F. Nur Unal

by M. Tarnowski, F.N. Ünal, N. Fläschner, B.S. Rem, A. Eckardt, K. Sengstock, C. Weitenberg.
Topology plays an important role in modern solid state physics describing intriguing quantum states such as topological insulators. It is an intrinsically non-local property and therefore challenging to access, often studied only via the resulting edge states. Here, we measure the topological index directly from the far-from-equilibrium dynamics of the bulk. We use the mapping of the Chern number to the linking number of dynamical vortex trajectories appearing after a quench to the Hamiltonian of interest. We thereby map out the topological phase diagram of quantum gases in optical lattices via a purely dynamical response. Such relations between two topological indices in static and dynamical properties could be also an important approach for exploring topology in the case of interactions.

Recent advances in realizing artificial gauge fields on optical lattices promise experimental detection of topologically non-trivial energy spectra. Self-similar fractal energy structures generally known as Hofstadter butterflies depend sensitively on the geometry of the underlying lattice, as well as the applied magnetic field. The recent demonstration of an adjustable lattice geometry [L. Tarruell \textit{et al.}, Nature 483, 302--305 (2012)] presents a unique opportunity to study this dependence. In this paper, we calculate the Hofstadter butterflies that can be obtained in such an adjustable lattice and find three qualitatively different regimes. We show that the existence of Dirac points at zero magnetic field does not imply the topological equivalence of spectra at finite field. As the real-space structure evolves from the checkerboard lattice to the honeycomb lattice, two square lattice Hofstadter butterflies merge to form a honeycomb lattice butterfly. This merging is topologically non-trivial, as it is accomplished by sequential closings of gaps. Ensuing Chern number transfer between the bands can be probed with the adjustable lattice experiments. We also calculate the Chern numbers of the gaps for qualitatively different spectra and discuss the evolution of topological properties with underlying lattice geometry.

The dynamics of a single impurity interacting with a many-particle background is one of the central problems of condensed-matter physics. Recent progress in ultracold-atom experiments makes it possible to control this dynamics by coupling an artificial gauge field specifically to the impurity. In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a single atom. We show that an external magnetic field coupled to the impurity is a versatile tool to probe the impurity dynamics. Using a Bethe ansatz, we calculate the eigenstates and corresponding energies exactly as a function of the flux through the trap. Adiabatic change of flux connects the ground state to excited states due to flux quantization. For repulsive interactions, the impurity disturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This drag transfers momentum from the impurity to the background and increases the effective mass. The effective mass saturates to the total mass of the system for infinitely repulsive interactions. For attractive interactions, the drag again increases the effective mass which quickly saturates to twice the mass of a single particle as a dimer of the impurity and one fermion is formed. For excited states with momentum comparable to number of particles, effective mass shows a resonant behavior. We argue that standard tools in cold-atom experiments can be used to test these predictions.