diff --git "a/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt" "b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/INE1T4oBgHgl3EQf_wYq/content/tmp_files/load_file.txt" @@ -0,0 +1,671 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf,len=670 +page_content='Reversal of quantised Hall drifts at non-interacting and interacting topological boundaries Zijie Zhu,∗ Marius G¨achter,∗ Anne-Sophie Walter, Konrad Viebahn,† and Tilman Esslinger Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland The transport properties of gapless edge modes at boundaries between topologically distinct domains are of fundamental and technological importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Therefore, it is crucial to gain a better understanding of topological edge states and their response to interparticle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Here, we experimentally study long-distance quantised Hall drifts in a harmonically confined topological pump of non-interacting and interacting ultracold fermionic atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We find that quantised drifts halt and reverse their direction when the atoms reach a critical slope of the confining potential, revealing the presence of a topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The drift reversal corresponds to a band transfer between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode, in agreement with the bulk-edge correspondence for non-interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We establish that a non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system, relying on a purely interaction-induced mechanism, in which pairs of fermions are split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The existence of individual edge modes at topo- logical boundaries plays a crucial role in quantum Hall physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' More specifically, a non-trivial topology in the bulk of a material ensures that its edge modes are gapless and chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Gaplessness is related to the bulk-edge corre- spondence, stating that the number of topological edge modes is equal to the difference in Chern number across an interface [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Consequently, a gapless mode should allow an adiabatic transfer from one band to another, resulting in a reflection of transverse bulk currents in the opposite direction if the two bands feature opposite Chern numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, the coherence time in most electronic materials is not sufficient to observe this effect, and edges are generally probed spectroscopically [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Moreover, studies of edge physics in engineered quantum systems, such as ultracold atoms and photonics, have so far been focussed on chirality [5–9] and localisation [10– 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A boundary reflection has not been detected [14, 15], or it was disregarded [16, 17], and to our knowledge it has never been studied for variable interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Here, we observe the reversal of quantised bulk drifts due to harmonic trapping in a topological Thouless pump, the temporal analogue of the quantum Hall effect [18– 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is a fundamental manifestation of confined topological matter and directly shows the gap- less nature of topological edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Going beyond the non-interacting regime, we discover the emergence of a second edge for repulsive Hubbard U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experiments are performed with ultracold fermionic potassium-40 atoms, which are loaded into the potential of a generalised optical lattice formed by a com- bination of standing and running waves of wavelength λ = 1064 nm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This creates an array of decoupled, one-dimensional tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Along the tube direction, the periodically modulated Rice-Mele-Hubbard Hamiltonian with harmonic confinement is realised, ˆH(τ) = − � j,σ � t + (−1)jδ(τ) � � ˆc† jσˆcj+1σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' � (1) + ∆(τ) � j,σ (−1)jˆc† jσˆcjσ + U � j ˆc† j↑ˆcj↑ˆc† j↓ˆcj↓ + � j,σ Vjˆc† jσˆcjσ , where ˆcjσ is the fermionic annihilation operator for spin σ ∈ {↑, ↓} on site j, and t denotes the average tun- nelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' An adiabatic modulation of bond dimerisa- tion δ(τ) = δ0 cos(2πτ/T) and sublattice offset ∆(τ) = ∆0 sin(2πτ/T) traces a closed trajectory in the δ–∆ plane around the origin, referred to as critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' There- fore, an insulator or homogeneously filled band at U = 0 describes a topological pump [18, 20] with T being the pump period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimentally, the topological pump manifests itself as a quantised drift of the atom position by one unit cell per pump cycle [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The harmonic confinement is characterised by the trap frequency ν, entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1 as Vj = 1 2m(2πνaj)2 ≡ V0j2 (a = λ/2, lattice spacing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' m, atomic mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the confinement, the atoms are initially localised at the centre of the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Topological pumping then leads to a quantised drift of atoms against the confining potential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Our measurements show that the quantised drift changes its direction at a certain distance from the trap centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We will demonstrate that this happens when the gradient of the confinement overcomes the band gap and a boundary between topological and trivial regions emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For repulsive interactions, we observe another reflection, closer to the trap centre, while a part of the atoms keeps drifting in the original direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the following, we develop a description of the re- flection in terms of gapless edge modes and the bulk- edge correspondence within the framework of the Harper- Hofstadter-Hatsugai (HHH) model with one real (x) and one synthetic (n) dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The model features bulk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='03583v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='quant-gas] 9 Jan 2023 2 Floquet Energy 0 π π Quasimomentum Gradient Interactions A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Reflection of quantised Hall drifts off a topo- logical interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) Topological Thouless