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: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = \sum_\alpha E_\alpha |E_\alpha \rangle \langle E_\alpha| }

• function of Hamiltonian: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(\hat{H}) = \sum_\alpha F(E_\alpha) |E_\alpha \rangle \langle E_\alpha| }

• Green operators: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{G}^{r,a} = [E\hat{I}-\hat{H} \pm i\eta]^{-1} }

Charge density

• charge density operator: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{n}(\mathbf{r}) = |\mathbf{r} \rangle \langle \mathbf{r}| }

• expectation value: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g(E) = \sum_\alpha \delta(E-E_\alpha) } (with possible normalization factors like Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2_s/V } )

• local density of states: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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(r,E) f(E) }

Nonequilibrium

• Expectation values:

• Current operator:

• Spin torque operator: