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__all__ = ['Lattice1d', 'cell_structures']
from scipy.sparse import (csr_matrix)
from qutip import (Qobj, tensor, basis, qeye, isherm, sigmax, sigmay, sigmaz)
import numpy as np
try:
import matplotlib.pyplot as plt
except:
pass
[docs]def cell_structures(val_s=None, val_t=None, val_u=None):
"""
Returns two matrices H_cell and cell_T to help the user form the inputs for
defining an instance of Lattice1d and Lattice2d classes. The two matrices
are the intra and inter cell Hamiltonians with the tensor structure of the
specified site numbers and/or degrees of freedom defined by the user.
Parameters
==========
val_s : list of str/str
The first list of str's specifying the sites/degrees of freedom in the
unitcell
val_t : list of str/str
The second list of str's specifying the sites/degrees of freedom in the
unitcell
val_u : list of str/str
The third list of str's specifying the sites/degrees of freedom in the
unitcell
Returns
-------
H_cell_s : list of list of str
tensor structure of the cell Hamiltonian elements
T_inter_cell_s : list of list of str
tensor structure of the inter cell Hamiltonian elements
H_cell : Qobj
A Qobj initiated with all 0s with proper shape for an input as
Hamiltonian_of_cell in Lattice1d.__init__()
T_inter_cell : Qobj
A Qobj initiated with all 0s with proper shape for an input as
inter_hop in Lattice1d.__init__()
"""
Er0_str = "At least one list of str necessary for using cell_structures!"
Er1_str = "val_s is required to be a list of str's or a str"
Er2_str = "val_t is required to be a list of str's or a str."
Er3_str = "val_u is required to be a list of str's or a str"
if val_s is None:
raise Exception(Er0_str)
if (not isinstance(val_s, list)) and (not isinstance(val_s, str)):
raise Exception(Er1_str)
if not all([isinstance(val, str) for val in val_s]):
raise(Er1_str)
if val_t is None:
lng0 = len(val_s)
row_i = np.arange(lng0).reshape(lng0, 1)
row_i_c = np.ones(lng0).reshape(1, lng0)
Rows = np.kron(row_i, row_i_c)
Rows = np.array(Rows, dtype=int)
H_cell_s = [[None for i in range(lng0)] for j in range(lng0)]
T_inter_cell_s = [[None for i in range(lng0)] for j in range(lng0)]
for ir in range(lng0):
for ic in range(lng0):
sst = val_s[Rows[ir][ic]]+" H "+val_s[Rows[ic][ir]]
H_cell_s[ir][ic] = "<" + sst + ">"
T_inter_cell_s[ir][ic] = "<cell(i):" + sst + ":cell(i+1) >"
if val_t is not None and val_u is None:
if not all([isinstance(val, str) for val in val_t]):
raise Exception(Er2_str)
lng0 = len(val_s)
lng1 = len(val_t)
p01 = lng0 * lng1
srow_i = np.kron(np.arange(lng0), np.ones(lng1)).reshape(p01, 1)
srow_i_c = np.ones(lng0 * lng1).reshape(1, p01)
sRows = np.kron(srow_i, srow_i_c)
sRows = np.array(sRows, dtype=int)
trow_i = np.kron(np.ones(lng0), np.arange(lng1)).reshape(p01, 1)
trow_i_c = np.ones(lng0 * lng1).reshape(1, p01)
t_rows = np.kron(trow_i, trow_i_c)
t_rows = np.array(t_rows, dtype=int)
H_cell_s = [[None for i in range(p01)] for j in range(p01)]
T_inter_cell_s = [[None for i in range(p01)] for j in range(p01)]
for ir in range(p01):
for jr in range(p01):
sst = []
sst.append(val_s[sRows[ir][jr]])
sst.append(val_t[t_rows[ir][jr]])
sst.append(" H ")
sst.append(val_s[sRows[jr][ir]])
sst.append(val_t[t_rows[jr][ir]])
sst = ''.join(sst)
H_cell_s[ir][jr] = "<" + sst + ">"
llt = []
llt.append("<cell(i):")
llt.append(sst)