pumping in the presence of confining potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the non-interacting case (top), a harmonic trap gives rise to topological trivial (C = 0) and non-trivial (C ̸= 0) regions, separated by a topological interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The atoms exhibit a quantised drift until they are reflected at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' With repulsive on-site interactions (bottom), the reflection happens closer to the centre, accom- panied by atoms still drifting in the original direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Using Floquet theory, the 1D Rice-Mele pump can be mapped to a 2D Harper-Hofstadter-Hatsugai model with a linear gra- dient along the synthetic dimension n which represents the photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The magnetic flux per plaquette is Φ = 1/2 in units of the magnetic flux quantum [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gradient along n leads to a transverse Hall drift along x (red arrows) due to the nontrivial topology of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (C) Schematic spec- trum of the mapped 2D Hofstadter model in a semi-infinite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The lowest two bands have C = ±1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The linear gradient induces Bloch oscillations in the synthetic reciprocal space (dashed arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A gapless edge mode (solid arrow) appears at the topological interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection of the Hall drift can be understood as atoms being transported from the lower band (C = 1) to the higher band (C = −1) via the topological edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Chern bands with C = +1 and C = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' An ex- act mapping between the non-interacting 1D Rice-Mele Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) and the two-dimensional (2D) HHH model can be obtained using Floquet theory, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1B (for derivation see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=', refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [19] and [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A linear gradient along the synthetic dimension n appears in the mapping since the state with n photons acquires an energy of −nℏω, where ω = 2π/T is the pump fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gradient along n or, equivalently, an exter- nal force causes Bloch oscillations along the synthetic re- ciprocal dimension kn which, in turn, lead to a Hall drift or ‘anomalous velocity’ along the transverse real direc- tion x [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The bulk Hall drift along x corresponds exactly to the quantised displacement measured in the topological pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The trap induces a boundary between topological (Ccentre = 1) and trivial (Cright = 0) regions and a single gapless edge mode emerges, according to the bulk-edge correspondence: Ccentre −Cright = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The edge modes connects two bands of opposite Chern invariant, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, a Bloch oscillation transfers the atoms from the ground to the first excited band via that edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the first excited band has Chern number −1 the atoms are now moving ‘backwards’, re- sulting in a reversal of the quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2 shows experimental in-situ images of the atomic cloud as a function of time τ at U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The data shows a quantised drift of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='00(1) × 2a/T up to about 60 T, which confirms the long coherence time of Bloch oscillations which induce the transverse drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At τ ≃ 75 T the atoms change their drift direction, which is a key observation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The expected topo- logical boundary (red dashed line) represents the posi- tion at which the local tilt from the external harmonic potential ∆ext (j) ≡ 1 2 |Vj − Vj−1| = V0 ��j − 1 2 �� equals the maximum sublattice offset ∆0, thus, xedge/ (2a) ≃ 1 2∆0/V0 = 92(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Beyond this position the total sub- lattice offset ceases to change sign, rendering the region outside xedge topologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The boundary caused by the harmonic confinement is not infinitely sharp, but smoothened over several lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This leads to a small T–dependence of the reflection position (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1), compared to its absolute value, and the calculation above should be understood as the outermost point of the re- flective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflected atoms exhibit a quantised drift of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='99(3) × 2a/T in the opposite direction, in agreement with a transfer to the first excited band with C = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The linear relation between the position of topological boundary xedge and the maximum sublattice offset ∆0 is further confirmed by measuring the reflection in different lattices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is observed under all parameter settings tested in this work, high- lighting that the existence of the topological boundary is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In addition to the reflection, we also observe a cloud of atoms temporarily remaining at the boundary before gradually dissolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This process can be understood via the presence of topologically trivial edge states, which hybridise with the gapless edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' To simplify the picture, let us consider a sharp domain wall between C = 1 and C = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' According to the bulk- edge correspondence, the topologically nontrivial region contributes exactly one gapless mode whereas the trivial region can contribute gapped edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to tunnel coupling at the interface, hybridisation takes place [34] and gaps on the order of the pump frequency 2π/T emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Bloch oscillations along kn can now lead to non- adiabatic ‘Landau-Zener’ transfers between topological and trivial edge modes, causing an incomplete transfer to the higher band, and atoms remaining at the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Subsequent Bloch oscillations will transfer atoms back into the topological domain, leading to the dissolu- 