# (p.133) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

# (p.133) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

The irreducible self-energy in the Hartree approximation constitutes a local single-particle potential which defines eigenstates fulfilling the self-consistent Schrödinger equation:

The one-particle Green function can be expressed as:

and making use of the Lehman representation:

(ℏ*w* _{F}—Fermi energy). Let us introduce the projector on the nth Landau level, Π_{n}, then

Positive values of η_{n} correspond to filled up Landau levels, negative values of η_{n} to empty levels. In the Eqn (5.84), the value

appears, in which summation goes over filled Landau levels and for equal space variables $iG(8,{8}^{+})=\overline{\rho}({\text{r}}_{\text{8}})$. Substituting the irreducible self-energy in the Hartree approximation (5.84) into the Schrödinger equation (B.l), the following expression is obtained

(p.134)
The first term in the above expression denotes exactly mean field Hamiltonian *H* _{0} and the second term disappears. Consequently, eigenstates of the system in the Hartree approximation are exactly Landau states and the self-consist ent Green function in the Hartree approximation becomes the Green function of the unperturbed state, *G* ^{0}. Let the vertex function in the Hartree approximation (5.92) be split up in the following way:

where

The two-particle correlation function depends only on the difference of coordinates *t* _{1} – *t* _{2}, thus

The same relationship refers to unperturbed function *F* _{0} which simplifies the Bethe-Salpeter equation (5.90). As the self-consistent one-particle Green function in the Hartree approximation becomes the Green function of unperturbed state, then

The above function constitutes the time-ordered correlation function of the unperturbed system. However, to study the linear response of the system, an information about the retarded correlation function is needed, given by the following expression:

The magnetoexciton basis serves well for detailed calculations. For two particles let us define coordinates of the centre of mass $\mathbf{\text{R=}}\frac{1}{2}\left({\mathbf{\text{r}}}_{\text{1}}+{\mathbf{\text{r}}}_{\text{2}}\right)$, and the relative
(p.135)
coordinates **r** = **r** _{1} – **r** _{2}. Then the Hamiltonian of a hole and a particle assumes the form:

where **P** = **p** _{1} + **p** _{2}, $\mathbf{\text{p=}}\frac{1}{2}\left({\mathbf{\text{p}}}_{\text{1}}+{\mathbf{\text{p}}}_{\text{2}}\right)$. One can note that for sucn a system of two particles, the following ‘momentum’ operator

commutes with the Hamiltonian. Consequently, an operator **Q**(**q**) eigenvalue is a good quantum number. In the magnetoexciton basis, states are numbered by two numbers *M = n—m*, *N* = min(*n, m*), while *n, m* number single-particle Landau states of both particles. Introducing complex coordinates of a position *Z _{j}* =

*x*(

_{j}+ iy_{j}*j*= 1, 2) and a ‘momentum’

*z*=

_{a}*iq*, the two-particle magnetoexciton wave function can be expressed as (

_{x}– q_{y}*a*

_{0}= 1) (Fetter and Hanna 1992

*a*):

where

The ‘momentum’ conservation for a particle-hole pair leads to the diagonality of the two-particle correlation function with respect to *z _{α}*. Moreover, the correlation function of the unperturbed system fulfils the following expression:

Consequently, the Bethe-Salpeter integral equation assumes the following form:

(p.136) After splitting the vertex function (B.7), it can be expressed as

and

where $\left(x=\frac{1}{2}{q}^{2}=\frac{1}{2}|{z}_{\alpha}{|}^{2}\right)$

Let us define the following expression:

which corresponds to analogous sums defined for anyon superconductor (6.1):

The following expression serves to find the correlation function in the real space:

Assuming identical space coordinates, ${{\mathbf{\text{r}}}^{\prime}}_{\text{1}}{=\mathbf{\text{r}}}_{\text{1},}{{\mathbf{\text{r}}}^{\prime}}_{\text{2}}{=\mathbf{\text{r}}}_{\text{2}}$ the density-density correlation function for an anyon gas is obtained. One can determine

where

In the same way, density-current and current-current correlation functions are determined and exactly the same matrix ${D}_{\text{R}}^{\text{RPA}}$, as in the random phase approximation is obtained. Calculating the difference between total and average currents in the Hartree approximation, the random phase approximation (RPA) is obtained (4.26).