Bose-Einstein Condensate on a Synthetic Topological Hall Cylinder

The interplay between matter particles and gauge fields in physical spaces with nontrivial geometries can lead to novel topological quantum matter. However, detailed microscopic mechanisms are often obscure, and unconventional spaces are generally challenging to construct in solids. Highly controllable atomic systems can quantum simulate such physics, even those inaccessible in other platforms. Here, we realize a Bose-Einstein condensate (BEC) on a synthetic cylindrical surface subject to a net radial synthetic magnetic flux. We observe a symmetry-protected topological band structure emerging on this Hall cylinder but disappearing in the planar counterpart. BEC's transport observed as Bloch oscillations in the band structure is analogous to traveling on a M\"obius strip in the momentum space, revealing topological band crossings protected by a nonsymmorphic symmetry. We demonstrate that breaking this symmetry induces a topological transition manifested as gap opening at band crossings, and further manipulate the band structure and BEC's transport by controlling the axial synthetic magnetic flux. Our work opens the door for using atomic quantum simulators to explore intriguing topological phenomena intrinsic in unconventional spaces.


INTRODUCTION
Physical spaces with nontrivial geometries can give rise to novel phenomena difficult to attain in planar spaces. Such unconventional spaces are studied in various disciplines such as general relativity and cosmology [1], photonics [2][3][4][5][6], and condensed matter physics [7][8][9][10]. For example, gravity stems from curved spacetimes in general relativity [1]. Superfluids on curved surfaces carry vortices with no counterpart in planar spaces [11,12]. Fractional quantum Hall states become degenerate on a torus, underlying the profound concept of topological order [13]. Highly controllable atomic systems offer opportunities to quantum simulate [14,15] various phenomena and uncover new physics inherent in unconventional spaces, including those challenging to study in conventional platforms. For instance, various analogues of cosmic phenomena [16][17][18] have been observed in tabletop experiments with a Bose-Einstein condensate (BEC), in which excitations such as phonons are in an effective curved spacetime [19,20]. Superfluid properties can be studied in detail with a BEC prepared in a ring trap [21,22].
Topological quantum matter [23][24][25][26] has received great attention across different areas because of its robust properties promising for reliable devices and quantum information processing [27]. Whereas topological phenomena in planar spaces have been studied extensively, those intrinsic in unconventional spaces remain substantially unexplored in experiments, because it is challenging to realize spaces that simultaneously incorporate the underlying novel geometries with crucial ingredients such as gauge fields and to manipulate the required Hamiltonians. For instance, threading a magnetic flux through a two-dimensional (2D) plane has led to the remarkable discovery of the quantum Hall effects and various topological quantum matter [23][24][25][26] for electrons. However, creating a net radial magnetic flux through the cylindrical surface of a nanotube is extremely difficult. Such a Hall cylinder is an important paradigm for many theoretical studies of topological physics [28][29][30][31][32][33], but its experimental exploration is largely lacking. The microscopic mechanisms of how changing the geometry of the underlying space may give rise to distinct topological matter require further research.
Synthetic spaces incorporating synthetic dimensions can bypass many constraints in the real space, holding promise for creating novel geometries with arbitrary dimensions. Synthetic dimensions can be constructed using atomic internal states [62,63], momentum states [64,65], time [66], or other degrees of freedom. Synthetic spaces have enabled experimental exploration of high-dimensional quantum matter, such as 4D quantum Hall systems [66] and a Yang monopole in a 5D parameter space [67]. Besides, manipulating boundary conditions is possible. This has allowed observations of edge states in synthetic 2D planes subject to magnetic fluxes [68][69][70][71][72]. In addition, there have been rich ideas exploiting the versatile nature of synthetic spaces for creating unconventional spaces [29-33, 63, 73], such as the surface of a cylinder, torus, or Möbius strip, with or without gauge fields. However, there remains very limited experimental exploration [74,75] of topological quantum matter and transport in unconventional spaces.
Here, we realize a BEC on a synthetic cylindrical surface, composed of a real spatial dimension and a curved synthetic dimension formed by cyclically coupled atomic spin states, subject to a net radial synthetic magnetic flux. We observe intriguing topological phenomena, such as emergent topological band crossings and topological Bloch oscillations, stemming from the interplay between gauge fields and nontrivial geometries of spaces. We also observe a topological transition manifested as gap opening at band crossings and further manipulate the band structure and BEC's transport via controlling the axial synthetic magnetic flux. In striking contrast, these phenomena emerging on the Hall cylinder vanish when we unzip the cylinder into a planar Hall strip, illustrating the crucial and intriguing role of topology and geometries of spaces in novel topological quantum phenomena.

