Imaging topology of Hofstadter ribbons.

Physical systems with non-trivial topological order find direct applications in metrology (Klitzing et al 1980 Phys. Rev. Lett. 45 494-7) and promise future applications in quantum computing (Freedman 2001 Found. Comput. Math. 1 183-204; Kitaev 2003 Ann. Phys. 303 2-30). The quantum Hall effect derives from transverse conductance, quantized to unprecedented precision in accordance with the system's topology (Laughlin 1981 Phys. Rev. B 23 5632-33). At magnetic fields beyond the reach of current condensed matter experiment, around 104 T, this conductance remains precisely quantized with values based on the topological order (Thouless et al 1982 Phys. Rev. Lett. 49 405-8). Hitherto, quantized conductance has only been measured in extended 2D systems. Here, we experimentally studied narrow 2D ribbons, just 3 or 5 sites wide along one direction, using ultracold neutral atoms where such large magnetic fields can be engineered (Jaksch and Zoller 2003 New J. Phys. 5 56; Miyake et al 2013 Phys. Rev. Lett. 111 185302; Aidelsburger et al 2013 Phys. Rev. Lett. 111 185301; Celi et al 2014 Phys. Rev. Lett. 112 043001; Stuhl etal 2015 Science 349 1514; Mancini et al 2015 Science 349 1510; An et al 2017 Sci. Adv. 3). We microscopically imaged the transverse spatial motion underlying the quantized Hall effect. Our measurements identify the topological Chern numbers with typical uncertainty of 5%, and show that although band topology is only properly defined in infinite systems, its signatures are striking even in nearly vanishingly thin systems.

Introduction

The importance of topology in physical systems is famously evidenced by the quantum Hall effect’s role as an ultra-precise realization of the von Klitzing constant R = h/e2 of resistance [1]. Although topological order is only strictly defined for infinite systems, the bulk properties of macroscopic topological systems closely resemble those of the corresponding infinite system. For 2D systems in a magnetic field B0, the topology is characterized by an integer invariant called the Chern number. Even at laboratory fields of tens of Tesla, crystalline materials have a small magnetic flux Φ = AB0 per individual lattice plaquette (with area A) compared to the flux quantum Φ0 = h/e. Superlattice [2-5] and ultracold atom [6-9] systems now realize 2D lattices in a regime where the magnetic flux per plaquette Φ is a significant fraction of Φ0.

Experimental signatures of Chern numbers generally leverage one of two physical effects: in condensed matter systems the edge-bulk correspondence allows the Chern number to be inferred from the quantized Hall conductivity σ = C/R, and in cold-atom experiments direct probes of the underlying band structure at every value of crystal momentum give access to the Chern number through either static [10,11] or dynamic [12-15] signatures. Both of these connections derive from the pioneering work of Thouless, Kohmoto, Nightingale, and denNijs [16], in the now famous TKNN paper. Going beyond these well known techniques, the TKNN paper showed that for rational flux Φ/Φ0 = P/Q (for relatively prime integers P and Q) the integer solutions s and C to the Diophantine equation uniquely[3] determine the Chern number C of the lowest band. Following theoretical work [11,17-20], we leveraged this TKNN equation to determine the Chern number of our system.

Experimental setup

We studied ultracold neutral atoms in a square lattice with a large magnetic flux per plaquette. As pictured in figure 1(a), our system consisted of a 2D lattice that was extremely narrow along one direction, just 3 or 5 sites wide—out of reach of traditional condensed matter experiments, with hard wall boundary conditions: a ribbon. Our system was qualitatively well described by the Harper-Hofstadter Hamiltonian in the Landau gauge [24,25] where j and m label lattice sites along e and e, with tunneling strengths t and t respectively. As shown in figure 1(a), tunneling along e was accompanied by a phase shift ei. Hopping around a single plaquette of this lattice imprints a phase ϕ, analogous to the Aharanov–Bohm phase, emulating a magnetic flux We implemented this 2D lattice by combining a 1D optical lattice defining sites along an extended direction e, with atomic spin states forming lattice sites along a narrow, synthetic [26-28,29] direction e. The exact Hamiltonian of the underlying atomic system differs from the Harper–Hofstadter Hamiltonian above in that t is non-uniform due to Clebsch–Gordan coefficients and there is a small m2 dependent potential term due to the quadratic Zeeman shift (see appendix D).

Figure 1.

Quantum Hall effect in Hofstadter ribbons. (a) 5-site wide ribbon with real tunneling coefficients along e and complex tunneling coefficients along e, creating a non-zero phase ϕ around each plaquette. (b) After applying a force along e for a time Δt, atomic populations shift transverselyalong e, signaling the Hall effect. (c), (d) TOF absorption images giving hybrid momentum/position density distributions n(k, m). Prior to applying the force (c),the m = 0 momentum peak is at k = 0, marked by the red cross. Then, in (d), the force directly changed q, evidenced by the displacement Δq of crystal momentum, andvia the Hall effect shifted population along e.

