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SumkDFT: analyse_block_structure_from_gf

This commit is contained in:
Gernot J. Kraberger 2018-02-27 19:54:33 +01:00
parent 8ebb0c3c29
commit 25218746f4

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@ -25,12 +25,15 @@ import numpy
import pytriqs.utility.dichotomy as dichotomy
from pytriqs.gf import *
import pytriqs.utility.mpi as mpi
from pytriqs.utility.comparison_tests import assert_arrays_are_close
from pytriqs.archive import *
from symmetry import *
from block_structure import BlockStructure
from sets import Set
from itertools import product
from warnings import warn
from scipy import compress
from scipy.optimize import minimize
class SumkDFT(object):
@ -848,6 +851,366 @@ class SumkDFT(object):
elif (ind1 < 0) and (ind2 < 0):
self.deg_shells[ish].append([block1, block2])
def analyse_block_structure_from_gf(self, G, threshold=1.e-5, include_shells=None, analyse_deg_shells = True):
r"""
Determines the block structure of local Green's functions by analysing
the structure of the corresponding non-interacting Green's function.
The resulting block structures for correlated shells are
stored in the :class:`SumkDFT.block_structure <dft.block_structure.BlockStructure>`
attribute.
This is a safer alternative to analyse_block_structure, because
the full non-interacting Green's function is taken into account
and not just the density matrix and Hloc.
Parameters
----------
G : list of BlockGf of GfImFreq or GfImTime
the non-interacting Green's function for each inequivalent correlated shell
threshold : real, optional
If the difference between matrix elements is below threshold,
they are considered to be equal.
include_shells : list of integers, optional
List of correlated shells to be analysed.
If include_shells is not provided all correlated shells will be analysed.
analyse_deg_shells : bool
Whether to call the analyse_deg_shells function
after having finished the block structure analysis
Returns
-------
G : list of BlockGf of GfImFreq or GfImTime
the Green's function transformed into the new block structure
"""
# make a GfImTime from the supplied G
if isinstance(G[0]._first(), GfImFreq):
gf = [BlockGf(name_block_generator = [(name, GfImTime(beta=block.mesh.beta,
indices=block.indices,n_points=len(block.mesh)+1)) for name, block in g_sh])
for g_sh in G]
for ish in range(len(gf)):
for name, g in gf[ish]:
g.set_from_inverse_fourier(G[ish][name])
else:
assert isinstance(G[0]._first(), GfImTime), "G must be a BlockGf of either GfImFreq or GfImTime"
gf = G
# initialize the variables
self.gf_struct_solver = [{} for ish in range(self.n_inequiv_shells)]
self.sumk_to_solver = [{} for ish in range(self.n_inequiv_shells)]
self.solver_to_sumk = [{} for ish in range(self.n_inequiv_shells)]
self.solver_to_sumk_block = [{}
for ish in range(self.n_inequiv_shells)]
# the maximum value of each matrix element of each block and shell
max_gf = [{name:numpy.max(numpy.abs(g.data),0) for name, g in gf[ish]} for ish in range(self.n_inequiv_shells)]
if include_shells is None:
# include all shells
include_shells = range(self.n_inequiv_shells)
for ish in include_shells:
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]:
n_orb = self.corr_shells[self.inequiv_to_corr[ish]]['dim']
# gives an index list of entries larger that threshold
maxgf_bool = (abs(max_gf[ish][sp]) > threshold)
# Determine off-diagonal entries in upper triangular part of the
# Green's function
offdiag = Set([])
for i in range(n_orb):
for j in range(i + 1, n_orb):
if maxgf_bool[i, j]:
offdiag.add((i, j))
# Determine the number of non-hybridising blocks in the gf
blocs = [[i] for i in range(n_orb)]
while len(offdiag) != 0:
pair = offdiag.pop()
for b1, b2 in product(blocs, blocs):
if (pair[0] in b1) and (pair[1] in b2):
if blocs.index(b1) != blocs.index(b2): # In separate blocks?