II. EXPERIMENTAL SETUP
In our experiments, a 87 Rb BEC is produced in an optical dipole trap [76]. As shown in Figs. 1(a) and 1(b), four internal spin states, |F, m F = |2, 2 , |2, 1 , |1, 0 , |1, 1 , respectively relabeled as |1 , |2 , |3 , |4 , form discrete sites in the synthetic dimension (the w direction), where F (m F ) is the hyperfine spin (the magnetic quantum number). The synthetic dimension w along with the real spatial dimension y together span a synthetic cylindrical space. Raman lasers along ±ŷ couple |1 and |2 as well as |3 and |4 with respective Raman couplings Ω R2 and Ω R1 . The Raman lasers' wavelength (λ ≈ 790 nm) defines the photon recoil momentum k r = 2π /λ and recoil energy E r = 2 k 2 r /(2m) respectively used for units of momentum and energy, where is the reduced Planck constant, and m is the mass of 87 Rb. Two microwave fields, with coupling strengths Ω 1 and Ω 2 , respectively couple |2 and |3 , and |1 and |4 . This setup delivers a cyclic coupling (a periodic boundary condition) in the w direction. Differently from other cyclic couplings for creating 2D spin-orbit couplings [77,78] and Yang monopoles [67], our scheme shown in Figs. 1(b) and 1(c) connects two edges of a 2D planar Hall strip (with a synthetic magnetic field discussed below) and thus synthesizes the y and the curved w dimensions into a Hall cylinder.
For either the Hall strip or the Hall cylinder, the syn-  Counter-propagating Raman lasers with orthogonal linear polarizations (parallel toẑ andx) and microwaves (propagating in the x-y plane) are applied to a BEC with typical atom number (1-2)×10 4 . Gravity "g" is towards −ŷ. (b) Internal spin states |1 , |2 , |3 , and |4 as discrete sites in the synthetic dimensionŵ are cyclically coupled by Raman couplings ΩR1,R2 and microwave couplings Ω1,2. Linear Zeeman splitting ωZ = ∆ωR is generated by a bias magnetic field, where ∆ωR is the Raman lasers' angular frequency difference. The quadratic Zeeman shift is ε0 ≈ 2.4 Er. (c) Connecting two edges of a Hall strip (left) gives rise to a Hall cylinder (right), a cylindrical surface threaded by synthetic magnetic fields (yellow arrows in shaded regions). The magnetic flux through a unit plaquette (highlighted area) formed by four maxima of the emergent BEC density modulations (wavy patterns with a periodicity of d/2; note that such density modulations do not occur in the Hall strip) corresponds to an Aharonov-Bohm phase of π. The phase (with + and − respectively denoting 0 and π at positions of maximum density) of each spin component has a periodicity of either d or d/2. thetic magnetic flux is identical and engineered by making atoms obtain a phase after completing a closed trajectory on the y-w surface. An atom at location y hopping along ±ŵ via a Raman transition obtains a net momentum of K along ±ŷ, acquiring a Raman laserimprinted y-dependent phase, ±Ky (Appendix A), where K = 2k r . For the shaded regions in Fig. 1(c), the phase acquired by an atom after traveling around an area of ∆y times one unit length alongŵ is ±K∆y. Such a phase is analogous to the Aharonov-Bohm phase acquired by charged particles in a magnetic field, and thus corresponds to an artificial magnetic field and flux (yellow arrows) [62,68,69,79,80].
Since the transverse x and z directions are decoupled from the cylinder, the single-particle Hamiltonian for atoms on the Hall cylinder [ Fig. 1(c)] is written in the basis of {|1 , |2 , |3 , |4 } as (Appendix A) wherep y = −i ∂ ∂y , I is the identity matrix, and ε 0 is the quadratic Zeeman shift. Here, the Raman-induced y-dependent phase factor, e ±iKy , cannot be gauged away due to our implemented periodic boundary condition, unlike open boundary conditions such as when Ω 2 = 0 (Appendix A). This makes H possess a translational symmetry under a translation of d = 2π/K. Furthermore, H has a nonsymmorphic symmetry [44]: a translation of d/2 alongŷ followed by a unitary transformation alonĝ w, |1 → |1 , |2 → − |2 , |3 → −|3 , |4 → |4 .

III. EMERGENCE OF BEC CRYSTALLINE ORDER AND TOPOLOGICAL BAND STRUCTURE
Even in the absence of an external lattice, the cylindrical surface (with the periodic boundary condition in theŵ direction) along with the radial synthetic magnetic flux cause the BEC to develop an emergent periodic or crystalline order in the y direction with an underlying nonsymmorphic symmetry (Appendix D). As sketched in Fig. 1(c) (see also Fig. 5 in Appendix F), the BEC density (squared amplitude of the wavefunction) alongŷ has a period of d/2 (half the period of H) while the phase of the BEC wavefunction of each spin component modulates with a period of either d or d/2. A plaquette [highlighted in Fig. 1(c)] formed by four maxima of the density modulations thus has a magnetic flux Φ/Φ 0 = (Kd/2)/(2π) = 1/2 corresponding to a phase of π, where Φ 0 = 2π /e is the magnetic flux quantum with q ≡ −e (e is the elementary charge) defined as the effective charge of a particle. Interestingly, here the emergent crystalline order corresponds to the generation of a subwavelength lattice having a periodicity of d/2 = λ/4 (Anderson et al. [81] reported a more detailed study on this context).
To gain further insights, Fig. 2(a) illustrates how states in momentum space are coupled by the light fields. These states are simply basis states for a generic lattice. However, they are coupled in a special way such that novel topological band structures can occur: there are two independent branches (solid and dashed circles, each branch connected by lines representing couplings), with Hamiltonians H i=1,2 (q y ), offset from each other by K, i.e. H 1 (q y ) = H 2 (q y ± K), where q y is the quasimomentum. Besides, each branch is invariant under a 2K trans-lation, i.e. H i (q y ) = H i (q y + n × 2K) where n is an integer, thus corresponding to the d/2 periodicity in the BEC density modulations. These two branches correspond to two groups of Bloch bands (each has a periodicity of 2K) that are also offset from each other by K and thus intersect at points occurring periodically by K (Appendix D). As shown in Fig. 2(b), the band structure possesses topological band crossing points that result from and are protected by the nonsymmorphic symmetry. Such band crossings are topologically robust against perturbations respecting the symmetry, such as variations of parameters in Equation (1), and have played important roles in topological quantum matter such as topological semimetals [44,82,83].

