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PhononLattice.py
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PhononLattice.py
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import itertools as it
import numpy as np
import qutip
from numpy import exp, sqrt, pi, dot
from scipy.linalg import sqrtm, eigh, inv, norm
from AbsHamiltonian import AbsPhononHamiltonian
from Lattice import Lattice
class PhononLattice(Lattice, AbsPhononHamiltonian):
def __init__(self, unit_cell, N, c_matrix, nfock=2, force_adj_only=True,
*args, **kwargs):
super(PhononLattice, self).__init__(
unit_cell=unit_cell, N=N, *args, **kwargs)
self.num_modes = self.dim_space * self.M
self._nfock = nfock
self.c_matrix = c_matrix
self._evals_dict = dict()
self._evect_dict = dict()
self._indices = list(
it.product(range(self.M), range(self.dim_space))
)
# self._force_adj = force_adj_only
self._force_adj = force_adj_only
self.A = set(self._gen_A())
self.B = set(self._gen_B())
def omega(self, q, v):
return sqrt(self.omega2(q, v))
def a(self, q, v):
if q in self.A:
x, px = self._x(q, v)
return 1/sqrt(2) * (x + 1j*px)
elif q in self.B:
x, px = self._x(q, v)
y, py = self._y(q, v)
return 1/2 * (x + 1j*px) + 1j/2 * (y + 1j*py)
def _iter_q(self):
return self.operator_q_vectors()
def _iter_v(self):
return range(self.num_modes)
def b_matrix(self):
return 2*pi * inv(self.unit_cell.a_matrix.conj().T)
def b_vectors(self):
b = self.b_matrix()
return [b[:, i] for i in range(self.dim_space)]
def d_matrix(self, kappa1, alpha1, kappa2, alpha2):
def _d(q):
d = 0
for p in self.unit_cells():
if self._force_adj and not self.are_adjacent(p, self.p0):
continue
dp = self.c_matrix(lattice=self,
kappa1=kappa1, alpha1=alpha1, p1=self.p0,
kappa2=kappa2, alpha2=alpha2, p2=p)
dp *= exp(1j * 2*pi * q.dot(p))
d += dp
d *= 1/sqrt(
self.unit_cell.mass(kappa1) * self.unit_cell.mass(kappa2))
return d
return _d
def e(self, kappa, alpha, v):
i = self._indices.index((kappa, alpha))
def _e(q):
if q not in self._evals_dict:
self._set_d_eigenvectors(q)
return self._evect_dict[q][v][i]
return _e
def omega2(self, q, v):
if q not in self._evals_dict:
self._set_d_eigenvectors(q)
return self._evals_dict[q][v]
def _gen_A(self):
for q in self.q_vectors():
if q == -q:
yield q
def _gen_B(self):
b = []
for q in self.q_vectors():
if q != -q and -q not in b:
b.append(q)
yield q
def operator_q_vectors(self):
return it.chain(self._gen_A(), self._gen_B())
def _ops(self, kappa, alpha, p, op):
dim = self.dim_space * self.M * self.Np
xop = [qutip.qeye(self._nfock)] * dim
idx = alpha + self.dim_space * kappa + self.dim_space * self.M * p[0]
for i in range(self.dim_space - 1):
idx += self.dim_space * self.M * self.N[i] * p[i+1]
xop.insert(idx, op)
xop.pop(idx+1)
return qutip.tensor(xop)
def _x_ops(self, kappa, alpha, p):
return self._ops(kappa, alpha, p, op=qutip.position(self._nfock))
def _p_ops(self, kappa, alpha, p):
return self._ops(kappa, alpha, p, op=qutip.momentum(self._nfock))
def _z(self, q, v):
real_z = 0+0j
imag_z = 0+0j
real_pz = 0+0j
imag_pz = 0+0j
for kappa, alpha, p in it.product(range(self.M), range(self.dim_space),
self.unit_cells()):
zi = exp(-1j * 2*np.pi * (q.dot(p)))
zi *= sqrt(self.unit_cell.mass(kappa) / self.unit_cell.mass(0))
zi *= np.conj(self.e(kappa, alpha, v)(q)) / sqrt(self.Np)