This system exhibits a Hall effect, where a longitudinal force —analogous to the electric force in electronic systems—drives a transverse ‘Hall’ current density for non-zero Φ/Φ0. A longitudinal force F would drive a change in the dimensionless crystal momentum and a transverse displacement Δm, giving a dimensionless Hall conductivity where G is the reciprocal lattice constant and N is the number of carriers per plaquette (see appendix C). Starting with Bose-condensed atoms in the lattice’s ground state (with transverse density shown in figure 1(a)) we applied a force along e and obtained Δm from site resolved density distributions [11] along e (figure 1(b)). Leveraging the TKNN equation (equation (1)), we further show that the force required to move the atoms a single lattice site signals the infinite system’s Chern number.

Our quantum Hall ribbons were created with optically trapped 87Rb Bose–Einstein condensates (BECs) in either the F = 1 or 2 ground state hyperfine manifold, creating 3 or 5 site-wide ribbons from the 2F + 1 states available in either manifold. We first loaded BECs into a 1D optical lattice along e formed by a retro-reflected λL = 1064 nm laser beam. This created a lattice with period a = λL/2 and depth 4.4(1) EL, giving tunneling strength t = 0.154(4)EL. Here, is the single photon recoil energy; is the single photon recoil momentum; and mRb is the atomic mass. We induced tunneling along e with either a spatially uniform rf magnetic field or two-photon Raman transitions. The tunneling strength was t = 1.97(8) t for Raman coupling in the F = 2 manifold, t = 0.97(8) t for Raman coupling in the F = 1 manifold, and t = 7.4(5) t for rf coupling in both manifolds. The rf-induced tunneling imparted at most only a spatially uniform tunneling phase, giving ϕ/2π = 0. In contrast the Raman coupling, formed by a pair of counter propagating laser beams with wavelength λR = 790 nm, imparted a phase factor exp(−2ikRx). Here, is the Raman recoil momentum, giving We then applied a force by shifting the center of the confining potential along e, effectively applying a linear potential. Using time-of-flight techniques [27], we measured hybrid momentum/position density distributions n(k, m), a function of momentum along e and position along e, as seen in figures 1(c), (d).

Hall conductivity measurement

We measured the Hall conductivity beginning with a BEC at q(t = 0) = 0 in the lowest band with transverse modal position = 0[4]. Figure 2(a) shows the band structure of a system similar to ours, but extended along the e direction, with 41 sites. The energy is plotted as a function of crystal momentum along e, with color indicating the expectation value of position along e, calculated by diagonalizing the full Hamiltonian with zero quadratic shift and uniform Clebsch–Gordan coefficients (see appendix D). Figure 2(b) shows the band structure of our experimental system, calculated from the full Hamiltonian for our experimental parameters (see appendix D). Note that the lowest, 3rd, and 5th bands of our system are akin to the three bulk bands of the extended system, while the 2nd and 4th bands of our system resemble the edge modes of the extended system.

Figure 2.

Band structure in an extended system and our 5-site wide ribbon. (a) Band structure of an extended system, with 41 sites along e, computed for a 4.4EL deep 1D lattice (λL = 1064 nm), 0.5EL Raman coupling strength (λR = 790 nm), and quadratic Zeeman shift ϵ = 0EL, and excluding anisotropy due to Clebsch–Gordan coefficients (see appendix D). The color indicates the expectation value of position along the synthetic direction (b) Band structure computed using full Hamiltonian for our experimental parameters of 4.4EL lattice depth, 0.5EL Raman coupling strength, and quadratic Zeeman shift ϵ = 0.02EL, giving (see appendix D). The black dot indicates the initial loading parameters. (c) TOF absorption images n(k, m) for varying longitudinal crystal momenta q.

We applied a force F = 0.106(5) EL/λL for varying times Δt, directly changing the longitudinal crystal momentum from 0 to a final q and giving a transverse Hall displacement from 0 to a final Figure 2(c) shows a collection of hybrid density distributions, where each column depicts n(k, m) for a specific final q, labeled by the overall horizontal axis. For each column, the change in crystal momentum is marked by the horizontal displacement of the diffraction orders relative to their location in the central q = 0 column. The transverse displacement is visible in the overall shift in density along m as a function of q, i.e. between columns.

