Categories: Physics, Quantum mechanics.

Matsubara Green’s function

The Matsubara Green’s function is an imaginary-time version of the real-time Green’s functions. We define as follows in the imaginary-time 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 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 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 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, 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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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