xop = self._x_ops(kappa, alpha, p)
pop = self._p_ops(kappa, alpha, p)
real_z += zi.real * xop
imag_z += zi.imag * xop
real_pz += zi.real * pop
imag_pz += zi.imag * pop
return real_z, imag_z, real_pz, imag_pz
def _l_inv(self, q, v):
m0 = self.unit_cell.mass(0)
# omega2 = self.omega2(q, v)+0j
omega2 = 1
return sqrt(2 * m0 * sqrt(omega2))
def _x(self, q, v):
real_z, imag_z, real_pz, imag_pz = self._z(q, v)
linv = self._l_inv(q, v)
if q in self.A:
return real_z/2 * linv, real_pz/2 * linv
elif q in self.B:
return real_z * linv, real_pz * linv
def _y(self, q, v):
real_z, imag_z, real_pz, imag_pz = self._z(q, v)
linv = self._l_inv(q, v)
if q in self.B:
return imag_z * linv, imag_pz * linv
def _get_matrix_rep_d(self, q):
d_mat = np.empty(shape=(self.num_modes, self.num_modes), dtype=np.complex)
for k1x1, i in zip(self._particles(), it.count()):
k1, x1 = k1x1
for k2x2, j in zip(self._particles(), it.count()):
k2, x2 = k2x2
d_mat[i, j] = self.d_matrix(
kappa1=k1, alpha1=x1, kappa2=k2, alpha2=x2)(q)
return d_mat
def _orthonormal_eigenvectors(self, dmat):
evals, evects = eigh(a=dmat, turbo=True)
on_evects = dot(evects, inv(sqrtm(dot(evects.conj().T, evects))))
return evals, [on_evects[:, i] for i in range(len(evals))]
def _set_d_eigenvectors(self, q):
dmat = self._get_matrix_rep_d(q)
evals, evects = self._orthonormal_eigenvectors(dmat=dmat)
self._evals_dict[q] = evals
self._evect_dict[q] = evects
class PhononLattice1D(PhononLattice):
def __init__(self, unit_cell, N_x, c_matrix, n_fock=2, *args, **kwargs):
super(PhononLattice1D, self).__init__(
unit_cell=unit_cell, N=[N_x], c_matrix=c_matrix, nfock=n_fock,
*args, **kwargs)
self.Nx = N_x
class PhononLattice2D(PhononLattice):
def __init__(self, unit_cell, N_x, N_y, c_matrix, n_fock=2, *args,
**kwargs):
super(PhononLattice2D, self).__init__(
unit_cell=unit_cell, N=[N_x, N_y], c_matrix=c_matrix, nfock=n_fock,
*args, **kwargs)
self.Nx = N_x
self.Ny = N_y
class PhononLattice3D(PhononLattice):
def __init__(self, unit_cell, N_x, N_y, N_z, c_matrix, n_fock=2,
*args, **kwargs):
super(PhononLattice3D, self).__init__(
unit_cell=unit_cell, N=[N_x, N_y, N_z], c_matrix=c_matrix,
nfock=n_fock, *args, **kwargs)
self.Nx = N_x
self.Ny = N_y
self.Nz = N_z
def get_c_matrix_simple_harmonic_interaction(k):
def c_matrix(lattice, kappa1, alpha1, p1, kappa2, alpha2, p2):
if (kappa1, alpha1, p1.all()) == (kappa2, alpha2, p2.all()):
return k * lattice.unit_cell.num_connections(kappa1)
elif alpha1 == alpha2 and lattice.are_connected(
kappa1=kappa1, p1=p1, kappa2=kappa2, p2=p2):
return -k
else:
return 0
return c_matrix
def get_c_matrix_coulomb_interaction(g):
def c_matrix(lattice, kappa1, alpha1, p1, kappa2, alpha2, p2):
if not lattice.are_connected(kappa1, p1, kappa2, p2):
return 0 # Only interact with neighbors
def sterm(kappa_i, p_i):
if (kappa_i, p_i.all()) == (kappa1, p1.all()):
return 0
disp = lattice.periodic_displacement_distance(
kappa1=kappa_i, p1=p_i, kappa2=kappa1, p2=p1)
t1 = -g * 3 * disp[alpha1] * disp[alpha2] / norm(disp, ord=2)**5
if alpha1 != alpha2:
return t1
else:
return t1 + g / norm(disp, ord=2)**3
if kappa1 == kappa2:
s = 0
for kappa_i, p_i in it.product(range(lattice.M), lattice.unit_cells()):
s += sterm(kappa_i, p_i)
return s
else:
return -sterm(kappa2, p2)
return c_matrix