic, with only a weak dependence on the correlation length of the disorder potential.

Anomalous Floquet systems are predicted to exhibit a rich phase diagram at large disorder strengths (schematic phase diagram: Fig. 4a). In particular, the interplay between periodic driving and disorder can stabilize genuine out-of-equilibrium topological phases known as the Anomalous Floquet Anderson insulator (AFAI) [31, 70, 33], where a fully localized bulk coexists with an extended topological edge mode. More generally, one expects a transition to the topologically trivial regime once disorder strengths become comparable to the bulk energy gap. Experimentally, this trivial phase is characterized by the disappearance of chiral motion, independent of the initial-state parameters of the BEC (Fig. 4b). To probe this phase diagram, we monitor the chiral signal while varying the disorder strength at fixed modulation parameters. Cuts through the phase diagram are illustrated in Fig. 4a. Starting near the phase transition in the Haldane regime we find a re-entrant behavior of chiral motion, signaling a transition from the Haldane to the anomalous regime at weak disorder (Fig. 4c). For large disorder the chiral signal gradually disappears, which we interpret as a transition to the topologically-trivial phase. This dataset was taken with the optimal parameters for the anomalous regime ($V_{s}$, $\omega_{s}$). Taking the same scan for a deep tweezer $\omega_{d}$ and shallow potential step $V_{d}$, we find a weak signal in the Haldane regime at low modulation frequencies as expected. The signal gradually disappears for larger modulation frequencies, further supporting that at large disorder the system enters the trivial phase.

In a second set of measurements, we study the behavior deep in the anomalous and Haldane regime where the experimental parameters have been chosen to maximize the signal in each respective regime (Fig. 4d). We find that in both cases the chiral signal gradually vanishes and it appears that the transition to the trivial regime happens already at much lower values for the Haldane compared to the anomalous regime despite the fact that the minimal energy gap in the Floquet spectrum is comparable (dashed vertical lines in Fig. 4d). Previous work indicates that the relevant energy scale in the anomalous regime is rather determined by the width of the Floquet Brillouin zone, i.e., $\omega/(2\pi)=7\,$kHz, which is much larger than the minimal gap. Note that the vertical scale is different for both data sets for better visibility. This observation further supports theoretical work suggesting an enhanced robustness of anomalous Floquet topological insulators (AFTIs) compared to conventional Chern insulators.

In conclusion, edge modes provide a powerful probe for exploring topological phase transitions. While energy-selective excitation of edge and bulk modes is relatively straightforward in photonic systems [61, 62], no practical counterpart exists for neutral atoms in optical lattices. In this work, we demonstrate a novel method for selectively preparing chiral edge modes by tailoring the properties of the initial wavepacket to enhance the overlap with the localized mode at the interface. We have benchmarked the robustness of this approach through numerical simulations and comparisons with the Bott index. We further presented the first experimental investigation of disorder-driven phase transitions between two distinct topological phases in synthetic quantum systems, which are inaccessible using conventional observables  [6]. In the future, these concepts could be extended through energy-resolved preparation schemes, where a cold reservoir is coupled to an initially empty lattice and the injection energy is tuned via the relative chemical potential [71]. When combined with microscopic control over initial-state preparation [7, 72], this opens exciting new avenues for studying the dynamics and robustness of topological edge modes, with particular relevance for applications in topological photonics [61, 62].

Exploring topological phases of matter in the presence of disorder further opens new avenues for investigating exotic Floquet topological phases. AFTIs share key features with conventional Chern insulators, such as quantized chiral edge states. However, AFTIs also support novel phases, where all bulk states are localized coexisting with extended edge modes, so-called AFAIs [31, 70, 33]. This unique behavior arises from the protection of topological properties by Floquet symmetry, making AFTIs intrinsically more robust than conventional Chern insulators. AFAIs exhibit remarkable resilience to temporal noise [73, 74], and initial theoretical work suggests this robustness may extend to interacting systems, potentially enabling many-body localized phases with topologically protected edge modes [75]. Neutral atoms offer an ideal platform to study these phenomena, by combining the observation of edge modes [29] with bulk expansion in homogeneous lattices [76, 77] and tunable interactions. Intriguingly, numerical studies suggest that the modulation scheme employed in our current work indeed supports the existence of an AFAI phase, as schematically illustrated in Fig. 4a. Adding an additional perturbation in the form of a homogeneous artificial magnetic field may enable experimental studies of a quantized bulk response [78] and open the door to experimental studies of the topological magnetoelectric effect of 3D topological insulators [79].