3 quasimomentum energy trivial nontrivial 0 36 108 72 144 time τ (T) 1st 2nd 2nd 2nd 2nd Brillouin zone Dimerisation Site ofset A C B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Measuring the reversal of a quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) The time-trace of atomic in-situ images shows a quantised drift along x before the atoms are reflected at the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Each density image is averaged over three individual measurements with the parameters V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='0148(9)t, ∆0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='7(1)t, and T = 3 ms = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='8(2)ℏ/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The red dashed line indicates the topological boundary xedge/ (2a) ≈ 1 2∆0/V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The white dashed lines are linear fits to the atom drift, yielding slopes of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='00(1) × 2a/T before, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='99(3) × 2a/T after the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Cloud positions, averaged over the transverse direction, are fitted using Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experimental Chern marker (lower panel, points) is determined by the velocity of the right-moving cloud at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The theoretical Chern marker (line) is calculated in a staggered potential Vj = V0 (−1)j j which has the same local tilt ∆ext(j) = V0 ��j − 1 2 �� as the harmonic trap [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In a local density approximation picture, the local tilt ∆ext shifts the δ–∆ pump trajectory upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Depending on whether or not the trajectory encloses the critical point, the pump is rendered topological or trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Measured band populations as a function of time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Each density image is averaged over six individual measurements with the parameters V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='0191(6)t, ∆0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='2(2)t and T = 3 ms = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='7(2)ℏ/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The total atom number remains constant, within error bars, throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the underlying honeycomb lattice geometry in the x–z plane, the first Brillouin zone has a diamond shape [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The band population is inverted when the bulk current is reflected off the topological interface, manifesting the gapless nature of the topological edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (C) Hybridisation of the edge modes at the topological interface due to tunnelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Bloch oscillations along kn can lead to Landau-Zener transfers between topologically trivial and nontrivial edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The population of trivial edge modes explains the atoms being left at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Hybridisation never changes the total number of gapless edge modes at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' tion of the cloud at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' While the harmonic confinement leads to a more complex level structure [21], the underlying process remains qualitatively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We support the in-situ images with measurements of band population before, during, and after the reflec- tion, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2B [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Before the reflection, we find a filled ground band, which is consistent with the observation of a quantised Hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At the reflection (τ ≃ 72 T), we observe an inversion of the population to the first excited band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' After the reflection, the inverted population remains almost unchanged while the atoms are travelling back, highlighting the absence of incoher- ent relaxation to the ground band, even after more than a hundred Bloch oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We further explore the effect of attractive and repul- sive interactions on the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For attrac- tive Hubbard U = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='27(7)∆0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3A), the quantised Hall drift is reversed at the same position as in the non-interacting situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This can be explained in terms of the Rice-Mele model in which fermions in the strongly attractive regime approach the limit of hard-core bosons [22], and the condition for the emergence of a topological boundary ∆ext (j) = ∆0 remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' For repulsive Hubbard U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3B), we observe a second reflection in addition to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Compared to the non-interacting case, this reflection ap- pears much closer to the trap centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The zoomed-in image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3C) shows that a proportion of the atoms start to move backwards after about 12 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In con- 4 Dimerisation Site ofset OD time т (T) B D E C A Time τ ( ) T τ0 τ0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='5 τ0 + 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Reflection of quantised Hall drifts from an interacting topological edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (A) An attractive Hubbard interaction of U = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t leads to the same reflection behaviour as observed for U = 0 (measurement parameters otherwise identical to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) Repulsive Hubbard interactions of U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='48(7)t lead to the emergence of a second reflection, closer to the trap centre, which we attribute to an interacting topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A zoom-in (C) shows that the early reflection happens after about twelve cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The white dashed lines are guides to the eye, calculated as linear fits to the cloud position, extracted as the sum of a skewed and a regular Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (D) Microscopic description of the interaction-induced reflection for repulsive Hubbard U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) becomes