We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This setup is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields -- which is not experimentally accessible for electrons in metals -- where a fractal band structure, the Hofstadter butterfly, characterizes the system.
The magnetic field generated Hofstadter butterfly in single-twist trilayer graphene (TLG) is investigated using circularly polarized light (CPL) and longitudinal light emanating from a waveguide. We show that single-twist TLG has two distinct chiral limits in the equilibrium state, and the central branch of the butterfly splits into two precisely degenerate components. The Hofstadter butterfly appears to be more discernible. We also discovered that CPL causes a large gap opening at the central branch of the Hofstadter butterfly energy spectrum and between the Landau levels, with a clear asymme…
We theoretically study the Hofstadter butterfly of a triangular network model in minimally twisted bilayer graphene (mTBLG). The band structure manifests periodicity in energy, mimicking that of Floquet systems. The butterfly diagrams provide fingerprints of the model parameters and reveal the hidden band topology. In a strong magnetic field, we establish that mTBLG realizes low-energy Floquet topological insulators (FTIs) carrying zero Chern number, while hosting chiral edge states in bulk gaps. We identify the FTIs by analyzing the nontrivial spectral flow in the Hofstadter butterfly, and by…
Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing wave cavity mode. Choosing a suitable pump geometry allows to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transit…
We develop a generic $\mathbf{k}\cdot \mathbf{p}$ open momentum space method for calculating the Hofstadter butterfly of both continuum (Moiré) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the Moiré model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hof…
Hierarchical sets such as the Pythagorean triplets ($\cal{PT}$) and the integral Apollonian gaskets ($\cal{IAG}$) are iconic mathematical sets made up of integers that resonate with a wide spectrum of inquisitive minds. Here we show that these abstract objects are related with a quantum fractal made up of integers, known as the {\it Hofstadter Butterfly}. The "butterfly fractal" describes a {\it physical system} of electrons in a crystal in a magnetic field, representing exotic states of matter known as {\it integer quantum Hall} states. Integers of the butterfly are the quanta of Hall conduct…
In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to non-trivial quantized responses. Here we study a particular invariant, the discrete shift $\mathscr{S}$, for the square lattice Hofstadter model of free fermions. $\mathscr{S}$ is associated with a $\mathbb{Z}_M$ classification in the presence of $M$-fold rotational symmetry and charge conservation. $\mathscr{S}$ gives quantized contributions to (i) the fractional charge bound to a lattice disclination, and (ii) the angular momentum of the ground state with an additional, symmetrical…
In a previous work [Phys. Rev. Lett. 123, 047202 (2019)] a translationally invariant framework called quantum-electrodynamical Bloch (QED-Bloch) theory was introduced for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized photon field in the optical limit. For such systems, we show that QED-Bloch theory predicts the existence of fractal polaritonic spectra as a function of the cavity coupling strength. In addition, for the energy spectrum as a function of the relative magnetic flux we find that a terahertz cavity can modify the standard H…
The Hofstadter butterfly is one of the first and most fascinating examples of the fractal and self-similar quantum nature of free electrons in a lattice pierced by a perpendicular magnetic field. However, the direct experimental verification of this effect on single-layer materials is still missing as very strong and inaccessible magnetic fields are necessary. For this reason, its indirect experimental verification has only been realized in artificial periodic 2D systems, like moiré lattices. The only recently synthesized 2D covalent-organic frameworks might circumvent this limitation: Due to …
High-field Josephson effect enabled by a moiré Hofstadter spectrum
Ludwig-Maximilians-Universität（路德维希-马克西米利安大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Nanomaterials and Nanotechnology Research Center (CINN-CSIC), Universidad de Oviedo (UO)（阿斯图里亚斯大学纳米材料与纳米技术研究中心（CINN-CSIC））；Yale University（耶鲁大学）；Aalto University（阿尔托大学）；National Institute for Materials Science（国立材料研究所）
A 2D Fock-state lattice (FSL is constructed from the many-body states of two interacting two-mode quantum gases. By periodically driving the interspecies interactions and pulsing the tunneling between the two modes of each gas, a synthetic gauge field is generated. We derive an effective Hamiltonian in the short pulse limit which resembles the Harper-Hofstadter Hamiltonian where the magnetic flux per plaquette is controlled by the ratio of the interaction energy and the driving frequency. The quasispectrum of the Floquet operator of the driving sequence shows the celebrated Hofstadter's butter…