IV. TOPOLOGICAL BAND CROSSINGS AND TOPOLOGICAL BLOCH OSCILLATIONS
To probe the band structure, we perform spin-and momentum-resolved quantum transport measurements. A BEC is initially prepared (Appendix J) around q y = 0 in either band 1 or band 2 [ Fig. 2(b)]. Then, the dipole trap is abruptly turned off, allowing atoms to fall under gravity (towards −ŷ) for various holding times, t hold , during which the Raman and microwave couplings remain on. In other words, the gravity induces transport of the BEC towards negative q y in Fig. 2(b) for various t hold . Subsequently, we immediately turn off all coupling fields to release the atoms for a 15-ms time of flight (TOF) including a 9-ms spin-resolved Stern-Gerlach process, and then perform absorption imaging. These TOF images unveil the mechanical momentum (alongŷ) and spin compositions of the atoms. We obtain atoms' average momentum by summing over all population-weighted mechanical momentum components (Appendix L). The (fractional) population of a spin component is obtained by summing over (fractional) populations of all mechanical momentum components corresponding to that spin state. Figure 2(c) shows select TOF images at various t hold and the corresponding quasimomentum (the relation between t hold and quasimomentum is determined by experimental calibration, see Fig. 7 in Appendix H). The extracted average momentum and spin population are shown as circles in Figs. 2(d) and 2(e), respectively. In Fig. 2(c), TOF images show the reoccurrence of similar patterns (indicated by colored rectangles) with a period of 2 K in quasimomentum (2.6 ms in time), consistent with the periodicity revealed in Figs. 2(d) and 2(e). These are Bloch oscillations [84] that possess twice the periodicity of the band structure ( K), analogous to traveling on a "momentum-space Möbius strip": atoms have to travel twice the period of the band structure to reach the same quantum state, because at a band touching point they undergo a diabatic transition from the ground to the first excited bands. Such period-multiplied topological Bloch oscillations can unveil the band topology, which is characterized by a symmetry-protected topological invariant, the period multiplier µ [85,86] . Here, the observed 2 K periodicity (µ = 2) is protected by the nonsymmorphic symmetry, and is consistent with the d/2 periodicity of the density modulations [ Fig. 1(c)]. Besides, similar TOF images for band 1 and band 2 are offset from each other by K or 1.3 ms, consistent with the out-of-phase Bloch oscillations in Fig. 2(d). These observed transport properties uncover the band crossings.
Unlike previous works using external optical lattices [62,68,69,74], here, the emergent BEC crystalline order and topological band structure result from "curving" the Hall strip into the Hall cylinder and vanish on the Hall strip (which we realize by setting one of the microwave couplings to zero, see Fig. 8 in Appendix I). Note that both the cylindrical surface and the net radial magnetic flux are essential for the emergent phenomena here, which disappear when either ingredient is absent (Appendix C).
We have performed calculations using similar experimental parameters to gain further insights into experi-mental observations. The calculation results are shown in Figs. 2(d) and 2(e), as explained below. First, we have calculated the average momentum and spin populations of the eigenstates of band 1 and band 2. Results of such eigenstate calculations (dashed lines) correspond to the transport of a noninteracting BEC. We have also solved the time-dependent 3D Gross-Pitaevskii (GP) equation for an interacting BEC (Appendix M), showing results as solid lines. The experimental results exhibit notable damping, consistent with GP simulations, but deviating from the eigenstate results that exhibit no damping. These results show that inter-particle interactions, which can broaden the momentum distribution of atoms, are important and can cause damping of the transport and Bloch oscillations (see Fig. 9 in Appendix M).

V. TOPOLOGICAL TRANSITION AND EFFECTS ON TRANSPORT
We can further break the nonsymmorphic symmetry that protects the band crossings by introducing a symmetry-breaking perturbation, a radio frequency (RF) wave coupling |1 and |2 as well as |3 and |4 with respective coupling strengths Ω RF2 and Ω RF1 [ Fig. 3(a)]. This perturbation induces a topological transition manifested as gap opening at band crossings and makes the band structure now depend on a synthetic axial magnetic flux Φ axial (no planar analogue) through the cylinder, as explained below. In the momentum space, such an RF wave changes how the basis states for a lattice are coupled. RF couplings merge the two independent branches in Fig. 2(a) into one that has a K periodicity [ Fig. 3(b)], breaking the nonsymmorphic symmetry (Appendix E). Besides, any two basis states are now connected by multiple pathways, distinct from Fig. 2(a) in which there is either one or zero pathways for any two states in the same or different branches, respectively. These multiple pathways cause an interference effect controlled by an axial phase θ axial , where θ axial = 2θ RF + θ 1 − θ 2 (Appendix B), θ RF is the phase associated with Ω RF1,RF2 (generated from one RF wave, thus sharing the same RF phase), and θ 1,2 is the phase associated with Ω 1,2 . This axial phase θ axial gives rise to Φ axial [Φ axial /Φ 0 = θ axial /2π, Fig. 3(c)], which can affect the band structure as well as transport properties. We can calibrate and precisely control Φ axial by performing quench experiments (see Fig. 10 in Appendix N).
To demonstrate band gap opening between the two lowest bands, a specific band structure with a relatively large gap induced by a large RF coupling is chosen [ Fig. 3(d)] and probed by the same type of quantum transport measurement, in which a BEC is initially prepared at the bottom of the ground band. Figure 3(e) presents select TOF images at various t hold and the corresponding quasimomentum. The extracted average momentum and spin population are shown as data points in Figs. 3(f) and 3(g), respectively. We observe that both momentum and spin oscillations exhibit half the period of those observed in Fig. 2. This directly indicates that most atoms stay in the ground band (i.e., the transport is nearly adiabatic) because of the large gap opened. That is, upon breaking the symmetry, the topological invariant µ changes from 2 to 1, signifying the topological transition manifested by the gap opening. The observed Bloch oscillations in Figs. 3(f) and 3(g) possess damping, again consistent with GP simulations (solid lines) that include effects of inter-particle interactions, but deviating from the non-interacting eigenstate calculations (dashed lines).

VI. CONTROLLING TRANSPORT VIA TUNING AXIAL SYNTHETIC MAGNETIC FLUX
Lastly, we demonstrate that controlling Φ axial allows manipulating both the band structure and transport properties. Whereas this axial flux can be gauged away when the nonsymmorphic symmetry protects the band crossings, it becomes important when RF couplings lift the symmetry and open the band gap, due to the previously discussed interference between pathways in the momentum space (see details in Appendix B). As shown in Fig. 4(a), we perform the same type of quantum transport with a BEC initially prepared around q y = 0 in band structures at moderate RF couplings (giving an intermediate gap size) with various Φ axial , focusing on measuring spin populations at q y = −1K (at t hold = 1.3 ms, indicated by dashed lines). Such spin populations are sensitive to the Φ axial -dependent gap size [ Fig. 4(b)] between the two lowest bands, because different gap sizes can cause notable differences in atoms' Landau-Zener tunneling probability [87] to the excited band. Figure 4(c) presents select TOF images for the measurements at q y = −1K with various Φ axial . The extracted spin population at various Φ axial , shown in Fig. 4(d), is consistent with both the GP simulations (solid lines) and the prediction (dashed lines) by the Landau-Zener formula (considering the gap size and eigenspin compositions of the two lowest bands at various Φ axial ). The observed strong dependence of a transport property (here spin population following transport) on the synthetic axial magnetic flux, reminiscent of a "magnetotransport" behavior [88], reflects the underlying Φ axial -dependent band structures.