smaller than the Hubbard interaction, formation of double occupancies is prohibited and one atom is left in the higher-energy site of a unit cell, which then drifts in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (E) The critical point in the δ–∆ plane is split into two in the presence of repulsive Hubbard U [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the pump trajectory encloses both critical points, a quantised drift is expected, as in the non-interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local tilt given by the external potential ∆ext shifts the trajectory along the ∆–axis, eventually enclosing just one critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since a single split critical point features half the topological charge of the orignal one, the material’s topology changes and a boundary emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' trast to the drift reversal in the non-interacting system, a large fraction of the atom cloud still undergoes quantised drifting in the original direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the following, we develop a microscopic picture of the interaction-induced partial reflection in the limiting case of two isolated spins (↑, ↓), which approximates our initial state in a unit cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As long as the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) is larger than the Hubbard U, the formation of a double occupancy is en- ergetically allowed and the quantised drift persists [22], even in the presence of an external potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, when ∆ext becomes larger, the energy offset between two neighbouring sites remains always smaller than U, dou- ble occupancy formation becomes prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the lat- ter case, one atom of a singlet pair is transferred to the energetically excited site of a unit cell, which will subse- quently drift in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The other atom, in the lower-energy site, will move onwards because on- site interactions become irrelevant if there is only one atom per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the underlying Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) is SU(2)–symmetric, spin–↑ and spin–↓ have equal probability of being reflected and they should remain cor- related after the splitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The full many-body description of the interaction- induced reversal requires the development of suitable topological invariants for smooth confinements and strong interactions, which goes beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Nevertheless, we obtain an intuition of the bound- ary’s topological origins using again the idea of shifted pump trajectories in the δ–∆ plane with a staggered po- tential (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Numerical simulations have shown that a repulsive Hubbard U can split the critical point at the origin into two separate ones [35], each retaining half the original topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The distance between the new critical points is approximately U up to a cor- rection on the order of the tunnelling t [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' When the trajectory encloses both critical points, quantised drift of two spins (↑, ↓) is expected, as in the non-interacting sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As the position-dependent local tilt ∆ext(j) shifts the trajectory upwards, it will enclose only one of the critical points beyond certain position along x (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 3E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This indicates a transition of topological properties and the emergence of an interacting topological edge in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The estimation of the interacting boundary at ∆ext(j) ≃ ∆0 − U/2 lies close to the centre and agrees with the microscopic picture discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Similar to the non-interacting case, this boundary should be con- sidered as the outermost position of the reflective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In conclusion, we have experimentally observed a re- versal of quantised Hall drifts at a topological boundary in a harmonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection is a direct mani- festation of the gapless nature of topological edge modes between Chern bands of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We explore the effect of Hubbard interactions, both attractive and re- 5 pulsive, and find an asymmetric behavior with respect to U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' While on the attractive side the topological boundary is unaffected, repulsive interactions lead to the emergence of a second interface, featuring a splitting of quantised drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, our experiments could en- able the realisation of circular current patterns for con- structing novel many-body phases [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' More broadly, our work allows the exploration of the bulk-edge corre- spondence in the presence of interactions [38], as well as the investigation of edge reconstruction [39] in the quan- tum Hall effect and in interacting topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank Jason Ho, Gian-Michele Graf, Thomas Ihn, Fabian Grusdt, Fabian Heidrich-Meisner, Armando Aligia, and Eric Bertok for valuable discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We also thank Julian L´eonard and Nur ¨Unal for comments on an earlier version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We would like to thank Alexander Frank for his contribu- tions to the electronic part of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We acknowledge funding by the Swiss National Science Foundation (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 182650, 212168, NCCR-QSIT, and TMAG-2 209376) and European Research Council advanced grant TransQ (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 742579).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' ∗ These authors contributed equally.' 