From f2970c55894b3c8d5fd2926a8918d166988109fe Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 12 Nov 2021 20:20:05 +0100 Subject: Expand knowledge base --- content/know/concept/heisenberg-picture/index.pdc | 8 +- content/know/concept/imaginary-time/index.pdc | 170 +++++++++ content/know/concept/interaction-picture/index.pdc | 6 +- .../concept/matsubara-greens-function/index.pdc | 396 +++++++++++++++++++++ 4 files changed, 576 insertions(+), 4 deletions(-) create mode 100644 content/know/concept/imaginary-time/index.pdc create mode 100644 content/know/concept/matsubara-greens-function/index.pdc (limited to 'content/know') diff --git a/content/know/concept/heisenberg-picture/index.pdc b/content/know/concept/heisenberg-picture/index.pdc index c49169f..03a386d 100644 --- a/content/know/concept/heisenberg-picture/index.pdc +++ b/content/know/concept/heisenberg-picture/index.pdc @@ -19,11 +19,15 @@ mechanics, and is equivalent to the traditionally-taught Schrödinger equation. In the Schrödinger picture, the operators (observables) are fixed (as long as they do not depend on time), while the state $\ket{\psi_S(t)}$ changes according to the Schrödinger equation, -which can be written using the generator of translations -$\hat{U}(t) = \exp\!(- i t \hat{H} / \hbar)$ like so: +which can be written using the generator of translations $\hat{U}(t)$ like so, +for a time-independent $\hat{H}_S$: $$\begin{aligned} \ket{\psi_S(t)} = \hat{U}(t) \ket{\psi_S(0)} + \qquad \quad + \boxed{ + \hat{U}(t) \equiv \exp\!\bigg(\!-\! i \frac{\hat{H}_S t}{\hbar} \bigg) + } \end{aligned}$$ In contrast, the Heisenberg picture reverses the roles: diff --git a/content/know/concept/imaginary-time/index.pdc b/content/know/concept/imaginary-time/index.pdc new file mode 100644 index 0000000..68e4e02 --- /dev/null +++ b/content/know/concept/imaginary-time/index.pdc @@ -0,0 +1,170 @@ +--- +title: "Imaginary time" +firstLetter: "I" +publishDate: 2021-11-11 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-05T15:19:29+01:00 +draft: false +markup: pandoc +--- + +# Imaginary time + + +Let $\hat{A}_S$ and $\hat{B}_S$ be time-independent in the Schrödinger picture. +Then, in the [Heisenberg picture](/know/concept/heisenberg-picture/), +consider the following expectation value +with respect to thermodynamic equilibium +(as found in [Green's functions](/know/concept/greens-functions/) for example): + +$$\begin{aligned} + \expval*{\hat{A}_H(t) \hat{B}_H(t')} + &= \frac{1}{Z} \Tr\!\Big( \exp\!(-\beta \hat{H}_{0,S}(t)) \: \hat{A}_H(t) \: \hat{B}_H(t') \Big) +\end{aligned}$$ + +Where the Hamiltonian $\hat{H}_{0,S}$ is time-independent. +Suppose a time-dependent $\hat{H}_{1,S}$ is added, +so that the total Hamiltonian is $\hat{H}_S = \hat{H}_{0,S} + \hat{H}_{1,S}$. +Then it is easier to consider the expectation value +in the [interaction picture](/know/concept/interaction-picture/): + +$$\begin{aligned} + \expval*{\hat{A}_H(t) \hat{B}_H(t')} + &= \frac{1}{Z} \Tr\!\Big( \exp\!(-\beta \hat{H}_S(t)) \: \hat{K}_I(0, t) \hat{A}_I(t) \hat{K}_I(t, t') \hat{B}_I(t') \hat{K}_I(t', 0) \Big) +\end{aligned}$$ + +Where $\hat{K}_I(t, t_0)$ is the time evolution operator of $\hat{H}_{1,S}$. +In front, we have $\exp\!(-\beta \hat{H}_S(t))$, +while $\hat{K}_I$ is an exponential of an integral of $\hat{H}_{1,I}$, so we are stuck. +Keep in mind that exponentials of operators +cannot just be factorized, i.e. in general +$\exp\!(\hat{A} \!+\! \hat{B}) \neq \exp\!(\hat{A}) \exp\!(\hat{B})$ + +To get around this, a useful mathematical trick is +to use an **imaginary time** variable $\tau$ instead of the real time $t$. +Fixing a $t$, we "redefine" the interaction picture along the imaginary axis: + +$$\begin{aligned} + \boxed{ + \hat{A}_I(\tau) + \equiv \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \: \hat{A}_S \: \exp\!\bigg( \!