VII. DISCUSSION
Our work differs from another Hall cylinder [74] recently realized in a number of fundamental ways. (1) Our setup does not use an external real-space optical lattice as in Ref. [74] (we also noted another recent experiment [75] exploring incommensurability-induced effects, where an external optical lattice is important). Rather, BEC crystalline order and topological band structure emerge due to curving the Hall strip into a Hall cylinder and thus are intrinsic properties of a Hall cylinder, not relying on an extra real-space lattice potential. (2) Topological band structures are distinct. We observe topological band crossings protected by a nonsymmorphic symmetry, while Ref. [74] revealed a gapped topological band protected by a generalized inversion symmetry. Based on (1) and (2), our work and Ref. [74] respectively reveal the in- trinsic and extrinsic topological properties of a Hall cylinder. Besides, once the emergent intrinsic crystalline order is incommensurate with the external lattice, intriguing localization phenomena are predicted to arise [89], where the axial flux would strongly influence the physical observables when near the localization-delocalization transitions. In fact, all these phenomena reflect the rich physics arising on a Hall cylinder yet absent in the planar counterpart (the Hall strip). This illustrates the crucial and intriguing role of geometries of spaces in novel topological phenomena. (3) We perform quantum transport measurement to probe the band structures as well as the topological transition and demonstrate topological Bloch oscillations. We also conduct quench experiments to calibrate the axial synthetic magnetic flux. In Ref. [74], quench dynamics is used to probe a band gap closing associated with a topological transition from triv-ial to topological gapped bands, where the transition is induced by varying one of the couplings between spin states. (4) We demonstrate the capability of tuning the axial synthetic magnetic flux to control the band structure as well as BEC's transport. References [33,75,89] point out that such a flux should play an important role but it is not considered in Ref. [74].

VIII. CONCLUSIONS AND OUTLOOK
In summary, engineering the geometry of space subject to a synthetic magnetic field has allowed us to observe an emergent topological band structure and a topological transition as probed by quantum transport measurements. Our work may offer valuable insights to exploring novel quantum matter intrinsic to unconventional spaces, a multidisciplinary direction in quantum science and engineering that is of high interest to broad communities of atomic and molecular physics, condensed matter physics, and photonics quantum simulations [48]. Future directions may include investigating topological transitions induced by the tunable axial magnetic flux [33], studying superfluidity on curved surfaces [11,12], implementing a Laughlin's charge pump [28] [e.g., by making Equation (1) time dependent], exploring the fractal energy spectrum of Hofstadter's butterfly as suggested in Ref. [62], and using Laguerre-Gaussian beams for the Raman lasers to create a Hall torus [32]. Even more possibilities arise if inter-particle interactions can be tuned by means such as optical lattices [34] or Feshbach resonances [90]. For example, it is interesting to study quantum many-body phases such as topologically ordered states, for example fractional quantum Hall-like states, on a Hall cylinder or torus [29-31, 72, 73] or in curved spaces such as hyperbolic surfaces [61].
Future experiments can also study how interactions can affect the quantum transport, such as Landau-Zener transitions [22,91,92]. Our GP simulations ( Fig. 11 in Appendix O) performed for a BEC on a synthetic Hall cylinder (but in a 1D trap different from our current experimental setup to enhance the interaction effects) have revealed that the Landau-Zener tunneling from ground to excited bands increases and approaches 1 with increasing interactions. This suggests that strong interactions could make the quantum transport of the BEC more diabatic, effectively experiencing a different topology (Möbius strip) in the momentum space, compared to a non-interacting BEC.
where ω Z is the effective linear Zeeman splitting and ε 0 is the effective quadratic Zeeman shift. In our experiments, ε 0 ∼ 2.4 E r , given by the applied bias magnetic field (about 5 gauss). We define the (two-photon) Raman In the following, we derive the single-particle Hamiltonians for various coupling schemes: (1) the Hamiltonian H and the corresponding momentum-space Hamiltonian H qy for the Hall cylinder with a nonsymmorphic symmetry; (2) the Hamiltonian H and the corresponding momentum-space Hamiltonian H qy for the Hall cylinder with a broken nonsymmorphic symmetry; (3) the momentum-space Hamiltonian H strip for the Hall strip.

Hall cylinder with the nonsymmorphic symmetry
The free atomic Hamiltonian taking into account the motion alongŷ is written as: where I is the identity matrix andp y = −i ∂ ∂y is the momentum operator alongŷ. In the rotating-wave approximation, the Hamiltonians describing the Raman (H Raman ) and microwave (H 1,2 ) couplings are respectively written as: where K = 2k r , H.c. stands for Hermitian conjugate, and the initial phases of these coupling fields are ignored temporarily but will be considered later.
We choose a rotating frame defined by the following unitary transformations to eliminate the time-dependent terms in Eqs. (A2-A4): In such a rotating frame (without tildes), where H Raman and H 1 become time independent. By further requiring , which is also time independent. Eq. (A10) is called the resonance condition for the cyclic coupling and is realized in this work [as depicted in Fig. 1(b)].
Therefore, in the rotating frame defined by Eq. (A5) and when the resonance condition in Eq. (A10) is fulfilled, H = H free +H Raman +H 1 +H 2 is time independent. In the basis of {|1 , |2 , |3 , |4 }, The above equation shows that a Raman transition corresponds to a y-dependent phase factor, e ±iKy , while a microwave transition does not lead to a positiondependent phase. Redefining all energies such that E 3 + ω 1 /2 = ε 0 + δ 1 /2 and using the definitions of ε 0 , δ R , δ 1 , δ 2 , and the resonance condition in Eq. (A10), we obtain and rewrite Eq. (A11) as This equation includes Raman and microwave detunings, which can be nonzero during the initial state preparation process as discussed later. After the initial state preparation, δ R = δ 1 = δ 2 = 0 is achieved and Eq. (A13) becomes Eq. (1). Thus, all the detunings are zero in the main text.
To calculate band structures, we derive the momentum-space Hamiltonian H qy by considering the coupling scheme in Fig. 2(a). The spin and mechanical momentum states comprise a plane-wave basis, denoted by where (q y + nK) is the mechanical momentum, m = 1, 2, 3, 4 label the spin, q y is the quasimomentum, and n is an integer. Then, H qy reads where the A n matrices are on the diagonal of H qy . Each A n is a 4-by-4 matrix written in the basis of {| (q y + nK); m }, where the four spin states have identical mechanical momentum (i.e., same n). Thus, A n only includes microwave couplings. When all the detunings are zero, Here B is a 4-by-4 matrix responsible for the Raman coupling between adjacent A n matrices:

