def test_pade():
"""Test pade Analytical Continuation for the semi-circular DOS"""
w_n = gf.matsubara_freq(200)
giw = gf.greenF(w_n)
omega = np.linspace(-0.7, 0.7, 100) # avoid semicircle edges
gw_ref = gf.greenF(-1j * omega + 1e-5)
pade_c = gf.pade_coefficients(giw, w_n)
gw_cont = gf.pade_rec(pade_c, omega, w_n)
assert np.allclose(gw_ref, gw_cont, 1e-3)
def pade_diag(gf_aa, gf_ab, w_n, w_set, w):
r"""Take diagonal and off diagonal Matsubara functions in the local
basis and return real axis functions in the symmetric and
anti-symmetric basis. Such that
⎡ⅈ⋅ωₙ + μ - t⟂ 0 ⎤ ⎡Σ_AA + Σ_AB 0 ⎤
G^{-1} = ⎢ ⎥ - ⎢ ⎥
⎣ 0 ⅈ⋅ωₙ + μ + t⟂ ⎦ ⎣ 0 Σ_AA - Σ_AB⎦
The Symmetric sum (Anti-bonding) returned first, Asymmetric is returned second
"""
gf_s = 1j * gf_aa.imag + gf_ab.real # Anti-bond
pc = gf.pade_coefficients(gf_s[w_set], w_n[w_set])
gr_s = gf.pade_rec(pc, w, w_n[w_set])
gf_a = 1j * gf_aa.imag - gf_ab.real # bond
pc = gf.pade_coefficients(gf_a[w_set], w_n[w_set])
gr_a = gf.pade_rec(pc, w, w_n[w_set])
return gr_s, gr_a
#
# Starting from the Metallic seed on the Bethe Lattice the DMFT
# equations are solved iteratively in by perturbation theory in the
# Matsubara axis. The resulting Green's function is then approximated
# by its Padé approximant which allows to evaluate it on the upper
# complex plane.
beta = 90.
U = 2.7
tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=1024))
g_iwn0 = greenF(w_n)
g_iwn, s_iwn = dmft_loop(U, 0.5, g_iwn0, w_n, tau, conv=1e-12)
x = int(2 * beta)
w_set = np.arange(x)
pc = pade_coefficients(g_iwn[w_set], w_n[w_set])
ax = plot_complex_gf(omega, np.linspace(1e-3, 1.2, 30), w_n[:17],
lambda z: pade_rec(pc, z, w_n[w_set]))
ax.view_init(15, -64)
ax.set_zlim3d([-2, 1])
###############################################################################
# For the insulating solution
U = 3.2
tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=1024))
g_iwn0 = greenF(w_n)
g_iwn, s_iwn = dmft_loop(U,
0.5,
1 / (1j * w_n - 1 / g_iwn0),
np.polyfit(w_n[:5], giw_s[i, 0, :5].imag, 1)[0] for i in range(len(tprr))
]
plt.plot(tprr, slopes, '+-', label='data')
plt.title(r'Slope of $Im G_{AA}$')
plt.xlabel(r'$t_\perp/D$')
plt.ylabel(r"$\Im m G'_{AA}(0)$")
###############################################################################
# Analytical Continuation
# -----------------------
w_set = np.concatenate((np.arange(8) * 256, np.arange(1, 241, 1)))
w = np.linspace(0, 3., 500)
plt.figure()
for i, tp in enumerate(tprr):
pc = gf.pade_coefficients(1j * giw_s[i, 0, w_set].imag, w_n[w_set])
plt.plot(w, -2 * tp - gf.pade_rec(pc, w, w_n[w_set]).imag)
plt.xlabel(r'$\omega$')
plt.ylabel(r'$A(\omega) - 2 t_\perp$')
###############################################################################
# Change of G_{AB}
# ----------------
for i in [1, 2, 4, 7, 11, 16]:
plt.plot(w_n,
giw_s[i, 1].real,
's-',
label=r'$t_\perp={}$'.format(tprr[i]))
plt.xlabel(r'$i\omega_n$')
plt.ylabel(r'$\Re e G_{AB} (i\omega_n)$')