def ftlan_rdm1s1c(qud,
hop,
v0,
T,
norb,
m=60,
Min_b=1e-15,
Min_m=2,
kB=1.,
E0=0.,
norm=np.linalg.norm):
beta = 1. / (kB * T)
E = 0.
v0 = v0 / norm(v0)
rdm1 = qud(v0)
a, b = [], []
krylov = []
krylov.append(v0)
Hv = hop(v0)
a.append(v0.dot(Hv))
v1 = Hv - a[0] * v0
b.append(norm(v1))
if b[0] < Min_b:
log.warning("Insufficient size of the Krylov space!")
return rdm1, 0., 1e-5
v1 = v1 / b[0]
Hv = hop(v1)
a.append(v1.dot(Hv))
krylov.append(v1)
for i in range(1, int(m - 1)):
v2 = Hv - b[i - 1] * v0 - a[i] * v1
b.append(norm(v2))
if abs(b[i]) < Min_b:
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return rdm1, 0., 1e-5
b.pop()
break
v2 = v2 / b[i]
krylov.append(v2)
Hv = hop(v2)
a.append(v2.dot(Hv))
v0 = v1.copy()
v1 = v2.copy()
a, b = np.asarray(a), np.asarray(b)
krylov = np.asarray(krylov).real
eps, phi = Tri_diag(a, b)
l = len(eps)
estate = krylov.T.dot(phi).real
coef = np.exp(-beta * (eps - E0) / 2.) * phi[0, :].real
exp_eps = np.exp(-beta * (eps - E0))
E = np.sum(exp_eps * eps * phi[0, :]**2.).real
Z = np.sum(exp_eps * phi[0, :]**2.).real
psi = estate.dot(coef.T)
rdm1 = qud(psi)
return np.asarray(rdm1), E, Z
def ftlan_E1c(hop,
v0,
T,
m=40,
Min_b=1e-15,
Min_m=3,
kB=1.0,
norm=np.linalg.norm,
E0=0.,
**kwargs):
beta = 1. / (T * kB)
E = 0.
a, b = [], []
v0 = v0 / norm(v0)
Hv = hop(v0)
a.append(v0.dot(Hv))
v1 = Hv - a[0] * v0
b.append(norm(v1))
if b[0] < Min_b:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
v1 = v1 / b[0]
Hv = hop(v1)
a.append(v1.dot(Hv))
for i in range(1, m - 1):
v2 = Hv - b[i - 1] * v0 - a[i] * v1
b.append(norm(v2))
if abs(b[i]) < Min_b:
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b.pop()
break
v2 = v2 / b[i]
Hv = hop(v2)
a.append(v2.dot(Hv))
v0 = v1.copy()
v1 = v2.copy()
a = np.asarray(a)
b = np.asarray(b)
eps, phi = Tri_diag(a, b)
l = len(eps)
# Eo = eps[0]
# eps = eps-Eo
exp_eps = np.exp(-beta * (eps - E0))
E = np.sum(exp_eps * eps * phi[0, :]**2.).real
Z = np.sum(exp_eps * phi[0, :]**2.).real
return E, Z
def ftlan_E(hop, v0, m=40, norm=np.linalg.norm, Min_b=1e-10, Min_m=3):
a, b = [], []
v0 = v0 / norm(v0)
Hv = hop(v0)
a.append(v0.dot(Hv))
v1 = Hv - a[0] * v0
b.append(norm(v1))
if b[0] < Min_b:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
v1 = v1 / b[0]
Hv = hop(v1)
a.append(v1.dot(Hv))
for i in range(1, m - 1):
v2 = Hv - b[i - 1] * v0 - a[i] * v1
b.append(norm(v2))
if abs(b[i]) < Min_b:
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b.pop()
break
v2 = v2 / b[i]
Hv = hop(v2)
a.append(v2.dot(Hv))
v0 = v1.copy()
v1 = v2.copy()
a = np.asarray(a)
b = np.asarray(b)
eps, phi = Tri_diag(a, b)
return eps[0]
def ftlan_mu1c_time(H_prod_v,
mu_prod_v,
v0,
T,
time_step,
m=40,
Min_b=1e-15,
Min_m=3,
kB=1.0,
norm=np.linalg.norm,
E0=0.,
**kwargs):
# H_prod_v - H*c function
# mu_prod_v - mu*c function
beta = 1. / (T * kB)
a_v, b_v, a_w, b_w = [], [], [], []
krylov_v, krylov_w = [], []
v0 = v0 / norm(v0)
w0 = mu_prod_v(v0)
w0 = w0 / norm(w0)
krylov_v.append(v0)
krylov_w.append(w0)
Hv = H_prod_v(v0)
Hw = H_prod_v(w0)
a_v.append(v0.dot(Hv))
a_w.append(w0.dot(Hw))
v1 = Hv - a_v[0] * v0
w1 = Hw - a_w[0] * w0
b_v.append(norm(v1))
b_w.append(norm(w1))
if b_v[0] < Min_b or b_w < Min_b:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
v1 = v1 / b_v[0]
w1 = w1 / b_w[0]
krylov_v.append(v1)
krylov_w.append(w1)
Hv = H_prod_v(v1)
Hw = H_prod_v(w1)
a_v.append(v1.dot(Hv))
a_w.append(w1.dot(Hw))
for i in range(1, m - 1):
v2 = Hv - b_v[i - 1] * v0 - a_v[i] * v1
b_v.append(norm(v2))
if abs(b_v[i]) < Min_b:
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b_v.pop()
