# -*- coding: utf-8 -*-
################################################################################
#
# TPRF: Two-Particle Response Function (TPRF) Toolbox for TRIQS
#
# Copyright (C) 2018 by The Simons Foundation
# Author: H. U.R. Strand
#
# TPRF is free software: you can redistribute it and/or modify it under the
# terms of the GNU General Public License as published by the Free Software
# Foundation, either version 3 of the License, or (at your option) any later
# version.
#
# TPRF 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 should have received a copy of the GNU General Public License along with
# TPRF. If not, see <http://www.gnu.org/licenses/>.
#
################################################################################
""" TRIQS, TPRF: Hartree-Fock static response function calculator
Author: Hugo U. R. Strand, hugo.strand@gmail.com (2018)
"""
# ----------------------------------------------------------------------
import sys
import itertools
import numpy as np
# ----------------------------------------------------------------------
class BaseResponse(object):
def __init__(self, solver, eps=1e-9):
print(self.logo())
self.solver = solver
self.eps = eps
self.beta = self.solver.beta
self.e_k = self.solver.e_k_MF.copy()
self.n_k = len(self.e_k.mesh)
self.norb = self.e_k.target_shape[0]
self.shape_ab = self.e_k.target_shape
self.shape_abcd = list(self.shape_ab) + list(self.shape_ab)
self.shape_AB = [self.norb**2, self.norb**2]
self.shape_kabcd = [self.n_k] + self.shape_abcd
self.shape_kAB = [self.n_k] + self.shape_AB
I_ab = np.eye(self.norb)
self.e_k.data[:] -= self.solver.mu * I_ab
print('norb =', self.norb)
print('shape_abcd =', self.shape_abcd)
print('shape_AB =', self.shape_AB)
print('beta =', self.beta)
# ------------------------------------------------------------------
def _compute_drho_dop(self, op):
drho_k = self._compute_drho_k_dop(op)
drho = np.sum(drho_k, axis=0) / self.n_k
return drho
# ------------------------------------------------------------------
def _compute_drho_k_dop(self, op):
beta, eps = self.beta, self.eps
E_p = self.e_k.data.copy() + eps*op[None, ...]
E_m = self.e_k.data.copy() - eps*op[None, ...]
e_p, U_p = np.linalg.eigh(E_p)
e_m, U_m = np.linalg.eigh(E_m)
fermi = lambda e, beta: 1./(np.exp(beta * e) + 1)
rho_k_p = np.einsum('kab,kb,kcb->kac', U_p, fermi(e_p, beta), np.conj(U_p))
rho_k_m = np.einsum('kab,kb,kcb->kac', U_m, fermi(e_m, beta), np.conj(U_m))
drho_k = (rho_k_p - rho_k_m) / (2. * eps)
return drho_k
# ----------------------------------------------------------------------
def _compute_chi0_ab(self):
chi0_ab = np.zeros(self.shape_ab, dtype=complex)
for a in range(self.norb):
F_a = np.zeros(self.shape_ab)
F_a[a, a] = 1.
chi0_b = -np.diag(self._compute_drho_dop(F_a))
chi0_ab[a] = chi0_b
return chi0_ab
# ----------------------------------------------------------------------
def _compute_R_abcd(self, field_prefactor=1.):
R_abcd = np.zeros(self.shape_abcd, dtype=complex)
for a, b in itertools.product(list(range(self.norb)), repeat=2):
F_ab = np.zeros(self.shape_ab, dtype=complex)
F_ab[a, b] += field_prefactor
F_ab[b, a] += np.conj(field_prefactor)
R_cd = -self._compute_drho_dop(F_ab)
R_abcd[b, a, :, :] = R_cd
return R_abcd
# ----------------------------------------------------------------------
def _compute_R_kabcd(self, field_prefactor=1.):
R_kabcd = np.zeros(self.shape_kabcd, dtype=complex)
for a, b in itertools.product(list(range(self.norb)), repeat=2):
F_ab = np.zeros(self.shape_ab, dtype=complex)
F_ab[a, b] += field_prefactor
F_ab[b, a] += np.conj(field_prefactor)
R_kcd = -self._compute_drho_k_dop(F_ab)
R_kabcd[:, b, a, :, :] = R_kcd
return R_kabcd
# ----------------------------------------------------------------------
def _compute_chi0_abcd(self):
chi0_kabcd = self._compute_chi0_kabcd()
chi0_abcd = np.sum(chi0_kabcd, axis=0) / self.n_k
return chi0_abcd
# ----------------------------------------------------------------------
def _compute_chi0_kabcd(self):
R_r_kabcd = self._compute_R_kabcd(field_prefactor=1.0)
R_i_kabcd = self._compute_R_kabcd(field_prefactor=1.j)
chi0_kabcd = 0.5 * (R_r_kabcd + R_i_kabcd.imag)
return chi0_kabcd
# ----------------------------------------------------------------------
def logo(self):
if 'UTF' in sys.stdout.encoding:
txt = """
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐ ┬─┐┌─┐┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤───├┬┘├─┘├─┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└ ┴└─┴ ┴ ┴
Triqs: Hartree-Fock Random Phase Approximation susceptibility
"""
else:
txt = r"""
_____ ___ ___ ___ ___ _ _ ___ ___ ___ _
|_ _| _ \_ _/ _ \/ __| | || | __|__| _ \ _ \/_\
| | | /| | (_) \__ \ | __ | _|___| / _/ _ \
|_| |_|_\___\__\_\___/ |_||_|_| |_|_\_|/_/ \_\
Triqs: Hartree-Fock Random Phase Approximation susceptibility
"""
return txt
# ----------------------------------------------------------------------
[docs]
class HartreeFockResponse(BaseResponse):
""" Hartree-Fock linear response calculator.
