triqs.gfs.semicirc.g_semicirc_tau

triqs.gfs.semicirc.g_semicirc_tau(tau, beta, D, p=12, n_levels=None)[source]

Semi-circular Green’s function on the imaginary-time axis.

Evaluates

\[G(\tau) = -\frac{2}{\pi}\int_0^1 \frac{e^{-\tau D\omega} + e^{(\tau-\beta)D\omega}} {1 + e^{-\beta D\omega}}\, \sqrt{1-\omega^2}\,d\omega\]

using dyadic panel quadrature on \([0,1]\):

Gauss-Jacobi (\(\alpha=1/2,\,\beta_J=0\)) on the panel \([1/2,\,1]\) to absorb the \(\sqrt{1-\omega}\) singularity.
Gauss-Legendre on all other panels, which are dyadically refined toward \(\omega = 0\).

Parameters:

taufloat or array_like: Imaginary time(s), \(0 \le \tau \le \beta\).
betafloat: Inverse temperature.
Dfloat: Half-bandwidth.
pint: Quadrature order per panel (default 12).
n_levelsint or None: Number of dyadic refinement levels. Default \(\lceil\log_2(\beta D)\rceil\) so the smallest panel width \(2^{-n} < 1/(\beta D)\).

Returns:

Gfloat or ndarray: Value of the Green’s function at the supplied imaginary times.