Figure 3 (left) quantifies this Hall effect by plotting the modal position as a function of q for Φ/Φ0 = 0, −4/3, and 4/3. The data is represented by gray dots, with uncertainty bars reflecting the propagated standard uncertainty from averaging six identical runs. For zero flux Φ/Φ0 = 0 (figure 3(a)), was independent of q; in contrast, for non-zero flux (figures 3(b), (c)), depends linearly on q with non-zero slope. These linear dependencies evoke our earlier discussion of the Hall conductance in which we anticipated slopes equal to the Chern number. Linear fits to the data give = 0.01(1), 0.87(3), and −0.85(3) for zero, negative and positive flux respectively, showing the expected qualitative behavior. The expected slopes, given by the Chern number, σH = 0, ±1 are indicated by black dashed lines in figure 3 (left).

Figure 3.

Hall displacement. Top: modal position is plotted as a function of q for the 5-site ribbon with flux (a) Φ/Φ0 = 0, (b) (c) Gray circles depict the measurements; black dashed lines are the prediction of our simple red curves are the expectation from the lowest band theory of our thin ribbon, and blue curves are the result of TDSE calculations for our experimental parameters including force. Bottom: extracted conductivity from the slope of a line of best fit to the data (gray circles), lowest band theory (red lines), and TDSE calculations (blue lines) as a function of maximum included in the fit range, for each flux value. As discussed in appendix B, the Φ/Φ0 = 0 data was compensated to account for non-adiabaticity in the loading procedure.

The red curves in figure 3 (top) show the expected behavior for our 5-site wide system for adiabatic changes in q as calculated from exact diagonalization of the full Hamiltonian (see appendix D), always within the lowest band (figure 2(a)), i.e. Bloch oscillations. This theory predicts a nearly linear slope for small q sharply returning to = 0 at the edges of the Brillouin zone. A linear fit to this theory produces and −0.6 for zero, negative and positive flux respectively, far from the Chern number. This discrepancy is resolved by recalling that Bloch oscillations require adiabatic motion. As the ribbon width grows, the band gaps at the edge of the Brillouin zone close (see figure 2(a)), making the Bloch oscillation model inapplicable. The departure of the data from the adiabatic theory at the edges of the Brillouin zone indicates a partial break down of adiabaticity was present in our data. To confirm this, we performed time-dependent Schroedinger equation (TDSE) calculations for our experimental parameters, including the magnitude of the force applied. This is displayed by the blue curves in figure 3. Note that the TDSE curves (blue) lie between the lowest band theory (red) and the large system limit (black dashed lines) at the edges of the Brillouin zone, confirming a partial breakdown of adiabaticity (see appendix C for further detail).

One might suspect that limiting the domain of the linear fit such that band edge effects are excluded would still provide a good measure of the Chern number. However, as shown in figure 3 (bottom), the slope of the best fit line for non-trivial topologies, and thus the measured conductivity, depends highly on the selected domain for both the theoretical (red), the experimental (black) and numerical (blue) data, and the appropriate choice of range is ambiguous. We conclude that for an extremely narrow system such as ours, a conductivity measurement is insufficient for determining the Chern number [20].

Chern number measurement via TKNN Diophantine equation

To better identify Chern numbers, we relate the TKNN equation (equation (1)) to the physical processes present in our system. Although the Hofstadter Hamiltonian in equation (2) is only invariant under m-translations that are integer multiples of Q, a so-called ‘magnetic-displacement’ by Δm = 1 along e accompanied by a displacement in crystal momentum along e by Δq/2kR = P/Q leaves equation (2) unchanged—this is the magnetic translation operator. Together, these symmetry operations give a Q-fold reduction of the Brillouin zone along e, and add a Q-fold degeneracy, as illustrated in figure 4(a) for Φ/Φ0 = 0, 1/3, and 2/5. Recalling that the Brillouin zone is 2 periodic along e, it follows that a displacement by 2kL/Q to the nearest symmetry related state involves an integer C magnetic displacements, shown in figure 4(b) for Φ/Φ0 = 1/3 and 2/5, given by solutions to 2kLs − 2kR C = 2kL/Q, where s counts the number oftimes the Brillouin zone was ‘wrapped around’ during the C vertical displacements. Because this is exactly the TKNN equation (1), we identify C as the Chern number. Both C and s directly relate to physical processes. First, each time the Brillouin zone is wrapped around implying a net change of momentum by 2 a pair of photons must be exchanged between the optical lattice laser beams. Second, each change of m by 1 must be accompanied by a 2 recoil kick imparted by the Raman lasers as they change the spin state. This physical motivation of the TKNN equation remains broadly applicable even for our narrow lattice, providing an alternate signature of the Chern number.

Figure 4.