llt.append(":cell(i+1) >")
llt = ''.join(llt)
T_inter_cell_s[ir][jr] = llt
if val_u is not None:
if not all([isinstance(val, str) for val in val_u]):
raise Exception(Er3_str)
lng0 = len(val_s)
lng1 = len(val_t)
lng2 = len(val_u)
p012 = lng0 * lng1 * lng2
srow_i = np.kron(np.arange(lng0), np.ones(lng1))
srow_i = np.kron(srow_i, np.ones(lng2))
srow_i = srow_i.reshape(p012, 1)
srow_i_c = np.ones(p012).reshape(1, p012)
sRows = np.kron(srow_i, srow_i_c)
sRows = np.array(sRows, dtype=int)
trow_i = np.kron(np.ones(lng0), np.arange(lng1))
trow_i = np.kron(trow_i, np.ones(lng2))
trow_i = trow_i.reshape(p012, 1)
trow_i_c = np.ones(P).reshape(1, p012)
t_rows = np.kron(trow_i, trow_i_c)
t_rows = np.array(t_rows, dtype=int)
urow_i = np.kron(np.ones(lng0), np.ones(lng1))
urow_i = np.kron(urow_i, np.arange(lng2))
urow_i = urow_i.reshape(p012, 1)
urow_i_c = np.ones(p012).reshape(1, p012)
uRows = np.kron(urow_i, urow_i_c)
uRows = np.array(uRows, dtype=int)
H_cell_s = [[None for i in range(p012)] for j in range(p012)]
T_inter_cell_s = [[None for i in range(p012)] for j in range(p012)]
for ir in range(p012):
for jr in range(p012):
sst = []
sst.append(val_s[sRows[ir][jr]])
sst.append(val_t[t_rows[ir][jr]])
sst.append(val_u[uRows[ir][jr]])
sst.append(" H ")
sst.append(val_s[sRows[jr][ir]])
sst.append(val_t[t_rows[jr][ir]])
sst.append(val_u[uRows[jr][ir]])
sst = ''.join(sst)
H_cell_s[ir][jr] = "<" + sst + ">"
llt = []
llt.append("<cell(i):")
llt.append(sst)
llt.append(":cell(i+1) >")
llt = ''.join(llt)
T_inter_cell_s[ir][jr] = llt
H_cell = np.zeros(np.shape(H_cell_s), dtype=complex)
T_inter_cell = np.zeros(np.shape(T_inter_cell_s), dtype=complex)
return (H_cell_s, T_inter_cell_s, H_cell, T_inter_cell)
[docs]class Lattice1d():
"""A class for representing a 1d crystal.
The Lattice1d class can be defined with any specific unit cells and a
specified number of unit cells in the crystal. It can return dispersion
relationship, position operators, Hamiltonian in the position represention
etc.
Parameters
----------
num_cell : int
The number of cells in the crystal.
boundary : str
Specification of the type of boundary the crystal is defined with.
cell_num_site : int
The number of sites in the unit cell.
cell_site_dof : list of int/ int
The tensor structure of the degrees of freedom at each site of a unit
cell.
Hamiltonian_of_cell : qutip.Qobj
The Hamiltonian of the unit cell.
inter_hop : qutip.Qobj / list of Qobj
The coupling between the unit cell at i and at (i+unit vector)
Attributes
----------
num_cell : int
The number of unit cells in the crystal.
cell_num_site : int
The nuber of sites in a unit cell.
length_for_site : int
The length of the dimension per site of a unit cell.
cell_tensor_config : list of int
The tensor structure of the cell in the form
[cell_num_site,cell_site_dof[:][0] ]
lattice_tensor_config : list of int
The tensor structure of the crystal in the
form [num_cell,cell_num_site,cell_site_dof[:][0]]
length_of_unit_cell : int
The length of the dimension for a unit cell.
period_bnd_cond_x : int
1 indicates "periodic" and 0 indicates "hardwall" boundary condition
inter_vec_list : list of list
The list of list of coefficients of inter unitcell vectors' components
along Cartesian uit vectors.
lattice_vectors_list : list of list
The list of list of coefficients of lattice basis vectors' components
along Cartesian unit vectors.
