# Source code for triqs.dos.hilbert_transform

# Copyright (c) 2013-2018 Commissariat à l'énergie atomique et aux énergies alternatives (CEA)
# Copyright (c) 2013-2018 Centre national de la recherche scientifique (CNRS)
# Copyright (c) 2018-2020 Simons Foundation
#
# This program is free software: you can redistribute it and/or modify
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You may obtain a copy of the License at
#
# Authors: Michel Ferrero, Olivier Parcollet, Nils Wentzell

from triqs.gf import *
import types, string, inspect, itertools
from triqs.dos import DOS, DOSFromFunction
import triqs.utility.mpi as mpi
import numpy

[docs] class HilbertTransform: r""" Computes the Hilbert Transform from a DOS object .. math:: \int_{-\infty}^\infty d \epsilon \rho(\epsilon) \Bigl( (\omega + \mu + I\eta)\mathbf{1} - \hat\varepsilon(\epsilon) - \text{field} - \Sigma(\epsilon) \Bigr)^{-1} """
[docs] def __init__(self, rho): """ :param rho: a DOS object. """ self.dos = rho assert isinstance(rho, DOS), "See Doc. rho must be a DOS" self.__normalize()
#------------------------------------------------------------- def __reduce__(self): return self.__class__, (self.rho) #------------------------------------------------------------- def __normalize(self): # normalisation. dos is not the value of the function, is the weight of the integrals R = numpy.array(self.dos.rho, copy=True) self.rho_for_sum = R eps = self.dos.eps R[0] *= (eps[1] - eps[0]) R[-1] *= (eps[-1] - eps[-2]) for i in range(1, eps.shape[0] - 1): R[i] *= (eps[i+1] - eps[i])/2+(eps[i] - eps[i-1])/2 R /= numpy.sum(R) #-------------------------------------------------------------
[docs] def __call__ (self, Sigma, mu=0, eta=0, field=None, epsilon_hat=None, result=None, n_points_integral=None, test_convergence=None): r""" Compute the Hilbert Transform Parameters ----------- mu: float eta: float Sigma: a GFBloc or a function epsilon-> GFBloc field: anything that can added to the GFBloc Sigma, e.g.: * an Array_with_GFBloc_Indices (same size as Sigma) * a GBloc epsilon_hat: a function that takes a 1d array eps[i] and returns 3d-array eps[i,:,:] where the:,: has the matrix structure of Sigma. Default: eps[i] * Identity_Matrix Used only when DOS is a DOSFromFunction: n_points_integral: How many points to use. If None, use the Npts of construction test_convergence: If defined, it will refine the grid until CV is reached starting from n_points_integral and multiplying by 2 Returns -------- Returns the result. If provided, use result to compute the result locally. """ # we suppose here that self.eps, self.rho_for_sum such that # H(z) = \sum_i self.rho_for_sum[i] * (z- self.eps[i])^-1 # Check Sigma # case 1) Sigma is a Gf if Sigma.__class__.__name__[0:2] == 'Gf': model = Sigma Sigma_fnt = False # case 2) Sigma is a function returning a Gf else: assert callable(Sigma), "If Sigma is not a Gf it must be a function" assert len(inspect.getargspec(Sigma)[0]) == 1, "Sigma must be a function of a single variable" model = Sigma(self.dos.eps[0]) Sigma_fnt = True # Check that Sigma is square N1, N2 = model.target_shape assert N1 == N2, "Sigma must be a square Gf" # Check result if result: assert result.target_shape == (N1, N2), "Size of result and Sigma mismatch" else: result = model.copy() if not(isinstance(self.dos, DOSFromFunction)): assert n_points_integral==None and test_convergence == None, " Those parameters can only be used with an dos_from_function" if field !=None: try: result += field except: assert 0, "field cannot be added to the Green function blocks !. Cf Doc" def HT(res): import triqs.utility.mpi as mpi # First compute the eps_hat array eps_hat = epsilon_hat(self.dos.eps) if epsilon_hat else numpy.array( [ x* numpy.identity (N1) for x in self.dos.eps] ) assert eps_hat.shape[0] == self.dos.eps.shape[0], "epsilon_hat function behaves incorrectly" assert eps_hat.shape[1] == eps_hat.shape[2], "epsilon_hat function behaves incorrectly (result not a square matrix)" assert N1 == eps_hat.shape[1], "Size of Sigma and of epsilon_hat mismatch" res.zero() # Perform the sum over eps[i] tmp, tmp2 = res.copy(), res.copy() tmp << iOmega_n + mu + eta * 1j if not(Sigma_fnt): tmp -= Sigma if field != None: tmp -= field # I slice all the arrays on the node. Cf reduce operation below. for d, e_h, e in zip(*[mpi.slice_array(A) for A in [self.rho_for_sum, eps_hat, self.dos.eps]]): tmp2.copy_from(tmp) tmp2 -= e_h if Sigma_fnt: tmp2 -= Sigma(e) tmp2.invert() tmp2 *= d res += tmp2 # sum the res GF of all nodes and returns the results on all nodes... # Cf Boost.mpi.python, collective communicator for documentation. # The point is that res is pickable, hence can be transmitted between nodes without further code... res << mpi.all_reduce(mpi.world, res, lambda x, y: x+y) mpi.barrier() # END of HT def test_distance(G1, G2, dist): def f(G1, G2): dS = max(abs(G1.data - G2.data).flatten()) aS = max(abs(G1.data).flatten()) return dS <= aS*dist #return reduce(lambda x, y: x and y, [f(g1, g2) for (i1, g1), (i2, g2) in izip(G1, G2)]) return f(G1, G2) # for block function, the previous one is for GF functions if isinstance (self.dos, DOSFromFunction): if not(n_points_integral): # if not defined, use the defaults given at construction of the dos n_points_integral= len(self.dos.eps) else: self.dos._DOS__f(n_points_integral) self.__normalize() HT(result) nloop, test = 1, 0 while test_convergence and nloop < 10 and (nloop == 1 or test > test_convergence): if nloop>1: self.dos._DOS__f(n_points_integral) self.__normalize() result_old = result.copy() result = DOS.HilbertTransform(self, Sigma, mu, eta, epsilon_hat, result) test = test_distance(result, result_old, test_convergence) n_points_integral *=2 else: # Ordinary DOS HT(result) return result