def plot_A_ev_utp(beta, urange, mu, tprange):
w = np.linspace(-2, 2, 1500) + 1j * 5e-3
for u_int, tp in zip(urange, tprange):
h_at, oper = dimer.hamiltonian(u_int, mu, tp)
eig_e, eig_v = op.diagonalize(h_at.todense())
gf = op.gf_lehmann(eig_e, eig_v, oper[0].T, beta, w)
plt.plot(w.real, u_int + gf.imag / gf.imag.min(), 'k-')
def test_hamiltonian_eigen_energies(u_int, mu, tp):
"""Test local basis and diagonal basis isolated dimer Hamiltonians
have same energy spectrum"""
h_loc, _ = dimer.hamiltonian(u_int, mu, tp)
h_dia, _ = dimer.hamiltonian_diag(u_int, mu, tp)
eig_e_loc, _ = op.diagonalize(h_loc.todense())
eig_e_dia, _ = op.diagonalize(h_dia.todense())
assert np.allclose(eig_e_loc, eig_e_dia)
def plot_A_ev_ru(beta, urange, mu, tp):
w = np.linspace(0, 2, 500) + 1j * 5e-3
for u_int in urange:
h_at, oper = dimer.hamiltonian(u_int, mu, tp)
eig_e, eig_v = op.diagonalize(h_at.todense())
gf = op.gf_lehmann(eig_e, eig_v, oper[0].T, beta, w)
plt.plot(w.real, u_int + gf.imag / gf.imag.min())
plt.plot(0.5 * np.sqrt(urange**2 + 16 * tp**2) - tp, urange, '*:',
label=r'$|GS\rangle \rightarrow \pm t_\perp + 1/2(U^2+16t^2_\perp)^{1/2}$')
plt.plot(0.5 * np.sqrt(urange**2 + 16 * tp**2) + tp, urange, '*:')
plt.plot(urange / 2 - tp, urange, 'x:',
label=r'$|T\rangle \rightarrow \pm t_\perp + U/2$')
plt.plot(urange / 2 + tp, urange, 'x:')
plt.xlim([min(w.real), max(w.real)])
def plot_eigen_spectra(U, mu, tp):
h_at, oper = dimer.hamiltonian(U, mu, tp)
eig_e = []
eig_e.append(LA.eigvalsh(h_at[1:5, 1:5].todense()))
eig_e.append(LA.eigvalsh(h_at[5:11, 5:11].todense()))
eig_e.append(LA.eigvalsh(h_at[11:15, 11:15].todense()))
plt.figure()
plt.title('Many particle Energy Spectra U={} $t_\perp={}$'.format(U, tp))
plt.plot(np.concatenate(eig_e), "o-")
plt.ylabel('Energy')
plt.xlabel('Eigenstate by N particle block')
plt.axvline(x=3.5)
plt.axvline(x=9.5)
def plot_A_ev_rtp(beta, u_int, mu, tprange):
w = np.linspace(-4, 4, 5500) + 1j * 5e-3
for tp in tprange:
h_at, oper = dimer.hamiltonian(u_int, mu, tp)
eig_e, eig_v = op.diagonalize(h_at.todense())
gfd = op.gf_lehmann(eig_e, eig_v, oper[0].T, beta, w)
gfo = op.gf_lehmann(eig_e, eig_v, oper[0].T, beta, w, oper[1])
gf = gfd + gfo
plt.plot(w.real, tp + gf.imag / gf.imag.min())
plt.title('Molecule exitation spectral function')
plt.xlabel(r'$\omega$')
plt.plot(0.5 * np.sqrt(u_int**2 + 16 * tprange**2) - tprange, tprange, '*:',
label=r'$|GS\rangle \rightarrow \pm t_\perp + 1/2(U^2+16t^2_\perp)^{1/2}$')
plt.plot(0.5 * np.sqrt(u_int**2 + 16 * tprange**2) + tprange, tprange, '*:')
plt.plot(u_int / 2 - tprange, tprange, 'x:',
label=r'$|T\rangle \rightarrow \pm t_\perp + U/2$')
plt.plot(u_int / 2 + tprange, tprange, 'x:')
plt.xlim([min(w.real), max(w.real)])
def molecule_sigma(omega, U, mu, tp, beta):
"""Return molecule self-energy in the given frequency axis"""
h_at, oper = dimer.hamiltonian(U, mu, tp)
oper_pair = [[oper[0], oper[0]], [oper[0], oper[1]]]
eig_e, eig_v = op.diagonalize(h_at.todense())
gfsU = np.array([
op.gf_lehmann(eig_e, eig_v, c.T, beta, omega, d) for c, d in oper_pair
])
invg = dimer.mat_inv(gfsU[0], gfsU[1])
plt.plot(omega.real, -gfsU[0].imag, label='Interacting')
plt.plot(omega.real, -dimer.mat_inv(omega, -tp)[0].imag, label='Free')
plt.xlabel(r'$\omega$')
plt.ylabel(r'$A(\omega)$')
plt.title(r'Isolated dimer $U={}$, $t_\perp={}$, $\beta={}$'.format(
U, tp, beta))
plt.legend(loc=0)
return [omega - invg[0], -tp - invg[1]]
def test_dimer_energies(u_int, mu, tp, beta=100):
h_loc, (a_up, b_up, a_dw, b_dw) = dimer.hamiltonian(u_int, mu, tp)
e_imp = (tp * (a_up.T * b_up + a_dw.T * b_dw + b_up.T * a_up +
b_dw.T * a_dw)).todense()
eig_e, eig_v = op.diagonalize(h_loc.todense())
w_n = gf.matsubara_freq(beta, 2**8) # n=2**7=256
gf_di = op.gf_lehmann(eig_e, eig_v, a_up.T, beta, 1j * w_n)
gf_of = op.gf_lehmann(eig_e, eig_v, a_up.T, beta, 1j * w_n, b_up)
ekin_gf = dimer.ekin(gf_di, gf_of, w_n, tp, beta, 0)
ekin_ed = op.expected_value(e_imp, eig_e, eig_v, beta)
assert abs(ekin_ed - ekin_gf) < 5e-4
epot_gf = dimer.epot(gf_di, w_n, beta, u_int**2 / 4 + tp**2, ekin_gf,
u_int)
docc = (a_up.T * a_up * a_dw.T * a_dw +
b_up.T * b_up * b_dw.T * b_dw).todense()
epot_ed = op.expected_value(docc, eig_e, eig_v, beta) * u_int
assert abs(epot_ed - epot_gf) < 1e-3