The Variational Cluster Approximation

This submodule provides functions that implements the Variational Cluster Approximation (VCA)

List of functions

pyqcm.vca.plot_GS_energy(param, prm, clus=0, file=None, plt_ax=None, **kwargs)

Draws a plot of the ground state energy as a function of a parameter param taken from the list prm. The results are going to be appended to ‘GS.tsv’

Parameters

  • param (str) – name of the parameter (independent variable)

  • prm ([float]) – list of values of the parameter

  • clus (int) – label of the cluster (starts at 0)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.vca.plot_sef(param, prm, file='sef.tsv', accur_SEF=0.0001, hartree=None, show=True, symmetrized_operator=None)

Draws a plot of the Potthoff functional as a function of a parameter param taken from the list prm. The results are going to be appended to ‘sef.tsv’

Parameters

  • param (str) – name of the parameter (independent variable)

  • prm ([float]) – list of values of the parameter

  • accur_SEF (float) – precision of the computation of the self-energy functional

  • hartree ((class hartree)) – Hartree approximation couplings (see pyqcm/hartree.py)

  • show (boolean) – if True, the plot is shown on the screen.

Returns

None

pyqcm.vca.transition_line(varia, P1, P1_range, P2, P2_range, delta, verb=False)

Builds the second-order transition line as a function of a control parameter P1. The results are written in the file transition.tsv

Parameters

  • varia (str) – variational parameter

  • P1 (str) – control parameter

  • P1_range ([float]) – an array of values of P1

  • P2 (str) – dependent parameter

  • P2_range ((float)) – 2-uple of values of P2 that bracket the transition

  • verb (boolean) – If True, prints progress

  • delta (float) – at each step, the new bracket for P2 will be P2c +/- delta, P2c being the previous critical value

Returns

None

pyqcm.vca.vca(var2sef=None, names=None, start=None, steps=None, accur=None, max=None, file='vca.tsv', accur_grad=1e-06, max_iter=30, max_iter_diff=None, method='NR', hartree=None, hartree_self_consistent=False, symmetrized_operator=None, var_max_start=None)

Performs a VCA with the QN or NR method

Parameters

  • var2sef – function that converts variational parameters to model parameters

  • names (str or [str]) – names of the variational parameters

  • start (float or [float]) – starting values

  • steps (float or [float]) – initial steps

  • accur (float or [float]) – accuracy of parameters (also step for 2nd derivatives)

  • max (float or [float]) – maximum values that are tolerated

  • accur_grad (float) – max value of gradient for convergence

  • max_iter (int) – maximum number of iterations in the procedure

  • max_iter_diff (float) – optional maximum value of the maximum step in the quasi-Newton method

  • method (str) – method used to optimize (‘SYMR1’, ‘NR’, ‘BFGS’, ‘altNR’, ‘Nelder-Mead’, ‘COBYLA’, ‘Powell’, ‘CG’, ‘minimax’)

  • hartree ((class hartree)) – Hartree approximation couplings (see pyqcm/hartree.py)

  • hartree_self_consistent (boolean) – True if the Hartree approximation is treated in the self-consistent, rather than variational, way.

  • symmetrized_operator (str) – name of an operator wrt which the functional must be symmetrized

  • var_max_start (int) – label of the first variable for which the function is a maximum (minimal vars first, maximal vars last)

Returns

None