Chern number from the TKNN equation. (a) Lowest band energy (left) and expected position along e, (right) within the Brillouin zone in an extended 2D system, where q and q are crystal momenta along e and e, respectively. Top. Φ/Φ0 = 0. Middle. Φ/Φ0 = 1/3: Brillouin zone shrinks by a factor of 3 and becomes three-fold degenerate, distance between adjacent energy minima spaced by 2kL/Q is labeled. Each minimum corresponds to a different expected position along e. Bottom. Φ/Φ0 = 2/5. (b) Fractional population in each spin state in the lowest band at q = 0. Top. Φ/Φ0 = 1/3. Bottom. Φ/Φ0 = 2/5. A momentum shift along e of 2kL/Q is accompanied by an integer number of spin flips C. A line connecting magnetic states separated by 2kL/Q, with slope C = 1 (top) and −2 (bottom), is indicated.

Figure 5 shows the full evolution of fractional population in each m site as a function of crystal momentum q in the lowest band. The black circles locate the peak of the fractional population in each spin state. We identify the locations of those peaks as the crystal momenta at which the atoms were displaced by a single lattice site along e starting at q = 0, similar to the suggestions in [19,20]. We associate the Chern number with the slope of a linear fit through the three peak locations. The dependence of Chern number extracted in this way on the strength of applied force is much weaker than the Hall conductivity approach (see appendix C). For the 3-site wide ribbon, we measured a Chern number of0.99(4), −0.98(5) for negative and positive flux respectively[5], in agreement with the exact theory as calculated from the full Hamiltonian (see appendix D), which predicts ±0.97 (1), with uncertainties reflecting fit uncertainty of peak locations. For the 5-site wide ribbon, we measured 1.11 (2) −0.97(4), close to the theoretical prediction of ±1.07(1).

Figure 5.

Chern number measurement. Lowest band fractional population measured as a function of crystal momentum in the e and position in the e. Darker color indicates higher fractional population. In the Raman-coupled cases, the points represent the fitted population maxima and the Chern number is extracted from the best fit line to those points. (a) 3-site (left) and 5-site (right) systems with positive flux. (b) 3-site (left) and 5-site (right) system with zero flux. (c) 3-site (left) and 5-site (right) systems with negative flux. The parameters for 3-site data were identical to those for 5-site data, see figure 3(a), except t = 0.98(8) t.

The deviation from unity results from a non-zero quadratic Zeeman shift, and t > t allowing hybridization of states in the vicinity of the edge (see figure 2 in [20]). The dependence of a Chern number inferred with this technique on both the size of the system along e and the tunneling t is studied in figure 6. Figure 6(a) shows the dependence of the Chern number on the width of the ribbon, from our experimental parameters of 3 and 5 sites to an extended system of 70 sites. As seen in the figure, as the system size grows the measured Chern number converges to the expected value of 1. These were calculated from the full Hamiltonian (appendix D), assuming uniform tunneling along e, with tunneling t = 0.5EL, synthetic direction coupling and no quadratic shift. Figure 6(b) shows the dependence of the measured Chern number on the coupling strength along the synthetic direction for lattice widths relevant to our experiment—3 and 5 sites. We used the same Hamiltonian and parameter values listed above. In the limit of vanishing tunneling, both the 3 and 5-site wide Chern number converge to the exact integer value of 1. This supports the hypothesis that deviation from unity at non-zero coupling strengths is a consequence of the hybridization of states in the vicinity of the edge, which is facilitated by stronger couplings.

Figure 6.

Dependence of Chern number inferred via the TKNN Diophantine equation method on system size and tunneling. (a) Chern number dependence on number of sites along e. (b) Chern number dependence on coupling strength along the synthetic direction for 3 and 5 site systems.

Conclusion

Our direct microscopic observations of topologically driven transverse transport demonstrate the power of combining momentum and site-resolved position measurements. With the addition of interactions, these systems have been shown to display chiral currents [30], and with many-body interactions are predicted to give rise to complex phase diagrams supporting vortex lattices and charge density waves [31-33]. Realizations of controlled cyclic coupling giving periodic boundary conditions [26] along e could elucidate the appearance of edge modes as the coupling between two of the three states is smoothly tuned to zero. In addition, due to the non-trivial topology as well assign the low heating afforded by synthetic dimensional systems, a quantum Fermi gas dressed similarly to our system would be a good candidate for realizing fractional Chern insulators [34].

Table D1.

Summary of experimental parameters.

Parameter	Value
V	4.4(1) EL
tx	0.154(4) EL
ts	3-site Raman: 0.97(8) tx; 5-site Raman: 1.97(8) tx; rf: 7.4(5) tx
ϵ	0.02 EL
δ	0.00(2) EL
Fx	0.106(5) EL/λL