(Anti-)Periodicity

Note that the Heisenberg representation the imaginary time dependence of creation and annihilation operators are not conjugated, i.e. \(c(\tau) \equiv e^{\tau H} c e^{-\tau H}\) and \(c^\dagger(\tau) \equiv e^{\tau H} c^\dagger e^{-\tau H}\)

\[G_{a\bar{b}}(\tau) \equiv - \langle \mathcal{T} a(\tau) \bar{b}(0) \rangle \equiv - \frac{1}{\mathcal{Z}} \textrm{Tr} \big[ \mathcal{T} e^{-\int_0^\beta d\bar{\tau} \, H(\bar{\tau})} a(\tau) \bar{b} \big]\]

To derive the boundary conditions we consider two cases. First, for \(0 < \tau < \beta\) we have

\[\begin{split}G_{a\bar{b}}(\tau) \Big|_{0 < \tau < \beta} & = - \langle a(\tau) \bar{b}(0) \rangle = - \frac{1}{\mathcal{Z}} \textrm{Tr} \big[ e^{-\beta H} e^{\tau H} a e^{-\tau H} \bar{b} \big] \\ & = - \frac{1}{\mathcal{Z}} \textrm{Tr} \big[ e^{-\beta H} \bar{b}(0) e^{(\tau - \beta) H} a e^{-(\tau - \beta) H} \big] = - \langle \bar{b}(0) a(\tau - \beta) \rangle \\ & = - \xi \langle \mathcal{T} a(\tau - \beta) \bar{b}(0) \rangle \Big|_{0 < \tau < \beta} = \xi G_{a\bar{b}}(\tau - \beta) \Big|_{0 < \tau < \beta}\end{split}\]

while for \(-\beta < \tau < 0\) one get

\[\begin{split}G_{a\bar{b}}(\tau) \Big|_{- \beta < \tau < 0} & = - \xi \langle \bar{b}(0) a(\tau) \rangle = - \frac{\xi}{\mathcal{Z}} \textrm{Tr} \big[ e^{-\beta H} \bar{b} e^{\tau H} a e^{-\tau H} \big] \\ & = - \frac{\xi}{\mathcal{Z}} \textrm{Tr} \big[ e^{-\beta H} e^{(\tau + \beta) H} a e^{-(\tau + \beta) H} \bar{b}(0) \big] = - \xi \langle a(\tau + \beta) \bar{b}(0) \rangle \\ & = - \xi \langle \mathcal{T} a(\tau + \beta) \bar{b}(0) \rangle \Big|_{-\beta < \tau < 0} = \xi G_{a\bar{b}}(\tau + \beta) \Big|_{-\beta < \tau < 0}.\end{split}\]

Thus we see that the single-particle Green’s function \(G_{a\bar{b}}(\tau)\) is \(\beta\) (anti-)periodic on \(\tau \in [\beta, -\beta]\):

\[\begin{split}G_{a\bar{b}}(\tau) \Big|_{0 < \tau < \beta} & = \xi G_{a\bar{b}}(\tau - \beta) \Big|_{0 < \tau < \beta}, \\ G_{a\bar{b}}(\tau) \Big|_{-\beta < \tau < 0} & = \xi G_{a\bar{b}}(\tau + \beta) \Big|_{-\beta < \tau < 0}.\end{split}\]

Two-particle Green’s functions

The (anti-)periodicity properties can be generalized to two-particle Green’s functions in imaginary time. Consider now \(G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}},\tau_b, \tau_{\bar{c}}, \tau_d=0)\), the cyclic property of the trace and the time-ordering operator (assuming that all operators are either fermionic or bosonic) then yield in the same way

\[G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}, \tau_d=0) \equiv \langle \mathcal{T} \bar{a}(\tau_{\bar{a}}) b(\tau_b) \bar{c}(\tau_{\bar{c}}) d(0) \rangle\]

As an example we take the case \(\beta > \tau_{\bar{a}} > \tau_b, \tau_{\bar{c}} > 0\):

\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}, \tau_d=0) & \equiv \langle \bar{a}(\tau_{\bar{a}}) \big[ \mathcal{T} b(\tau_b) \bar{c}(\tau_{\bar{c}}) \big] d(0) \rangle \\ & = \langle \big[ \mathcal{T} b(\tau_b) \bar{c}(\tau_{\bar{c}}) \big] d(0) \bar{a}(\tau_{\bar{a}} - \beta) \rangle \\ & = \xi \langle \mathcal{T} \bar{a}(\tau_{\bar{a}} - \beta) b(\tau_b) \bar{c}(\tau_{\bar{c}}) d(0) \rangle \\ & = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}} - \beta, \tau_b, \tau_{\bar{c}}, 0).\end{split}\]

