From fran@physics.rutgers.edu Thu Oct 26 17:47:08 2000 Date: Thu, 26 Oct 2000 13:10:06 -0400 (EDT) From: Fran Delucia To: fran@physics.rutgers.edu, kotliar@physics.rutgers.edu Subject: file Gabi: Here is the hint for problem set 3 fran \documentstyle[12pt]{article} \begin{document} \begin{center} {\bf Problem Set 3} \end{center} $H_{o} = \sum\limits_{k} \varepsilon\limits_{k} c^{\dagger}_{k \sigma} c_{k \sigma}$ \bigskip $N = \sum\limits_{k} c_{k \sigma}^{\dagger} c_{k \sigma}$ \bigskip $\mid \psi_{o} \rangle = \prod\limits_{k < k _{F}} c_{k \uparrow}^{\dagger} c_{k \downarrow}^{\dagger} \mid o \rangle$ \bigskip $\rho_{q} \equiv \sum\limits_{k \sigma} \ \ c_{t g \sigma}^{\dagger} c_{k \sigma}$ \bigskip $\Delta_{q} \equiv \sum\limits_{k} c_{-ktq \sigma \uparrow}^{\dagger} c_{k \downarrow}^{\dagger}$ \bigskip a) zero temperature.\\ Calculate the Fourrier transform of $G_{\rho}(t,q) G_{p}(t,q)$ in time.\\ \bigskip $G_{\rho}(t,q) = \langle T(\rho_{q}(t) \rho_{-q}(0)) \rangle$ \bigskip $G_{p}(t,q) = \langle T(\Delta_{q}(t)\Delta_{+q}(o))\rangle$ \bigskip in 3 dimensions and in 2 dimensions. \bigskip b) Finite temperature - if $\tau$ is the imaginary time it is convenient to use Fourrier series. $G_{\rho}(i \nu_{n},g)= \int_{o}^{\beta} e^{i \nu_{n}\tau} G_{\rho}(\tau,_{t}^{o})d \tau$ $G_{p}(i \nu_{n},q) = \int_{o}^{\beta} e^{i \nu_{n}} \tau G_{p}(\tau,_{t}^{o}) d \tau$ \bigskip \underline{Evaluate them} in 3 and 2 dimensions. Now G$_{g}$ and G$_{p}$ are defined by \smallskip $G_{\rho}(\tau,q)$ $= tr e^{-\beta(H_{o} -\mu N)}$ $\rho_{t} ^{o}(\rho_{q}(o))$ \smallskip $G_{p}(\tau, q) = t_{r}$ $e^{-\beta(H_{o} -\mu N)}$ $\Delta_{q}(\rho) \Delta_{t q}(o)$ \bigskip c) What happens in 1 dimension? \newpage Hints to solve problem set 3\\ Define $\rho_{g} (t, t^{o}) = -i \ \ \langle T (\rho q (t) \rho t q^{t} (o)) \rangle$\\ $\rho_{\Delta}(t, q) = -i \ \ \langle (\Delta_{q} (t) \ \Delta_{t}^{t} (o) \rangle)$ \smallskip $\rho_{q}^{(t)} = \sum\limit_{k} c_{-k q \uparrow} c_{k \downarrow}$ \bigskip a) Use Wick's theorem to express $\rho_{g} (t,q)$ and $\rho_{\Delta}(t,q)$ in terms of\\ $-i \langle T (c_{k}(t) c_{k}^{t} (o)) \rangle = G(kt)$ \bigskip b) Compute G(k,w) = $\int_{\infty}^{\infty} e^{iwt} G(k,t) dt$ - show it as given by\\ $G(k,w) = \frac{1}{iw - \varepsilon_{k} ti \rho sign (\mid k \mid-k_{F})}$ $\rho \rangle 0$ \bigskip c) Combine (a) + (b) and use the fact that the Fourrier transfer of the product is the solution of the Fourrier transfer to show that $\rho_{g} (w,q) = \int_{-\infty}^{\infty} dt e^{iwt} \rho_{g} (t,q)$\\ $\rho_{\Delta} (w,q) = \int_{-\infty}^{\infty}dt e^{iwt} \rho{\Delta} (t,q)$\\ are given Bg expressions line\\ $\rho_{g} (w,q) ~ -i \int dw' \sum_{k} G(w'tw, ke) G (w',k)$\\ $\rho_{\Delta} (w,q) ~ -i \int dw' \sum_{k} G(-w'tw, ktq) G(w',k)$\\ \bigskip d) Now carry out the frequency integral using the residue theorem. Get expressions of the fermi\\ \bigskip $\rho_{g}(w,q) ~ \sum_{k} \frac{[f(\varepsilon_{ktq}) - f(\varepsilon_{k})]}{\varepsilon_{ktq} - \varepsilon_{k} -U - i \sigma} \ \ w > o$\\ $\rho_{\Delta}(w,q) ~ \sum_{k} \frac{[f(\varepsilon_{ktq}) - f(\varepsilon_{k})]}{\varepsilon_{ktq} + \varepsilon_{k} - w -i \sigma} w > o $f(\eplison_{k}) = {1 lf k