We quantum-simulated the 2D Harper-Hofstadter (HH) lattice model in a highly elongated tube geometry -- three sites in circumference -- using an atomic Bose-Einstein condensate. In addition to the usual transverse (out-of-plane) magnetic flux, piercing the surface of the tube, we threaded a longitudinal flux $Φ_{\rm L}$ down the axis of the tube This geometry evokes an Aharonov-Bohm interferometer, where noise in $Φ_{\rm L}$ would readily decohere the interference present in trajectories encircling the tube. We observe this behavior only when transverse flux is a rational fraction of the flux-quantum, and remarkably find that for irrational fractions the decoherence is absent. Furthermore, at rational values of transverse flux, we show that the time evolution averaged over the noisy longitudinal flux matches the time evolution at nearby irrational fluxes. Thus, the appealing intuitive picture of an Aharonov-Bohm interferometer is insufficient. Instead, we quantitatively explain our observations by transforming the HH model into a collection of momentum-space Aubry-André models.
We show how the weakly trapped Harper-Hofstadter model can be mapped onto a Harper-Hofstadter model in momentum space. In this momentum-space model, the band dispersion plays the role of the periodic potential, the Berry curvature plays the role of an effective magnetic field, the real-space harmonic trap provides the momentum-space kinetic energy responsible for the hopping, and the trap position sets the boundary conditions around the magnetic Brillouin zone. Spatially local interactions translate into nonlocal interactions in momentum space: within a mean-field approximation, we show that i…
We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.
The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number, $C$. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with $|C|>1$. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor, $ν$, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors $ν= r/(r|C| +1)$ for bosons, or $ν= r/(2r|C| +1)$ for fe…
The interplay of magnetic fields and interacting particles can lead to exotic phases of matter exhibiting topological order and high degrees of spatial entanglement. While these phases were discovered in a solid-state setting, recent techniques have enabled the realization of gauge fields in systems of ultracold neutral atoms, offering a new experimental paradigm for studying these novel states of matter. This complementary platform holds promise for exploring exotic physics in fractional quantum Hall systems due to the microscopic manipulation and precision possible in cold atom systems. Howe…
We study the stroboscopic and non-stroboscopic dynamics in the Floquet realization of the Harper-Hofstadter Hamiltonian. We show that the former produces the evolution expected in the high-frequency limit only for observables which commute with the operator to which the driving protocol couples. On the contrary, non-stroboscopic dynamics is capable of capturing the evolution governed by the Floquet Hamiltonian of any observable associated with the effective high-frequency model. We provide exact numerical simulations for the dynamics of the density operator following a quantum cyclotron orbit …
We study the ground-state phase diagram of the strongly interacting Harper-Hofstadter-Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in $y$-direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogues of fractional quantum Hall phases at fillings $ν=1/2$ and $3/2$, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings - where in the full two-dimensional system the ground-state is e…
We study the recent Floquet-realisation of the Harper-Hofstadter model in a gas of cold bosonic atoms. We study in detail the scattering processes in this system in the weakly interacting regime due to the interplay of particle interactions and the explicit time dependence of the Floquet states that lead to band transitions and heating. We focus on the experimentally used parameters and explicitly model the transverse confining direction. Based on transition rates computed within the Floquet-Fermi golden rule we obtain band population dynamics which are in agreement with the dynamics observed …
We study the effects of disorder on the topological Chern insulating phase in the Harper--Hofstadter--Hatsugai (HHH) model. The model with half flux has a bulk band gap and thus exhibits a nontrivial topological phase. We consider two typical types of disorder: on-site random disorder and the Aubry--Andre type quasi-periodic potential. Using the coupling matrix method, we clarify the global topological phase diagram in terms of the Chern number. The disorder modifies the gap closing behavior of the system. This modification induces the Chern insulating phase even in the trivial phase parameter…
We derive the reciprocal cluster mean-field method to study the strongly-interacting bosonic Harper-Hofstadter-Mott model. The system exhibits a rich phase diagram featuring band insulating, striped superfluid, and supersolid phases. Furthermore, for finite hopping anisotropy we observe gapless uncondensed liquid phases at integer fillings, which are analyzed by exact diagonalization. The liquid phases at fillings 1 and 3 exhibit the same band fillings as the fermionic integer quantum Hall effect, while the phase at filling 2 is CT-symmetric with zero charge response. We discuss how these phas…