In the same way the three periodicity relations read

\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_{\bar{a}} > \tau_b, \tau_{\bar{c}} > 0} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}} - \beta, \tau_b, \tau_{\bar{c}}, 0) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_{b} > \tau_{\bar{a}}, \tau_{\bar{c}} > 0} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}}, \tau_b - \beta, \tau_{\bar{c}}, 0) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_{\bar{c}} > \tau_{\bar{a}}, \tau_{b} > 0} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}}, \tau_b, \tau_{\bar{c}} - \beta, 0)\end{split}\]

and the second triple of relations become

\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_b, \tau_{\bar{c}} > 0 > \tau_{\bar{a}} > -\beta} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}} + \beta, \tau_b, \tau_{\bar{c}}, 0) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_{\bar{a}}, \tau_{\bar{c}} > 0 > \tau_{b} > -\beta} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}}, \tau_b + \beta, \tau_{\bar{c}}, 0) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}}) \Big|_{\beta > \tau_{\bar{a}}, \tau_{b} > 0 > \tau_{\bar{c}} > -\beta} = \xi G^{(2)}_{\bar{a}b\bar{c}d}( \tau_{\bar{a}}, \tau_b, \tau_{\bar{c}} + \beta, 0).\end{split}\]

Kubo-Martin-Schwinger (KMS) boundary conditions

The boundary conditions in imaginary time for the Green’s functions are generated by the commutation relation \([a, \bar{b}]_{-\xi} = a\bar{b} - \xi \bar{b}a = \delta_{ab}\), where \(\xi = \pm 1\) for bosons and fermions respectively

\[\begin{split}G_{a\bar{b}}(0^+) = -\langle \mathcal{T} a(0^+) \bar{b}(0) \rangle = -\langle a \bar{b} \rangle, \\ G_{a\bar{b}}(0^-) = -\langle \mathcal{T} a(0^-) \bar{b}(0) \rangle = - \xi \langle \bar{b} a \rangle,\end{split}\]

so that the boundary condition at \(\tau = 0^\pm\) is

\[G_{a\bar{b}}(0^+) - G_{a\bar{b}}(0^-) = -\langle a \bar{b} - \xi \bar{b} a \rangle = -\langle [a, \bar{b}]_{-\xi} \rangle = -\delta_{ab}.\]

Using the periodicity relation \(G_{a\bar{b}}(0^-) = \xi G_{a\bar{b}}(\beta^-)\) we finally arrive at the boundary condition restricted to \(\beta > \tau > 0\)

\[G_{a\bar{b}}(0^+) - \xi G_{a\bar{b}}(\beta^-) = - \delta_{ab}.\]

Note

The anomalous Green’s functions has the simpler boundary condition

\[\begin{split}G_{ab}(0^+) - \xi G_{ab}(\beta^-) = 0 \\ G_{\bar{a}\bar{b}}(0^+) - \xi G_{\bar{a}\bar{b}}(\beta^-) = 0\end{split}\]

since \([a, b]_{-\xi} = 0\) and \([\bar{a}, \bar{b}]_{-\xi} = 0\).

Generalization to two-particle Green’s functions

For the two-particle Green’s function the KMS boundary conditions generalize to relations incorporating the single particle Green’s function.

\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(0^+,\tau_b, \tau_{\bar{c}}) - \xi G^{(2)}_{\bar{a}b\bar{c}d}(\beta^-,\tau_b, \tau_{\bar{c}}) = \xi \delta_{ad} G_{b\bar{c}}(\tau_b - \tau_{\bar{c}}) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, 0^+) - \xi G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau_b, \beta^-) = \delta_{cd} G_{b\bar{a}}(\tau_b - \tau_{\bar{a}}) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, 0^+, \tau_{\bar{c}}) - \xi G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \beta^-, \tau_{\bar{c}}) = 0\end{split}\]

Thus the discontinuities at \(\tau_{\bar{a}}=0\) and \(\tau_{\bar{c}}=0\) are non-trivial and given by the single-particle Green’s function.

The two additional discontinuities in \(\tau_{\bar{a}}, \tau_b, \tau_{\bar{c}} \in [\beta, 0]\) are the three equal time planes \(\tau_{\bar{a}} = \tau_b\), \(\tau_b = \tau_{\bar{c}}\), and \(\tau_{\bar{a}} = \tau_{\bar{c}}\).

\[\begin{split}G^{(2)}_{\bar{a}b\bar{c}d}(\tau^+, \tau^-, \tau_{\bar{c}}) - G^{(2)}_{\bar{a}b\bar{c}d}(\tau^-, \tau^+, \tau_{\bar{c}}) = -\delta_{bc} G_{d\bar{a}}(\beta - \tau_{\bar{a}}) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau^+, \tau^-) - G^{(2)}_{\bar{a}b\bar{c}d}(\tau_{\bar{a}}, \tau^-, \tau^+) = - \delta_{bc} G_{d\bar{a}}( \beta - \tau_{\bar{a}} ) \\ G^{(2)}_{\bar{a}b\bar{c}d}(\tau^+, \tau_b, \tau^-) - G^{(2)}_{\bar{a}b\bar{c}d}(\tau^-, \tau_b, \tau^+) = 0\end{split}\]