H_intra : qutip.Qobj
The Qobj storing the Hamiltonian of the unnit cell.
H_inter_list : list of Qobj/ qutip.Qobj
The list of coupling terms between unit cells of the lattice.
is_real : bool
Indicates if the Hamiltonian is real or not.
"""
def __init__(self, num_cell=10, boundary="periodic", cell_num_site=1,
cell_site_dof=[1], Hamiltonian_of_cell=None,
inter_hop=None):
self.num_cell = num_cell
self.cell_num_site = cell_num_site
if (not isinstance(cell_num_site, int)) or cell_num_site < 0:
raise Exception("cell_num_site is required to be a positive \
integer.")
if isinstance(cell_site_dof, list):
l_v = 1
for i, csd_i in enumerate(cell_site_dof):
if (not isinstance(csd_i, int)) or csd_i < 0:
raise Exception("Invalid cell_site_dof list element at \
index: ", i, "Elements of cell_site_dof \
required to be positive integers.")
l_v = l_v * cell_site_dof[i]
self.cell_site_dof = cell_site_dof
elif isinstance(cell_site_dof, int):
if cell_site_dof < 0:
raise Exception("cell_site_dof is required to be a positive \
integer.")
else:
l_v = cell_site_dof
self.cell_site_dof = [cell_site_dof]
else:
raise Exception("cell_site_dof is required to be a positive \
integer or a list of positive integers.")
self._length_for_site = l_v
self.cell_tensor_config = [self.cell_num_site] + self.cell_site_dof
self.lattice_tensor_config = [self.num_cell] + self.cell_tensor_config
# remove any 1 present in self.cell_tensor_config and
# self.lattice_tensor_config unless all the eleents are 1
if all(x == 1 for x in self.cell_tensor_config):
self.cell_tensor_config = [1]
else:
while 1 in self.cell_tensor_config:
self.cell_tensor_config.remove(1)
if all(x == 1 for x in self.lattice_tensor_config):
self.lattice_tensor_config = [1]
else:
while 1 in self.lattice_tensor_config:
self.lattice_tensor_config.remove(1)
dim_ih = [self.cell_tensor_config, self.cell_tensor_config]
self._length_of_unit_cell = self.cell_num_site*self._length_for_site
if boundary == "periodic":
self.period_bnd_cond_x = 1
elif boundary == "aperiodic" or boundary == "hardwall":
self.period_bnd_cond_x = 0
else:
raise Exception("Error in boundary: Only recognized bounday \
options are:\"periodic \",\"aperiodic\" and \"hardwall\" ")
if Hamiltonian_of_cell is None: # There is no user input for
# Hamiltonian_of_cell, so we set it ourselves
H_site = np.diag(np.zeros(cell_num_site-1)-1, 1)
H_site += np.diag(np.zeros(cell_num_site-1)-1, -1)
if cell_site_dof == [1] or cell_site_dof == 1:
Hamiltonian_of_cell = Qobj(H_site, type='oper')
self._H_intra = Hamiltonian_of_cell
else:
Hamiltonian_of_cell = tensor(Qobj(H_site),
qeye(self.cell_site_dof))
dih = Hamiltonian_of_cell.dims[0]
if all(x == 1 for x in dih):
dih = [1]
else:
while 1 in dih:
dih.remove(1)
self._H_intra = Qobj(Hamiltonian_of_cell, dims=[dih, dih],
type='oper')
elif not isinstance(Hamiltonian_of_cell, Qobj): # The user
# input for Hamiltonian_of_cell is not a Qobj and hence is invalid
raise Exception("Hamiltonian_of_cell is required to be a Qobj.")
