--- title: "Matsubara Green's function" sort_title: "Matsubara Green's function" date: 2021-11-12 categories: - Physics - Quantum mechanics layout: "concept" --- The **Matsubara Green's function** is an [imaginary-time](/know/concept/imaginary-time/) version of the real-time [Green's functions](/know/concept/greens-functions/). We define as follows in the imaginary-time [Heisenberg picture](/know/concept/heisenberg-picture/): $$\begin{aligned} \boxed{ C_{AB}(\tau, \tau') \equiv -\frac{1}{\hbar} \Expval{\mathcal{T} \big\{ \hat{A}(\tau) \hat{B}(\tau') \big\}} } \end{aligned}$$ Where the expectation value $$\Expval{}$$ is with respect to thermodynamic equilibrium, and $$\mathcal{T}$$ is the [time-ordered product](/know/concept/time-ordered-product/) pseudo-operator. Because the Hamiltonian $$\hat{H}$$ cannot depend on the imaginary time, $$C_{AB}$$ is a function of the difference $$\tau \!-\! \tau'$$ only: $$\begin{aligned} C_{AB}(\tau, \tau') &= - \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} \hat{A}(\tau) \hat{B}(\tau') \Big) \\ &= - \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{\tau \hat{H} / \hbar} \hat{A} e^{-\tau \hat{H} / \hbar} e^{\tau' \hat{H} / \hbar} \hat{B} e^{-\tau' \hat{H} / \hbar} \Big) \\ &= - \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B} \Big) \end{aligned}$$ For $$\tau > \tau'$$, we see by expanding in the many-particle eigenstates $$\Ket{n}$$ that we need to demand $$\hbar \beta > \tau \!-\! \tau'$$ to prevent $$C_{AB}$$ from diverging for increasing temperatures: $$\begin{aligned} C_{AB}(\tau \!-\! \tau') &= - \frac{1}{\hbar Z} \sum_{n} \Matrixel{n}{e^{-\beta \hat{H}} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B}}{n} \\ &= - \frac{1}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{A} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B}}{n} e^{-\beta E_n} e^{(\tau - \tau') E_n / \hbar} \end{aligned}$$ And likewise, for $$\tau < \tau'$$, we must demand that $$\tau \!-\! \tau' > -\hbar \beta$$ for the same reason: $$\begin{aligned} C_{AB}(\tau \!-\! \tau') &= \mp \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} \hat{B}(\tau') \hat{A}(\tau) \Big) \\ &= \mp \frac{1}{\hbar Z} \Tr\!\Big( e^{-\beta \hat{H}} e^{-(\tau - \tau') \hat{H} / \hbar} \hat{B} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A} \Big) \\ &= \mp \frac{1}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{B} e^{(\tau - \tau') \hat{H} / \hbar} \hat{A}}{n} e^{-\beta E_n} e^{- (\tau - \tau') E_n / \hbar} \end{aligned}$$ With $$-$$ for bosons, and $$+$$ for fermions, due to the time-ordered product for $$\tau > \tau'$$. On this domain $$[-\hbar \beta, \hbar \beta]$$, the Matsubara Green's function $$C_{AB}$$ obeys a useful shift relation: it is $$\hbar \beta$$-periodic for bosons, and $$\hbar \beta$$-antiperiodic for fermions: $$\begin{aligned} \boxed{ C_{AB}(\tau \!-\! \tau') = \begin{cases} \pm C_{AB}(\tau \!-\! \tau' \!+\! \hbar \beta) & \mathrm{if\;} \tau \!-\! \tau' < 0 \\ \pm C_{AB}(\tau \!-\! \tau' \!-\! \hbar \beta) & \mathrm{if\;} \tau \!-\! \tau' > 0 \end{cases} } \end{aligned}$$