Key equations from quantum statistical tools

Equilibrium

Expectation values

$A=\mathrm {Tr} [{\hat {\rho }}_{\mathrm {eq} }{\hat {A}}]$

Density matrix of fermions in equilibrium

• using spectral decomposition:

${\hat {\rho }}_{\mathrm {eq} }=\sum _{\alpha }f(E_{\alpha })|E_{\alpha }\rangle \langle E_{\alpha }|=f({\hat {H}}-\mu {\hat {I}})$

• using Green functions:

${\hat {\rho }}_{\mathrm {eq} }=-{\frac {1}{\pi }}\int dE\,\mathrm {Im} G^{r}f(E)$

• Fermi-Dirac distribution function: $f(E)=1/[\exp((E-\mu )/k_{B}T)+1]$

• Hamiltonian and its spectral decomposition: ${\hat {H}}=\sum _{\alpha }E_{\alpha }|E_{\alpha }\rangle \langle E_{\alpha }|$

• function of Hamiltonian: $F({\hat {H}})=\sum _{\alpha }F(E_{\alpha })|E_{\alpha }\rangle \langle E_{\alpha }|$

• Green operators:

${\hat {G}}^{r,a}(E)=[E{\hat {I}}-{\hat {H}}\pm i\eta ]^{-1}$

$\mathrm {Im} {\hat {G}}^{r}=({\hat {G}}^{r}-{\hat {G}}^{a})/2i$

Charge density

• charge density operator: ${\hat {n}}(\mathbf {r} )=|\mathbf {r} \rangle \langle \mathbf {r} |$

• expectation value: $n(\mathbf {r} )=\mathrm {Tr} [{\hat {\rho }}_{\mathrm {eq} }|\mathbf {r} \rangle \langle \mathbf {r} |]=\langle \mathbf {r} |{\hat {\rho }}_{\mathrm {eq} }|\mathbf {r} \rangle$ (in some discrete representation these is just diagonal matrix element)

Density of states

• definition of total DOS: $g(E)=\sum _{\alpha }\delta (E-E_{\alpha })$ (with possible normalization factors like $2_{s}/V$)

• definition of LDOS: $g(E)=\int d^{3}\mathbf {r} g(\mathbf {r} ,E)$

• LDOS using wavefunctions: $n(\mathbf {r} )=\mathrm {Tr} [{\hat {\rho }}_{\mathrm {eq} }|\mathbf {r} \rangle \langle \mathbf {r} |]=\sum _{\alpha }|\Psi _{\alpha }(\mathbf {r} )|^{2}f(E_{\alpha })=\int dE\left[\sum _{\alpha }|\Psi _{\alpha }(\mathbf {r} )|^{2}\delta (E-E_{\alpha })\right]f(E)=\int dE\,g(\mathbf {r} ,E)f(E)$

• LDOS using Green functions:

$g(\mathbf {r} ,E)=-{\frac {1}{\pi }}\langle \mathbf {r} |\mathrm {Im} {\hat {G}}^{r}(E)|\mathbf {r} \rangle$

• total DOS using Green functions:

$g(E)=-{\frac {1}{\pi }}\mathrm {Tr} [{\hat {G}}^{r}(E)]=-{\frac {1}{\pi }}\int d^{3}\mathbf {r} \,\langle \mathbf {r} |\mathrm {Im} {\hat {G}}^{r}(E)|\mathbf {r} \rangle$

Nonequilibrium

• Expectation values:

• Current operator: