A perturbation theory approach to the ground state exciton energy in the limit of a weak magnetic field in anomalous exciton Hall effect

The anomalous exciton Hall effect is a phenomenon that occurs in a quantum well in the presence of an external magnetic field applied perpendicular to the surface due to the interaction of the exciton dipole moment with an electric field, formed by the charged impurities. The effect was fully described in [1] for different magnetic field regimes. In this paper, we focus on the way the perturbation method was used for finding the ground state energy of an exciton in the limit of a weak magnetic field.


Introduction
Excitons, propagating in the presence of an external magnetic field orthogonal to their velocity, acquire an electric dipole moment perpendicular to both the magnetic field and their propagation direction [2]. The existence of a dipole moment causes the excitons to have a nonzero asymmetric scattering rate while interacting with charged impurities. The T -matrix formalism was exploited to describe the symmetric and asymmetric contributions to the scattering rates. The problem then boiled down to the calculation of T -matrix elements, which satisfy the Lippmann-Schwinger equation: By virtue of the smallness of the magnetic field, one may desire to solve the Lippmann-Schwinger equation perturbatively: Here (g) is the ground state energy of an exciton with a wave vector g in the presence of a magnetic field; ν = |∂ (k)/∂k| −1 k/(2π);V is the exciton-impurity scattering potential.
However, the perturbation theory approach turned out to be invalid in this case, since the calculated first-order correction was of the same order as the main contribution. Nevertheless, being inapplicable for solving the Lippmann-Schwinger equation, the method can be used for the calculation of the ground state energy (g) and the matrix elements of the scattering potential.
In the following sections, the steps are presented for finding an analytical expression of the ground state energy of an exciton in the limit of a weak magnetic field using the perturbation method. During the calculations, several infinite sums appear, which are computed using an elegant method from [3].

Formulation of the problem
(i) We start by writing the unperturbed Hamiltonian of the relative electron-hole motion: Its wave functions Φ m 1 ,m 2 (r) = Φ m (r) = |m consist of a coordinate (R m 1 (r)) and an angular (Θ m 2 (θ)) parts and can be found in [4]. The ground state wave function and energy read (a B = 4πε 0 ε 2 /(µe 2 ) is the Bohr radius): The magnetic field affects the system through the perturbation potentialV , which assumed to be small:V (iii) The goal of this paper is to find the ground state energy (k) using a perturbation theory approach.
3. Calculation of the ground state energy (i) Let us expand the ground energy up to the second order in the magnetic field strength B: (ii) One may notice, that: m 0 |xk y −ŷk x |m 0 = 0 and m|L z |m 0 = 0. Thus (we omit dependence on k in the energy for brevity; B Oz): One can see that the analytical solution requires a calculation of several infinite sums. The smart way to perform that is presented in next subsections.

First auxiliary task
First, we want to concentrate on the following problem: how a hermitian operatorb 1 , satisfying the relation iµ[Ĥ 0 ,b 1 ] =x, acts on Φ m 0 (r).
(i) Let us introduce b 1 (r) asb 1 Φ m 0 (r) = b 1 (r)Φ m 0 (r). Then: The solution for the equation above is: with C being an arbitrary constant.

Second auxiliary task
Now let us focus on the similar problem: how a hermitian operatorb 2 , satisfying the relation iµ[Ĥ 0 ,b 2 ] =r 2 , acts on Φ m 0 (r).
. Then, following exactly the same steps, we obtain the equation on b 2 (r): (ii) The solution reads: with C being an arbitrary constant.

Calculation of the infinite sums
Using the results of the previous section, we can find analytical expressions for several sums, which appear during energy calculation.