\section{(27 February 2009)} \subsection*{Transport coefficients, cont.} Recall: \begin{align*} \mathcal{L} & = [H, \, \cdot \, ] & \rho & = \text{ equilibrium density matrix} & & \\ \mathcal{L}_{\text{per}} & = \mathcal{L} + \vec{\lambda} \cdot \vec{\delta} \, , & \vec{\delta} & = ( \delta_{1} , \ldots , \delta_{n} ) \quad \text{commuting derivations:} & [\delta_{i} , \delta_{j} ] & =0 \\ & & \vec{\lambda} & = (\lambda_{1} , \ldots , \lambda_{n} ) \quad \text{complex constants} & & \end{align*} Given the perturbation \( \vec{\lambda} \cdot \vec{\delta} \) of \( \mathcal{L} \) there is a response: \( \langle \vec{\delta} H \rangle_{\text{per}} = \tau ( \rho_{\text{per}} \vec{\delta} H ) \), according to our philosophy. Hypothesis: \begin{align*} \rho_{\text{per}} & = \text{``}\lim_{t\rightarrow\infty}\text{''} \mathrm{e}^{\mi t \mathcal{L}_{\text{per}}} ( \rho ) \\ & \hspace{2em} \uparrow \hspace{3em} \uparrow \\ & \hspace{2em} \uparrow \hspace{1em} \text{{\scriptsize equation of motion}} \\ & \text{{\scriptsize very often must be regularized}} \end{align*} Assumptions: \begin{itemize} \item[1)] \( \tau \circ \delta = 0 \) Then \begin{align*} \langle \vec{\delta} H \rangle_{\text{per}} & = \tau ( \rho_{\text{per}} \vec{\delta} H ) = \lim_{t \rightarrow \infty} \tau ( \mathrm{e}^{\mi t \mathcal{L}_{\text{per}}} (\rho) \vec{\delta} \cdot \vec{H} ) \\ & (\text{formally, } \, \tau(\mathcal{L}_{\text{per}}(\rho)A) = -\tau(\rho \mathcal{L}_{\text{per}} (A))) \\ & = \lim_{t \rightarrow \infty} \tau(\rho \mathrm{e}^{-\mi t \mathcal{L}_{\text{per}}}\vec{\delta} \cdot \vec{H}) \tag{\( \ast \)} \end{align*} \item[2)] Bold assumption: use the Dyson expansion for \( \mathrm{e}^{-\mi t \mathcal{L}_{\text{per}}} \) \begin{align*} (\ast) & \overset{\text{Dyson}}{\underset{\text{power series in }\lambda}{=}} \tau ( \rho \, \vec{\delta} \cdot \vec{H} ) + \underbrace{\lim_{t \rightarrow \infty}\tau \left( \rho \int_{\underset{ s_{0} + s_{1} \, = 1}{s_{0} , s_{1} \, \geq \, 0}} \mathrm{e}^{-\mi s_{0} \mathcal{L}} \vec{\lambda} \cdot \vec{\delta} \mathrm{e}^{-\mi s_{1} \mathcal{L}} \vec{\delta} \vec{H} \right)} + \bigoh ( \lambda^{2} ) \\ & \hspace{8em} \rule[0em]{0.5pt}{2em} \hspace{12em} 1^{\text{st}}\text{ order term} \\ & \hspace{5em} \lambda^{0}\text{-term; mostly} = 0 \\ & \hspace{20em} \text{translate this thing} \end{align*} \item[3)] \( 1^{\text{st}} \) order term. Assumption: \( \bigoh (\lambda^{2}) \) is really negligible. \[ \langle \vec{\delta} H \rangle_{\text{per}} = \sigma^{\delta} \vec{\lambda} \quad (\sigma^{\delta} \text{ is a matrix; a tensor in the case of the higher order terms.}) \] where \[ \sigma^{\delta}_{\nu \mu} = \lim_{t \rightarrow \infty} \overset{\sigma_{\nu \mu}(t)}{\overbrace{\tau \left( \rho \int_{0}^{t} \mathrm{e}^{-\mi (t - s_{1}) \mathcal{L}} \delta_{\mu} \mathrm{e}^{-\mi s_{1} \mathcal{L}} \delta_{\nu} H \right) \mathrm{d} s_{1}}} \] is the tensor of transport coefficients in the \( 1^{\text{st}} \) order approximation (higher orders; higher order tensors). We have: \begin{align*} \sigma_{\nu \mu}(t) & = \tau \left( \underset{= \rho \text{ by invariance}}{\underbrace{\mathrm{e}^{\mi (t - s_{1})\mathcal{L}} (\rho)}} \delta_{\mu} \int_{0}^{t} \mathrm{e}^{-s_{1}\mathcal{L}} \delta_{\nu} H \mathrm{d}s_{1} \right) \\ & \overset{\footnotemark}{=} \tau \left( -\delta_{\mu} (\rho) \int_{0}^{t} \mathrm{e}^{-\mi s \mathcal{L}} \delta_{\nu} H \mathrm{d}s \right) \end{align*} \footnotetext{Using the fact that \( \delta_{\mu} \) is a derivation and that \( \tau \) is invariant under \( \delta \).