break
v2 = v2 / b_v[i]
Hv = H_prod_v(v2)
krylov_v.append(v2)
a_v.append(v2.dot(Hv))
v0 = v1.copy()
v1 = v2.copy()
for i in range(1, m - 1):
w2 = Hw - b_w[i - 1] * w0 - a_w[i] * w1
b_w.append(norm(w2))
if abs(b_w[i] < Min_b):
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b_w.pop()
break
w2 = w2 / b_w[i]
krylov_w.append(w2)
Hw = H_prod_v(w2)
a_w.append(w2.dot(Hw))
w0 = w1.copy()
w1 = w2.copy()
a_v, b_v, a_w, b_w = np.asarray(a_v), np.asarray(b_v), np.asarray(
a_w), np.asarray(b_w)
krylov_v, krylov_w = np.asarray(krylov_v), np.asarray(krylov_w)
eps_v, phi_v = Tri_diag(a_v, b_v)
eps_w, phi_w = Tri_diag(a_w, b_w)
estate_v = krylov_v.T.dot(phi_v)
estate_w = krylov_w.T.dot(phi_w)
coef_v = np.exp((-beta - 1.j * time_step) * eps_v) * (phi_v[0, :].conj())
coef_w = np.exp(-1.j * time_step * eps_w) * (phi_w[0, :].conj())
Z = np.sum(np.exp(-beta * (eps_v)) * (phi_v[0, :] * phi_v[0, :].conj()))
# Eo = eps[0]
# eps = eps-Eo
bra = estate_v.dot(coef_v.T)
ket = estate_w.dot(coef_w.T)
C_t = bra.T.conj().dot(mu_prod_v(ket))
return C_t, Z
def ftlan_mu1c_freq(H_prod_v,
mu_prod_v,
v0,
T,
freq_list,
m=40,
Min_b=1e-15,
Min_m=3,
kB=1.0,
norm=np.linalg.norm,
E0=0.,
**kwargs):
# H_prod_v - H*c function
# mu_prod_v - mu*c function
# a list of w grid i.e. (5.0, 20) means the range of w is (-5.0, 5.0), and 20 grids are made
beta = 1. / (T * kB)
a_v, b_v, a_w, b_w = [], [], [], []
krylov_v, krylov_w = [], []
v0 = v0 / norm(v0)
w0 = mu_prod_v(v0)
w0 = w0 / norm(w0)
krylov_v.append(v0)
krylov_w.append(w0)
Hv = H_prod_v(v0)
Hw = H_prod_v(w0)
a_v.append(v0.dot(Hv))
a_w.append(w0.dot(Hw))
v1 = Hv - a_v[0] * v0
w1 = Hw - a_w[0] * w0
b_v.append(norm(v1))
b_w.append(norm(w1))
if b_v[0] < Min_b or b_w < Min_b:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
v1 = v1 / b_v[0]
w1 = w1 / b_w[0]
krylov_v.append(v1)
krylov_w.append(w1)
Hv = H_prod_v(v1)
Hw = H_prod_v(w1)
a_v.append(v1.dot(Hv))
a_w.append(w1.dot(Hw))
for i in range(1, m - 1):
v2 = Hv - b_v[i - 1] * v0 - a_v[i] * v1
b_v.append(norm(v2))
if abs(b_v[i]) < Min_b:
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b_v.pop()
break
v2 = v2 / b_v[i]
Hv = H_prod_v(v2)
krylov_v.append(v2)
a_v.append(v2.dot(Hv))
v0 = v1.copy()
v1 = v2.copy()
for i in range(1, m - 1):
w2 = Hw - b_w[i - 1] * w0 - a_w[i] * w1
b_w.append(norm(w2))
if abs(b_w[i] < Min_b):
if i < Min_m:
log.warning("Insufficient size of the Krylov space!")
return 0., 1e-10
b_w.pop()
break
w2 = w2 / b_w[i]
krylov_w.append(w2)
Hw = H_prod_v(w2)
a_w.append(w2.dot(Hw))
w0 = w1.copy()
w1 = w2.copy()
a_v, b_v, a_w, b_w = np.asarray(a_v), np.asarray(b_v), np.asarray(
a_w), np.asarray(b_w)
krylov_v, krylov_w = np.asarray(krylov_v), np.asarray(krylov_w)
eps_v, phi_v = Tri_diag(a_v, b_v)
eps_w, phi_w = Tri_diag(a_w, b_w)
estate_v = krylov_v.T.dot(phi_v)
estate_w = krylov_w.T.dot(phi_w)
coef_v = np.exp(-beta * eps_v) * (phi_v[0, :].conj())
coef_w = phi_w[0, :].conj()
bra = np.einsum('ij,j -> ij', estate_v, coef_v)
ket = np.einsum('ij,j -> ij', estate_w, coef_w)
half_Nstep = freq_list[1]
w_max = freq_list[0]
lstep = float(w_max) / half_Nstep
C_omega = np.zeros(2 * half_Nstep, dtype=np.complex64)
for i in range(len(krylov_v)):
for j in range(len(krylov_w)):
delta_e = eps_w[j] - eps_v[i]
if abs(delta_e) > w_max:
continue
if delta_e < 0:
idx = int(delta_e / lstep) + half_Nstep - 1
else:
idx = int(delta_e / lstep) + half_Nstep
C_omega[idx] = bra[:, i].T.conj().dot(mu_prod_v(ket[:, j]))
Z = np.sum(np.exp(-beta * (eps_v)) * (phi_v[0, :] * phi_v[0, :].conj()))
return C_omega, Z