else: # We check if the user input Hamiltonian_of_cell have the
# right shape or not. If approved, we give it the proper dims
# ourselves.
r_shape = (self._length_of_unit_cell, self._length_of_unit_cell)
if Hamiltonian_of_cell.shape != r_shape:
raise Exception("Hamiltonian_of_cell does not have a shape \
consistent with cell_num_site and cell_site_dof.")
self._H_intra = Qobj(Hamiltonian_of_cell, dims=dim_ih, type='oper')
is_real = np.isreal(self._H_intra.full()).all()
if not isherm(self._H_intra):
raise Exception("Hamiltonian_of_cell is required to be Hermitian.")
nSb = self._H_intra.shape
if isinstance(inter_hop, list): # There is a user input list
inter_hop_sum = Qobj(np.zeros(nSb))
for i in range(len(inter_hop)):
if not isinstance(inter_hop[i], Qobj):
raise Exception("inter_hop[", i, "] is not a Qobj. All \
inter_hop list elements need to be Qobj's. \n")
nSi = inter_hop[i].shape
# inter_hop[i] is a Qobj, now confirmed
if nSb != nSi:
raise Exception("inter_hop[", i, "] is dimensionally \
incorrect. All inter_hop list elements need to \
have the same dimensionality as Hamiltonian_of_cell.")
else: # inter_hop[i] has the right shape, now confirmed,
inter_hop[i] = Qobj(inter_hop[i], dims=dim_ih)
inter_hop_sum = inter_hop_sum + inter_hop[i]
is_real = is_real and np.isreal(inter_hop[i].full()).all()
self._H_inter_list = inter_hop # The user input list was correct
# we store it in _H_inter_list
self._H_inter = Qobj(inter_hop_sum, dims=dim_ih, type='oper')
elif isinstance(inter_hop, Qobj): # There is a user input
# Qobj
nSi = inter_hop.shape
if nSb != nSi:
raise Exception("inter_hop is required to have the same \
dimensionality as Hamiltonian_of_cell.")
else:
inter_hop = Qobj(inter_hop, dims=dim_ih, type='oper')
self._H_inter_list = [inter_hop]
self._H_inter = inter_hop
is_real = is_real and np.isreal(inter_hop.full()).all()
elif inter_hop is None: # inter_hop is the default None)
# So, we set self._H_inter_list from cell_num_site and
# cell_site_dof
if self._length_of_unit_cell == 1:
inter_hop = Qobj([[-1]], type='oper')
else:
bNm = basis(cell_num_site, cell_num_site-1)
bN0 = basis(cell_num_site, 0)
siteT = -bNm * bN0.dag()
inter_hop = tensor(Qobj(siteT), qeye(self.cell_site_dof))
dih = inter_hop.dims[0]
if all(x == 1 for x in dih):
dih = [1]
else:
while 1 in dih:
dih.remove(1)
self._H_inter_list = [Qobj(inter_hop, dims=[dih, dih],
type='oper')]
self._H_inter = Qobj(inter_hop, dims=[dih, dih], type='oper')
else:
raise Exception("inter_hop is required to be a Qobj or a \
list of Qobjs.")