Materials and Correspondence: Correspondence and requests for materials should be addressed to monika.aidelsburger@mpq.mpg.de

Supplemental Material

• S.I

Experimental sequence S.I

• S.II

Disorder potential S.II

• S.III

Disorder calibration S.III

• S.IV

Tweezer alignment routine S.IV

• S.V

Measuring the phase transition for zero S.V

disorder

• S.VI

Data evaluation S.VI

• S.VII

Bott index evaluation S.VII

• S.VIII

Numerical edge state propagation S.VIII

S.I Experimental sequence

To optimize the signal when probing the edge states in both the Haldane as well as in the anomalous regime, it is crucial to optimize the number of atoms transferred into the edge state, as well as the edge state velocity. The differences in the preparation protocol for the edge mode in the anomalous (left) and in the Haldane (right) regime are illustrated in Fig. S1. In both cases, we load the atoms from a BEC of $\mathrm{{}^{39}K}$ into the optical tweezer within $210\,$\mathrm{ms}. Subsequently, within $30\,$\mathrm{ms} the tweezer power is briefly increased, while the dipole trap power is reduced, such that atoms not trapped in the tweezer are expelled. Afterwards, the tweezer power is ramped down to its final value, i.e., to a radial confinement frequency $\omega_{s}/(2\pi)=$1.3(1)\text{,}\mathrm{kHz} or $\omega_{d}/(2\pi)=$2.0(1)\text{\,}\mathrm{kHz} depending on the measurement, as discussed in the main text. From now on the tweezer potential is kept constant, while the other potentials are being ramped up.

First, we ramp up both the potential step used to generate the topological interface $V_{0}$ generated by a DMD simultaneously with the disorder potential $\overline{V}_{D}$ within $30\,$\mathrm{ms}. As the edge state velocity depends on the height of the topological interface, we ramp it to $V_{d}/h=$2.8\text{\,}\mathrm{kHz} for the Haldane regime, and to $V_{s}/h=$13.2\text{,}\mathrm{kHz}$$ for the anomalous regime, to maximize the edge state velocity and hence the signal-to-noise [29].

Subsequently, we ramp up the optical lattice to a lattice depth of $5.9{E_{R}}$ within $30\text{\,}\mathrm{ms}$. Finally, we ramp the amplitude of the lattice modulation $m$ to its final value within five modulation cycles. As a last step, we reduce the scattering length one $\mathrm{ms}$ before releasing the atoms from $13a_{0}$ to $6a_{0}$ when preparing the edge state in the Haldane regime, while it was constant at $6a_{0}$ for the preparation scheme in the anomalous regime. Keeping the scattering length at a higher value reduces atom loss due to the stronger confinement at a trap frequency of $2$ $\mathrm{kHz}$. To initiate the dynamics in the periodically-modulated lattice, we abruptly switch off the tweezer potential.

S.II Disorder potential

The disorder potential is generated by projecting the Fourier plane of a diffuser onto the atoms, yielding a speckle pattern. For this, a holographic diffuser (Edmund Optics #35-693, with a $2\text{\,}\mathrm{\SIUnitSymbolDegree}$ diffusing angle) is illuminated at 532\,$$\mathrm{nm} with the uncollimated output of an optical fiber (NKT Photonics aeroGUIDE POWER, seeded by a Lighthouse Sprout-G-15W). The diffuser is mounted in a rotation mount, such that it is illuminated off-center. Rotating the mount allows to illuminate different phase patches on the diffuser, yielding different speckle patterns.

The image on the diffuser is Fourier-transformed by L1 ($f_{1}=100\,$\mathrm{mm}), and subsequently demagnified by a factor 45 by two telescopes ($f_{2}=150\,$\mathrm{mm}, $f_{3}=100\,$\mathrm{mm}, and $f_{4}=750\,$\mathrm{mm} together with our objective with focal length $f_{\mathrm{obj}}=25\,$\mathrm{mm}$$).

