Fermionic edge states and new physics

Abstract

We investigate the properties of the Dirac operator on manifolds with boundaries in presence of the Atiyah-Patodi-Singer boundary condition. An exact counting of the number of edge states for boundaries with isometry of a sphere is given. We show that the problem with the above boundary condition can be mapped to one where the manifold is extended beyond the boundary and the boundary condition is replaced by a delta function potential of suitable strength. We also briefly highlight how the problem of the self-adjointness of the operators in the presence of moving boundaries can be simplified by suitable transformations which render the boundary fixed and modify the Hamiltonian and the boundary condition to reflect the effect of moving boundary.Comment: 24 pages, 3 figures. Title changed, additional material in the Introduction section, the Application section and in the Discussion section highlighting some recent work on singular potentials, several references added. Conclusions remain unchanged. Version matches the version to appear in PR

Hyperspherical δ-δ′ potentials

“…The theory of selfadjoint extensions for symmetric operators has been well known to mathematicians for many years. However, it only became a valuable tool for modern quantum physics after the seminal works of Asorey et al [15,17,18], in which the problem was re-formulated in terms of physically meaningful quantities for relevant operators in quantum mechanics and quantum field theory [26][27][28][29][30][31].…”

Dirac Green function for δ potentials

“…Moreover, point interactions of the form δ(x), or even δ (x), have been used recently to analyze perturbations of a free kinetic Schrödinger Hamiltonian [16], the harmonic oscillator [17][18][19], a constant electric field [17], the infinite square well [20,21], the conical oscillator [22,23], and even the semi-oscillator, which has been used as a simple toy model potential showing resonance phenomena [24]. Finally, we mention that in black-hole theory and the 't Hooft brick wall model [25] the addition of δ(x) and δ (x) interactions to the Hamiltonian is relevant at least in three different ways: first, it can help in introducing time-dependent boundaries [26], second, the δ (x) term is needed when fermions are considered in order to build self-adjoint extensions of a Dirac operator with a δ(x) potential [27], and third, it can also serve to model membrane mechanisms for certain precise black-hole horizons [26,28].…”

Embedding into flat spacetime and black hole thermodynamics

“…In particular, It is well known that in manifolds with boundary, the Laplacian operator has self adjoint extensions and the edge states localized at the boundary arises naturally. They may serve as models for accounting the microstates associated with a given black hole geometry [32][33][34]. We will explore this avenue further in this work.…”