def AkBkExplicit(ky,
kz,
N=None,
J=None,
S=None,
h=None,
eps=None,
a=None,
mu=None,
Nr=4,
Ng=4):
xx = Dkxx(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yy = Dkyy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz = Dkzz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xy = Dkxy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xx_table = xx.table(ky, kz, N)
yy_table = yy.table(ky, kz, N)
zz_table0 = zz.table(10**-6, 10**-6, N)
xy_table = xy.table(ky, kz, N)
Atemp = np.zeros((N, N), dtype=np.complex)
Btemp = np.zeros((N, N), dtype=np.complex)
for i in range(N):
for j in range(N):
if i == j:
Atemp[i, j] = h + S * np.sum(zz_table0[i:i + N])
Atemp[i, j] += S * Jk(i, j, ky, kz, N=N, a=a, J=J)
Atemp[i, j] -= S / 2 * (xx_table[i - j + N - 1] +
yy_table[i - j + N - 1])
Btemp[i, j] = -0.5 * S * (xx_table[i - j + N - 1] -
2j * xy_table[i - j + N - 1] -
yy_table[i - j + N - 1])
return Atemp, Btemp
def AkBk(ky,
kz,
N=None,
J=None,
S=None,
h=None,
eps=None,
a=None,
mu=None,
Nr=4,
Ng=4):
xx = Dkxx(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yy = Dkyy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz = Dkzz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xy = Dkxy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xx_table = xx.table(ky, kz, N)
yy_table = yy.table(ky, kz, N)
zz_table0 = zz.table(10**-6, 10**-6, N)
xy_table = xy.table(ky, kz, N)
Atemp = np.zeros((N, N), dtype=np.complex)
Btemp = np.zeros((N, N), dtype=np.complex)
Atemp += np.diag([h + S * np.sum(zz_table0[i:i + N]) for i in range(N)])
Atemp += np.diag(
np.ones(N) * S * J * (6 - 2 * np.cos(ky * a) - 2 * np.cos(kz * a)))
Atemp[0, 0] -= S * J
Atemp[N - 1, N - 1] -= S * J
Atemp += np.diag(np.ones(N - 1), -1) * -J * S
Atemp += np.diag(np.ones(N - 1), 1) * -J * S
for i in range(N):
Atemp[i, :] -= .5 * S * np.flip(xx_table[i:i + N] + yy_table[i:i + N])
Btemp[i, :] -= .5 * S * np.flip(xx_table[i:i + N] - 2j *
xy_table[i:i + N] - yy_table[i:i + N])
return Atemp, Btemp
def AkBkAngle(ky,
kz,
phi,
alpha,
N=None,
J=None,
S=None,
h=None,
eps=None,
a=None,
mu=None,
Nr=4,
Ng=4):
xx = Dkxx(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yy = Dkyy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz = Dkzz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xy = Dkxy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xz = Dkxz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yz = Dkyz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz_table0 = zz.table(10**-6, 10**-6, N)
xx_table0 = xx.table(10**-6, 10**-6, N)
xz_table0 = xz.table(10**-6, 10**-6, N)
xx_table = xx.table(ky, kz, N)
yy_table = yy.table(ky, kz, N)
xy_table = xy.table(ky, kz, N)
zz_table = zz.table(ky, kz, N)