From images recorded in an intermediary image plane we extract a radial correlation length of $7.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$, which after demagnification by a factor 30 is consistent with a numerically simulated correlation length of $296\,$\mathrm{nm} in the atomic plane. For the numerical estimate, we initialize the beam in the back focal plane of the objective and propagate it to the entrance pupil using a Fresnel propagation kernel. We then clip the wings of the beam according to the aperture of the objective and Fourier transform it to obtain the field distribution in the atomic plane. The simulation takes two input parameters, the envelope and the correlation length in the back focal plane, which determine the properties of the speckle. The envelope was directly measured in the back focal plane, and is compatible with the correlation length of the images obtained in the intermediary image plane. The correlation length of the beam in the back focal plane is inferred from the envelope on the images taken in the intermediary image plane. We attribute an error of $\pm$2\text{\,}\mathrm{mm} to the measurement of the beam waist in the back focal plane, which results in a correlation length $\sigma_{r}=296^{+11}_{-7}$nm.

To estimate the correlation length of the pattern along the optical axis of the objective, we determine an effective numerical aperture $\mathrm{NA}_{\mathrm{eff}}$ of our objective that corresponds to the computed correlation length $\sigma_{r}=$296\text{,}\mathrm{nm}$$ using the standard relation [47]

[TABLE]

Using the relation between the radial and the axial correlation length via the effective numerical aperture

[TABLE]

we obtain $\sigma_{z}=$2.4\text{,}\mathrm{\SIUnitSymbolMicro m}$$.

S.III Disorder calibration

To determine the strength of the disorder in our experiment, we devised a calibration scheme for the speckle potential that directly measures the average potential strength $\overline{V}_{D}$ in the atomic plane, which is crucial for a quantitative analysis of the disorder-driven shift of the topological phase transition. We start out with a BEC consisting of $\approx$2\text{\times}{10}^{5} atoms of $\mathrm{{}^{39}K}$ at scattering length $6a_{0}$ in a crossed optical dipole trap with a horizontal trap frequency on the order of $\omega_{r}/(2\pi)=$25\text{\,}\mathrm{Hz}, and a vertical trap frequency of $\omega_{z}/(2\pi)=$0.33(3)\text{,}\mathrm{kHz}$$. In analogy to Kapitza-Dirac diffraction of a BEC in a pulsed optical lattice [80, 81], we abruptly switch on the speckle potential for a short interaction time $\tau$ between 10 $\mathrm{\SIUnitSymbolMicro s}$ and 230 $\mathrm{\SIUnitSymbolMicro s}$. Subsequently, the atoms are held in the crossed dipole trap for a time of $8\text{\,}\mathrm{ms}$. One can now observe an exponential decay in atom number from the unscattered atoms as a function of pulse length, which we quantify by performing a two-component fit as described below. By comparing this with the results obtained by numerically simulating a single-particle wavefunction evolving under the pulsed potential, the average potential depth of the speckle is inferred.

During the pulse, the atoms are coupled to different momenta according to the Fourier transform $\widetilde{V}$ of the disorder potential

[TABLE]

For short pulses ($\tau\overline{V}_{D}\sim 1$), the final wavefunction in momentum-space is bimodal, which is apparent in the time-of-flight absorption images that are shown in the top panel of Fig. S3. The strong, central peak can be attributed to a remnant, unscattered fraction of the initial wavefunction, which we refer to as zeroth order contribution. It is surrounded by a broad distribution of scattered atoms that originates from the interaction with the speckle potential. In the bottom panel of Fig. S3, we plot the evolution of the zeroth order contribution as a function of pulse duration extracted by performing a two-component fit of a broad Gaussian for the scattered, and a narrower Thomas-Fermi function for the unscattered atoms. It decays exponentially with the duration of the pulse, where the decay rate $\Gamma$ depends on the average potential strength.

To identify the scaling between the decay rate and the disorder strength, we use exact diagonalization to simulate the dynamics of a single-particle wavefunction evolving in the presence of a speckle potential in two dimensions. For the computation, the initial wavefunction in the form of a Thomas-Fermi profile is expanded in a 2D Fourier series that is truncated for large wavevectors. The resolution in $k$-space is adjusted to the radius of the BEC $\Delta k/(2\pi)=1/2r_{\mathrm{BEC}}$ and the cutoff is justified by the suppression of couplings to higher momentum states through the kinetic energy offset. The final grid size of the expansion in $k$-space is 143x143, such that the dimension of the Hilbert space that is diagonalized is 20449. The unscattered fraction after time-evolution is extracted via the parameter $A$ of a two-component fit to the modulus square of the resulting wavefunction

[TABLE]

with 3 independent fitting parameters $A,B$ and $\sigma_{k}$. After variation of the average speckle strength, we extract the curve $\Gamma(\overline{V}_{D})=\overline{V}_{D}^{a+1}\cdot\left(b\cdot\overline{V}_{D}^{a}+c\right)^{-1}$ by fitting the three free parameters $a,b,c$ to the simulated decay rates.

