Energy formulas

Kinetic Energy

This energy is the weigthed average of non-interacting kinetic energy. Assuming a non-interacting Hamiltonian in momentum space the kinetic average is.

\langle T \rangle = Tr \frac{1}{\beta} \sum_{k,n} \epsilon_k^0 G(k, i\omega_n)

It can be transformed into a treatable form relying on local quantities

\langle T \rangle = Tr \frac{1}{\beta} \sum_{k,n} \left( \epsilon_k^0 G(k, i\omega_n) + G(k, i\omega_n)^{-1}G(k, i\omega_n) - G^{free}(k, i\omega_n)^{-1}G^{free}(k, i\omega_n) \right)

= Tr \frac{1}{\beta} \sum_{k,n} \left( \epsilon_k^0 G(k, i\omega_n) + (i\omega_n - \epsilon_k^0 - \Sigma(i\omega_n))G(k, i\omega_n) - (i\omega_n - \epsilon_k^0)G^{free}(k, i\omega_n) \right)

= Tr \frac{1}{\beta} \sum_{k,n} \left( i\omega_n \left( G(k, i\omega_n)- G(k, i\omega_n)^{free} \right) - \Sigma(i\omega_n) G(k, i\omega_n) + \epsilon_k^0G^{free}(k, i\omega_n) \right)

The first two terms can be summed in reciprocal space to yield a local the quantities that come out of the DMFT self-consistency and the last term as it belongs to the non-interacting system is trivially solvable

\langle T \rangle = Tr \frac{1}{\beta} \sum_n \left( i\omega_n \left( G(i\omega_n)- G(i\omega_n)^{free} \right) - \Sigma(i\omega_n)G(i\omega_n) \right) + \int_{-\infty}^\infty \epsilon\rho(\epsilon)n_F(\epsilon-\mu) d\epsilon

It is also possible to take a simpler approac by introducing a zero to the frequecy sum, with a constant factor. In this case one takes from

\langle T \rangle = Tr \frac{1}{\beta} \sum_{k,n} \left( \epsilon_k^0 G(k, i\omega_n) + G(k, i\omega_n)^{-1}G(k, i\omega_n) \right)

= Tr \frac{1}{\beta} \sum_{k,n} \left( \epsilon_k^0 G(k, i\omega_n) + (i\omega_n - \epsilon_k^0 - \Sigma(i\omega_n))G(k, i\omega_n) \right)

But the local self-energy can be expresed by

\Sigma(i\omega_n) = \mathcal{G}^{0, -1} - G(i\omega_n)^{-1} = i\omega_n - h_{loc} - \Delta(i\omega_n) - G(i\omega_n)^{-1}

Where h_{loc} is the momentum independent part of the hamiltonian. Then the expression transforms into.

\langle T \rangle = Tr \frac{1}{\beta} \sum_{k,n} \left(h_{loc} + \Delta(i\omega_n)\right) G(k, i\omega_n) = Tr \frac{1}{\beta} \sum_n \left(h_{loc} + \Delta(i\omega_n)\right) G(i\omega_n)

Potential energy

According to [Fetter-Walecka] in equation 23.14 then transformed to Matsubara frequencies the potential energy can be described by:

\langle V \rangle = \frac{1}{\beta} \sum_{k,n} \frac{1}{2}\left( i\omega_n - \epsilon_k^0 \right)Tr G(k, i\omega_n)

And expressing it in local quantities with the DMFT approximation that the Self-Energy is local

(1)\langle V \rangle = \frac{1}{\beta} \sum_{n} \frac{1}{2} Tr(\Sigma(i\omega_n)G(i\omega_n))

References

[Fetter-Walecka]Fetter, Walecka, Quantum Theory of many-particle systems