Find the poles of the Green's function G_{c0} for the physical electron
G_{c0}(k\omega) = \frac{1}{4N} \sum_{q} \left[
\left(1+\frac{\lambda}{\omega_q}\right) \left(1+\frac{\xi_h(q-k)}{E_h(q-k)}\right) \frac{n_q + f(E_h(q-k))}{i\omega - \omega_q + E_h(q-k)}
+ \left(1+\frac{\lambda}{\omega_q}\right) \left(1-\frac{\xi_h(q-k)}{E_h(q-k)}\right) \frac{n_q + 1 - f(E_h(q-k))}{i\omega - \omega_q - E_h(q-k)}
- \left(1-\frac{\lambda}{\omega_q}\right) \left(1+\frac{\xi_h(q-k)}{E_h(q-k)}\right) \frac{n_q + 1 - f(E_h(q-k))}{i\omega + \omega_q + E_h(q-k)}
- \left(1-\frac{\lambda}{\omega_q}\right) \left(1-\frac{\xi_h(q-k)}{E_h(q-k)}\right) \frac{n_q + f(E_h(q-k))}{i\omega + \omega_q - E_h(q-k)} \right]
lambda is the boson chemical potential - treat as a free parameter (scaled with t0?)
at zero temperature, need to fix D1, mu, and F0 by self-consistent equations used before
at higher temperatures, beta is a free parameter; fix beta_c by looking at point where F0 = 0
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self-consistent equation:
-interface function "absError(val float64, env Environment) float64" gives lhs - rhs of equation (not abs(lhs - rhs) to avoid singularity in derivative)
-"solve(env Environment) Environment" finds the root of absError by building a closure around absError with env and passing the closure to a root-finder. returned Environment has the appropriate variable set to the root of absError.
self-consistent system:
-interface function "solve(env Environment) Environment" solves all associated self-consistent equations iteratively. returns Environment after system has been solved.
-implementing type holds a slice of self-consistent equations. those with lowest indexes are highest priority to solve. start with eq[0]. then do eq[1] followed by eq[0], iterating until both are solved. continue to eq[2]->1->0, etc.
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requires packages gsl and gsl-devel to build on fedora