Experimentally, the power of the speckle beam is stabilized onto a photodiode located immediately next to the fiber output on the experimental table. The average intensity and thus also disorder strength in the atomic plane is directly proportional to the photodiode voltage according to $\overline{V}_{D}/h=\alpha\cdot U_{\mathrm{PD}}$, with $U_{\mathrm{PD}}$ being the photodiode voltage. To find $\alpha$, we perform a single parameter fit of the experimental data to the numerically calculated calibration curve (Fig. S4). This procedure corresponds to a rescaling of the horizontal axis of the plot, while leaving the vertical axis with the measured decay rates unchanged. For the specific case of our optical setup, we obtain $\alpha=$9.24\text{,}\mathrm{kHz}\text{,}{\mathrm{V}}^{-1}$$.

To estimate the error, we consider multiple factors: First, we notice that the decay of the zeroth-order in the simulation slows down and is no longer captured well by a single exponential. Therefore, we extract a systematic error on the determination of the decay constant by varying the time window considered for the exponential fit. Starting from a fit that captures the initial, fast decay ($\tau\overline{V}_{D}\in[0.2,0.6]$ depending on the disorder strength), we increase the window size until the full decay is taken into account for the fit. In this manner, we find the largest and smallest possible decay constants for every disorder strength. We then fit the experimental data to these limits to estimate the error $\Delta_{\mathrm{decay}}=\left(\begin{subarray}{c}+0.64\\ -0.51\end{subarray}\right)\,$\mathrm{kHz}\text{,}{\mathrm{V}}^{-1}. The estimate of the correlation length of the speckle pattern is based on the waist of the speckle beam in the back focal plane of the objective and measurements of the correlation length in the intermediary image plane, as described in [S.II](https://arxiv.org/html/2508.20154v1#S2). As the correlation length of the speckle pattern slightly impacts the decay rates, we perform the numerical simulations for the upper and lower estimate of the correlation lengths to extract an error on the calibration $\Delta_{\mathrm{corr}}=\left(\begin{subarray}{c}+0.24\\ -0.05\end{subarray}\right)\,$\mathrm{kHz}\text{\,}{\mathrm{V}}^{-1}. Another factor influencing the decay curve is the diameter of the BEC probing the speckle pattern. We vary the size of the BEC by $\pm$2\text{,}\mathrm{\SIUnitSymbolMicro m}, and perform the calibration on the calibration curves obtained this way, yielding $\Delta_{\mathrm{BEC}}=\pm 0.05\,$\mathrm{kHz}\text{\,}{\mathrm{V}}^{-1}. Finally, the enveloping of the speckle pattern in the atomic plane leads to a varying potential strength due to the movement of the atoms. We estimate this error to be $\Delta_{\mathrm{envelope}}=\pm 0.11$\mathrm{kHz}\text{,}{\mathrm{V}}^{-1}$$.

Combining these factors, we obtain the final calibration constant $\alpha=\left(9.24\pm\left(0.25\right)_{\mathrm{stat}}\left(\begin{subarray}{c}+0.93\\ -0.61\end{subarray}\right)_{\mathrm{sys}}\right)\,$\mathrm{kHz}\text{,}{\mathrm{V}}^{-1}$$. Here, $\Delta_{\mathrm{decay}}$, $\Delta_{\mathrm{corr}}$ and $\Delta_{\mathrm{BEC}}$ contribute as systematic errors to the measurement, and $\Delta_{\mathrm{envelope}}$ together with the fit error of $\alpha$ onto the calibration curve (which originates from the uncertainty in estimating the unscattered atom fraction, captured via bootstrapping) contribute as statistic errors.