# Merge two blocks
b1.extend(blocs.pop(blocs.index(b2)))
break # Move on to next pair in offdiag
# Set the gf_struct for the solver accordingly
num_blocs = len(blocs)
for i in range(num_blocs):
blocs[i].sort()
self.gf_struct_solver[ish].update(
[('%s_%s' % (sp, i), range(len(blocs[i])))])
# Construct sumk_to_solver taking (sumk_block, sumk_index) --> (solver_block, solver_inner)
# and solver_to_sumk taking (solver_block, solver_inner) -->
# (sumk_block, sumk_index)
for i in range(num_blocs):
for j in range(len(blocs[i])):
block_sumk = sp
inner_sumk = blocs[i][j]
block_solv = '%s_%s' % (sp, i)
inner_solv = j
self.sumk_to_solver[ish][(block_sumk, inner_sumk)] = (
block_solv, inner_solv)
self.solver_to_sumk[ish][(block_solv, inner_solv)] = (
block_sumk, inner_sumk)
self.solver_to_sumk_block[ish][block_solv] = block_sumk
# transform G to the new structure
full_structure = BlockStructure.full_structure(
[{sp:range(self.corr_shells[self.inequiv_to_corr[ish]]['dim'])
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]}
for ish in range(self.n_inequiv_shells)],None)
G_transformed = [
self.block_structure.convert_gf(G[ish],
full_structure, ish, beta=G[ish].mesh.beta, show_warnings=threshold)
for ish in range(self.n_inequiv_shells)]
if analyse_deg_shells:
self.analyse_deg_shells(G_transformed, threshold, include_shells)
return G_transformed
def analyse_deg_shells(self, G, threshold=1.e-5, include_shells=None):
r"""
Determines the degenerate shells of local Green's functions by analysing
the structure of the corresponding non-interacting Green's function.
The results are stored in the
:class:`SumkDFT.block_structure <dft.block_structure.BlockStructure>`
attribute.
Parameters
----------
G : list of BlockGf of GfImFreq or GfImTime
the non-interacting Green's function for each inequivalent correlated shell
threshold : real, optional
If the difference between matrix elements is below threshold,
they are considered to be equal.
include_shells : list of integers, optional
List of correlated shells to be analysed.
If include_shells is not provided all correlated shells will be analysed.
"""
# initialize
self.deg_shells = [[] for ish in range(self.n_inequiv_shells)]
# helper function
def null(A, eps=1e-15):
""" Calculate the null-space of matrix A """
u, s, vh = numpy.linalg.svd(A)
null_mask = (s <= eps)
null_space = compress(null_mask, vh, axis=0)
return null_space.conjugate().transpose()
# make a GfImTime from the supplied G
if isinstance(G[0]._first(), GfImFreq):
gf = [BlockGf(name_block_generator = [(name, GfImTime(beta=block.mesh.beta,
indices=block.indices,n_points=len(block.mesh)+1)) for name, block in g_sh])
for g_sh in G]
for ish in range(len(gf)):
for name, g in gf[ish]:
g.set_from_inverse_fourier(G[ish][name])
else:
assert isinstance(G[0]._first(), GfImTime), "G must be a BlockGf of either GfImFreq or GfImTime"
gf = G
if include_shells is None:
# include all shells
include_shells = range(self.n_inequiv_shells)
# We consider two blocks equal, if their Green's functions obey
# maybe_conjugate1( v1^dagger G1 v1 ) = maybe_conjugate2( v2^dagger G2 v2 )
# where maybe_conjugate is a function that conjugates the Green's
# function if the flag 'conjugate' is set and the v are unitary
# matrices
#
# for each pair of blocks, we check whether there is a transformation
# maybe_conjugate( T G1 T^dagger ) = G2
# where our goal is to find T
# we just try whether there is such a T with and without conjugation
for ish in include_shells:
for block1 in self.gf_struct_solver[ish].iterkeys():
for block2 in self.gf_struct_solver[ish].iterkeys():
if block1==block2: continue
# check if the blocks are already present in the deg_shells
ind1 = -1
ind2 = -2
for n, ind in enumerate(self.deg_shells[ish]):
if block1 in ind:
ind1 = n
v1 = ind[block1]
if block2 in ind:
ind2 = n
v2 = ind[block2]
# if both are already present, go to the next pair of blocks
if ind1 >= 0 and ind2 >= 0:
continue
gf1 = gf[ish][block1]
gf2 = gf[ish][block2]
# the two blocks have to have the same shape
if gf1.target_shape != gf2.target_shape:
continue
# Instead of directly comparing the two blocks, we
# compare its eigenvalues. As G(tau) is Hermitian,
# they are real and the eigenvector matrix is unitary.
# Thus, if the eigenvalues are equal we can transform
# one block to make it equal to the other (at least
# for tau=0).
e1 = numpy.linalg.eigvalsh(gf1.data[0])
e2 = numpy.linalg.eigvalsh(gf2.data[0])
if numpy.any(abs(e1-e2) > threshold): continue
for conjugate in [False,True]:
if conjugate:
gf2 = gf2.conjugate()
# we want T gf1 T^dagger = gf2
# while for a given tau, T could be calculated
# by diagonalizing gf1 and gf2, this does not
# work for all taus simultaneously because of
# numerical imprecisions
# rather, we rewrite the equation to
# T gf1 = gf2 T
# which is the Sylvester equation.
# For that equation, one can use the Kronecker
# product to get a linear problem, which consists
# of finding the null space of M vec T = 0.
M = numpy.kron(numpy.eye(*gf1.target_shape),gf2.data[0])-numpy.kron(gf1.data[0].transpose(),numpy.eye(*gf1.target_shape))
N = null(M, threshold)
# now we get the intersection of the null spaces
# of all values of tau
for i in range(1,len(gf1.data)):
M = numpy.kron(numpy.eye(*gf1.target_shape),gf2.data[i])-numpy.kron(gf1.data[i].transpose(),numpy.eye(*gf1.target_shape))
# transform M into current null space
M = numpy.dot(M, N)
N = numpy.dot(N, null(M, threshold))
if numpy.size(N) == 0:
break
# no intersection of the null spaces -> no symmetry
if numpy.size(N) == 0: continue
# reshape N: it then has the indices matrix, matrix, number of basis vectors of the null space
N = N.reshape(gf1.target_shape[0], gf1.target_shape[1], -1).transpose([1, 0, 2])
"""
any matrix in the null space can now be constructed as
M = 0
for i in range(N.shape[-1]):
M += y[i]*N[:,:,i]
with coefficients (complex numbers) y[i].
We want to get a set of coefficients y so that M is unitary.
Unitary means M M^dagger = 1.
Thus,
sum y[i] N[:,:,i] y[j].conjugate() N[:,:,j].conjugate().transpose() = eye.
The object N[:,:,i] N[:,:,j] is a four-index object which we call Z.
"""
Z = numpy.einsum('aci,bcj->abij', N, N.conjugate())
"""
function chi2
This function takes a real parameter vector y and reinterprets it as complex.
Then, it calculates the chi2 of
sum y[i] N[:,:,i] y[j].conjugate() N[:,:,j].conjugate().transpose() - eye.
