def plot_Kitaev_green():
''' plot the Det(G) in the plane u-w'''
L = 8
num = 51
u_num = 20
ws = np.linspace(-3, 3, num)
Gd = np.zeros((num, u_num), dtype=np.complex)
for j, u in enumerate(np.linspace(0, 2, u_num)):
H = Kitaev(L, t=1., u=u, d=1., pbc=False)
evals, evecs = func.solve_hamiltonian(H)
Gwk = np.zeros(num, dtype=np.complex)
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
for i in range(num):
Gwk[i] = det(Gij[i])
Gd[:, j] = Gwk
plt.figure("poles")
fig = plt.imshow(np.real(Gd), interpolation='nearest', aspect=0.4)
fig.set_clim(-1, 1)
plt.colorbar(ticks=[-1, 0, 1], label="Det(G)")
plt.xticks([0.5, u_num / 2., u_num - 0.5], [0, 1, 2], fontsize=20)
plt.yticks([0.5, num / 2., num - 0.5], [3, 0, -3], fontsize=20)
plt.xlabel('u', fontsize=20)
plt.ylabel('w', fontsize=20)
def plot_SSH_green():
''' plot the Det(G) in the plane dt-w'''
L = 8
num = 51
d_num = 20
ws = np.linspace(-3, 3, num)
Gd = np.zeros((num, d_num), dtype=np.complex)
for j, d in enumerate(np.linspace(-1, 1, d_num)):
H = SSH(L, t=1., d=d, pbc=False)
evals, evecs = func.solve_hamiltonian(H)
Gwk = np.zeros(num, dtype=np.complex)
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
for i in range(num):
Gwk[i] = det(Gij[i])
Gd[:, j] = Gwk
# plt.figure("Green u=%.2f"%d)
# plt.plot(ws, np.real(Gwk),'o-')
# plt.ylim([-1,1])
# print np.min( np.imag(Gwk) ), np.max( np.imag(Gwk) )
plt.figure("poles")
fig = plt.imshow(np.real(Gd), interpolation='nearest', aspect=0.4)
fig.set_clim(-1, 1)
plt.colorbar(ticks=[-1, 0, 1], label="Det(G)")
plt.xticks([0.5, d_num / 2., d_num - 0.5], [-1, 0, 1], fontsize=20)
plt.yticks([0.5, num / 2., num - 0.5], [3, 0, -3], fontsize=20)
plt.xlabel('d', fontsize=20)
plt.ylabel('w', fontsize=20)
def plot_SSH_green_wk(dt=1.):
''' plot the greens function in the k-w plane '''
L = 12
num = 51 # number of energies to calculate
ws = np.linspace(-3, 3, num)
H = SSH(L, t=1., d=dt, pbc=True)
evals, evecs = func.solve_hamiltonian(H)
f, ax = plt.subplots(2, 2) #, sharex='col', sharey='row')
d_num = L / 2 + 1
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
lim = 2
cdict = {
'red': [(0.0, 0.0, 0.0), (0.5, 1.0, 1.0), (1.0, 1.0, 1.0)],
'green': [(0.0, 0.0, 0.0), (0.5, 1.0, 1.0), (1.0, 0.0, 0.0)],
'blue': [(0.0, 1.0, 1.0), (0.5, 1.0, 1.0), (1.0, 0.0, 0.0)]
}
from matplotlib.colors import LinearSegmentedColormap
cmap = LinearSegmentedColormap('mycm', cdict)
for x1 in range(2):
for x2 in range(2):
Gd = np.zeros((num, d_num), dtype=np.complex)
for i in range(num):
Gd[i, :] = func.fourier(Gij[i][x1, x2::2])
# plt.figure("%d-%d"%(x1,x2))
im = ax[x1][x2].imshow(np.real(Gd),
interpolation='nearest',
origin='lower',
extent=(0, np.pi * 2, -3., 3.),
aspect='auto',
cmap=cmap)
im.set_clim(-lim, lim)
def plot_Kitaev_green():
''' plot the Det(G) in the plane u-w'''
L = 8
num = 51
u_num = 20
ws = np.linspace(-3,3,num)
Gd = np.zeros((num, u_num), dtype=np.complex)
for j, u in enumerate(np.linspace(0,2,u_num)):
H = Kitaev(L, t=1., u=u, d=1., pbc=False)
evals, evecs = func.solve_hamiltonian(H)
Gwk = np.zeros(num, dtype=np.complex)
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
for i in range(num):
Gwk[i] = det(Gij[i])
Gd[:,j] = Gwk
plt.figure("poles")
fig = plt.imshow(np.real(Gd), interpolation='nearest', aspect=0.4)
fig.set_clim(-1,1)
plt.colorbar(ticks=[-1,0,1],label="Det(G)")
plt.xticks([0.5,u_num/2.,u_num-0.5], [0,1,2], fontsize=20)
plt.yticks([0.5,num/2.,num-0.5], [3,0,-3], fontsize=20)
plt.xlabel('u', fontsize=20)
plt.ylabel('w', fontsize=20)
def plot_SSH_green():
''' plot the Det(G) in the plane dt-w'''
L = 8
num = 51
d_num = 20
ws = np.linspace(-3,3,num)