xz_table = xz.table(ky, kz, N)
yz_table = yz.table(ky, kz, N)
Atemp = np.zeros((N, N), dtype=np.complex)
Btemp = np.zeros((N, N), dtype=np.complex)
Atemp += np.diag([
h * np.cos(phi - alpha) + S *
np.sum(zz_table0[i:i + N] * np.cos(phi)**2 + xx_table0[i:i + N] *
np.sin(phi)**2 + xz_table0[i:i + N] * np.sin(phi) * np.cos(phi))
for i in range(N)
])
Atemp += np.diag(
np.ones(N) * S * J * (6 - 2 * np.cos(ky * a) - 2 * np.cos(kz * a)))
Atemp[0, 0] -= S * J
Atemp[N - 1, N - 1] -= S * J
Atemp += np.diag(np.ones(N - 1), -1) * -J * S
Atemp += np.diag(np.ones(N - 1), 1) * -J * S
for i in range(N):
Atemp[i, :] -= np.flip(
S / 2 * (xx_table[i:i + N] * np.cos(phi)**2 + yy_table[i:i + N] +
zz_table[i:i + N] * np.sin(phi)**2) -
2 * xz_table[i:i + N] * np.sin(phi) * np.cos(phi))
Btemp[i, :] -= 0.5 * S * np.flip(
xx_table[i:i + N] * np.cos(phi)**2 - yy_table[i:i + N] +
zz_table[i:i + N] * np.sin(phi)**2 - 2 * xz_table[i:i + N] *
np.sin(phi) * np.cos(phi) + 2j * xy_table[i:i + N] * np.cos(phi) -
2j * yz_table[i:i + N] * np.sin(phi))
return Atemp, Btemp
def AkBkAngleExplicit(ky,
kz,
phi,
alpha,
N=None,
J=None,
S=None,
h=None,
eps=None,
a=None,
mu=None,
Nr=4,
Ng=4):
xx = Dkxx(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yy = Dkyy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz = Dkzz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xy = Dkxy(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
xz = Dkxz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
yz = Dkyz(eps=eps, a=a, mu=mu, Nr=Nr, Ng=Ng)
zz_table0 = zz.table(10**-6, 10**-6, N)
xx_table0 = xx.table(10**-6, 10**-6, N)
xz_table0 = xz.table(10**-6, 10**-6, N)
xx_table = xx.table(ky, kz, N)
yy_table = yy.table(ky, kz, N)
xy_table = xy.table(ky, kz, N)
zz_table = zz.table(ky, kz, N)
xz_table = xz.table(ky, kz, N)
yz_table = yz.table(ky, kz, N)
A = np.zeros((N, N), dtype=np.complex)
B = np.zeros((N, N), dtype=np.complex)
for i in range(N):
for j in range(N):
if i == j:
A[i, j] = h * np.cos(phi - alpha) + S * np.sum(
zz_table0[i:i + N] * np.cos(phi)**2 +
xx_table0[i:i + N] * np.sin(phi)**2 +
xz_table0[i:i + N] * np.sin(phi) * np.cos(phi))
A[i, j] += S * Jk(i, j, ky, kz, N=N, a=a, J=J)
A[i,
j] -= (S / 2 * (xx_table[i - j + N - 1] * np.cos(phi)**2 +
yy_table[i - j + N - 1] +
zz_table[i - j + N - 1] * np.sin(phi)**2) -
2 * xz_table[i - j + N - 1] * np.sin(phi) * np.cos(phi))
B[i, j] = -0.5 * S * (
xx_table[i - j + N - 1] * np.cos(phi)**2 -
yy_table[i - j + N - 1] +
zz_table[i - j + N - 1] * np.sin(phi)**2 -
2 * xz_table[i - j + N - 1] * np.sin(phi) * np.cos(phi) +
2j * xy_table[i - j + N - 1] * np.cos(phi) -
2j * yz_table[i - j + N - 1] * np.sin(phi))
return A, B
def spin_momentum_nonlinear(ev, ky, kz, N, a=None, mu=None, S=None):
