def loop_bandwidth(drange, tp, beta, seed='mott gap'):
"""Solves IPT dimer and return Im Sigma_AA, Re Simga_AB
returns list len(betarange) x 2 Sigma arrays
"""
s = []
g = []
w = np.linspace(-6, 6, 2**13)
dw = w[1] - w[0]
gssi = gf.semi_circle_hiltrans(w + 5e-3j - tp - 1)
gsai = gf.semi_circle_hiltrans(w + 5e-3j + tp + 1)
nfp = gf.fermi_dist(w, beta)
for D in drange:
print('D/U: ', D, 'tp/U: ', tp, 'Beta', beta)
(gss, gsa), (ss, sa) = ipt.dimer_dmft(1,
tp,
nfp,
w,
dw,
gssi,
gssi,
conv=1e-2,
t=(D / 2))
g.append((gss, gsa))
s.append((ss, sa))
return np.array(g), np.array(s), w, nfp
def ss_dmft_loop(gloc, w, u_int, beta, conv):
"""DMFT Loop for the single band Hubbard Model at Half-Filling
Parameters
----------
gloc : complex 1D ndarray
local Green's function to use as seed
w : real 1D ndarray
real frequency points
u_int : float
On site interaction, Hubbard U
beta : float
Inverse temperature
conv : float
convergence criteria
Returns
-------
gloc : complex 1D ndarray
DMFT iterated local Green's function
sigma : complex 1D ndarray
DMFT iterated self-energy
"""
dw = w[1] - w[0]
eta = 2j * dw
nf = gf.fermi_dist(w, beta)
converged = False
while not converged:
gloc_old = gloc.copy()
# Self-consistency
g0 = 1 / (w + eta - .25 * gloc)
# Spectral-function of Weiss field
A0 = -g0.imag / np.pi
# Second order diagram
isi = ph_hf_sigma(A0, nf, u_int) * dw * dw
isi = 0.5 * (isi + isi[::-1])
# Kramers-Kronig relation, uses Fourier Transform to speed convolution
hsi = -signal.hilbert(isi, len(isi) * 4)[:len(isi)].imag
sigma = hsi + 1j * isi
# Semi-circle Hilbert Transform
gloc = gf.semi_circle_hiltrans(w - sigma)
converged = np.allclose(gloc, gloc_old, atol=conv)
return gloc, sigma
def optical_cond(ss, sa, tp, w, beta):
E = np.linspace(-1, 1, 61)
dos = np.exp(-2 * E**2) / np.sqrt(np.pi / 2)
de = E[1] - E[0]
dosde = (dos * de).reshape(-1, 1)
nf = gf.fermi_dist(w, beta)
lat_Aa = (-1 / np.add.outer(-E, w + tp - sa)).imag / np.pi
lat_As = (-1 / np.add.outer(-E, w - tp - ss)).imag / np.pi
a = optical_conductivity(lat_Aa, lat_Aa, nf, w, dosde)
b = optical_conductivity(lat_As, lat_As, nf, w, dosde)
c = optical_conductivity(lat_Aa, lat_As, nf, w, dosde)
d = optical_conductivity(lat_As, lat_Aa, nf, w, dosde)
return np.sum((a, b, c, d), axis=0)
def optical_conductivity(beta, ss, sa, omega, tp, eps_k, eta=4e-2, crop=True):
"""Calculate the contributions to the optical_conductivity in the dimer
Parameters
----------
beta : float - Inverse Temperature
ss : 1D complex ndarray - SYM real-freq self-energy
sa : 1D complex ndarray - ASYM real-freq self-energy
omega : 1D float ndarray - real frequency grid (equispaced)
tp : float - Dimer Hybridization
eps_k : 1D float ndarray - Energy level bandwidth (equispaced)
eta : float - broadening for A(eps_k, w)
crop : bool - Return only positive frequencies
Returns
-------
intra_band : Intra band Optical response
inter_band : Inter band Optical response
"""
nfp = gf.fermi_dist(omega, beta)
pos_freq = omega > 0
rho = np.exp(-2 * eps_k**2) / np.sqrt(np.pi / 2)
de = eps_k[1] - eps_k[0]
rhode = (rho * de).reshape(-1, 1)
def lat_gf(eps_k, w_sig):
return 1 / np.add.outer(-eps_k, w_sig)
lat_As = -lat_gf(eps_k, omega - tp - ss + eta * 1j).imag / np.pi
lat_Aa = -lat_gf(eps_k, omega + tp - sa + eta * 1j).imag / np.pi
intra_band = opt_sig(lat_Aa, lat_Aa, nfp, omega, rhode)
intra_band += opt_sig(lat_As, lat_As, nfp, omega, rhode)
inter_band = opt_sig(lat_Aa, lat_As, nfp, omega, rhode)