Gd = np.zeros((num, d_num), dtype=np.complex)
for j, d in enumerate(np.linspace(-1,1,d_num)):
H = SSH(L, t=1., d=d, pbc=False)
evals, evecs = func.solve_hamiltonian(H)
Gwk = np.zeros(num, dtype=np.complex)
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
for i in range(num):
Gwk[i] = det(Gij[i])
Gd[:,j] = Gwk
# plt.figure("Green u=%.2f"%d)
# plt.plot(ws, np.real(Gwk),'o-')
# plt.ylim([-1,1])
# print np.min( np.imag(Gwk) ), np.max( np.imag(Gwk) )
plt.figure("poles")
fig = plt.imshow(np.real(Gd), interpolation='nearest', aspect=0.4)
fig.set_clim(-1,1)
plt.colorbar(ticks=[-1,0,1],label="Det(G)")
plt.xticks([0.5,d_num/2.,d_num-0.5], [-1,0,1], fontsize=20)
plt.yticks([0.5,num/2.,num-0.5], [3,0,-3], fontsize=20)
plt.xlabel('d', fontsize=20)
plt.ylabel('w', fontsize=20)
def plot_SSH_green_wk(dt=1.):
''' plot the greens function in the k-w plane '''
L = 12
num = 51 # number of energies to calculate
ws = np.linspace(-3,3,num)
H = SSH(L, t=1., d=dt, pbc=True)
evals, evecs = func.solve_hamiltonian(H)
f, ax = plt.subplots(2,2)#, sharex='col', sharey='row')
d_num = L/2+1
Gij = func.green(ws, op.fermion_c(), L, evals, evecs)
lim = 2
cdict = {'red': [(0.0, 0.0, 0.0), (0.5, 1.0, 1.0), (1.0, 1.0, 1.0)],
'green': [(0.0, 0.0, 0.0), (0.5, 1.0, 1.0), (1.0, 0.0, 0.0)],
'blue': [(0.0, 1.0, 1.0), (0.5, 1.0, 1.0), (1.0, 0.0, 0.0)]}
from matplotlib.colors import LinearSegmentedColormap
cmap = LinearSegmentedColormap('mycm', cdict)
for x1 in range(2):
for x2 in range(2):
Gd = np.zeros((num, d_num), dtype=np.complex)
for i in range(num):
Gd[i, :] = func.fourier(Gij[i][x1,x2::2])
# plt.figure("%d-%d"%(x1,x2))
im = ax[x1][x2].imshow(np.real(Gd), interpolation='nearest', origin='lower',
extent=(0,np.pi*2,-3.,3.), aspect='auto', cmap=cmap)
im.set_clim(-lim,lim)
def Kitaev(L, t=1., u=1., d=1., pbc=False):
''' - t * c^d * c + 2u * n + d * c * c
= ( -t * c^d + d * c) * c + h.c. + 2u * n '''
print "\n--> Construct the Hamiltonian for the Kitaev model of size L=%d ..." % (
L)
store = [[None, None, None] for i in range(L + 1)]
store[1][0] = op.fermion_c().hermitian().mult(op.fermion_c()).times(u)
store[1][1] = op.fermion_c().hermitian().times(-t).plus(
op.fermion_c().times(d))
store[1][2] = op.fermion_c()
def iterate(l):
if store[l] != store[0]: return store[l]
le = int(l / 2)
lH, llo, lro = iterate(le)
rH, rlo, rro = iterate(l - le)
lid = lH.identity()
rid = rH.identity()
store[l][0] = lH.kron(rid)
temp = lid.kron(rH)
store[l][0].plus(temp)
store[l][0].plus(lro.kron(rlo))
store[l][1] = llo.kron(rid)
store[l][2] = lid.kron(rro)
return store[l]
H, lo, ro = iterate(L)
if pbc:
edge = operator(L, [store[1][1], store[1][2]], [0, L - 1])
edge.times(-d)
H.plus(edge)
temp = H.hermitian()
H.plus(temp)
print "--> Hamiltonian constructed."
H.basis = op.basis(2)
H.L = L
return H
def SSH(L, t=1., d=1., pbc=False):
''' - (t-d) * c^d * c - (t+d) * c^d * c '''
print "\n--> Construct the Hamiltonian for the SSH model of size L=%d ..." % (
L)
ope_z = op.fermion_c().times(0.0)
ope_c = op.fermion_c()
ope_cd = op.fermion_c().hermitian()
def iterate(a, b):
if a == b: return [ope_z, ope_c, ope_cd]
ab = a + int((b - a + 1) / 2)
lH, llo, lro = iterate(a, ab - 1)
rH, rlo, rro = iterate(ab, b)
lid = lH.identity()
rid = rH.identity()
H = lH.kron(rid)
temp = lid.kron(rH)
H.plus(temp)
td = -t + d
if ab % 2: td = -t - d
H.plus(lro.kron(rlo).times(td))
return [H, llo.kron(rid), lid.kron(rro)]
H, lo, ro = iterate(1, L)
if pbc:
edge = operator(L, [ope_cd, ope_c], [0, L - 1])
edge.times(t + d)
H.plus(edge)
temp = H.hermitian()
H.plus(temp)
print "--> Hamiltonian constructed."
H.basis = op.basis(2)
H.L = L
return H