Hall cylinder with a broken nonsymmorphic symmetry
An RF wave whose angular frequency equals ∆ω R couples |1 and |2 , and |3 and |4 , with respective coupling strengths Ω RF2 and Ω RF1 . Note that |Ω RF1 | = |Ω RF2 | (Ω RF1 > 0 and Ω RF2 < 0) due to the different Clebsch-Gordan coefficients associated with different transitions. The corresponding Hamiltonian H is obtained by adding the RF couplings to H as Let δ R = δ 1 = δ 2 = 0 for simplicity. Since the RF wave only couples spin states that have the same mechanical momentum, the corresponding momentum-space Hamiltonian H qy has the same form as Eq. (A15) but with a modified A n denoted by A n : , with Ω 2 = 0 and δ R = δ 1 = δ 2 = 0. Noting thatp 2 y /(2m)I = 2 q 2 y /(2m)I for the plane-wave basis, this transformation gauges away the y-dependent phase factor in H and leads to H strip . In the basis of {|1 , |2 , |3 , |4 }, It is important to realize that if Ω 2 = 0, the transformation cannot gauge away the y-dependent phase factor becausê U 0 HÛ 0 −1 would still have the y-dependent terms Ω 2 e −2iKy and Ω 2 e 2iKy .

Appendix B: Phases in Hamiltonians and synthetic axial magnetic flux
Now, we take into account phases associated with the Raman, microwave, and RF couplings, denoting them respectively as θ R , θ 1,2 , and θ RF . Without loss of generality, we consider Hamiltonians without detunings, δ R = δ 1 = δ 2 = 0.

Hall cylinder with the nonsymmorphic symmetry
We refer the reader to Fig. 1(b). In this case, we show that θ R and θ 1,2 can be gauged away and thus have no effect on the band structure. The Hamiltonian H in Eq. (A13) becomes and ∆θ t = θ R − θ 1 /2 + θ 2 /2. Let y = y − ∆θ t /K. The final Hamiltonian is then independent of all the phases and equivalent to Eq. (A13). Note that ∆θ t is simply half of the accumulated phase acquired by an atom completing a close trajectory counterclockwise in the synthetic dimension [ Fig. 1(b)], corresponding to a synthetic axial magnetic flux Φ axial /Φ 0 = ∆θ t /π. In conclusion, Φ axial (∆θ t ) here can only lead to a spatial translation in y and has no effect on the band structure.

Hall cylinder with a broken nonsymmorphic symmetry
We refer the reader to Fig. 3(a). In this case, we show that θ R only causes a spatial translation in y and can still be gauged away, while an axial phase θ axial = 2θ RF + θ 1 − θ 2 cannot be gauged away and thus can affect the band structure. The Hamiltonian H in Eq. (A18) becomes which does not depend on θ R . Apply a unitary transformationÛ 2 to H : where The transformed Hamiltonian depends on a single phase, θ axial . Thus, here the synthetic axial magnetic flux Φ axial /Φ 0 = θ axial /2π is crucial as it can affect the band structure.
Appendix C: Importance of the synthetic radial magnetic flux As mentioned in the main text, the net radial magnetic flux is key to many phenomena emerging on the Hall cylinder, which otherwise disappear. To understand this, for example, one can realize a periodic boundary condition by replacing the Raman couplings in Fig. 1(b) with RF couplings, which do not change the momentum of an atom. Such a cyclic coupling delivers a cylinder without any magnetic field on the cylindrical surface. The corresponding Hamiltonian is similar to H but without the y-dependent phase factors, i.e. e ±iKy = 1. The corresponding dispersion remains parabolic and non-periodic. Consequently, there are no Bloch oscillations and those observed phenomena in the main text vanish.
Interestingly, one may realize another periodic boundary condition by applying two different pairs of Raman lasers in Fig. 1(b) such that the matrix element 3|H|4 ( 4|H|3 ) in Eq. (1) changes from e −iKy (e iKy )to e iKy (e −iKy ). Consequently, a cylinder is penetrated by magnetic fields, but the net radial magnetic flux is zero. In this case, the y-dependent phase factors in the corresponding Hamiltonian can be gauged away, and the observed phenomena in the main text disappear. This again uncovers the essence of the net radial magnetic flux for the emergence of the observed phenomena.
Appendix D: Symmetries of the Hamiltonian H
Recall that H = H 0 + H 0 . The band structure of H is simply that of H 0 (q y ) plus H 0 (q y ). For a small but finite δΩ, H 0 can be treated as a perturbation, and its contribution to H can be estimated by the first-order perturbation theory. At q y , such an energy contribution to H is δE 0 (q y ) = ψ(q y )|H 0 (q y )|ψ(q y ) , while at −q y it is opposite, i.e., δE 0 (−q y ) = Î −1 ψ(q y )|H 0 (q y )|Î −1 ψ(q y ) = −δE 0 (q y ). Therefore, the band structure of H is not inversion symmetric even though H 0 (q y ) is inversion symmetric, unless when Ω R1 = Ω R2 (δΩ = 0) is fulfilled such that H 0 vanishes.

