Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the realization of one, two, and three dimensional Dirac equations as well as the detection of some relativistic effects, including particularly the well-known Zitterbewegung effect and Klein tunneling. The realization of quantum anomalous Hall effects is also briefly discussed.

We introduce a generalized joint density of states (GJDOS), which incorporates the coherent factor into the JDOS, to study quasiparticle interference (QPI) in superconductors. The intimate relation between the Fourier-transformed local density of states and GJDOS is revealed: they correspond respectively to the real and imaginary parts of a generalized impurity-response function, and particularly share the same angular factors and singular boundaries, as seen from our approximate analytic results for d-wave superconductors. Remarkably, our numerical GJDOS analysis agrees well with the QPI patten of d-wave cuprates and s±-wave iron-based superconductors. Moreover, we illustrate that the present GJDOS scenario can uncover the sign features of the superconducting gap and thus can be used to explore the possible pairing symmetry of the KxFe2−ySe2 superconductors. PACS numbers:In superconductors, scattering of Bogoliubov quasiparticles by impurities can form quasiparticle interference (QPI) patterns which can be examined by the Fourier transformed scanning tunneling spectroscopy (FT-STS) 1,2 . For high-T c cuprates the characteristic wave vectors as well as their dispersions measured by the FT-STS experiments are interpreted quite well by the octet model 2,3 , which is based on the joint density of states (JDOS) argument. However, JDOS alone is incapable of explaining the variation of the QPI patterns when the magnetic field is applied. Indeed it was observed that QPI peaks corresponding to certain octet wave vectors are suppressed or enhanced by the magnetic field depending on whether the sign of the SC gaps is reversed or preserved 4 . The reason lies in that the probability of the elastic scattering of Bogoliubov quasiparticles on the constant contour of energy (CCE) is governed not only by their JDOS but also by the matrix element of the impurity Hamiltonian. Considering that Bogoliubov quasiparticles in superconductors carry proper coherent factors reflecting their particle-hole dualism, this scattering matrix element will bear a particular combination of the coherent factors relying on the type of the scatterers 5 . Taking into account an important role of coherent factors, the sign-sensitive effect of the QPI pattern on the magnetic field is naturally explained in the sense that the Andreev scattering of quasiparticles is enhanced by the pair-breaking impurities (mainly vortices) induced by the field. Notably, this sign-sensitive effect has also been applied to verify the s ± pairing symmetry for iron-based superconducting (SC) material Fe(Se,Te) 6 .The insufficiency of the JDOS argument motivates us to propose the generalized JDOS (GJDOS) in this paper, which intrinsically incorporates the coherence factors. We find that the GJDOS and the Fourier-transformed local density of states (FT-LDOS) can be derived from the same quantity. The close connection between GJ-DOS and the Fourier-transformed local density of states (FT-LDOS) are established, which confirmed by our analytic results of d-wave superconductors u...

We find that a model Hamiltonian of s-wave superconductors in the presence of spin-orbit interactions and a Zeeman field is exactly solvable. Most intriguingly, based on the exact solutions, an unconventional type of Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) ground state is rigorously revealed, in which the center-of-mass momentum of the fermion pair is proportional to the Zeeman field. We also elaborate on the drifting effect of the Zeeman field on the spin-orbit-coupled Bose-Einstein condensate.

We propose to realize Majorana fermions (MFs) on an edge of a two-dimensional topological insulator in the proximity with s-wave superconductors and in the presence of transverse exchange field h. It is shown that there appear a pair of MFs localized at two junctions and that a reverse in direction of h can lead to permutation of two MFs. With decreasing h, the MF states can either be fused or form one Dirac fermion on the π-junctions, exhibiting a topological phase transition. This characteristic can be used to detect physical states of MFs when they are transformed into Dirac fermions localized on the π-junction. A condition of decoupling two MFs is also given.