We study two-component bosons on the Harper-Hofstadter model with two legs. The synthetic magnetic fields for the two types of bosons point to either the same direction or opposite directions. The bosons have hardcore intra-species interaction such that there can be no more than one boson of the same type on each lattice site. For certain filling factors in the absence of inter-species interaction, each component realizes a vortex Mott insulator with rung current or a Meissner superfluid without rung current. The system undergoes phase transitions to other phases as inter-species interaction i…
Quantized Transport of Gap Solitons in the Harper-Hofstadter Model
School of Physics and Astronomy, University of Birmingham（伯明翰大学物理与天文学院）；Blackett Laboratory, Imperial College London（帝国理工学院布莱克特实验室）；Extreme Light Infrastructure – Nuclear Physics (ELI-NP) and Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH)（极轻基础设施-核物理（ELI-NP）和霍里亚·胡鲁贝伊核物理与核工程研发国家研究所（IFIN-HH））；Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica, Università di Trento（皮塔耶夫斯基玻色-爱因斯坦凝聚中心、意大利国家研究委员会光学研究所及特伦托大学物理系）
We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force $F_x(t)$ is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching-on of complex time-dependent hopping amplitudes $\mathrm{e}^{-\frac{i}{\hbar}\mathcal{A}_x(t)}$ in the $\hat{\mathbf{x}}$-direction such that $\partial_t \mathcal{A}_x(t)=F_x(t)$. The switching-on can be sudden, $F_x(t)=θ(t) F$, where $F$ is the steady driving force, or more generally smooth $F_x(t)=f(t…
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number $|C|>1$. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit $n_ϕ\to 1/|C|$. Near these points, the Hofstadter model realises large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective co…
Using an efficient cluster approach, we study the physics of two-dimensional lattice bosons in a strong magnetic field in the regime where the tunneling is much weaker than the on-site interaction strength. We study both dilute, hard core bosons at filling factors much smaller than unity occupation per site, and the physics in the vicinity of the superfluid-Mott lobes as the density is tuned away from unity. For hardcore bosons, we carry out extensive numerics for a fixed flux per plaquette $ϕ=1/5$ and $ϕ= 1/3$. At large flux, the lowest energy state is a strongly correlated superfluid, analog…
We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $ν=1/2$ fractional quantum Hall effect on the lattice. We address the robustness of the ground state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice biparti…
We demonstrate that all of the salient features of the Harper-Hofstadter model can be implemented with ultracold atoms trapped in a bichromatic ring-shaped lattice. Using realistic sinusoidal lattice potentials rather than assume the idealized tight-binding picture, we determine the optimal conditions necessary to realize the critical point where the spectrum becomes fractal, and identify the nature and cause of the departures from the discrete model predictions. We also show that even with a commensurate ring with a few lattice sites, the Aubry-André localization transition can be realized. L…
Shaking optical lattices in a resonant manner offers an efficient and versatile method to devise artificial gauge fields and topological band structures for ultracold atomic gases. This was recently demonstrated through the experimental realization of the Harper-Hofstadter model, which combined optical superlattices and resonant time-modulations. Adding inter-particle interactions to these engineered band systems is expected to lead to strongly-correlated states with topological features, such as fractional Chern insulators. However, the interplay between interactions and external time-periodi…
The experimental realization of the Harper-Hofstadter model in ultra-cold atomic gases has placed fractional states of matter in these systems within reach---a fractional Chern insulator state (FCI) is expected to emerge for sufficiently strong interactions when half-filling the lowest band. The experimental setups naturally allow to probe the dynamics of this topological state, yet little is known about its out-of-equilibrium properties. We explore, using density matrix renormalization group (DMRG) simulations, the response of the FCI state to spatially localized perturbations. After confirmi…
Perfect elliptic dichroism: Probing the metric of anisotropic quantum Hall droplets
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等理工学院）；Advanced Institute for Materials Research (WPI-AIMR), Tohoku University（东北大学先进材料研究所（WPI-AIMR））；Université Paris-Saclay, CNRS, LPTMS（巴黎萨克雷大学，法国国家科学研究中心，理论与实验物理实验室）；Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-Université PSL, Sorbonne Université（卡斯勒-布罗塞尔实验室，法兰西公学院，法国国家科学研究中心，巴黎文理研究大学，索邦大学）；International Solvay Institutes（国际索尔维研究所）；Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles（布鲁塞尔自由大学非线性现象与复杂系统中心）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））
Fast Hamiltonian engineering from cut polytope geometry
Institute for Quantum Inspired and Quantum Optimization, Hamburg University of Technology（量子启发与量子优化研究所，汉堡工业大学）；Center for Theoretical Physics -- a Leinweber Institute, Massachusetts Institute of Technology（理论物理中心——莱因韦伯研究所，麻省理工学院）
