Topological Invariants and Ground-State Wave functions of Topological Insulators on a Torus

Abstract

We define topological invariants in terms of the ground-state wave functions on a torus. This approach leads to precisely defined formulas for the Hall conductance in four dimensions and the topological magnetoelectric θ term in three dimensions, and their generalizations in higher dimensions. They are valid in the presence of arbitrary many-body interactions and disorder. These topological invariants systematically generalize the two-dimensional Niu-Thouless-Wu formula and will be useful in numerical calculations of disordered topological insulators and strongly correlated topological insulators, especially fractional topological insulators.

Holographic topological semimetals

“…In figure 20 different colors are used to distinguish different bands. Furthermore, since −G −1 (0, k) can be treated as a topological Hamiltonian [43,45], the spectral density plot should qualitatively agree with the plot for eigenvalues in the ω-k x plane. • From figure 20 we can see that between each two adjacent poles bands I and II always and only intersect once in the upper frequency plane.…”

Topological nodal line semimetals in holography

“…4, which is taken from [32]. In the framework of topological systems, we can treat −G −1 (0, k) as a topological Hamiltonian [3,33] which essentially determines the topological behavior of the system and the eigenvalues plot would agree qualitatively with the spectral density plot in the ω-k x plane. Different from the weakly coupled band structure in Fig.…”

Topological invariants for holographic semimetals

“…In [18,16] it was shown that the zero frequency Green function G(0, k) already contains all the topological information. One could define an effective topological Hamiltonian H t (k) = −G −1 (0, k) (4.6) and define eigenvectors using this effective topological Hamiltonian.…”

“…However, the topological invariants defined from Green functions usually require an integral in the imaginary frequency axis, which is extremely time consuming when we only have numerical results for the Green functions. In [16,17,18], a method called topological Hamiltonian was developed, which states that topological invariants of a strongly coupled system could be calculated from the eigenstates of an effective Hamiltonian in the same way as in the weakly coupled theory.…”