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Interact- ing Hofstadter Interface, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 122, 010406 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Chklovskii, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Shklovskii, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Glazman, Electrostatics of edge channels, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' B 46, 4026 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 7 SUPPLEMENTAL MATERIALS Dependence of the drift reversal on experimental parameters The expected drift reversal happens when the maxi- mum local site offset over one pump-cycle ∆0 is equal to the local tilt given by the harmonic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This po- sition given by xedge ≃ ∆0a/V0 with a = λ/2 and V0 = 1 2m(2πνa)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By measuring the reflection point in lattices with different ∆0, we verify the relevant scaling xedge ∝ ∆0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1A marks the theoretically expected xedge with the un- certainty propagated from the uncertainty of the trap frequency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The disagreement between theory and ex- periment for larger values of ∆0 can be explained by the finite waist of the lattice beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In order to explore the edge in our system, the atoms are pumped by almost 100 unit cells (∼ 100 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the Gaussian envelope of the transverse beams, which are essential to realise the pump, the lattice is effectively shallower far away from the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, ∆0 decreases towards the edge and atoms are reflected sooner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We also find a small dependence of the reflection point on the pump period, compared to its absolute value, which spans roughly 10 unit cells when changing T from 2 ms to 10 ms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This can be understood by con- sidering the energy spectrum of the Harper-Hofstadter- Hatsugai (HHH) model in a harmonic potential, which will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimental sequence We start by preparing a degenerate cloud of fermionic 40K in a crossed dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We have a spin mixture of mF = {−9/2, −7/2} except for the measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1, where a spin-polarised cloud in the magnetic state F = 9/2, mF = −9/2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The spin- polarised cloud is directly loaded into the pumping lat- tice, while the spin mixture is first loaded into an inter- mediate chequerboard lattice with strongly attractive in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The two-step loading precludes the presence of atoms in the higher band and gives a larger fraction of atoms in doubly occupied unit cells [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' After pumping the system for varying times, we either take a in-situ absorption image to measure the density or detect the band population with band-mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The latter is implemented with an exponential ramp to switch off the lattice beam in 500 µs, followed by a time-of-flight expansion of 25 ms before absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimental dependence of the reflection position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The reflection point is expected to depend linearly on the maximal site-offset per pump cycle ∆0 which is experi- mentally verified in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Deviations for large values of ∆0 can be explained by the finite waist of the laser beams forming the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (B) shows the period dependence of the reflec- tion position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Changing the pump period T over an order of magnitude only changes the reflection point by about 10 unit cells, which is a result of the smooth boundary of a harmonic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Realisation of a Thouless pump in the Rice-Mele model The lattice setup is comprised of non-interfering stand- ing waves in x, y, and z directions, together with in- terfering laser beams in the x–z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' All the lattice beams come from a single laser source at wavelength λ = 1064 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' These potential combine to form a honey- comb lattice in the x–z plane, which can be considered as isolated tubes of one-dimensional superlattices along x in the limit of deep transverse lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In each 1D tube the potential can be modeled by a one-dimensional superlattice with two sites per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' With this setup, we realise the Rice-Mele model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the tight-binding limit, the Rice-Mele model can be described with three numbers: the offset energy ∆ between the two sites of a unit cell, the averaged nearest-neighbour tunneling t and the bond dimerisation δ which gives half the differ- ence between the inter- and intra-dimer tunnellings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By having a dynamical control of the relative phase ϕ be- tween the laser beams generating the interfering and the non-interfering lattice, we manage to shift the two with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This shift modulates ∆ and δ pe- riodically, which can be depicted as a closed trajectory in the ∆-δ coordinate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In the adiabatic limit, this realises a Thouless pump with its hallmark quan- 8 tised transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In this case, the atomic displacement is given by the number of revolutions around the origin of the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' φ = 0 φ = ∏/2 φ = ∏ φ = 3∏/2 x E x E x E x E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Realisation of a Thouless pump in the Rice- Mele model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In our system the 1D lattice potential can be modelled by a superlattice with two sites per unit cell, which is depicted in the bottom part of this figure as a function of relative phase ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local site offset ∆ as well as the bond dimerisation δ is depicted in the ∆-δ plots corresponding to the respective potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The resulting pump displacement corresponds to the number of revolutions around the origin in the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Mapping a 1D Thouless pump to a 2D Hofstadter Model with quantum Hall response A 1D Thouless pump with a period of T, as realised in our experiment, can be mapped to a 2D topological tight-binding (HHH) model with an applied electric field E = 2πℏ qT where q can be thought of as a fictitious charge of netural atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the topological bandstructure, this electric field leads to a transverse current Itrans = q T of one atom per period, when considering a fully occupied band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The 2D model therefore has a quantised transverse conductance σtrans = Itrans/E = q2 2πℏ analogous to the Hall conductance in the Quantum Hall Effect (QHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The time-periodicity of