def gf(w, U, mu, beta):
"""Calculate by Lehmann representation the green function"""
H, d_up, d_dw = hamiltonian(U, mu)
e, v = diagonalize(H.todense())
g_up = gf_lehmann(e, v, d_up.T, beta, w)
g_dw = gf_lehmann(e, v, d_dw.T, beta, w)
return g_up, g_dw
def test_gf_lehmann2():
"""Verifies the lehmann representation of a random Hamiltonian"""
w = np.linspace(-1.5, 1.5, 500)
d_up = fermion.destruct(6, 0)
H = np.random.rand(*d_up.shape)
H += H.T
e, v = operators.diagonalize(H)
g_up_diag = operators.gf_lehmann(e, v, d_up.T, 400, w)
g_up_cross = operators.gf_lehmann(e, v, d_up.T, 400, w, d_up)
assert np.allclose(g_up_diag, g_up_cross)
def test_gf_lehmann2():
"""Verifies the lehmann representation of a random Hamiltonian"""
w = np.linspace(-1.5, 1.5, 500)
d_up = fermion.destruct(6, 0)
H = np.random.rand(*d_up.shape)
H += H.T
e, v = operators.diagonalize(H)
g_up_diag = operators.gf_lehmann(e, v, d_up.T, 400, w)
g_up_cross = operators.gf_lehmann(e, v, d_up.T, 400, w, d_up)
assert np.allclose(g_up_diag, g_up_cross)
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 solve(self, e_c, u_int, hyb):
"""Solves the impurity problem"""
self.update_H(e_c, u_int, hyb)
d_up_dag = self.oper[0].T
self.GF['Imp G'] = gf_lehmann(self.eig_energies, self.eig_states,
d_up_dag, self.beta, self.omega)
self.GF['Imp G$_0$'] = self.imp_free_gf(e_c, hyb)
self.GF[r'$\Sigma$'] = 1 / self.GF['Imp G$_0$'] - 1 / self.GF['Imp G']
def test_gf_lehmann(U, mu, beta):
"""Verifies the lehmann representation of the atom"""
w = np.linspace(-1.5*U, 1.5*U, 500)
d_up, d_dw = [fermion.destruct(2, sigma) for sigma in range(2)]
sigma_z = d_up.T*d_up - d_dw.T*d_dw
H = - U/2 * sigma_z * sigma_z - mu * (d_up.T*d_up + d_dw.T*d_dw)
e, v = operators.diagonalize(H.todense())
g_up = operators.gf_lehmann(e, v, d_up.T, beta, w)
g_up_cross = operators.gf_lehmann(e, v, d_up.T, beta, w, d_up)
Z = 1+2*np.exp(beta*(U/2+mu)) + np.exp(2*beta*mu)
g_up_ref = ((1+np.exp(beta*(U/2+mu)))/(w + mu + U/2.) +
(np.exp(beta*(U/2+mu)) + np.exp(2*beta*mu))/(w + mu - U/2.))/Z
assert np.allclose(g_up, g_up_ref)
assert np.allclose(g_up_cross, g_up_ref)
def solve(self, e_c, u_int, hyb):
"""Solves the impurity problem"""
self.update_H(e_c, u_int, hyb)
self.GF['Imp G'] = np.asarray([gf_lehmann(self.eig_energies, self.eig_states,
d.T, self.beta, self.omega) for d in self.oper[:2]])
self.GF['Imp G$_0$'] = self.imp_free_gf(e_c, hyb)
self.GF[r'$\Sigma$'] = 1/self.GF['Imp G$_0$'] - 1/self.GF['Imp G']
def solve(self, e_c, u_int, hyb):
"""Solves the impurity problem"""
self.update_H(e_c, u_int, hyb)
d_up_dag = self.oper[0].T
self.GF['Imp G'] = gf_lehmann(self.eig_energies, self.eig_states,
d_up_dag, self.beta, self.omega)
self.GF['Imp G$_0$'] = self.imp_free_gf(e_c, hyb)
self.GF[r'$\Sigma$'] = 1/self.GF['Imp G$_0$'] - 1/self.GF['Imp G']
def test_gf_lehmann(U, mu, beta):
"""Verifies the lehmann representation of the atom"""
w = np.linspace(-1.5 * U, 1.5 * U, 500)
d_up, d_dw = [fermion.destruct(2, sigma) for sigma in range(2)]
sigma_z = d_up.T * d_up - d_dw.T * d_dw
H = -U / 2 * sigma_z * sigma_z - mu * (d_up.T * d_up + d_dw.T * d_dw)
e, v = operators.diagonalize(H.todense())
g_up = operators.gf_lehmann(e, v, d_up.T, beta, w)
g_up_cross = operators.gf_lehmann(e, v, d_up.T, beta, w, d_up)
Z = 1 + 2 * np.exp(beta * (U / 2 + mu)) + np.exp(2 * beta * mu)
g_up_ref = ((1 + np.exp(beta * (U / 2 + mu))) / (w + mu + U / 2.) +
(np.exp(beta * (U / 2 + mu)) + np.exp(2 * beta * mu)) /
(w + mu - U / 2.)) / Z
assert np.allclose(g_up, g_up_ref)
assert np.allclose(g_up_cross, g_up_ref)
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 plot_A_ev_utp(beta, urange, mu, tprange):
w = np.linspace(-2, 3.5, 1500) + 1j * 1e-2
Aw = []
for u_int, tp in zip(urange, tprange):
h_at, oper = dimer.hamiltonian_diag(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)
aw = gf.imag / gf.imag.min()
Aw.append(aw)
return np.array(Aw)
def solve(self, e_c, u_int, hyb):
"""Solves the impurity problem"""
self.update_H(e_c, u_int, hyb)
self.GF['Imp G'] = np.asarray([
gf_lehmann(self.eig_energies, self.eig_states, d.T, self.beta,
self.omega) for d in self.oper[:2]
])
self.GF['Imp G$_0$'] = self.imp_free_gf(e_c, hyb)
self.GF[r'$\Sigma$'] = 1 / self.GF['Imp G$_0$'] - 1 / self.GF['Imp G']
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
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 molecule_sigma_d(omega, U, mu, tp, beta):
"""Return molecule self-energy in the given frequency axis"""
h_at, oper = dimer.hamiltonian_diag(U, mu, tp)
oper_pair = [[oper[0], oper[0]], [oper[1], 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
])
plt.plot(omega.real, -(gfsU[1]).imag, label='Anti-Bond')
plt.plot(omega.real, -(gfsU[0]).imag, label='Bond')
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 + tp - 1 / gfsU[0], omega - tp - 1 / gfsU[1]]
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]]