Parameters
----------
hartree_fock_solver : HartreeFockSolver instance
Converged Hartree-Fock solver.
eps : float
Step size in finite difference linear response calculation.
"""
def __init__(self, hartree_fock_solver, eps=1e-9):
super(HartreeFockResponse, self).__init__(hartree_fock_solver)
self.hfs = self.solver
I_AB = np.matrix(np.eye(self.shape_AB[0]))
U_AB = self._to_matrix_AB(self.hfs.U_abcd)
chi0_AB = self._to_matrix_AB(self._compute_chi0_abcd())
chi_AB = np.linalg.inv(I_AB - chi0_AB * U_AB) * chi0_AB
self.chi0_abcd = self._to_tensor_abcd(chi0_AB)
self.chi_abcd = self._to_tensor_abcd(chi_AB)
#self.chi0_kabcd = self._compute_chi0_kabcd()
#np.testing.assert_array_almost_equal(
# self.chi0_abcd, np.sum(self.chi0_kabcd, axis=0)/self.n_k)
def mode_decomposition(self):
U_AB = self._to_matrix_AB(self.hfs.U_abcd)
chi0_AB = self._to_matrix_AB(self.chi0_abcd)
e = np.linalg.eigvals(np.asmatrix(chi0_AB) * np.asmatrix(U_AB))
idx = np.argsort(e.real)
e = e[idx]
e = 1./(1. - e.real)
return e
def _to_matrix_AB(self, tensor_abcd):
matrix_AB = np.matrix(tensor_abcd.reshape(self.shape_AB))
return matrix_AB
def _to_tensor_abcd(self, matrix_AB):
tensor_abcd = np.array(matrix_AB).reshape(self.shape_abcd)
return tensor_abcd
def __check_op(self, op):
assert( op.shape == self.shape_ab )
def bare_response(self, op1, op2):
self.__check_op(op1)
self.__check_op(op2)
chi0_op1op2 = np.einsum('ab,abcd,cd->', op1, self.chi0_abcd, op2)
return chi0_op1op2
def response(self, op1, op2):
self.__check_op(op1)
self.__check_op(op2)
chi_op1op2 = np.einsum('ab,abcd,cd->', op1, self.chi_abcd, op2)
return chi_op1op2
def compute_chi0_k(self):
from triqs.gf import Gf
chi0_k = Gf(mesh=self.e_k.mesh, target_shape=self.shape_abcd)
chi0_k.data[:] = self.chi0_kabcd
return chi0_k
# ----------------------------------------------------------------------
[docs]
class HartreeResponse(BaseResponse):
""" Hartree linear response calculator.
Parameters
----------
hartree_solver : HartreeSolver instance
Converged Hartree solver.
eps : float
Step size in finite difference linear response calculation.
"""
def __init__(self, hartree_solver, eps=1e-9):
super(HartreeResponse, self).__init__(hartree_solver)
I_ab = np.eye(self.norb)
U_ab = np.asmatrix(self.extract_dens_dens(self.solver.U_abcd))
chi0_ab = np.asmatrix(self._compute_chi0_ab())
chi_ab = chi0_ab * np.linalg.inv(I_ab - U_ab * chi0_ab)
self.chi0_ab = np.array(chi0_ab)
self.chi_ab = np.array(chi_ab)
def __check_op(self, op):
""" Operators have to be diagonal in the Hartree approx """
assert( op.shape == self.e_k.target_shape )
np.testing.assert_almost_equal(
op - np.diag(np.diag(op)), np.zeros_like(op))
def bare_response(self, op1, op2):
self.__check_op(op1)
self.__check_op(op2)
chi0_op1op2 = np.einsum('aa,ab,bb->', op1, self.chi0_ab, op2)
return chi0_op1op2
def response(self, op1, op2):
self.__check_op(op1)
self.__check_op(op2)
chi_op1op2 = np.einsum('aa,ab,bb->', op1, self.chi_ab, op2)
return chi_op1op2
def extract_dens_dens(self, chi_abcd):
norb = chi_abcd.shape[0]
chi_ab = np.zeros((norb, norb), dtype=complex)
for i1, i2 in itertools.product(list(range(norb)), repeat=2):
chi_ab[i1, i2] = chi_abcd[i1, i1, i2, i2]
return chi_ab
# ----------------------------------------------------------------------