Key equations from quantum statistical tools

Equilibrium

Expectation values

$A=\mathrm{T}\mathrm{r}[{\hat{\rho }}_{\mathrm{e}\mathrm{q}}\hat{A}]$

Density matrix of fermions in equilibrium

• using spectral decomposition:

${\hat{\rho }}_{\mathrm{e}\mathrm{q}}=\sum _{\alpha }f(E_{\alpha })|E_{\alpha }⟩⟨E_{\alpha }|=f(\hat{H}-\mu \hat{I})$

• using Green functions:

${\hat{\rho }}_{\mathrm{e}\mathrm{q}}=-\frac{1}{\pi }\int dE\mathrm{I}\mathrm{m}{G}^{r}f(E)$

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

• Hamiltonian and its spectral decomposition: $\hat{H}=\sum _{\alpha }{E}_{\alpha }|E_{\alpha }⟩⟨E_{\alpha }|$

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

• Green operators:

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

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

Charge density

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

• expectation value: $n(\mathbf{𝐫})=\mathrm{T}\mathrm{r}[{\hat{\rho }}_{\mathrm{e}\mathrm{q}}|\mathbf{𝐫}⟩⟨\mathbf{𝐫}|]=⟨\mathbf{𝐫}|{\hat{\rho }}_{\mathrm{e}\mathrm{q}}|\mathbf{𝐫}⟩$ (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: $n(\mathbf{𝐫})=\mathrm{T}\mathrm{r}[{\hat{\rho }}_{\mathrm{e}\mathrm{q}}|\mathbf{𝐫}⟩⟨\mathbf{𝐫}|]=\sum _{\alpha }|{\mathrm{\Psi }}_{\alpha }(\mathbf{𝐫}){|}^{2}f(E_{\alpha })=\int dE[\sum _{\alpha }|{\mathrm{\Psi }}_{\alpha }(\mathbf{𝐫}){|}^2\delta (E-E_{\alpha })]f(E)=\int dEg(\mathbf{𝐫},E)f(E)$

• LDOS using Green functions:

$g(\mathbf{𝐫},E)=-\frac{1}{\pi }⟨\mathbf{𝐫}|\mathrm{I}\mathrm{m}{\hat{G}}^r(E)|\mathbf{𝐫}⟩$

• total DOS using Green functions:

$g(E)=-\frac{1}{\pi }\mathrm{T}\mathrm{r}[{\hat{G}}^r(E)]=-\frac{1}{\pi }\int d^3\mathbf{𝐫}⟨\mathbf{𝐫}|\mathrm{I}\mathrm{m}{\hat{G}}^r(E)|\mathbf{𝐫}⟩$

Nonequilibrium

• Expectation values:

• Current operator: