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.