Nonsymmorphic symmetry and band crossings
The Hamiltonian H in Eq. (A13) as well as Eq. (1) Here, ApplyingŜ to Eqs. (D5, D6), one can verify that ψ p and ψ m are eigenfunctions ofŜ with the corresponding eigenvalues ±e iqyd/2 . With Eqs. (D7, D8), we also see that ψ p (q y ) and ψ m (q y ) are still Bloch waves labeled by q y .
Consider two sets of eigenfunctions {ψ p (q y ), ψ m (q y )} and {ψ p (q y + K), ψ m (q y + K)}. Their corresponding eigenvalues of the operatorŜ are {e iqyd/2 , −e iqyd/2 } and {−e iqyd/2 , e iqyd/2 }. Thus, one obtains ψ p (q y ) = ψ m (q y + K) and ψ m (q y ) = ψ p (q y + K). This suggests two properties associated with the nonsymmorphic symmetry: (I) both ψ p (q y ) and ψ m (q y ) have a periodicity of 2K in q y , and (II) ψ p (q y ) and ψ m (q y ) are offset from each other by K in q y . Denote the corresponding eigenenergies (eigenvalues of H) for ψ p (q y ) and ψ m (q y ) as E p and E m , the energy spectrum also possesses properties associated with the nonsymmorphic symmetry, corresponding to properties (I) and (II) above. Corresponding to (I), we have This suggests that the band structure has crossing points at some q y . Recall that the Hamiltonian H with δ R = δ 1 = δ 2 = 0 and Ω R1 = Ω R2 possesses a generalized inversion symmetry in Eq. (D1). Given the relations in Eq. (D1) and Eq. (D9), we obtain Consequently, for q y = (2n+1)K/2 where n is an integer, E p is equal to E m , corresponding to a degenerate point (band crossing) in the band structure. Such a degeneracy at q y = (2n + 1)K/2 is protected by the nonsymmorphic symmetry and the generalized inversion symmetry. If any of δ R , δ 1 , δ 2 is nonzero or if Ω R1 = Ω R2 , the generalized inversion symmetry is broken while the nonsymmorphic symmetry is retained, the two branches still cross but at q y = (2n + 1)K/2. Furthermore, the two independent branches in the spin-mechanical momentum coupling scheme in Fig. 2(a) implies that the plane wave basis {| (q y + nK); m } in Eq. (A14) can also be decomposed into two subsets based on the nonsymmorphic symmetry. These two branches can be written in the following form: φ p (q y ) = n (c 1,n |q y + 2nK; 1 + c 2,n |q y + 2nK + K; 2 +c 3,n |q y + 2nK + K; 3 + c 4,n |q y + 2nK; 4 ) (D11) and φ m (q y ) = n (d 1,n |q y + 2nK + K; 1 + d 2,n |q y + 2nK; 2 +d 3,n |q y + 2nK; 3 + d 4,n |q y + 2nK + K; 4 ). From Eqs. (D11, D12), we readily see that φ p (q y ) and φ m (q y + K) are identical if one equates d 1,n with c 1,n+1 , d 4,n with c 4,n+1 , d 2,n with c 2,n , and d 3,n with c 3,n . Thus, Eqs. (D11, D12) respectively correspond to the band 1 and band 2 in Fig. 2 in the main text, providing another way to understand band crossings due to the nonsymmorphic symmetry.
Appendix E: Symmetries of the Hamiltonian H When the RF wave is applied, the corresponding Hamiltonian H [Eq. (A18)] is obtained by adding the RF couplings to the Hamiltonian H. These y-independent RF terms are added to the Raman terms in the 1|H|2 , 2|H|1 , 3|H|4 , 4|H|3 matrix elements of H. Upon d/2 translation byĜ, the Raman terms flip sign but the y-independent RF terms do not; therefore, the nonsymmorphic symmetry is broken in H . However, a d translation still leaves H invariant. Similar to H, the generalized inversion symmetry is also broken in H and thus the band structure is asymmetric.

Appendix F: Calculations of BEC wavefunctions in the real space
We solve the eight-spin version (Appendix P) of the Hamiltonians H qy or H qy [see Eqs. (A15) and (A19)] to obtain the probability amplitude (b qy n,m ) of the constituent plane waves of the form b qy n,m e i(qy+nK)y |m , whose superposition gives the BEC wavefunction in the real space. From the wavefunction, we obtain the variations of the density and phase in the real space for each spin state. For example, we perform such calculations for a BEC at q y = 0 in the ground or first excited bands in two exemplary cases (in units of E r ). (1) A band structure with band crossings at Ω R1 = 2.3, Ω 1 = 2.3, and other Raman and microwave couplings obtained by the corresponding scaling relations. No RF couplings. (2) A band structure with gap opening at Ω R1 = 2.3, Ω 1 = 2.3, Ω RF1 = 0.8, θ axial = 0, and other Raman, microwave, and RF couplings obtained by the corresponding scaling relations (Appendix P). The results shown below focus on spin states |1 , |2 , |3 , and |4 . spectively for a BEC in the ground and the first ex-cited bands. The red line, green dashed line, blue circles and black squares respectively correspond to |1 , |2 , |3 , and |4 . The BEC wavefunction corresponding to Fig. 5(a)[(b)] can be described by φ p (q y = 0) [φ m (q y = 0)] in Eq. (D11) [Eq. (D12)], an eigenfunction of theŜ operator with an eigenvalue of e iqyd/2 = 1 (−e iqyd/2 = −1). Because of the nonsymmorphic symmetry, we find that (I) the calculated population of each spin state has a periodicity of d/2; (II) for the ground band, the phase of |1 and |4 (|2 and |3 ) have a period of d/2 (d). For the first excited band, the phase of |1 and |4 (|2 and |3 ) have a period of d (d/2). In general, for q y = 0, the phase of two spin states would have a periodicity of d while the phase of the other two would have a periodicity of d/2. This is because a nonzero q y only introduces an overall phase to the spin states at q y = 0.
(2) Case 2 The calculated population and phase in the real space for each spin shown in Figs. 6(a) and 6(b), respectively, correspond to a BEC in the ground and first excited bands. The periodicity of the population and phase for each spin is identical to the periodicity of the Hamiltonian, d. The maximum population of the ground state sits at y = ±nd, where n is an integer, because of the s-wave nature of the ground state. On the other hand, for the first excited state, the maximum population of |1 and |4 sits at y = ±(2n+1)d/2 rather than y = ±nd. Besides, there is also local peak population appearing at y = ±nd. The quasimomentum of the BEC at t hold can be measured by the displacement of the mechanical momentum components of, say |1 or |4 , at t hold relative to those at t hold = 0 (q y = 0). Fig. 7 shows the calibrated relation between quasimomentum and t hold . The corresponding slope, d( q y )/d(t hold ), is obtained by the linear fit to the data. The average slope, 0.751 K/ms, serves as the calibration used for calculations. The calculated physical quantities that are functions of quasimomentum can then be converted to functions of t hold . Note that the calibrated slope is slightly smaller than the predicted 0.843 K/ms which is obtained by using g = 9.81 m/s 2 for the gravity. This small difference may be due to the presence of small background (e.g. magnetic) fields that counteract the gravity during experiments. is pierced by the same magnetic field as for the cylinder. Nonetheless, as shown in Eqs. (A20) and (A21), the Raman-imprinted phase factor e ±iKy can now be gauged away, resulting in a non-periodic single-particle dispersion [ Fig. 8(b)]. We probe this dispersion by performing the same type of quantum transport measurement with a BEC initially prepared around the minimum of the right well in the ground band [ Fig. 8(b)]. Figure 8(c) presents select TOF images at various t hold and the corresponding quasimomentum. The extracted average momentum is shown in Fig. 8(d), in which circles are experimental data and solid lines are eigenstate calculations. In this case, BEC's average momentum keeps increasing due to the gravity. No Bloch oscillations or periodic occurrences are observed. (1) Band 1 and band 2 in Fig. 2

