We propose an experimental scheme to simulate and observe relativistic Dirac fermions with cold atoms in a hexagonal optical lattice. By controlling the lattice anisotropy, one can realize both massive and massless Dirac fermions and observe the phase transition between them. Through explicit calculations, we show that both the Bragg spectroscopy and the atomic density profile in a trap can be used to demonstrate the Dirac fermions and the associated phase transition.PACS numbers: 05.30.Fk,03.65.Pm,31.30.Jv,73.43.Nq Control of ultracold atoms in an optical lattice opens up many avenues to explore some fundamental phenomena at the forefront of condensed matter physics [1,2,3,4,5]. By designing configurations of this atomic system, one can simulate effective theories that are very different from the microscopic atomic physics. In this paper, we add an unusual example to the avenues of quantum simulation by showing that ultracold atoms in an optical lattice can be used to investigate physics associated with relativistic Dirac fermions. The ultracold atomic gas, as the coldest setup in the universe, is one of the most non-relativistic systems. Nevertheless, we will see that effective theories for the quasiparticles in this system can become relativistic under certain conditions.We simulate Dirac fermions with single-component cold atoms in a two-dimensional hexagonal lattice. This lattice can be formed through interference of three laser beams, as we show below. The physics here is closely related to the properties of electrons in the graphene material formed with a single layer of carbon atoms [6,7,8,9,10,11,12]. The graphene, with its emergent relativistic massless quasiparticles, has recently raised strong interest in condensed-matter physics [6,7,9,10,11,12]. Compared with the graphene, the system with cold atoms in an optical lattice may offer more controllability. For instance, we show that one can realize both massive and massless Dirac fermions by controlling anisotropy of the optical lattice. This anisotropy can be conveniently tuned through variation of the trapping laser intensity. Under such a tuning, one can also observe a quantum phase transition in this system. This phase transition is not associated with any usual symmetry breaking, but instead it is characterized by a topological change of the fermi surfaces [13,14]. To detect the massive and the massless Dirac fermions and the phase transition between them, we calculate the Bragg spectrum for this system as well as its atomic density profile in a trap. From this calculation, we show that the conventional atomic detection techniques based on the Bragg spectroscopy [15] or the density profile measurement [1,16,17] can be used to demonstrate the Dirac fermions and the phase transition.For cold atoms, one realizes an effectively twodimensional system by raising the potential barrier of the optical lattice along the z direction to suppress the vertical tunneling between different planes. Then, in the x-y plane, one can form a hexagonal optical lattic...

Non-Hermiticity from non-reciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization in a non-reciprocal quasiperiodic lattice, dubbed nonreciprocal Aubry-André model, and a rescaled transition point is exactly proved. The non-reciprocity can induce not only the NHSE, but also the asymmetry in localized states with two Lyapunov exponents for both sides. Meanwhile, this transition is also topological, characterized by a winding number associated with the complex eigenenergies under periodic boundary conditions (PBCs), establishing a bulk-bulk correspondence. This interplay can be realized by an elaborately designed electronic circuit with only linear passive RLC devices instead of elusive non-reciprocal ones, where the transport of a continuous wave undergoes a transition between insulating and amplifying. This initiative scheme can be immediately applied in experiments to other non-reciprocal models, and will definitely inspires the study of interplay of NHSEs and more other quantum/topological phenomena.