| In condensed matter physics, the origins of superconductivity and pseudogap in cuprates have been long-standing problems. Especially, there are some debates about abnormal phenomena in pseudogap region. It is believed that competing phases are responsible for the pseudogap normal state. Several schemes, ranging from spin and charge density waves (SDW and CDW) at finite momentum to the more exotic loop-current order (LCO) at zero momentum, are proposed to explain the phenomena ob-served in experiments. Especially, the LCO suggested by Varma based on mean field theory is an ambitious proposal which breaks time-reversal symmetry and rotation symmetry but preserves translational symmetry.Because LCO is beyond the scope of the effective single orbital Zhang-Rice model, a three orbital model in terms of the copper dx2-y2 orbital and the oxygen px and py orbitals is necessary. Aiming at treating all possible orderings in a unified framework and capturing the essential physics of pseudogap, here we perform functional renor-malization group calculations for the effect of electron correlations.We find that in the parameter regime given by experiments, SDW and d-wave superconductivity are the dominant instabilities which have contributions mainly from copper dx2-y2-orbital. It suggests that the Zhang-Rice model is sufficient to describe the system in low energy scale. If the Coulomb interaction between nearest copper and oxygen is assumed to be very strong, an intra-unitcell CDW describing charge imbalance between copper and oxygen sites emerges. It seems that a large Coulomb interaction between nearest copper and oxygen tends to recover the charge transfer gap between cooper and oxygen orbital. However, in all energy scale the LCO is too weak, and should be irrelevant in cuprates.Searching for Majorana fermion (MF) is another topic attracting many attention in condensed matter physics. Especially, the idea of topological materials renews the research for Majorana fermion. In a recent experiment, signatures of MF were found in the vortex core threading a heterostructure composed of n layers of topological in-sulator (TI) deposited on a bulk s-wave superconductor (SC). Here we provide strong theoretical support to the experiment.Firstly, we demonstrate that MF modes are bounded in the vortex core on both top and bottom layers of TI, and are well separated for n≥6. However, as n increases, the top MF becomes more extended and is difficult to observe, so we believe that there is an optimal number nopt of TI layers. While the number of TI layers n=nopt, the signatures of MF is clear. This result agrees with the experiment.Secondly, we show both analytically and numerically that the MF mode centered in vortex core is always accompanied by another finite energy bound state. It leads to a zero-bias peak and a side peak in the local density of states (LDOS). As stepping away from the vortex core, both of them disappear gradually and the superconducting gap recovers.However, a local scalar impurity at the core can get rid of the accompanying side-peak state but leaving the zero-energy MF mode intact. Consequently, the LDOS becomes symmetric about the fermi level, in perfect agreement with the experiment.Finally we try to study the robustness of MF at the vortex core. We find that the MF is extremely stable against a single local magnetic/scalar impurity. In contrast, the stability in terms of the critical impurity strength is reduced drastically for a mod-erate concentration of impurities(e.g.10%), so a high-quality sample is necessary for experiments or application in the future. |
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