Source code for kdotp_symmetry._symmetric_hamiltonian

# © 2017-2018, ETH Zurich, Institut für Theoretische Physik
# Author:  Dominik Gresch <>
Defines functions to construct the basis of the symmetry-constrained Hamiltonian.

import sympy as sp
from sympy.physics.quantum import TensorProduct
import numpy as np
import scipy.linalg as la
from fsc.export import export

from ._expr_utils import expr_to_vector, matrix_to_expr_operator
from ._repr_utils import hermitian_to_vector, hermitian_basis, repr_to_matrix_operator, check_orthogonal, frobenius_product
from ._linalg import intersection_basis, nullspace_blocked
from ._to_matrix import to_matrix
from ._logging_setup import LOGGER

[docs]@export def symmetric_hamiltonian( *symmetry_operations, expr_basis, repr_basis='auto', check_repr_basis=False ): r""" Calculates the basis of the symmetric Hamiltonian for a given set of symmetry operations. :param symmetry_operations: The symmetry operations that the Hamiltonian should respect. :type symmetry_operations: :py:class:`symmetry_representation.SymmetryOperation` :param expr_basis: The basis for the :math:`\mathbf{k}`-functions that are considered. :type expr_basis: :py:class:`list` of :py:mod:`sympy` expressions :param repr_basis: The basis for the hermitian matrices, with the same size as the representations. By default, the :py:func:`.hermitian_basis` of the appropriate size is used. :type repr_basis: :py:class:`list` of :py:mod:`sympy` matrices :param check_repr_basis: Flag to enable explicitly checking the orthogonality of ``repr_basis``. :type check_repr_basis: bool :returns: Basis for the symmetric Hamiltonian, as a :py:class:`list` of :py:mod:`sympy` matrix expressions. """ if any(sym_op.numeric for sym_op in symmetry_operations): raise ValueError( 'Symmetry operations used in kdotp-symmetry can not be numeric.' ) expr_dim = len(expr_basis) # for sympy or numpy matrices try: repr_matrix_size = symmetry_operations[0].repr.matrix.shape[0] # for plain lists -- this doesn't work for sympy matrices because # their 'len' is the total number of elements except AttributeError: repr_matrix_size = len(symmetry_operations[0].repr.matrix) if repr_basis == 'auto': repr_basis = hermitian_basis(repr_matrix_size) if check_repr_basis: check_orthogonal(repr_basis) repr_dim = len(repr_basis) repr_basis_norm_squares = [frobenius_product(b, b) for b in repr_basis] full_dim = expr_dim * repr_dim full_basis = [ sp.Matrix(x) for x in np.outer(expr_basis, repr_basis). reshape(full_dim, repr_matrix_size, repr_matrix_size).tolist() ] invariant_bases = [] for sym_op in symmetry_operations:'Calculating matrix form of expression.') expr_mat = to_matrix( operator=matrix_to_expr_operator( sym_op.rotation_matrix, repr_has_cc=sym_op.repr.has_cc ), basis=expr_basis, to_vector_fct=expr_to_vector )'Calculating matrix form of representation.') repr_mat = to_matrix( operator=repr_to_matrix_operator( sym_op.repr.matrix, complex_conjugate=sym_op.repr.has_cc ), basis=repr_basis, to_vector_fct=hermitian_to_vector, to_vector_kwargs=dict(basis_norm_squares=repr_basis_norm_squares) ) # outer product'Calculating outer product.') full_mat = TensorProduct(expr_mat, repr_mat) # get Eig(F \ocross G, 1) basis mat = full_mat - sp.eye(full_dim)'Calculating nullspace.') curr_basis = np.array(nullspace_blocked(mat, simplify=sp.nsimplify) ).tolist() if len(curr_basis) != _numeric_nullspace_dim(mat): raise ValueError( 'Analytic and numeric dimensions of the nullspace of the matrix {mat} do not match' .format(mat=mat) ) invariant_bases.append(curr_basis)'Calculating basis intersection.') basis_vectors = intersection_basis(*invariant_bases)'Expanding basis vectors.') basis_vectors_expanded = [] for vec in basis_vectors: basis_vectors_expanded.append( sum((v * b for v, b in zip(vec, full_basis)), sp.zeros(repr_matrix_size)) ) return basis_vectors_expanded
def _numeric_nullspace_dim(mat): """Numerically computes the nullspace dimension of a matrix.""" mat_numeric = np.array(mat.evalf().tolist(), dtype=complex) eigenvals = la.eigvals(mat_numeric) return np.sum(np.isclose(eigenvals, np.zeros_like(eigenvals)))