Orthogonal basis

Résumé

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. As coordinates Any orthogonal basis can be used to define a system of orthogonal coordinates V. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds. In functional analysis In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars. Extensions Symmetric bilinear form The concept of an orthogonal basis is applicable to a vector space V (over any field) equipped with a symmetric bilinear form \langle \cdot, \

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Pseudopotential formalism for fractional Chern insulators

Ronny Thomale

Recently, generalizations of fractional quantum Hall (FQH) states known as fractional quantum anomalous Hall or, equivalently, fractional Chern insulators states, have been realized in lattice models. Ideal wave functions such as the Laughlin wave function, as well as their corresponding trial Hamiltonians, have been vital to characterizing FQH phases. The Wannier function representation of fractional Chern insulators recently proposed [X.-L. Qi, Phys. Rev. Lett. 107, 126803 (2011)] defines an approach to generalize these concepts to fractional Chern insulators. In this paper, we apply the Wannier function representation to develop a systematic pseudopotential formalism for fractional Chern insulators. The family of pseudopotential Hamiltonians is defined as the set of projectors onto asymptotic relative angular momentum components which forms an orthogonal basis of two-body Hamiltonians with magnetic translation symmetry. This approach serves both as an expansion tool for interactions and as a definition of positive-semidefinite Hamiltonians for which the ideal fractional Chern insulator wave functions are exact null-space modes. We compare the short-range two-body pseudopotential expansion of various fractional Chern insulator models at filling mu = 1/3 in phase regimes where a Laughlin-type ground state is expected to be realized. We also discuss the effect of inhomogeneous Berry curvature which leads to components of the Hamiltonian that can not be expanded into pseudopotentials, and elaborate on their role in determining low-energy theories for fractional Chern insulators. Finally, we generalize our Chern pseudopotential approach to interactions involving more than two bodies with the goal of facilitating the identification of non-Abelian fractional Chern insulators.

Amer Physical Soc2013

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