ev_HP = ev_in_HP_basis(ev)
ev_S = ev_HP_to_S(ev_HP, S=S)
eps = a**(-2)
xx = Dkxx(eps=eps, a=a, mu=mu).table(ky, kz, N)
yy = Dkyy(eps=eps, a=a, mu=mu).table(ky, kz, N)
zz = Dkzz(eps=eps, a=a, mu=mu).table(ky, kz, N)
xy = Dkxy(eps=eps, a=a, mu=mu).table(ky, kz, N)
xz = Dkxz(eps=eps, a=a, mu=mu).table(ky, kz, N)
yz = Dkyz(eps=eps, a=a, mu=mu).table(ky, kz, N)
res = [] # first calculate per x_i point
for i in range(N):
Si = ev_S[:, i]
temp_res = []
for j in range(N):
# delta = i - j + N - 1
if i != j:
continue
Sj = ev_S[:, j]
Sx = Sj[0]
Sy = Sj[1]
Sz = Sj[2]
x = Sx * xx[i - j + N - 1] + Sy * xy[i - j + N -
1] + Sz * xz[i - j + N - 1]
y = Sy * yy[i - j + N - 1] + Sz * yz[i - j + N -
1] + Sx * xy[i - j + N - 1]
z = Sz * zz[i - j + N - 1] + Sy * yz[i - j + N -
1] + Sx * xz[i - j + N - 1]
G = np.stack((x, y, z))
temp_res.append(np.cross(G, Si))
temp_res = np.array(temp_res)
res.append(np.sum(temp_res, axis=0))
res = np.array(res)
return res
def Dkxy_uni(ky, kz, x, theta=None, mu=None, a=None):
k = np.sqrt(ky**2 + kz**2)
if x == 0:
return Dkxy(eps=a**(-2), a=a, mu=mu).run(0, ky, kz)
return 1j * 2 * np.pi * np.sin(theta) * mu**2 / (
a**2) * np.abs(k) * x / np.abs(x) * np.exp(-np.abs(k) * np.abs(x))
def Dkxy_uni(k, x, theta=None, mu=None, a=None):
return 1j * 2 * np.pi * np.sin(theta) * mu**2 / (
a**2) * np.abs(k) * x / np.abs(x) * np.exp(-np.abs(k) * np.abs(x))
if __name__ == "__main__":
H = 700.0
h = mu * H
eps = a**(-2)
K = Dkxx(eps, a, mu, Nr=10, Ng=10)
funs = [
Dkxx(eps, a, mu, Nr=10, Ng=10),
Dkyy(eps, a, mu, Nr=10, Ng=10),
Dkxy(eps, a, mu, Nr=10, Ng=10)
]
uni_funs = [
partial(Dkxx_uni, mu=mu, a=a),
partial(Dkyy_uni, mu=mu, a=a),
partial(Dkxy_uni, mu=mu, a=a)
]
parse_funs = [np.real, np.real, np.imag]
titles = ['Dkxx', 'Dkyy', 'Dkxy']
for K, f, parse, tit in zip(funs, uni_funs, parse_funs, titles):
plt.figure()
plt.suptitle(tit)
for i, ky in enumerate(np.logspace(3, 6, 4)):
kz = 10**4
theta = np.arctan2(ky, kz)
absk = np.sqrt(ky**2 + kz**2)
def spin_momentum_linear_old(ev,
ky,
kz,
N,
phi=None,
a=None,
mu=None,
S=None,
J=None,
h=None,
alpha=None):
ev = ev_in_HP_basis(ev)
sqrS = .5 * np.sqrt(2 * S)
eps = a**(-2)
xx = Dkxx(eps=eps, a=a, mu=mu).table(ky, kz, N)
yy = Dkyy(eps=eps, a=a, mu=mu).table(ky, kz, N)
zz = Dkzz(eps=eps, a=a, mu=mu).table(ky, kz, N)
xy = Dkxy(eps=eps, a=a, mu=mu).table(ky, kz, N)
xz = Dkxz(eps=eps, a=a, mu=mu).table(ky, kz, N)
yz = Dkyz(eps=eps, a=a, mu=mu).table(ky, kz, N)
zz0 = Dkzz(eps=eps, a=a, mu=mu).table(10**(-10), 10**(-10), N)
Q = np.sin(phi)
C = np.cos(phi)
res = [] # first calculate per x_i point
for i in range(N):
bi = ev[i]
temp_res = []
for j in range(N):
# delta = i - j + N - 1
if i != j:
