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Quantum Hall systemsBraid groups, composite fermions and fractional charge$

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

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(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

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

The one-particle Green function can be expressed as:

(B.2) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

and making use of the Lehman representation:

(B.3) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.4) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

where

(B.8) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

(B.9) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.10) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

where

(B.16) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.18) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.19) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

and

(B.20) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

where ( x = 1 2 q 2 = 1 2 | z α | 2 )

(B.21) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

(B.22) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

Let us define the following expression:

(B.23) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.24) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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

(B.25) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

where

(B.27) Appendix B Correlation Function For An Anyon Gas In The Self-Consistent Hartree Approximation

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).