June 15, 2008

Fig. 1: The dimensionless part of the transverse
current response function. (a) Both real (blue solid) and
imaginary (red solid) parts of the response function are
plotted for x=1. (b)Both real (blue dashed) and
imaginary parts (red dashed) of the response function are
plotted for x=3. The computer code that generated
this plot is available here. |

In this study, we investigate the free electron gas current response function. The charge current operator is defined as

where **A** is the vector potential and
*&psi*(**r**) is the electron annihilation operator at **r**.
Note that this definition of the current operator above is gauge
invariant. We obtain

Now, we are ready to calculation the current response function. In linear response regime, the perturbative Hamiltonian is given by

where the Coulomb gauge is selected and the
**A**^{2} term is neglected. Note that in *H*' it is
**j**, not **J**, that couples to **A**. In terms of
*c*_{k}'s, which are defined as

the currect **j** is given by

Thus, we are ready to obtain the current response

where

and *E _{f}* is the Fermi energy of the
electron gas.

Due to the spherical symmetry of the electron gas, the response function can be decomposed into longitudinal and transverse parts:

The longitudinal part is related to the density response by virtue of the current continuity equation since the particle number is conserved. The continuity equation in the Coulomb gauge is given by

which indicates that

Since in the previous report the density response function has been computed, we here only need to compute the transverse part of the current response function as follows:

whose imaginary part can be evaluated in a closed form. We denote that

Thus, the transverse response function is given by

where

This complicated integral has not closed form. However, its imaginary part is simple:

Thus, the imaginary part of the transverse current response function is given by

Once we obtain the imaginary part of the transverse response function, its real part can be obtained by the Kramers-Kronig relation:

where *P* indicates the principle part of the
integral.

© 2008 H. Yao. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] J. R. Schrieffer, *Theory of
Superconductivity* (Perseus, 1999).

[2] A. L. Fetter and J. D. Walecka, *Quantum Theory
of Many-Particle Systems* (Dover, 2003).

[3] X.-G. Wen, *Quantum Field Theory of Many-body
Systems* (Oxford, 2007).