Abstract:
We study one-component fermions in chain lattices with proximity-induced superconducting gap and interparticle short-range interaction, capable of hosting Majorana fermions. By systematically tracking various physical quantities, we show that topological states and topological phase transitions in the system can be identified by multiple signatures in thermodynamic quantities and pair-condensate properties, in good agreement with the known signatures in the ground-state energy and entanglement spectrum. We find the disappearance of the topological phase in a largely attractive regime, in which the system undergoes a first-order transition between two topologically trivial states. In addition, the stability of the signatures against finite size, disorder, and inhomogeneity is analyzed. Our results provide additional degrees of freedom for the characterization of topological states with interaction and for the experimental detection of emergent Majorana fermions.

Abstract:
Ternary flavor mixtures of ultracold fermionic atoms in an optical lattice are studied in the case of equal, repulsive on-site interactions U>0. The corresponding SU(3) invariant Hubbard model is solved numerically exactly within dynamical mean-field theory using multigrid Hirsch-Fye quantum Monte Carlo simulations. We establish Mott transitions close to integer filling at low temperatures and show that the associated signatures in the compressibility and pair occupancy persist to high temperatures, i.e., should be accessible to experiments. In addition, we present spectral functions and discuss the properties of a ``semi-compressible'' state observed for large U near half filling.

Abstract:
Recently, concepts of topological phases of matter are extended to non-equilibrium systems, especially periodically driven systems. In this paper, we construct an example which shows non-equilibrium topological phase transitions using ultracold fermions in optical lattices. We show that the Rabi oscillation has the possibility to induce non-equilibrium topological phases which are classified into time-reversal-invariant topological insulators for a two-orbital model of alkaline-earth-metal atoms. Furthermore we study the non-equilibrium topological phases using time-dependent Schrieffer-Wolff-type perturbation theory, and we obtain an analytical expression to describe the topological phase transitions from a high-frequency limit of external driving fields.

Abstract:
Three-dimensional Weyl fermions are found to emerge from simple cubic lattices with staggered fluxes. The mechanism is to gap the quadratic band touching by time-reversal-symmetry-breaking hoppings. The system exhibits rich phase diagrams where the number of Weyl fermions and their topological charge are tunable via the plaquette fluxes. The Weyl semimetal state is shown to be the intermediate phase between non-topological semimetal and quantum anomalous Hall insulator. The transitions between those phases can be understood through the evolution of the Weyl points as Berry flux insertion processes. As the Weyl points move and split (or merge) through tuning the plaquette fluxes, the Fermi arcs and surface states undergo significant manipulation. We also propose a possible scheme to realize the model in ultracold fermions in optical lattices with artificial gauge fields.

Abstract:
Ultracold Fermi gases trapped in honeycomb optical lattices provide an intriguing scenario, where relativistic quantum electrodynamics can be tested. Here, we generalize this system to non-Abelian quantum electrodynamics, where massless Dirac fermions interact with effective non-Abelian gauge fields. We show how in this setup a variety of topological phase transitions occur, which arise due to massless fermion pair production events, as well as pair annihilation events of two kinds: spontaneous and strongly-interacting induced. Moreover, such phase transitions can be controlled and characterized in optical lattice experiments.

Abstract:
Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter-Hubbard model. We include an additional staggered lattice potential and an artificial Rashba--type spin-orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interaction-induced transition from normal to topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba--type spin-orbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.

Abstract:
We discuss the thermodynamic signatures for the topological phase transitions into Majorana and Weyl superfluid phases in ultracold Fermi gases in two and three dimensions in the presence of Rashba spin-orbit coupling and a Zeeman field. We analyze the thermodynamic properties exhibiting the distinct nature of the topological phase transitions linked with the Majorana fermions (2D Fermi gas) and Weyl fermions (3D Fermi gas) which can be observed experimentally, including pressure, chemical potential, isothermal compressibility, entropy, and specific heat, as a function of the interaction and the Zeeman field at both zero and finite temperatures. We conclude that among the various thermodynamic quantities, the isothermal compressibility and the chemical potential as a function of the artificial Zeeman field have the strongest signatures of the topological transitions in both two and three dimensions.

Abstract:
Motivated by recent observations of superfluidity of ultracold fermions in optical lattices, we investigate the stability of superfluid flow of paired fermions in the lowest band of a strong optical lattice. For fillings close to one fermion per site, we show that superflow breaks down via a dynamical instability leading to a transient density wave. At lower fillings, there is a distinct dynamical instability, where the superfluid stiffness becomes negative; this evolves, with increasing pairing interaction, from the fermion pair breaking instability to the well-known dynamical instability of lattice bosons. Our most interesting finding is the existence of a transition, over a range of fillings close to one fermion per site, from the fermion depairing instability to the density wave instability as the strength of the pairing interaction is increased.

Abstract:
A gas of electrons in a one-dimensional periodic potential can be transported even in the absence of a voltage bias if the potential is modulated slowly and periodically in time. Remarkably, the transferred charge per cycle is only sensitive to the topology of the path in parameter space. Although this so-called Thouless charge pump has first been proposed more than thirty years ago, it has not yet been realized. Here we report the first demonstration of topological Thouless pumping using ultracold atoms in a dynamically controlled optical superlattice. We observe a shift of the atomic cloud as a result of pumping and extract the topological invariance of the pumping process from this shift. We demonstrate the topological nature of the Thouless pump by varying the topology of the pumping path and verify that the topological pump indeed works in the quantum region by varying speed and temperature.

Abstract:
We study the effects of anisotropic hopping amplitudes on quantum phases of ultracold fermions in optical lattices described by the repulsive Fermi-Hubbard model. In particular, using dynamical mean-field theory (DMFT) we investigate the dimensional crossover between the isotropic square and the isotropic cubic lattice. We analyze the phase transition from the antiferromagnetic to the paramagnetic state and observe a significant change in the critical temperature: Depending on the interaction strength, the anisotropy can lead to both a suppression or increase. We also investigate the localization properties of the system, such as the compressibility and double occupancy. Using the local density approximation in combination with DMFT we conclude that density profiles can be used to detect the mentioned anisotropy-driven transitions.