The Variational Cluster Approximation

This submodule provides functions that implements the Variational Cluster Approximation (VCA). The class VCA implements the VCA method. Defining an instance of this class performs the VCA procedure.

class VCA(model, var2sef=None, varia=None, start=None, steps=0.01, accur=0.0001, max=100, 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, consistency_check=False, verb=True)

class containing the elements of a VCA computation. The constructor executes the computation.

Parameters:

  • model (lattice_model) – the (unique) model on which the computation is based

  • var2sef – function that converts variational parameters to model parameters

  • varia (str or [str]) – variational parameters (list or tuple)

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

  • hartree_self_consistent (bool) – 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)

  • verb (bool) – if True (default) prints ample progress messages

Returns:

None

Variables:

  • model (lattice_model) – (unique) model on which the computation is based

  • I – current model instance (changes in the course of the computation)

  • hessian – Hessian matrix of the Potthoff functional (matrix of second derivatives)

Other VCA-related functions

plot_sef(model, param, prm, file='sef.tsv', accur_SEF=0.0001, hartree=None, show=True, symmetrized_operator=None, consistency_check=False)

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:

  • model (lattice_model) – the lattice model

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

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

  • model – the lattice model on which the computation is based

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

  • file (str) – name of the file to which intermediate results are written

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

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

  • symmetrized_operator (str) – name of an operator with respect to which the functional is symmetrized

  • consistency_check (bool) – if True, checks the ground state consistency at each point

Returns:

None

plot_GS_energy(model, 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:

  • model (lattice_model) – the lattice model on which the computation is based

  • 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 (matplotlib.axes.Axes) – 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

transition_line(model, 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:

  • model (lattice_model) – the lattice model on which the computation is based

  • 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 (bool) – 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