self.positions_of_sites = [(i/self.cell_num_site) for i in
range(self.cell_num_site)]
self._inter_vec_list = [[1] for i in range(len(self._H_inter_list))]
self._Brav_lattice_vectors_list = [[1]] # unit vectors
self.is_real = is_real
def __repr__(self):
s = ""
s += ("Lattice1d object: " +
"Number of cells = " + str(self.num_cell) +
",\nNumber of sites in the cell = " + str(self.cell_num_site) +
",\nDegrees of freedom per site = " +
str(
self.lattice_tensor_config[2:len(self.lattice_tensor_config)]) +
",\nLattice tensor configuration = " +
str(self.lattice_tensor_config) +
",\nbasis_Hamiltonian = " + str(self._H_intra) +
",\ninter_hop = " + str(self._H_inter_list) +
",\ncell_tensor_config = " + str(self.cell_tensor_config) +
"\n")
if self.period_bnd_cond_x == 1:
s += "Boundary Condition: Periodic"
else:
s += "Boundary Condition: Hardwall"
return s
[docs] def Hamiltonian(self):
"""
Returns the lattice Hamiltonian for the instance of Lattice1d.
Returns
----------
Qobj(Hamil) : qutip.Qobj
oper type Quantum object representing the lattice Hamiltonian.
"""
D = qeye(self.num_cell)
T = np.diag(np.zeros(self.num_cell-1)+1, 1)
Tdag = np.diag(np.zeros(self.num_cell-1)+1, -1)
if self.period_bnd_cond_x == 1 and self.num_cell > 2:
Tdag[0][self.num_cell-1] = 1
T[self.num_cell-1][0] = 1
T = Qobj(T)
Tdag = Qobj(Tdag)
Hamil = tensor(D, self._H_intra) + tensor(
T, self._H_inter) + tensor(Tdag, self._H_inter.dag())
dim_H = [self.lattice_tensor_config, self.lattice_tensor_config]
return Qobj(Hamil, dims=dim_H)
[docs] def basis(self, cell, site, dof_ind):
"""
Returns a single particle wavefunction ket with the particle localized
at a specified dof at a specified site of a specified cell.
Parameters
-------
cell : int
The cell at which the particle is to be localized.
site : int
The site of the cell at which the particle is to be localized.
dof_ind : int/ list of int
The index of the degrees of freedom with which the sigle particle
is to be localized.
Returns
----------
vec_i : qutip.Qobj
ket type Quantum object representing the localized particle.
"""
if not isinstance(cell, int):
raise Exception("cell needs to be int in basis().")
elif cell >= self.num_cell:
raise Exception("cell needs to less than Lattice1d.num_cell")
if not isinstance(site, int):
raise Exception("site needs to be int in basis().")
elif site >= self.cell_num_site:
raise Exception("site needs to less than Lattice1d.cell_num_site.")
if isinstance(dof_ind, int):
dof_ind = [dof_ind]
if not isinstance(dof_ind, list):
raise Exception("dof_ind in basis() needs to be an int or \
list of int")
if np.shape(dof_ind) == np.shape(self.cell_site_dof):
for i in range(len(dof_ind)):
if dof_ind[i] >= self.cell_site_dof[i]:
raise Exception("in basis(), dof_ind[", i, "] is required\
to be smaller than cell_num_site[", i, "]")
else:
raise Exception("dof_ind in basis() needs to be of the same \
dimensions as cell_site_dof.")
doft = basis(self.cell_site_dof[0], dof_ind[0])
for i in range(1, len(dof_ind)):
doft = tensor(doft, basis(self.cell_site_dof[i], dof_ind[i]))
vec_i = tensor(
basis(self.num_cell, cell), basis(self.cell_num_site, site),
doft)
ltc = self.lattice_tensor_config
vec_i = Qobj(vec_i, dims=[ltc, [1 for i, j in enumerate(ltc)]])
return vec_i
[docs] def distribute_operator(self, op):
"""
A function that returns an operator matrix that applies op to all the
cells in the 1d lattice
Parameters
-------
op : qutip.Qobj
Qobj representing the operator to be applied at all cells.
Returns
----------
op_H : qutip.Qobj
Quantum object representing the operator with op applied at all
cells.