-\! \frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) + } +\end{aligned}$$ + +Ironically, $\tau$ is real; the point is that this formula +comes from the real-time definition by replacing $t \to -i \tau$. +The Heisenberg and Schrödinger pictures can be redefined in the same way. + +In fact, by substituting $t \to -i \tau$, +all the key results of the interaction picture can be updated, +for example the Schrödinger equation for $\ket{\psi_S(\tau)}$ becomes: + +$$\begin{aligned} + \hbar \dv{t} \ket{\psi_S(\tau)} + = - \hat{H}_S \ket{\psi_S(\tau)} + \quad \implies \quad + \ket{\psi_S(\tau)} + = \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar} \bigg) \ket{\psi_H} +\end{aligned}$$ + +And the interaction picture's time evolution operator $\hat{K}_I$ +turns out to be given by: + +$$\begin{aligned} + \boxed{ + \hat{K}_I(\tau, \tau_0) + = \mathcal{T} \bigg\{ \exp\!\bigg( \!-\! \frac{1}{\hbar} \int_{\tau_0}^\tau \hat{H}_{1,I}(\tau') \dd{\tau'} \bigg) \bigg\} + } +\end{aligned}$$ + +Where $\mathcal{T}$ is the +[time-ordered product](/know/concept/time-ordered-product/) +with respect to $\tau$. +This operator works as expected: + +$$\begin{aligned} + \ket{\psi_I(\tau)} + = \hat{K}_I(\tau, \tau_0) \ket{\psi_I(\tau_0)} +\end{aligned}$$ + +Where $\ket{\psi_I(\tau)}$ is related to +the Schrödinger and Heisenberg pictures as follows: + +$$\begin{aligned} + \ket{\psi_I(\tau)} + \equiv \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \ket{\psi_S(\tau)} + = \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar}\bigg) \ket{\psi_H} +\end{aligned}$$ + +It is interesting to combine this definition +with the action of time evolution $\hat{K}_I(\tau, \tau_0)$: + +$$\begin{aligned} + \ket{\psi_I(\tau)} + &= \hat{K}_I(\tau, \tau_0) \ket{\psi_I(\tau_0)} + \\ + \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar}\bigg) \ket{\psi_H} + &= \hat{K}_I(\tau, \tau_0) \exp\!\bigg(\frac{\tau_0 \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau_0 \hat{H}_S}{\hbar}\bigg) \ket{\psi_H} +\end{aligned}$$ + +Rearranging this leads to the following useful +alternative expression for $\hat{K}_I(\tau, \tau_0)$: + +$$\begin{aligned} + \boxed{ + \hat{K}_I(\tau, \tau_0) + = \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) + \exp\!\bigg(\!-\! \frac{(\tau \!-\! \tau_0) \hat{H}_{S}}{\hbar}\bigg) + \exp\!\bigg(\!-\! \frac{\tau_0 \hat{H}_{0,S}}{\hbar}\bigg) + } +\end{aligned}$$ + +Returning to our initial example, +we can set $\tau = \hbar \beta$ and $\tau_0 = 0$, +so $\hat{K}_I(\tau, \tau_0)$ becomes: + +$$\begin{aligned} + \hat{K}_I(\hbar \beta, 0) + &= \exp\!\big(\beta \hat{H}_{0,S}\big) \exp\!\big(\!-\! \beta \hat{H}_{S}\big) + \\ + \implies \quad + \exp\!\big(\!-\! \beta \hat{H}_{S}\big) + &= \exp\!\big(\!-\! \beta \hat{H}_{0,S}\big) \hat{K}_I(\hbar \beta, 0) +\end{aligned}$$ + +Using the easily-shown fact that +$\hat{K}_I(\hbar \beta, 0) \hat{K}_I(0, \tau) = \hat{K}_I(\hbar \beta, \tau)$, +we can therefore rewrite the thermodynamic expectation value like so: + +$$\begin{aligned} + \expval*{\hat{A}_H(\tau) \hat{B}_H(\tau')} + &= \frac{1}{Z} \Tr\!\Big(\! \exp\!(-\beta \hat{H}_{0,S}) \hat{K}_I(\hbar \beta, \tau) + \hat{A}_I(\tau) \hat{K}_I(\tau, \tau') \hat{B}_I(\tau') \hat{K}_I(\tau', 0) \!\Big) +\end{aligned}$$ + +Assuming $\tau > \tau'$, +we introduce a time-ordering $\mathcal{T}$, +allowing us to reorder the operators inside, +and thereby reduce the expression considerably: + +$$\begin{aligned} + \expval*{\hat{A}_H \hat{B}_H} + &= \frac{1}{Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{K}_I(\hbar \beta, \tau) \hat{K}_I(\tau, \tau') \hat{K}_I(\tau', 0) + \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp\!(-\beta \hat{H}_{0,S}) \Big) + \\ + &= \frac{1}{Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp\!(-\beta \hat{H}_{0,S}) \Big) +\end{aligned}$$ + +Where $Z = \Tr\!\big(\exp\!(-\beta \hat{H}_S)\big) = \Tr\!\big(\hat{K}_I(\hbar \beta, 0) \exp\!(-\beta \hat{H}_{0,S})\big)$. +For another application of imaginary time, +see e.g. the [Matsubara Green's function](/know/concept/matsubara-greens-function/). + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. diff --git a/content/know/concept/interaction-picture/index.pdc b/content/know/concept/interaction-picture/index.pdc index 1ce330d..45950ff 100644 --- a/content/know/concept/interaction-picture/index.pdc +++ b/content/know/concept/interaction-picture/index.pdc @@ -40,8 +40,10 @@ With $\hat{H}_S(t)$ the full Schrödinger Hamiltonian. We define the unitary conversion operator: $$\begin{aligned} - \hat{U}(t) - \equiv \exp\!