continue
bj = ev[j]
x = 0
z = 0
d = i - j + N - 1
SxSx = S**2 * Q**2 + 2 * sqrS * S * C * Q * (bj + bj.conj())
SxSy = -1j * sqrS * S * (bj - bj.conj()) * Q
SxSz = sqrS * S * (bj + bj.conj()) * (C**2 - Q**2) + S**2 * Q * C
SySy = 0
SySz = -1j * sqrS * S * (bj - bj.conj()) * C
SzSz = S**2 * C**2 - 2 * sqrS * S * C * Q * (bj + bj.conj())
sum_z0 = 0
for n in range(N):
sum_z0 += zz0[i - n + N - 1]
# y = SxSx * xz[d] + SxSy * yz[d] + SxSz * zz[d]
y = SxSx * xz[d] + SxSy * yz[d] + SxSz * sum_z0
y -= SxSz * xx[d] + SySz * xy[d] + SzSz * xz[d]
# y = sqrS * 2 * S * ((xx[d] - np.sum(zz0[i:i + N])) *
# (bi + bi.conj()) + 1j * xy[d] *
# (bi - bi.conj()))
J0 = S * J * (6 - 2 * (np.cos(ky * a) + np.cos(kz * a)))
if i == 0:
J0 -= S * J
if i == N - 1:
J0 -= S * J
# y += J0 * sqrS * (bi + bi.conj()) + h
temp_res.append([x, y, z])
temp_res = np.array(temp_res)
res.append(np.sum(temp_res, axis=0))
res = np.array(res)
return res
def spin_momentum_linear(ev,
ky,
kz,
N,
phi=None,
a=None,
mu=None,
S=None,
J=None,
h=None,
alpha=None):
ev = ev_in_HP_basis(ev)
eps = a**(-2)
xx = Dkxx(eps=eps, a=a, mu=mu)
yy = Dkyy(eps=eps, a=a, mu=mu)
zz = Dkzz(eps=eps, a=a, mu=mu)
xy = Dkxy(eps=eps, a=a, mu=mu)
xz = Dkxz(eps=eps, a=a, mu=mu)
yz = Dkyz(eps=eps, a=a, mu=mu)
zz_table0 = zz.table(np.finfo('float64').eps, np.finfo('float64').eps, N)
xx_table0 = xx.table(np.finfo('float64').eps, np.finfo('float64').eps, N)
xz_table0 = xz.table(np.finfo('float64').eps, np.finfo('float64').eps, N)
xx_table = xx.table(ky, kz, N)
yy_table = yy.table(ky, kz, N)
xy_table = xy.table(ky, kz, N)
zz_table = zz.table(ky, kz, N)
xz_table = xz.table(ky, kz, N)
yz_table = yz.table(ky, kz, N)
temp = []
for i in range(N):
bi = ev[i]
dt_bi = []
for j in range(N):
A = 0
if i != j:
continue
if i == j:
A += h * np.cos(phi - alpha) + S * np.sum(
zz_table0[i:i + N] * np.cos(phi)**2 +
xx_table0[i:i + N] * np.sin(phi)**2 +
xz_table0[i:i + N] * np.sin(phi) * np.cos(phi))
A += S * Jk(i, j, ky, kz, N=N, a=a, J=J)
A -= (S / 2 * (xx_table[i - j + N - 1] * np.cos(phi)**2 +
yy_table[i - j + N - 1] +
zz_table[i - j + N - 1] * np.sin(phi)**2) -
2 * xz_table[i - j + N - 1] * np.sin(phi) * np.cos(phi))
B = -0.5 * S * (xx_table[i - j + N - 1] * np.cos(phi)**2 -
yy_table[i - j + N - 1] +
zz_table[i - j + N - 1] * np.sin(phi)**2 - 2 *
xz_table[i - j + N - 1] * np.sin(phi) * np.cos(phi)
+ 2j * xy_table[i - j + N - 1] * np.cos(phi) -
2j * yz_table[i - j + N - 1] * np.sin(phi))
dt_bi.append(-1j * (bi * A + bi.conj() * B.conj()))
dt_bi = np.array(dt_bi)
y = -1j / 2 * (dt_bi.sum() - dt_bi.sum().conj())
x = 1 / 2 * (dt_bi.sum() + dt_bi.sum().conj())
z = -dt_bi.sum() * dt_bi.sum().conj()
temp.append([x, y, z])
temp = np.array(temp)
return temp