S.IV Tweezer alignment routine

Before each experimental run, the position of the tweezer was aligned to the potential step by varying the tweezer position with respect to the step, optimizing the strength of the edge state signal. As different wall heights are required to obtain a large signal in the Haldane- and anomalous regime, the tweezer position was optimized separately to deliver a strong signal at $\omega/(2\pi)=7$ $\mathrm{kHz}$ for the shallow tweezer and at $\omega/(2\pi)=16$ $\mathrm{kHz}$ for the deep tweezer, as at these modulation frequencies the energy gaps are reasonably large. In the following, data was taken at the location, where the absolute value of the difference signal $\absolutevalue{\Delta\mathrm{OD}}$ integrated in a region close to the edge of the system was maximized. Here, for data taken with the shallow tweezer settings $(\omega_{s},V_{s})$ the position was chosen where the edge state signal at $\omega/(2\pi)=$7\text{,}\mathrm{kHz} was maximized, and respectively for the deep tweezer settings $(\omega_{d},V_{d})$ the position where the edge state signal at $\omega/(2\pi)=$16\text{\,}\mathrm{kHz} was maximized.

Furthermore, data was taken for varying tweezer positions to ensure that this alignment procedure does not introduce systematic bias into the data taken. An exemplary measurement is shown in (Fig. S5), illustrating good agreement of the observed shift of the phase transition between the anomalous and Haldane regime for different positions of the tweezer perpendicular to the wall.

S.V Measuring the phase transition for zero disorder

To benchmark the numerical simulations of the Bott index as well as the location of the phase transition extracted from the edge state evolution, we compare it to a gap-closing-measurement at $\overline{V}_{D}=0$ $\mathrm{kHz}$. For this, we employ Stückelberg interferometry [68, 69, 20]:

We load a bulk BEC at scattering length $6a_{0}$ into our optical lattice, and, while ramping up the amplitude modulation, nonadiabatically sweep the phase of two lattice beams to accelerate the atoms to the next $\Gamma$ point in quasimomentum space. This nonadiabatic sweep leads to a splitting of the atom population onto the lowest two bands of our system. Subsequently, we hold the atoms at this quasimomentum for an integer number of modulation periods, and afterwards ramp lattice phase and amplitude modulation back to transfer the atoms to zero quasimomentum (at the $\Gamma$-point). Ramping the phase back nonadiabatically recombines the population in the two bands, leading to oscillations in the band population due to the acquired phase difference, as illustrated on the left in Fig. S6.

This means, that in the vicinity of a phase transition at $\Gamma$, we can observe the size of the gap reduce to zero, and increase again (right side in Fig. S6). From this we extract the phase transition for vanishing disorder \overline{V}_{D}=0\,$$\mathrm{kHz} at $\omega_{c}/(2\pi)=$11.13(8)\text{,}\mathrm{kHz}$$, where the error bar denotes the fit error. From a 6-band model calculation, we determine a transition frequency of $10.92\text{\,}\mathrm{kHz}$.

S.VI Data evaluation

To track the movement of the edge states, we calculate the center-of-mass (COM) position of the atoms. To this end, we calculate the weighted average of the position, using the optical density extracted from absorption pictures as a weight:

[TABLE]

where $y_{i}$ denotes the position along the topological interface each camera pixel $i$ corresponds to in the atomic plane, and $w_{i}$ is the optical density (OD) at this pixel. The absorption pictures have been calibrated to account for saturation effects due to the non-negligible intensity of the imaging beam with respect to the saturation intensity of the atoms [82].

To track the chiral movement from this, we evaluate the difference between the position obtained for modulation with chirality $\kappa=1$ and for modulation with chirality $\kappa=-1$ parallel to the step of the hard-wall potential.

To estimate the error in this determination, we employ bootstrapping: From the set of all absorption pictures taken with the same parameters, we choose random sets of the same length as the initial set, such that images can be selected multiple times. We perform the evaluation for 100 of such sets, and take the standard deviation in the results to be the error.

S.VII Bott index evaluation

For disordered systems, lacking translational invariance, conventional topological invariants defined in terms of geometric properties of Bloch states in momentum space are no longer practical. The Bott index, introduced by Loring and Hastings [66], is a real-space topological marker that can be evaluated on a disordered lattice to quantify whether topological features prevent the existence of a set of localized wavefunctions spanning a band’s eigenspace. For vanishing disorder, the Bott index of a given energy band has been shown to coincide with its Chern number in the thermodynamic limit [67]. In the following, we adopt the procedure outlined in Ref. [66] to evaluate the Bott index for the lower band of the effective Hamiltonian resulting from our periodic modulation scheme.