"""
def chi2(y):
# reinterpret y as complex number
y = y.view(numpy.complex_)
ret = 0.0
for a in range(Z.shape[0]):
for b in range(Z.shape[1]):
ret += numpy.abs(numpy.dot(y, numpy.dot(Z[a, b], y.conjugate()))
- (1.0 if a == b else 0.0))**2
return ret
# use the minimization routine from scipy
res = minimize(chi2, numpy.ones(2 * N.shape[-1]))
# if the minimization fails, there is probably no symmetry
if not res.success: continue
# check if the minimization returned zero within the tolerance
if res.fun > threshold: continue
# reinterpret the solution as a complex number
y = res.x.view(numpy.complex_)
# reconstruct the T matrix
T = numpy.zeros(N.shape[:-1], dtype=numpy.complex_)
for i in range(len(y)):
T += N[:, :, i] * y[i]
# transform gf1 using T
G_transformed = gf1.copy()
G_transformed.from_L_G_R(T, gf1, T.conjugate().transpose())
# it does not make sense to check the tails for an
# absolute error because it will usually not hold;
# we could just check the relative error
# (here, we ignore it, reasoning that if the data
# is the same, the tails have to coincide as well)
try:
assert_arrays_are_close(G_transformed.data, gf2.data, threshold)
except (RuntimeError, AssertionError):
# the symmetry does not hold
continue
# Now that we have found a valid T, we have to
# rewrite it to match the convention that
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2),
# where C conjugates if the flag is True
# For each group of degenerate shells, the list
# SK.deg_shells[ish] contains a dict. The keys
# of the dict are the block names, the values
# are tuples. The first entry of the tuple is
# the transformation matrix v, the second entry
# is the conjugation flag
# the second block is already present
# set v1 and C1 so that they are compatible with
# C(T gf1 T^dagger) = gf2
# and with
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
if (ind1 < 0) and (ind2 >= 0):
if conjugate:
self.deg_shells[ish][ind2][block1] = numpy.dot(T.conjugate().transpose(), v2[0].conjugate()), not v2[1]
else:
self.deg_shells[ish][ind2][block1] = numpy.dot(T.conjugate().transpose(), v2[0]), v2[1]
# the first block is already present
# set v2 and C2 so that they are compatible with
# C(T gf1 T^dagger) = gf2
# and with
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
elif (ind1 >= 0) and (ind2 < 0):
if conjugate:
self.deg_shells[ish][ind1][block2] = numpy.dot(T.conjugate(), v1[0].conjugate()), not v1[1]
else:
self.deg_shells[ish][ind1][block2] = numpy.dot(T, v1[0]), v1[1]
# the blocks are not already present
# we arbitrarily choose v1=eye and C1=False and
# set v2 and C2 so that they are compatible with
# C(T gf1 T^dagger) = gf2
# and with
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
elif (ind1 < 0) and (ind2 < 0):
d = dict()
d[block1] = numpy.eye(*gf1.target_shape), False
if conjugate:
d[block2] = T.conjugate(), True
else:
d[block2] = T, False
self.deg_shells[ish].append(d)
def density_matrix(self, method='using_gf', beta=40.0):
"""Calculate density matrices in one of two ways.
@ -1616,3 +1979,38 @@ class SumkDFT(object):
def __set_deg_shells(self,value):
self.block_structure.deg_shells = value
deg_shells = property(__get_deg_shells,__set_deg_shells)
# a helper function
def conjugate_in_tau(gf_im_freq, in_place=False):
""" Calculate the conjugate in tau of a GfImFreq
Parameters
----------
gf_im_freq : GfImFreq of BlockGf
the Green's function
in_place : whether to modify the gf_im_freq object (True) or return a copy (False)
Returns
-------
ret : GfImFreq of BlockGf
the Green's function that has been FT to G(tau), conjugated, and
FT back
"""
if in_place:
ret = gf_im_freq
else:
ret = gf_im_freq.copy()
if isinstance(ret, BlockGf):
for name, gf in ret:
conjugate_in_tau(gf, in_place=True)
else:
""" there is an easier way to do this, namely to make
ret.data[:,:,:] = gf_im_freq.data[::-1,:,:].conjugate()
ret.tail.data[:,:,:] = gf_im_freq.tail.data.conjugate()
but this relies on symmetric Matsubara meshes and is maybe
not safe enough"""
G_tau = GfImTime(beta=gf_im_freq.mesh.beta,
indices=gf_im_freq.indices,n_points=len(gf_im_freq.mesh)+1)
G_tau.set_from_inverse_fourier(gf_im_freq)
ret.set_from_fourier(G_tau.conjugate())
return ret