(b)
We note that the eigenstate around q y = 0 in band 1 (band 2) has dominant populations in |4 and |1 (|3 and |2 ). Thus, to load a BEC around q y = 0 in band 1 (band 2), a BEC is first prepared at |4 (|3 ) with δ R , δ 1 < −2.5 E r . The value of δ 2 is inferred from Eq. (A12). For band 1, we then ramp on the Raman and microwave couplings Ω R1,R2 and Ω 1,2 from zero to final values while ramping δ R and δ 1 to zero in 15 ms.
For band 2, we then ramp on the microwave couplings Ω 1,2 from zero to final values while ramping both δ R and δ 1 to around −0.6 E r in 15 ms. Subsequently, while keeping Ω 1,2 at the final values, we ramp on the Raman couplings Ω R1,R2 from zero to final values in 5 ms during which we ramp δ R and δ 1 to zero in 3 ms and then hold δ R and δ 1 at zero for the remaining 2 ms.

3(d) and 4(a)
A BEC is first prepared at |4 with δ R , δ 1 < −5 E r . Then, we ramp on the Raman and microwave couplings Ω R1, R2 and Ω 1,2 from zero to final values while ramping δ R and δ 1 to zero in 15 ms. At the very beginning (at which δ R , δ 1 < −5 E r , so the RF wave is off resonant) of this 15ms ramp, the RF couplings Ω RF1,RF2 are abruptly turned on to the final values. Then, Ω RF1,RF2 are held at the same final values while δ R and δ 1 are ramped to zero in 15 ms.
(3) Ground band in Fig. 8(a) To prepare a BEC around the minimum of the right well, a BEC is first prepared at |4 with δ R , δ 1 < −2.5 E r . Then, we ramp on the Raman and microwave couplings Ω R1,R2 and Ω 1 from zero to final values while ramping δ R and δ 1 to zero in 15 ms. In this case, Ω 2 is zero throughout the experiment.

Appendix K: Condensate fraction
In experiments, the condensate fraction before loading atoms to band structures is 90%, decreases to 50-70% after a 15-ms loading procedure, and then drops to 40-60% after another 7-ms holding time for transport. We find that the decreasing of the condensate fraction (with a typical time scale of 40-60 ms) is mainly due to the finite lifetime of atoms in the F = 2 hyperfine manifold in our system.

Appendix L: Imaging analysis
For each spin state in TOF images, a proper window is chosen to enclose the corresponding atomic clouds. We sum the optical density of each pixel over the entire window to obtain the atom number N i of spin state i. The average mechanical momentum p i of atoms in spin state i is determined by the difference in average pixel position between the atoms and a BEC that has zero mechanical momentum. Such a difference is then converted to mechanical momentum based on the calibrated conversion between K and image pixels. The average mechanical momentum p of all atoms is the weighted average of the mechanical momentum of each spin state, i.e., p = N i p i /( i N i ).