} \end{itemize} Now \emph{a priori} there is a singularity if \( t \rightarrow \infty \) in case \( \delta_{\nu} \not\in \krn \mathcal{L}^{\perp} \). Suppose that the temperature \( T \) is very low and the Fermi energy \( E_{F} \in \mathrm{Gap}(H) \). Hence \( \rho =P_{F} = \) the spectral projection of \( H \) to states below the Fermi energy: \begin{figure*}[ht] \setlength{\unitlength}{10pt} \begin{center} \begin{picture}(12, 10)(-2, -2) % \multiput(0, 2)(2, 0){6}{\circle*{0.2}} \put(0, 2){\line(1, 0){10}} % \put(2, 2.2){\line(0, 1){0.4}} % \put(0.4, 0.8){\( \inf (H) \)} % \put(6, 2.2){\line(0, 1){0.4}} % \put(5.4, 0.8){\( E_{0}\)} \put(8, 1.8){\line(0, 1){0.4}} \put(7.4, 0.8){\( E_{F} \)} \put(7.6, 7){\( \rho = P_{F} \quad (T \rightarrow 0 )\)} % \put(10, 2.2){\line(0, 1){0.4}} % \put(9.4, 0.8){\( E_{1} \)} % \put(-0.4, 7.75){\( 1 \cdots \)} \thicklines % \put(-1, 2){\line(1,0){1}} % \qbezier(0, 2)(0.5, 2)(1, 5) % \qbezier(1, 5)(1.5, 8)(2, 8) \put(0, 8){\line(1,0){6}} \qbezier(6, 8)(7, 8)(8, 5) \qbezier(8, 5)(9, 2)(10, 2) \put(10, 2){\line(1,0){2}} \thinlines \multiput(0, 1.8)(0.2, 0){31}{\line(0, 1){0.4}} \qbezier(4, 8)(6, 8)(8, 5) \qbezier(8, 5)(10, 2)(12, 2) \put(3, 5){\( T >> 0 \)} \put(5, 6){\vector(1, 1){1}} \multiput(10, 1.8)(0.2, 0){11}{\line(0, 1){0.4}} \end{picture} \end{center} \end{figure*} \begin{remark*} If \( p = p^{2} \) and \( \delta \) is any derivation: \begin{align*} \delta (p^{2}) & = p \delta (p) + \delta (p) p \, , \text{ and} \\ || \, \, \, & \rule[0em]{0em}{0.1em} \\ \delta (p) & = p \delta (p) + p^{\perp} \delta (p) \quad (p^{\perp} = 1 - p) \\ || \, \, \, & \rule[0em]{0em}{0.1em} \\ \delta (p) & = \delta (p) p + \delta (p) p^{\perp} \quad (\text{ditto}) \\ \Leftrightarrow \delta (p) p & = p^{\perp} \delta (p) \text{ and } p \delta (p) = \delta (p) p^{\perp} \\ \text{put this together } \overset{\footnotemark}{\Rightarrow } \delta (p) & = p \delta (p) p^{\perp} + p^{\perp} \delta (p) p \tag{\( \ast \)} \end{align*} \footnotetext{\( \delta (p) = p(p \delta (p)) + p^{\perp} (p^{\perp} \delta (p)) = p \delta (p) p^{\perp} + p^{\perp} \delta (p) p\)} \end{remark*} \emph{The result marked \( (\ast) \) is an extremely important algebraic calculation.} Suppose that we have an eigenbasis of \( H \) by \( \{ \psi_{i} \}_{i} \) (perhaps generalized eigenvectors). Then \begin{align*} \tau \left( \delta_{\mu} (P_{F}) \int_{0}^{t} \mathrm{e}^{-\mi s \mathcal{L}} \delta_{\nu} H \mathrm{d}s \right) & = \sum_{i,j} \underset{X_{ij}}{\underbrace{\langle \psi_{i} \, | \, \delta_{\mu} P_{F} \, | \, \psi_{j} \rangle}} \langle \psi_{j} \, | \, \int_{0}^{t} \mathrm{e}^{-\mi s \mathcal{L}}\delta_{\nu} H \, | \, \psi_{i} \rangle \\ \text{where } X_{ij} & = 0 \text{ if both } \psi_{i} \text{ and } \psi_{j} \text{ belong to either } \\ & \quad \im P_{F} \text{ or its orthogonal