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

<math>E^{(1)} = \frac{1}{2}\int d^{3}r d^{3}r^{\prime} v(\vec{r}-\vec{r}^{ \prime})[n^{2}-\sum_{\sigma}G_{\sigma}(\vec{r}-\vec{r}^{\prime})^{2}]

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