\Big( i \frac{\hat{H}_{0,S} t}{\hbar} \Big) + \boxed{ + \hat{U}(t) + \equiv \exp\!\bigg( i \frac{\hat{H}_{0,S} t}{\hbar} \bigg) + } \end{aligned}$$ The interaction-picture states $\ket{\psi_I(t)}$ and operators $\hat{L}_I(t)$ diff --git a/content/know/concept/matsubara-greens-function/index.pdc b/content/know/concept/matsubara-greens-function/index.pdc new file mode 100644 index 0000000..a89ec43 --- /dev/null +++ b/content/know/concept/matsubara-greens-function/index.pdc @@ -0,0 +1,396 @@ +--- +title: "Matsubara Green's function" +firstLetter: "M" +publishDate: 2021-11-12 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-09T14:20:16+01:00 +draft: false +markup: pandoc +--- + +# Matsubara Green's function + +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: + +$$\begin{aligned} + \boxed{ + C_{AB}(\tau \!-\! \tau') = + \begin{cases} + \pm C_{AB}(\tau \!-\! \tau' \!+\! \hbar \beta) + & \mathrm{if\;} \tau \!-\! \tau' < 0 + \\ + \pm C_{AB}(\tau \!-\! \tau' \!-\! \hbar \beta) + & \mathrm{if\;} \tau \!-\! \tau' > 0 + \end{cases} + } +\end{aligned}$$ + +
+ + + +
+ +Due to this limited domain $\tau \in [-\hbar \beta, \hbar \beta]$, +the [Fourier transform](/know/concept/fourier-transform/) +of $C_{AB}(\tau)$ consists of discrete frequencies +$k_n \equiv n \pi / (\hbar \beta)$. +The forward and inverse Fourier transforms +are therefore defined as given below (with $\tau' = 0$). +It is convention to write $C_{AB}(i k_n)$ instead of $C_{AB}(k_n)$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + C_{AB}(i k_n) + &\equiv \frac{1}{2} \int_{-\hbar \beta}^{\hbar \beta} C_{AB}(\tau) \: e^{i k_n \tau} \dd{\tau} + \\ + C_{AB}(\tau) + &= \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty C_{AB}(i k_n) e^{-i k_n \tau} + \end{aligned} + } +\end{aligned}$$ + +
+ + + +
+ +Let us now define the **Matsubara frequencies** $\omega_n$ +as a species-dependent subset of $k_n$: + +$$\begin{aligned} + \boxed{ + \omega_n \equiv + \begin{cases} + \displaystyle\frac{2 n \pi}{\hbar \beta} + & \mathrm{bosons} + \\ + \displaystyle\frac{(2 n + 1) \pi}{\hbar \beta} + & \mathrm{fermions} + \end{cases} + } +\end{aligned}$$ + +With this, we can rewrite the definition of the forward Fourier transform as follows: + +$$\begin{aligned} + \boxed{ + C_{AB}(i \omega_n) + = \int_0^{\hbar \beta} C_{AB}(\tau) \: e^{i \omega_n \tau} \dd{\tau} + = \int_{-\hbar \beta}^0 C_{AB}(\tau) \: e^{i \omega_n \tau} \dd{\tau} + } +\end{aligned}$$ + +
+ + + +
+ +If we actually evaluate this, +we obtain the following form of $C_{AB}$, +which is almost identical to the +[Lehmann representation](/know/concept/lehmann-representation/) +of the "ordinary" retarded and advanced Green's functions: + +$$\begin{aligned} + \boxed{ + C_{AB}(i \omega_m) + = \frac{1}{Z} \sum_{n n'} \frac{\matrixel*{n}{\hat{A}}{n'} \matrixel*{n'}{\hat{B}}{n}}{i \hbar \omega_m + E_n - E_{n'}} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + } +\end{aligned}$$ + +
+ + + +
+ +This gives us the primary use of the Matsubara Green's function $C_{AB}$: +calculating the retarded $C_{AB}^R$ and advanced $C_{AB}^A$. +Once we have an expression for Matsubara's $C_{AB}$, +we can recover $C_{AB}^R$ and $C_{AB}^A$ by substituting +$i \omega_m \to \omega \!+\! i \eta$ and $i \omega_m \to \omega \!-\! i \eta$ respectively. + +In general, we can define the **canonical Green's function** $C_{AB}(z)$ +on the complex plane: + +$$\begin{aligned} + C_{AB}(z) + = \frac{1}{Z} \sum_{n n'} \frac{\matrixel*{n}{\hat{A}}{n'} \matrixel*{n'}{\hat{B}}{n}}{z + E_n - E_{n'}} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) +\end{aligned}$$ + +This is a [holomorphic function](/know/concept/holomorphic-function/), +except for poles on the real axis. +It turns out that $C_{AB}(z)$ must have these properties +for the substitution $i \omega_n \to \omega \!\pm\! i \eta$ to be valid. + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. -- cgit v1.2.3