Self-consistent Hartree-Fock approach to Phase Transitions

Hartree-Fock approach to Phase Transitions

Hartree-Fock in second quantized form

The interaction energy operator for a two body system with pairwise interactions is given by

$\nu ={\frac {1}{2}}\sum _{\sigma \sigma ^{\prime }}\int d^{3}rd^{3}r^{\prime }v({\vec {r}}-{\vec {r}}^{\prime })\psi _{\sigma }^{\dagger }({\vec {r}})\psi _{\sigma ^{\prime }}^{\dagger }({\vec {r}}^{\prime })\psi _{\sigma ^{\prime }}({\vec {r}}^{\prime })\psi _{\sigma }({\vec {r}})$

In the above operator, it is important to note the order of the operators. Including the kinetic energy term we can write the Hamiltonian for particles of mass m with pairwise interactions in second quantized formalism as

$H=\sum _{\sigma \sigma ^{\prime }}\int d^{3}r{\frac {1}{2m}}\nabla \psi _{\sigma }^{\dagger }({\vec {r}})\cdot \nabla \psi _{\sigma }({\vec {r}})+{\frac {1}{2}}\sum _{\sigma \sigma ^{\prime }}\int d^{3}rd^{3}r^{\prime }v({\vec {r}}-{\vec {r}}^{\prime })\psi _{\sigma }^{\dagger }({\vec {r}})\psi _{\sigma ^{\prime }}^{\dagger }({\vec {r}}^{\prime })\psi _{\sigma ^{\prime }}({\vec {r}}^{\prime })\psi _{\sigma }({\vec {r}})$

The first-order correction to the energy induced by the interaction is

$E^{(1)}={\frac {1}{2}}\int d^{3}rd^{3}r^{\prime }v({\vec {r}}-{\vec {r}}^{\prime })\sum _{\sigma \sigma ^{\prime }}<\Phi _{0}|\psi _{\sigma }^{\dagger }({\vec {r}})\psi _{\sigma ^{\prime }}^{\dagger }({\vec {r}}^{\prime })\psi _{\sigma ^{\prime }}({\vec {r}}^{\prime })\psi _{\sigma }({\vec {r}})|\Phi _{0}>$

Define $G_{\sigma }({\vec {r}}-{\vec {r}}^{\prime })=<\Phi _{0}|\psi _{\sigma }^{\dagger }({\vec {r}})\psi _{\sigma }({\vec {r}}^{\prime })|\Phi _{0}>$

Then, we find that

$E^{(1)}={\frac {1}{2}}\int d^{3}rd^{3}r^{\prime }v({\vec {r}}-{\vec {r}}^{\prime })[n^{2}-\sum _{\sigma }G_{\sigma }({\vec {r}}-{\vec {r}}^{\prime })^{2}]$ Second Quantization

This is just a practice.

Let's write an integral -

$\int \psi ^{\dagger }\psi \left({\bar {r}}-{\bar {r}}^{'}\right)$

(You can find intermediate steps in "Lectures on Quantum Mechanics" by Gordon Baym).

The first term in $E^{(1)}$ is called the Hartree energy or direct energy. The second term is called the Fock term and it corresponds to the energy due to exchange interactions.