No abstract

No abstract

We investigate a one-dimensional two-component system in an optical lattice of attractive interactions under a spin-dependent external potential. Based on the density-matrix renormalization group methods, we obtain its phase diagram as a function of the external potential imbalance and the strength of the attractive interaction through the analysis on the density profiles and the momentum pair correlation functions. We find that there are three different phases in the system, a coexisted fully polarized and Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase, a normal polarized phase, and a Bardeen–Cooper–Schrieffer (BCS) phase. Different from the systems of spin-independent external potential, where the FFLO phase is normally favored by the attractive interactions, in the present situation, the FFLO phases are easily destroyed by the attractive interactions, leading to the normal polarized or the BCS phase.

The observables of continuous eigenvalues are defined in an infinite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and electrons in the vacuum. From the derivation of the Casimir force without an artificial regulator we determine the explicit expression of the spectrum density factor for the photon field to be an exponential function. The undetermined constant in the function is fixed by the experimental data for the Lamb shift. With that, we predict that there exists a correction to the Casimir force.