inter_band += opt_sig(lat_As, lat_Aa, nfp, omega, rhode)
if crop:
intra_band = intra_band[pos_freq]
inter_band = inter_band[pos_freq]
return intra_band, inter_band
def plot_optical_cond(sigma_iw, ur, tp, w_n, w, w_set, beta, seed):
nuv = w[w > 0]
zerofreq = len(nuv)
dw = w[1] - w[0]
E = np.linspace(-1, 1, 61)
dos = np.exp(-2 * E**2) / np.sqrt(np.pi / 2)
de = E[1] - E[0]
dosde = (dos * de).reshape(-1, 1)
nf = gf.fermi_dist(w, beta)
eta = 0.8
for U, (sig_d, sig_o) in zip(ur, sigma_iw):
ss, sa = dimer.pade_diag(sig_d, sig_o, w_n, w_set, w)
lat_Aa = (-1 / np.add.outer(-E, w + tp + 4e-2j - sa)).imag / np.pi
lat_As = (-1 / np.add.outer(-E, w - tp + 4e-2j - ss)).imag / np.pi
#lat_Aa = .5 * (lat_Aa + lat_As)
#lat_As = lat_Aa
a = optical_conductivity(lat_Aa, lat_Aa, nf, w, dosde)
a += optical_conductivity(lat_As, lat_As, nf, w, dosde)
b = optical_conductivity(lat_Aa, lat_As, nf, w, dosde)
b += optical_conductivity(lat_As, lat_Aa, nf, w, dosde)
#b *= tp**2 * eta**2 / 2 / .25
sigma_E_sum_a = .5 * a[w > 0]
plt.plot(nuv, sigma_E_sum_a, 'k--')
sigma_E_sum_i = .5 * b[w > 0]
plt.plot(nuv, sigma_E_sum_i, 'k:')
sigma_E_sum = .5 * (a + b)[w > 0]
plt.plot(nuv, sigma_E_sum)
# To save data manually at some point
np.savez('opt_cond{}'.format(seed), nuv=nuv, sigma_E_sum=sigma_E_sum)
return sigma_E_sum_a, sigma_E_sum_i, sigma_E_sum, nuv
import numpy as np
import matplotlib.pylab as plt
from dmft.ipt_imag import dmft_loop
from dmft.common import greenF, tau_wn_setup, pade_continuation, fermi_dist
from dmft.utils import optical_conductivity
U = 2.7
BETA = 100.
tau, w_n = tau_wn_setup(dict(BETA=BETA, N_MATSUBARA=2**9))
omega = np.linspace(-4, 4, 600)
eps_k = np.linspace(-1, 1, 61)
dos = np.exp(-2 * eps_k**2) / np.sqrt(np.pi / 2)
de = eps_k[1] - eps_k[0]
dosde = (dos * de).reshape(-1, 1)
nf = fermi_dist(omega, BETA)
###############################################################################
# Insulator Calculations
# ----------------------
ig_iwn, is_iwn = dmft_loop(U,
0.5,
-1.j / (w_n - 1 / w_n),
w_n,
tau,
conv=1e-12)
igw = pade_continuation(ig_iwn, w_n, omega, np.arange(100))
isigma_w = pade_continuation(is_iwn, w_n, omega, np.arange(100))
lat_gf = 1 / (np.add.outer(-eps_k, omega + 4e-2j) - isigma_w)
lat_Aw = -lat_gf.imag / np.pi
def low_en_qp(ss):
glp = np.array([0.])
sigtck = splrep(w, ss.real, s=0)
sig_0 = splev(glp, sigtck, der=0)[0]
dw_sig0 = splev(glp, sigtck, der=1)[0]
quas_z = 1 / (1 - dw_sig0)
return quas_z, sig_0, dw_sig0
w = np.linspace(-4, 4, 2**12)
dw = w[1] - w[0]
BETA = 512.
nfp = gf.fermi_dist(w, BETA)
def plot_low_energy(u_int, tp):
gss = gf.semi_circle_hiltrans(w + 5e-3j)
gsa = gf.semi_circle_hiltrans(w + 5e-3j)
(gss, _), (ss, _) = ipt_real.dimer_dmft(u_int,
tp,
nfp,
w,
dw,
gss,
gsa,
conv=1e-3)
# author: Óscar Nájera
from __future__ import division, absolute_import, print_function
import numpy as np
import matplotlib.pyplot as plt
import dmft.common as gf
import dmft.ipt_real as ipt
w = np.linspace(-4, 4, 2**12)
dw = w[1] - w[0]
beta = 800.
nfp = gf.fermi_dist(w, beta)
###############################################################################
# The :math:`t_\perp/D=0.3` scenario
# ==================================
#
tp = 0.3
gss = gf.semi_circle_hiltrans(w + 5e-3j - tp)
gsa = gf.semi_circle_hiltrans(w + 5e-3j + tp)
urange = np.arange(0.2, 3.3, 0.3)
urange = [0.2, 1., 2., 3., 3.47, 3.5]
plt.close('all')
for i, U in enumerate(urange):
(gss, gsa), (ss, sa) = ipt.dimer_dmft(U,
tp,
nfp,