The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by 2×2 Dirac Hamiltonians are discussed. In one dimension, the correlation function measures the charge-polarization correlation between Wannier states at different positions, while in two dimensions it measures the itinerant-circulation correlation between Wannier states. The correlation function is nonzero in both the topologically trivial and nontrivial states, and allows to extract a correlation length that diverges at topological phase transitions. The correlation length and the curvature function that defines the topological invariants are shown to have universal critical exponents, allowing the notion of universality classes to be introduced. Particularly in two dimensions, the universality class is determined by the orbital symmetry of the Dirac model. The scaling laws that constrain the critical exponents are revealed, and are predicted to be satisfied even in interacting systems, as demonstrated in an interacting topological Kondo insulator.

We construct a lattice model for a cubic Kondo insulator consisting of one spin-degenerate d and f orbital at each lattice site. The odd-parity hybridization between the two orbitals permits us to obtain various trivial and topological insulating phases, which we classify in the presence of cubic symmetry. In particular, depending on the choice of our model parameters, we find a strong topological insulator phase with a band inversion at the X point, modeling the situation potentially realized in SmB6, and a topological crystalline insulator phase with trivial Z2 indices but nonvanishing mirror Chern numbers. Using the Kotliar-Ruckenstein slave-boson scheme, we further demonstrate how increasing interactions among f electrons can lead to topological phase transitions. Remarkably, for fixed band parameters, the f -orbital occupation number at the topological transitions is essentially independent of the interaction strength, thus yielding a robust criterion to discriminate between different phases.

The recent discovery of topological Kondo insulators has triggered renewed interest in the well-known Kondo insulator samarium hexaboride, which is hypothesized to belong to this family. In this Letter, we study the spin texture of the topologically protected surface states in such a topological Kondo insulator. In particular, we derive close relationships between (i) the form of the hybridization matrix at certain high-symmetry points, (ii) the mirror Chern numbers of the system, and (iii) the observable spin texture of the topological surface states. In this way, a robust classification of topological Kondo insulators and their surface-state spin texture is achieved. We underpin our findings with numerical calculations of several simplified and realistic models for systems like samarium hexaboride.

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We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an "entanglement Fermi surface." Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and Z2 topological insulators.