Appendix M: Gross-Pitaevskii (GP) simulations
We solve the time-dependent Gross-Pitaevskii equation to simulate BEC's transport with an atom number N of 15000. We have checked that the small spin-dependent interaction has negligible effect in our calculated results, so a spin-independent scattering length a s = 93.2467a B is used for all the scattering lengths, where a B is the Bohr radius. The effective Hamiltonian reads H eff = H+V +U , where H, V , and U respectively correspond to the kinetic energy plus light (Raman, microwave, RF) couplings, harmonic trapping potential, and interactions. The interaction term is U = 4π 2 as m n( r), where n( r) is the density of all the spin states and is normalized to N , i.e., r n( r)dr = N . Recursively applying the Trotter formula, e (A+B)δt = e Aδt/2 e Bδt e Aδt/2 + O(δt 3 ), we obtain an approximation of the short time propagator that is a product of the exponentials of H, V , and U . The kinetic energy propagator e Kδt is evaluated in momentum space and the rest in real space.
(1) Initial state preparation and time evolution To prepare the initial state of a BEC starting in the ground band in Figs. 2, 3, 4, we use imaginary time evolution to find the ground state of H eff . That is, an arbitrary initial state ψ trial is evolved by applying the operator e H eff δt/ until the normalized wave function does not change. We apply e −iH eff δt/ consecutively to propagate the wavefunction in real time.
To prepare the initial state of a BEC starting in the first excited band in Fig. 2, we follow the same preparation method used in the experiment. In this case, more than 90% of atoms is loaded to the first excited band.
(2) Comparison between GP and eigenstate calculations Figure 9 presents a comparison between eigenstate and GP (interacting and non-interacting) calculations for the transport in Figs. 2 and 3. Figure 9(a) compares these calculations of the average momentum versus t hold for the transport in band 1 and band 2 in Fig. 2(b). The spin population versus t hold corresponding to the transport in band 1 is shown in Fig. 9(b). Figure 9(c) compares these calculations of the average momentum versus t hold for the transport in Fig. 3(d). The corresponding spin population versus t hold is shown in Fig. 9(d). By comparing the results of interacting and non-interacting GP, we see that interactions lead to damping of both the momentum and spin oscillations. In addition, results of non-interacting GP and eigenstate calculations are similar, almost overlapping with each other except for transport in band 2 in Fig. 9(a).
For the transport in band 2, the results of interacting GP are only slightly more damped than that of the non-interacting GP. Besides, these GP results are notably different from that of the eigenstate calculation. This is probably because the ramping procedure used in GP for the initial state preparation in this case is relatively fast compared to the time scales of both interactions and adiabatic loading. This gives rise to small interaction effects and non-adiabatic loading to band 2. To calibrate the axial phase θ axial and axial magnetic flux Φ axial (Φ axial /Φ 0 = θ axial /2π) for the experiment in Fig. 4, we perform quench experiments. We refer the reader to Fig. 3(a). We first prepare a BEC at |4 , and then suddenly turn on only microwave and RF couplings [Ω 1(2) = 2.2(3.2) E r and Ω RF1(RF2) = 1.1(−1.6) E r ] with corresponding phases θ 1,2 and θ RF for 90 µs. This results in spin dynamics that depends on a single phase θ axial = 2θ RF + θ 1 − θ 2 , as explained in Eq. (B6). The parameters are chosen such that the sensitivity of spin dynamics to θ axial is measurable and allows the phase cal- Quasimomentum ħq y (ħK) Quasimomentum ħq y (ħK)   Fig. 2(b). The spin population versus t hold corresponding to the transport in band 1 is shown in (b). (c) Calculations of the average momentum versus t hold for the transport in Fig. 3(d).
The corresponding spin population versus t hold is shown in (d). ibration. Specifically, at the end of such a pulse, we measure spin-|4 population as a function of θ axial . The experimental results are then compared with numerical re-sults obtained by solving the time-dependent Schrödinger equation to calibrate θ axial . Fig. 10 presents a typical calibration, demonstrating our capability of controlling θ axial (Φ axial ).
To independently control θ RF and θ 1 − θ 2 , we use an RF function generator with two outputs. One output channel generates the desired RF field with phase θ RF at atoms. Another output channel sends an RF signal to a microwave mixer, which mixes this RF signal with a microwave (from a microwave generator) and then outputs the desired microwave fields 1 and 2, where θ 1 − θ 2 is controlled by the phase of the RF signal sent to the mixer. We thus can control θ axial . Our RF generator can control the phases of the RF signals within ±1.75 milliradian. For daily experiments, the stability of θ axial is estimated to be within ±0.035π (including the fitting uncertainty of ±0.025π) obtained by monitoring the drift of the experimental curve (Fig. 10) in a day.

Appendix O: Effects of inter-particle interactions on
Landau-Zener tunneling While in our experiment it is difficult to control the inter-particle interactions of 87 Rb by adjusting its scattering length, we have performed GP simulations to explore how (repulsive) inter-particle interactions may af- Calculated PLZ increases and eventually approaches 1 with increasing interaction energy per particle. In the simulation, 15000 87 Rb atoms are prepared in a shallow 1D harmonic trap, where the trap frequency is 25 Hz. The single-particle gap size is 0.14 Er. Here dqy/dt = 0.188 K/ms. fect atoms' quantum transport in a gapped band when RF couplings are applied. In the simulation, 15000 87 Rb atoms at zero temperature are prepared in a shallow 1D harmonic trap (along the Raman beam or transport directionŷ), where the trap frequency is 25 Hz. Such a 1D trap is not available in our current experimental setup but is chosen for our simulation setup to enhance and demonstrate the interaction effects that could be explored in future experiments, and for the relative ease of simulation (where we can use the 1D GP equation rather than the more time-consuming 3D GP equation but still demonstrate the essential physics). In principle, the 1D trap can be experimentally realized by applying a 2D optical lattice that realizes parallel 1D tubes, where inter-particle interactions can be tuned by adjusting the lattice confinement. To reduce computation time and without loss of generality, this simulation adopts the 1D GP equation in a four-spin model (see below for details of four-spin and eight-spin models) with otherwise similar parameters used for the simulation in Fig. 3 except that the RF couplings are applied such that the single-particle band gap is 0.14 E r . In the simulation the interaction energy per particle is varied by varying an effective 1D scattering length (or coupling parameter g 1D ), which could, for example, be varied by the confinement potential if the 1D tubes are realized by a 2D optical lattice. Applying a force leads to quantum transport of the atoms (with dq y /dt = 0.188 K/ms), which are initially prepared at the ground band and now tunnel to the excited band with a probability of P LZ [ Fig. 11(a)]. After the atoms pass the gap, we calculate the fraction of atoms in the excited band, i.e., P LZ . As shown in Fig. 11(b), the calculated P LZ increases and eventually approaches 1 with increasing interaction energy per particle. This means that a strong interaction could change the topology underlying the transport in the momentum space, making atoms effectively experience a Möbius strip (band crossings) rather than a regular strip (gapped band structure) experienced by a non-interacting BEC.  6|H 8×8 |7 = 7|H 8×8 |6 * = ΩR5 2 e −iKy e iθR + ΩRF5 2 e −iθRF , 7|H 8×8 |7 = ε 7 , 8|H 8×8 |8 = ε 8 .
Similarly, we can extend all the previous four-spin Hamiltonians to the corresponding eight-spin Hamilto-nians. For the parameter regime in this work, we find that including extra spins can notably modify the shape of band structures and thus atoms' transport. This is the main reason why we need to use eight-spin Hamiltonians for both the eigenstate and GP calculations. On the other hand, in both experiments and calculations, the fractional occupation of extra spins is typically small, with an estimated maximum of 5%. Therefore, for all the calculated results shown in figures, we calculate the fractional population of spin i (i = 1, 2, 3, 4) as n i = N i /( i=1,..,4 N i ) and the average momentum as i=1,..,4 n i p i , where N i and p i are respectively the population and mechanical momentum of spin i.
In summary, the presence of extra spins mainly modifies the band structures and atoms' transport quantitatively. On the other hand, the occupation of these extra spin states is typically small. The four-spin model is a good approximation and adequately captures the key physics we study here.