## Lucjan Jacak, Piotr Sitko, Konrad Wieczorek, and Arkadiusz Wójs

Print publication date: 2003

Print ISBN-13: 9780198528708

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198528708.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 17 January 2017

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

Source:
Quantum Hall systems
Publisher:
Oxford University Press

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

(B.1)

The one-particle Green function can be expressed as:

(B.2)

and making use of the Lehman representation:

(B.3)

(ℏw F—Fermi energy). Let us introduce the projector on the nth Landau level, Πn, then

(B.4)

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

(B.5)

appears, in which summation goes over filled Landau levels and for equal space variables $i G ( 8 , 8 + ) = ρ ¯ ( r 8 )$. Substituting the irreducible self-energy in the Hartree approximation (5.84) into the Schrödinger equation (B.l), the following expression is obtained

(B.6)

(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:

(B.7)

where

(B.8)

(B.9)

The two-particle correlation function depends only on the difference of coordinates t 1t 2, thus

(B.10)

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

(B.11)

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:

(B.12)

The magnetoexciton basis serves well for detailed calculations. For two particles let us define coordinates of the centre of mass $R= 1 2 ( r 1 + r 2 )$, and the relative (p.135) coordinates r = r 1r 2. Then the Hamiltonian of a hole and a particle assumes the form:

(B.13)

where P = p 1 + p 2, $p= 1 2 ( p 1 + p 2 )$. One can note that for sucn a system of two particles, the following ‘momentum’ operator

(B.14)

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 Zj = xj + iyj (j = 1, 2) and a ‘momentum’ za = iqx – qy, the two-particle magnetoexciton wave function can be expressed as (a 0 = 1) (Fetter and Hanna 1992a):

(B.15)

where

(B.16)

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

(B.17)

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

(B.18)

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

(B.19)

and

(B.20)

where $( x = 1 2 q 2 = 1 2 | z α | 2 )$

(B.21)

(B.22)

Let us define the following expression:

(B.23)

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

(B.24)

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

(B.25)

Assuming identical space coordinates, $r ′ 1 = r 1 , r ′ 2 = r 2$ the density-density correlation function for an anyon gas is obtained. One can determine

(B.26)

where

(B.27)

In the same way, density-current and current-current correlation functions are determined and exactly the same matrix $D R RPA$, 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).