The Casimir interaction between topological insulators with opposite topological magnetoelectric polarizabilities and finite surface band gaps has been investigated. For a large surface band-gap limit (m$\rightarrow${}$\infty${}), we can obtain results given by Grushin and Cortijo [Phys. Rev. Lett. 106, 020403 (2011)]. For a small surface band-gap limit (m$\rightarrow${}0), the Casimir interaction between topological insulators is attractive and in analogy to the ideal mental model in a short separation limit. Generally, there is a critical value mc, and when the surface band gap is greater than the critical value, the Casimir force is repulsive in an intermediate separation region. We estimate the critical surface band gap mc$\sim${}1/(2a), in which a is a critical separation where the Casimir force vanishes.

We propose a new mechanism to generate the Casimir-Lifshitz torque between Weyl semimetals arising from the chiral anomaly. For short distances ranging from a nanometer to a few tens of nanometers, chiral anomaly is manifested via a Casimir-Lifshitz torque $\sim \sin (\theta )$ with $\theta$ being the twisting angle. As the distance between Weyl semimetals increases from a submicrometer to a few micrometers, chiral-anomaly-driven Casimir-Lifshitz torque between Weyl semimetals is remarkably large, which is comparable with that of conventional birefringent materials.

We generalize the calculation of the Casimir interaction between topological insulators with opposite topological magnetoelectric polarizabilities and finite surface band gaps to finite-temperature cases. We find that finite temperature quantitatively depresses the repulsive peak and enlarges the critical surface gap $m_c$ for the repulsive Casimir force. However, the universal property $m_{c}a\sim 1/2$ is still valid for various oscillator strengths, temperature regions, and topological magnetoelectric polarizabilities. DOI: http://dx.doi.org/10.1103/PhysRevB.85.115102 ©2012 American Physical Society

In this work, we study the magnetic impurity resonance states in the superconducting phase of ‘magic’ angle twisted bilayer graphene for different pairing symmetries. Using a two-orbital model on the emergent honeycomb lattice, we find that the resonance states are dramatically different for -wave pairing and topological nontrivial pairings. When the magnetic impurity is located at one site of the emergent honeycomb lattice, i.e. the center of the AB spot of the moiré pattern, the spacial distributions of the resonance states will break both the threefold and twofold rotation symmetries of point group for pairing symmetries which belong to the irreducible representations of this point group. When the magnetic impurity is located at the center of the emergent honeycomb lattice i.e. the center of the AA spot of the moiré pattern, the appearance of resonance peak at the position close to the impurity can be considered as a strong evidence of non- -wave pairing.

We apply mean-field theory and the Hirsch-Fye quantum Monte Carlo method to study the spin-spin interaction in the bulk of three-dimensional topological insulators. We find that the spin-spin interaction has three different components: longitudinal, transverse, and transverse Dzyaloshinskii-Moriya-like terms. When the Fermi energy is located in the bulk gap of a topological insulator, the spin-spin interaction decays exponentially due to the Bloembergen-Rowland interaction. The longitudinal correlation is antiferromagnetic and the transverse correlations are ferromagnetic. When the chemical potential is in the conduction or valence band, the spin-spin interaction follows a power-law decay, and the isotropic ferromagnetic interaction dominates in the short separation limit.