The Propagator

Many observables are expectation values of fermionic operators. Once normal-ordered, those operators may be replaced by a sum of Wick contractions. Each pair of contracted fermionic operators yields a propagator.

tdg.observable.UU.G(ensemble)

The equal-time propagator that is the contraction of \(\tilde{\psi}^\dagger_{a\sigma} \tilde{\psi}_{b\tau}\) where \(a\) and \(b\) are sites and \(\sigma\) and \(\tau\) are spin indices.

\[\mathcal{G}^{\sigma\tau}_{ab} = [ \mathbb{U} (\mathbb{1} + \mathbb{U})^{-1} ]_{ba}^{\tau\sigma}\]

A five-axis tensor: configurations slowest, then \(a\), \(b\), \(\sigma\), and \(\tau\).

tdg.observable.UU.G_momentum(ensemble)

The equal-time propagator that is the contraction of \(N_x^{-2} \tilde{\psi}^\dagger_{k\sigma} \tilde{\psi}_{q\tau}\) where \(k\) and \(q\) are integer momenta and \(\sigma\) and \(\tau\) are spin indices.

\[\mathcal{G}^{\sigma\tau}_{kq} = \frac{1}{N_x^2} \sum_{xy} e^{+2\pi i k x / N_x} \mathcal{G}^{\sigma\tau}_{xy} e^{-2\pi i q y / N_x}\]

A five-axis tensor: configurations slowest, then \(k\), \(q\), \(\sigma\), and \(\tau\).