the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1 with ˆH(τ) = ˆH(τ + T) allows us to use Floquet’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Solutions of the time-dependent Schr¨odinger equation iℏ∂τ |Ψ(τ)⟩ = H(τ) |Ψ(τ)⟩ (S1) can thus be written as |Ψ(τ)⟩ = e−iϵτ/ℏ |u(τ)⟩ (S2) with |u(τ + T)⟩ = |u(τ)⟩ and ϵ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Due to the time- periodicity of u(τ) we expand it as a Fourier series, |u(τ)⟩ = � n e−iωnτ |un⟩ , (S3) where ω = 2π/T is the pump frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The change from the time-domain into the Fourier-domain is the key ingredient to map the 1D Thouless pump to a 2D tight- binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The index n is also called the photon number of the mode |un⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Using a multi-index α = (j, σ) we write the T-periodic 1D Hamiltonian for U = 0 in the Fourier-basis: ˆH(τ) = � α,β hαβ(τ) |α⟩ ⟨β| (S4) = � α,β,m e−imωτhm αβ |α⟩ ⟨β| with hm αβ = 1 T � T 0 eimωτhαβ(τ)dτ and |α⟩ corresponding to an atom localised on site j with spin σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Likewise, we use Fourier decomposition to express the solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1 as |Ψ(τ)⟩ = e−iϵτ/ℏ � n,α e−inωτun,α |α⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (S5) where un,α = ⟨α|un⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, we obtain an eigen- value equation for un,α: ϵun,α = −nℏωun,α + � β,m hm αβun−m,β ∀n, α (S6) which can be understood as a time independent Schr¨odinger equation of a 2D tight-binding model with a tilted potential energy along one axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By explicitly evaluating the hm αβ, we get H2D = Hreal + Hsynth + Hdiag + HV + Htilt, (S7) with Hreal = −t � j,n,σ (ˆc† j,n,σˆcj+1,n,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), (S8) Hdiag = −δ0 2 � j,n,σ e−iπj(ˆc† j,n,σˆcj+1,n+1,σ + ˆc† j,n,σˆcj+1,n−1,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), Hsynth = −∆0 2 � j,n,σ e−iπj(iˆc† j,n,σˆcj,n+1,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content='), HV = � j,n,σ V (j)ˆc† j,n,σˆcj,n,σ, Htilt = − � j,n,σ ℏωnˆc† j,n,σˆcj,n,σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Hreal and Hsynth describe tunneling along the real (x) and synthetic (n) dimension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The diagonal tunnelling terms in Hdiag are crucial be- cause they open a bandgap between the ground band and the first excited band, characterised by the topological Chern number C which is further related to the quantised Hall conductance via σtrans = q2 2πℏC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The terms in HV 9 describe the external potential along the real-space direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Htilt corresponds to a linear tilt in potential energy along the synthetic dimension which can be thought of as originating from an electric field E = 2πℏ qT pointing along n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Edge modes and their reflection properties To illustrate the topological edge modes in the presence of an external potential, we evaluate the spectrum of H2D in the adiabatic limit (ω → 0) for different potentials V (j) = 1 2m(2πνa)2jκ (S9) with m being the mass of 40K, trap frequency ν = 134 Hz, lattice spacing a = 532 nm, and lattice site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The pa- rameter κ, an even integer, characterises steepness of the trap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' the limit κ → ∞ corresponds to the textbook case of infinitely sharp walls [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A-C shows the nu- merically calculated energy spectra, omitting states on localised to the left edge for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A shows the spectrum for a box-like potential with κ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In this case there is a family of topologi- cal edge states, marked in red, which connect the lower and the upper band (black), separated from topologically trivial states above 5 kHz (also in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' All red states are localised along the right edge in x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The lower and upper band have Chern number 1 and −1, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Considering the dynamics in this model, an applied electric field along n as defined in Htilt leads to Bloch oscillations with a period T along kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' At the same time, the center-of-mass of the atoms moves by one unit cell per Bloch oscillation period along x, which corre- sponds to the quantised bulk Hall drift [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This drift can be evaluated in the numerics by following the eigen- states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Since the red edge states are gapped from the next higher-lying trivial (black) states, the atoms ‘Bloch-oscillate’ from the ground to the excited band via the red-marked edge modes over several periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Once they are in the excited band they are transported backwards along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3B shows the situation for κ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' It behaves similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3A, except that there are more lo- calised states marked in red, compared to κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Like- wise, these states are transported along x as they undergo Bloch oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As before, this family of edge states is gapped from trivial states and connects right-moving to left-moving states, which leads to the reflection phe- nomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The experimentally relevant case is a harmonic trap with κ = 2 (see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [23, 29–31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3C shows that for κ = 2 the number of localised sates outside of the bands is even larger than for κ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As before, we adiabatically follow these localised states along kn A B C D FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Energy spectra for different confining poten- tials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' States localised to the left edge are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Energy spectra of H2D in the limit ω → 0 for κ = 24 (A), κ = 10 (B), and κ = 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Topological edge modes which connect the two bands with Chern number 1 and −1 in (A) and (B) are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The upper inset in (C) marks the topological boundary where the reflection is observed as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The