complement} \end{align*} \begin{quote}\emph{Notes are incomplete from here,} \[ \langle \psi_{j} \, | \, H \delta_{\nu} H \, | \, \psi_{i} \rangle - \langle \psi_{j} \, | \, \delta_{\nu} (H) H \, | \, \psi_{i} \rangle \] \[ H | \psi_{i} \rangle = E_{i} | \psi_{i} \rangle \] \[ \Rightarrow |E_{i} - E_{j}| \geq |\tilde{E}_{1} - \tilde{E}_{0}| \] \emph{to here} \end{quote} So \[ \langle \psi_{j} | \int_{0}^{t} \mathrm{e}^{-\mi s \mathcal{L}} \delta_{\nu}|\psi_{i} \rangle \mathrm{d}s = \int_{0}^{t} \mathrm{e}^{-\mi s (E_{j} - E_{i})} \mathrm{d}s \, \langle \psi_{j} \, | \, \delta_{\nu} H \, | \, \psi_{i} \rangle \] \begin{align*} \lim_{t \rightarrow \infty} \int_{0}^{t} \mathrm{e}^{-\mi s \omega} \mathrm{d}s & = \text{FT of the Heaviside function} \\ & = \underset{\text{principal value}}{PV\left(\frac{1}{\mi \omega}\right)} + \frac{\pi}{2} \delta(\omega) \text{ in the distribution space} \end{align*} Hence \[ \frac{1}{\mi (E_{j} - E_{i})} \langle \psi_{j} \, | \, \delta_{\nu} H \, | \, \psi_{i} \rangle = \frac{1}{\mi} \langle \psi_{j} \, | \, \mathcal{L}^{-1} \delta_{\nu} H \, | \, \psi_{i} \rangle \] Drawing everything together, \begin{proposition*} \( \sigma^{\delta}_{\nu \mu} = \underset{\text{\textnormal{not at all invertible so need} } P_{F}}{\underset{\hspace{2.4em} |}{-\frac{1}{\mi}\tau \left( \delta_{\mu}(P_{F}) \mathcal{L}^{-1} \delta_{\nu} H \right)}} \) \end{proposition*} Using the extremely important \( (\ast) \) twice (as well as commuting properties of \( \mathcal{L} \) with the projections), \begin{gather*} -\frac{1}{\mi}\tau \left( \delta_{\mu}(P_{F}) \mathcal{L}^{-1} \delta_{\nu} H \right) = -\frac{1}{\mi}\tau \left( P_{F} \delta_{\mu}(P_{F}) P_{F}^{\perp} \mathcal{L}^{-1} \delta_{\nu} H + P_{F}^{\perp} \delta_{\mu}(P_{F}) P_{F} \mathcal{L}^{-1} \delta_{\nu} H \right) \\ = -\frac{1}{\mi}\tau \left( P_{F} \delta_{\mu}(P_{F}) P_{F}^{\perp} \mathcal{L}^{-1}( \delta_{\nu} (H) )P_{F} + P_{F}^{\perp} \delta_{\mu}(P_{F}) P_{F} \mathcal{L}^{-1} (\delta_{\nu} (H) )P_{F}^{\perp} \right) \end{gather*} \begin{lemma*} \begin{align*} P_{F}^{\perp} \mathcal{L}^{-1} (\delta_{\nu}(H)) P_{F} & = -P_{F}^{\perp} \delta_{\nu}(P_{F}) P_{F} \\ \text{\textnormal{and} } P_{F} \mathcal{L}^{-1} (\delta_{\nu}(H)) P_{F}^{\perp} & = P_{F} \delta_{\nu}(P_{F}) P_{F}^{\perp} \end{align*} \end{lemma*} \begin{proof} Apply \( \mathcal{L} \) to right-hand side: \begin{align*} \mathcal{L} \left( P_{F}^{\perp} \delta_{\nu} (P_{F}) P_{F} \right) & = P_{F}^{\perp} \!\!\!\!\!\!\!\! \underset{\delta_{\nu} \underset{=0 - [ \delta_{\nu} (H), P_{F} ]}{[H, P_{F}] - [ \delta_{\nu} (H), P_{F} ]}}{\underbrace{[ H, \delta_{\nu} (P_{F}) ]}} \!\!\!\!\!\!\!\! P_{F} \, (\text{commutator in } H \text{ commutes with } (\delta_{\nu} H)) \\ & = -P_{F}^{\perp} [ \delta_{\nu} (H), P_{F} ] P_{F} \\ & = -P_{F}^{\perp} \delta_{\nu} (H) P_{F} \end{align*} This proves the first equality. The second is proved in the same sort of way. \end{proof} Final result: \begin{proposition*} \( \sigma^{\delta}_{\nu \mu} = -\mi \tau \left( P_{F} \delta_{\mu} (P_{F}) \delta_{\nu} (P_{F}) - \delta_{\nu} (P_{F}) \delta_{\mu} (P_{F}) \right) \) \end{proposition*} This is a non-commutative Chern character\,!!!