Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight

Information about the pairing mechanism for superconductivity is contained in the spectral weight for the anomalous (Gorkov) Green's function. In the most general case, this spectral weight can change sign on the positive real axis or even be complex in the presence of broken time-reversal symmetry. This renders impossible the direct analytic continuation with maximum entropy methods of numerical results obtained for the anomalous Green's function either in Matsubara frequency or in imaginary time. Here we show that one can define auxiliary Green's functions that allow one to extract the anomalous spectral weight in the most general multiorbital spin-singlet or triplet cases with no particular symmetry. As a simple example, we treat the case of lead in Eliashberg theory: Maximum entropy analytic continuation with auxiliary Green's functions (MaxEntAux) allows us to recover spectral functions that change sign on the positive real axis. The approach can be extended to transport quantities.

Figure 1

Solid magenta and green lines: real and imaginary parts of the gap function extracted from Ref. [10]. Black solid lines: spline interpolations of the data. Note that these interpolations are extrapolated within the gap with a polynomial function that satisfies the continuity of the gap function and its derivative, and has a zero slope at $\omega =0$. $\mathrm{Im}\mathrm{\Delta }$ is odd in $\omega$ and $\mathrm{Re}\mathrm{\Delta }$ even.

Figure 2

Red solid lines: spline interpolations of the real and imaginary parts of the gap function extracted from Ref. [10], before performing the maximum-entropy analytic continuation. Blue solid lines: real and imaginary parts of the gap function obtained from Eq. (13), after performing the maximum-entropy analytic continuation.

Figure 3

Red solid lines: anomalous spectral function and cumulative order parameter extracted from Eq. (8), before performing the maximum-entropy analytic continuation. Blue solid lines: anomalous spectral function and cumulative order parameter obtained from Eq. (12), after performing the maximum-entropy analytic continuation. Dashed lines: frequency of the maximum of each cumulative order parameter. Green dotted line: value of the superconducting order parameter ${\phi }_{SC}$.