inset in the center of (C) shows the tiny avoided crossings which can lead to a slight period dependence of the observed reflection point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' (D) Energy spectrum for a linearly increasing staggered po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The gapless, topological edge mode is marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 10 and evaluate their centre-of-mass along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' By numer- ically observing these drifts we confirm that the states describe quantised drifting in a large region, which man- ifests their nontrivial topological nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Thus, the κ = 2 case is ideal to observe the reflection after long-distance quantised Hall drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' However, the gaps between topo- logical (right-moving and left-moving) and trivial (sta- tionary) states become smaller, compared to the κ = 24 and κ = 10 cases, as shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3C (κ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As a result, the reflection point for κ = 2 is spread out over several unit cells but the reflection itself remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Faster pumping leads to non-adiabatic crossings of the energy gaps between right-moving and left-moving states, causing the reflection to happen later in time and further up in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We confirm this depen- dence experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S1B, which shows a later reversal for smaller pump periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S3D shows the spectrum for the linearly increasing staggered potential, described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This potential allows a straightforward identification of the gapless edge mode (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The states correspond- ing to this gapless edge mode are localised around the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' x E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Linearly increasing staggered potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' To elucidate the topology in our system, a linearly increasing staggered potential is considered (blue): V (j) = jV0(−1)j with V0 = 1/2 × m(2πνa)2, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' It is chosen such that the local tilt always equals the tilt from the harmonic poten- tial (orange), but alternates in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The staggered potential allows a simple pictorial representation of the emergence of the topological boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' In a local density approximation the trap linearly shifts the pump trajectory upwards in the ∆-δ plane, as depicted in the upper part of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' As soon as the trajectory ceases to enclose the critical point, a topological–trivial boundary develops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Staggered potential Another possibility to identify the topological bound- ary in our system makes use of a staggered potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' First, we consider a potential with uniform staggering, given by Vc(j) = V (−1)j, where j indexes the lattice-site and 2V corresponds to the energy difference between ad- jacent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Adding such a potential to the Rice-Mele Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 1) changes its trajectory in the ∆-δ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The onsite energy in such a system is given by (∆(τ) + V )(−1)j, which ranges from −∆ + V to ∆ + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The tunnellings are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Therefore, the trajectory remains circular and it is simply shifted upwards by an amount V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' A topological boundary emerges for a linearly increas- ing staggered potential, given by V (j) = jV0(−1)j, with V0 = 1 2m(2πνa)2 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' V (j) is chosen such that it has the same local tilt as the harmonic trap in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Within the local density approximation we assign a ∆-δ trajectory locally to each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The trajectories are thus linearly shifted upwards as func- tion of j (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S4), describing a change of topology in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' We expect the local density approximation to be valid since the atomic eigenstates in the exper- iment are strongly localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Similar models with lin- early increasing staggered potential have been studied in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' [29, 32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Local Chern marker The mathematical formulation of the Chern number as a topological invariant requires translational invariance, which does not apply to realistic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Instead, we use a local quantity, known as Chern marker [8, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local Chern marker depends on the real-space posi- tion and it is defined by: c(rγ) = −4π Ac Im � s=A,B ⟨rγs| ˆP ˆx ˆQˆy ˆP |rγs⟩ , (S10) where rγ is the position of the unit cell γ with sub-lattice- sites at positions rγA and rγB, |rγs⟩ = c† γs |0⟩ is the state localised on the corresponding lattice site , Ac is the area of a real-space unit cell, ˆQ = 1− ˆP and ˆP is the projector onto the ground band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Defining ˆP is not unambiguously possible in our system (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' S7) because of the energy shift from the harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Instead, we use a linearly increasing staggered potential, as described in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' This model leaves the bands in- tact and a ground band can be unambiguously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Experimentally, a local probe of the band topology is the velocity of the Hall drift, plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' Theory and experiment agree approximately with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The local velocity is extracted from the atomic positions by fitting linear functions to groups of three adjacent dat- apoints in ten pump cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'} +page_content=' The resulting velocities are plotted against position and smoothed through a running average of width three (ten cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQf_wYq/content/2301.03583v1.pdf'}