The sinusoidal drive of the intensity of the three laser beams interfering to form the optical honeycomb lattice results in the following time dependence for the nearest-neighbor tunneling amplitudes:

[TABLE]

where $\phi_{i}=\frac{2\pi}{3}(i-1),i\in\{1,2,3\}$ and $A$, $B$ and $C$ are parameters that depend on the modulation amplitude $m$. For $m=0.25$, as set for all experimental results discussed in the main text, $A\approx 0.220E_{R}$, $B\approx 0.767$ and $C\approx-0.065E_{R}$, with the recoil energy $E_{R}=h^{2}/(2\lambda_{L}^{2}2m_{\textrm{K}})\approx h\times 9.23\,$\mathrm{kHz}$$. Next-nearest neighbor tunneling amplitudes are about one order of magnitude smaller, and thereby neglected in the following. In order to evaluate the effective Hamiltonian numerically, the modulation period is discretized into $N$ timesteps to evaluate $J_{i}(t_{j})$ and the corresponding instantaneous Hamiltonian $\hat{H}(t_{j})$ for $t_{j}=\frac{T}{N}j,\>j=0,1,...,N-1$. For all numerical results discussed in the main text, the number of timesteps is set to $N=21$, resulting in the discretization of the tunneling rates along the three lattice directions shown in Fig. S7. At each timestep, $\hat{H}(t_{j})$ is perturbed by an additional term $\hat{V}_{D}$, modeling on-site energy offsets $\langle x_{i},y_{i}|\hat{V}_{D}|x_{i},y_{i}\rangle$ sampled by the hexagonal lattice on a numerical speckle potential, as displayed in Fig. S8. To ensure correct ordering of eigenstates upon diagonalization, we subtract the mean of all $N_{s}$ on-site energy offsets from each diagonal entry of $\hat{H}(t_{j})$.

Ultimately, the time-evolution operator over one modulation period reads

[TABLE]

from which the effective Hamiltonian is readily obtained:

[TABLE]

To evaluate the Bott index for a given disorder realization, we enforce periodic boundary conditions on a rectangular system consisting of $N_{s}$ lattice sites, construct $\hat{H}_{\text{eff}}$ in real space and diagonalize it to obtain its eigenbasis $\{|\psi_{n}\rangle,\>n=1,...,N_{s}\}$. In a second step, we construct the projector onto the lower-band eigenspace $\hat{P}=\sum_{1\leq n\leq N_{s}/2}|\psi_{n}\rangle\langle\psi_{n}|$ to evaluate the following band-projected position operators:

[TABLE]

with $L_{\alpha}$ denoting the system size in $x$- or $y$-direction.

Finally, the Bott index of the lower band is given by

[TABLE]

where $\hat{\mathcal{G}}_{\alpha}=\hat{G}_{\alpha}+(\hat{\mathbb{1}}-\hat{P})$.

In the absence of disorder ($\overline{V}_{D}=0$), the transition from $\mathcal{B}=0$, corresponding to the anomalous regime, to $\mathcal{B}=1$, corresponding to the Haldane regime, takes place at a modulation frequency $\omega/(2\pi)\approx 10.44$ $\mathrm{kHz}$ for $m=0.25$, in agreement with the numerical evaluation of the corresponding Chern number in a two-band model with the same modulation scheme. Numerical estimates for transition frequencies provided by a two-band model, which do not capture higher-band corrections, systematically underestimate the transition frequency predicted by a six-band calculation, namely $\omega/(2\pi)\approx 10.92$ $\mathrm{kHz}$, and the one measured experimentally, as shown in Fig. 3 in the main text and studied in detail in Ref. [20].

The number of lattice sites set for the Bott index evaluations discussed in the main text is $N_{s}=576$. Figure S9 shows the Bott index across the transition between the anomalous and the Haldane regime for varying system size $N_{s}$ and fixed disorder strength $\overline{V}_{D}/h=1$ $\mathrm{kHz}$, averaged over $N_{\text{avg}}=100$ disorder realizations. For system sizes larger than $N_{s}=400$, the average Bott index converges to the thermodynamic limit, which justifies the value chosen for $N_{s}$ and rules out finite-size effects.

