We propose to utilize the sub-system fidelity (SSF), defined by comparing a pair of reduced density matrices derived from the degenerate ground states, to identify and/or characterize symmetry protected topological (SPT) states in one-dimensional int
eracting many-body systems. The SSF tells whether two states are locally indistinguishable (LI) by measurements within a given sub-system. Starting from two polar states (states that could be distinguished on either edge), the other combinations of these states can be mapped onto a Bloch sphere. We prove that a pair of orthogonal states on the equator of the Bloch sphere are LI, independently of whether they are SPT states or cat states (symmetry-preserving states by linear combinations of states that break discrete symmetries). Armed with this theorem, we provide a scheme to construct zero-energy exitations that swap the LI states. We show that the zero mode can be located anywhere for cat states, but is localized near the edge for SPT states. We also show that the SPT states are LI in a finite fraction of the bulk (excluding the two edges), whereas the symmetry-breaking states are distinguishable. This can be used to pinpoint the transition from SPT states to the symmetry-breaking states.
We investigate the pairing symmetry in heavily overdoped Ba$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ based on the spin-fluctuation mechanism. The exotic octet nodes of the superconducting gap and the unusual evolution of the gap with doping observed by the rec
ent experiments are well explained in a unified manner. We demonstrate that the scatterings of electrons on the Fermi patches is mainly responsible for the incommensurate spin fluctuations and consequently the Fermi-surface-dependent multi-gap structure, since the Fermi level is close to the flat band. In addition, we find that a $d$-wave pairing state will prevail over the s-wave pairing state around the Lifshitz transition point.
Based on an effective two-band model and using the fluctuation-exchange (FLEX) approach, we explore spin fluctuations and unconventional superconducting pairing in Fe-based layer superconductors. It is elaborated that one type of interband antiferrom
agnetic (AF) spin fluctuation stems from the interband Coulomb repulsion, while the other type of intraband AF spin fluctuation originates from the intraband Coulomb repulsion. Due to the Fermi-surface topology, a spin-singlet extended s-wave superconducting state is more favorable than the nodal $d_{XY}$-wave state if the interband AF spin fluctuation is more significant than the intraband one, otherwise vice versa. It is also revealed that the effective interband coupling plays an important role in the intraband pairings, which is a distinct feature of the present two-band system.
Based on a single band Hubbard model and the fluctuation exchange approximation, the effective mass and the energy band renormalization in Na$_{0.33}$CoO$_2$ is elaborated. The renormalization is observed to exhibit certain kind of anisotropy, which
agrees qualitatively with the angle-resolved photoemission spectroscopy (ARPES) measurements. Moreover, the spectral function and density of states (DOS) in the normal state are calculated, with a weak pseudogap behavior being seen, which is explained as a result of the strong Coulomb correlations. Our results suggest that the large Fermi surface (FS) associated with the $a_{1g}$ band plays likely a central role in the charge dynamics.