S.VIII Numerical edge state propagation

The procedure to obtain $\hat{U}(T)$ outlined in the previous section can be employed on a lattice with closed boundary conditions to investigate the propagation dynamics of wavepackets localized to the system’s numerical edges for increasing disorder strength. Repeated action of $\hat{U}(T)$ on an initial state $\ket{\varphi_{0}}$ provides access to the same observables probed experimentally, i.e., the COM distance. After $n$ modulation periods, the initial state is mapped onto

[TABLE]

Depending on the overlap $|\langle\varphi_{0}|\psi_{n}\rangle|^{2}$ of the initial state with the eigenstates $\{|\psi_{n}\rangle,\>n=1,...,N_{s}\}$ of the effective Hamiltonian in a given regime, the wavefunction population will propagate along the numerical boundary of the system or scatter towards its bulk. Both in the anomalous and in the Haldane regime, for a zigzag edge geometry, only the outermost lattice sites have a non-zero population for an infinitely high and steep boundary. However, as described in Ref. [29], the edge state wave functions display a uniform phase pattern in the anomalous regime and an alternating phase pattern in the Haldane regime, such that $\ket{\varphi_{0}}$ can be initialized to have a good overlap with the edge states in the anomalous regime and vanishing overlap with the edge state in the Haldane regime, providing a numerical equivalent to the regime-sensitive state preparation achieved experimentally by varying the tweezer depth.

To closely reproduce the experimental setup, we generate lattice strips for which the system boundary in the $y$-direction, along which we initialize and time-propagate numerical wavepackets, displays a zigzag termination geometry. Suitable system sizes are chosen depending on the largest expansion time of interest such that edge modes do not propagate past the system’s corner. This ensures that edge dynamics can be solely investigated by means of the center-of-mass in the $y$-direction and the time-evolution thereof. For all numerical results displayed in Fig. 2 of the main text, probing edge propagation deep in the Haldane regime up to expansion times of $80T$, numerical strips consist of $N_{s}=3360$ lattice sites, with a ratio between the lengths of the system in the $y$- and $x$-direction of $L_{y}/L_{x}\approx 2.5$. On the other hand, for Fig. 3 of the main text, the fixed expansion time at which the COM distance decay is evaluated across the disorder-shifted transition amounts to $20T$, such that smaller systems with $N_{s}=952$ lattice sites suffice, with a slightly larger ratio of $L_{y}/L_{x}\approx 2.8$.

Intuitively, to numerically populate edge modes in the $\pi$-gap of the anomalous regime and in the [math]-gap of the Haldane regime, $\ket{\varphi_{0}}$ should be prepared to closely match the corresponding wavefunctions, i.e., such that only lattice sites along the boundary, belonging to the same sublattice and spaced by $\sqrt{3}a$, are populated with uniform and alternating phase patterns, respectively [29], as displayed in the left panel of Fig. S10a,c. For the purposes of the numerics discussed in Fig. 3 of the main text, however, the wavefunction in Fig. S10a is seen to display a nonvanishing overlap with the edge state wavefunctions in the $\pi$-gap of a system deep in the Haldane regime, $\omega/(2\pi)=16$ $\mathrm{kHz}$ (rightmost panel of Fig. S10a), resulting in a residual chiral signal in the Haldane regime. Moreover, we find that the dynamics of such an initial wavepacket is strongly influenced by the initial phase of the modulation. In contrast, we find that a slightly extended wavefunction that more closely resembles the situation realized experimentally (leftmost panel of Fig. S10b) provides a more robust observable with higher contrast. Here, also sites slightly away from the interface (on the other sublattice) are populated. This initial state displays good overlap with the edge mode of interest in the anomalous regime and vanishing overlap with the edge mode in the Haldane regime, providing a more accurate probe for the disorder-driven shift of the transition.

In the Haldane regime the transversal width of the initial state $\ket{\varphi_{0}}$ does not seem to play a significant role. We perform a similar comparison using the initial wavefunctions shown in the leftmost panels of Fig. S10c,d. Both yield similar results in terms of overlap with eigenstates at $\omega/(2\pi)=16$ $\mathrm{kHz}$. In principle, both initial states could be used for the numerical simulations shown in Fig. 2d. As it more closely resembles the ideal wavefunction of the [math]-gap edge state, we employ the first option, distributing the population over eight sites from the same sublattice to further suppress overlap with bulk states.