"""
nSb = self._H_intra.shape
if not isinstance(op, Qobj):
raise Exception("op in distribute_operator() needs to be Qobj.\n")
nSi = op.shape
if nSb != nSi:
raise Exception("op in distribute_operstor() is required to \
have the same dimensionality as Hamiltonian_of_cell.")
cell_All = list(range(self.num_cell))
op_H = self.operator_at_cells(op, cells=cell_All)
return op_H
[docs] def x(self):
"""
Returns the position operator. All degrees of freedom has the cell
number at their correspondig entry in the position operator.
Returns
-------
Qobj(xs) : qutip.Qobj
The position operator.
"""
nx = self.cell_num_site
ne = self._length_for_site
# positions = np.kron(range(nx), [1/nx for i in range(ne)]) # not used
# in the current definition of x
# S = np.kron(np.ones(self.num_cell), positions)
# xs = np.diagflat(R+S) # not used in the
# current definition of x
R = np.kron(range(0, self.num_cell), np.ones(nx*ne))
xs = np.diagflat(R)
dim_H = [self.lattice_tensor_config, self.lattice_tensor_config]
return Qobj(xs, dims=dim_H)
[docs] def k(self):
"""
Returns the crystal momentum operator. All degrees of freedom has the
cell number at their correspondig entry in the position operator.
Returns
-------
Qobj(ks) : qutip.Qobj
The crystal momentum operator in units of 1/a. L is the number
of unit cells, a is the length of a unit cell which is always taken
to be 1.
"""
L = self.num_cell
kop = np.zeros((L, L), dtype=complex)
for row in range(L):
for col in range(L):
if row == col:
kop[row, col] = (L-1)/2
# kop[row, col] = ((L+1) % 2)/ 2
# shifting the eigenvalues
else:
kop[row, col] = 1/(np.exp(2j * np.pi * (row - col)/L) - 1)
qkop = Qobj(kop)
[kD, kV] = qkop.eigenstates()
kop_P = np.zeros((L, L), dtype=complex)
for eno in range(L):
if kD[eno] > (L // 2 + 0.5):
vl = kD[eno] - L
else:
vl = kD[eno]
vk = kV[eno]
kop_P = kop_P + vl * vk * vk.dag()
kop = 2 * np.pi / L * kop_P
nx = self.cell_num_site
ne = self._length_for_site
k = np.kron(kop, np.eye(nx*ne))
dim_H = [self.lattice_tensor_config, self.lattice_tensor_config]
return Qobj(k, dims=dim_H)
[docs] def operator_at_cells(self, op, cells):
"""
A function that returns an operator matrix that applies op to specific
cells specified in the cells list
Parameters
----------
op : qutip.Qobj
Qobj representing the operator to be applied at certain cells.
cells: list of int
The cells at which the operator op is to be applied.
Returns
-------
Qobj(op_H) : Qobj
Quantum object representing the operator with op applied at
the specified cells.
"""
if isinstance(cells, int):
cells = [cells]
if isinstance(cells, list):
for i, cells_i in enumerate(cells):
if not isinstance(cells_i, int):
raise Exception("cells[", i, "] is not an int!elements of \
cells is required to be ints.")
else:
raise Exception("cells in operator_at_cells() need to be an int or\
a list of ints.")
nSb = self._H_intra.shape
if (not isinstance(op, Qobj)):
raise Exception("op in operator_at_cells need to be Qobj's. \n")
nSi = op.shape
if (nSb != nSi):
raise Exception("op in operstor_at_cells() is required to \
be dimensionaly the same as Hamiltonian_of_cell.")
(xx, yy) = op.shape
row_ind = np.array([])
col_ind = np.array([])
data = np.array([])
nS = self._length_of_unit_cell
nx_units = self.num_cell
ny_units = 1
for i in range(nx_units):
lin_RI = i
if (i in cells):
for k in range(xx):
for l in range(yy):
row_ind = np.append(row_ind, [lin_RI*nS+k])
col_ind = np.append(col_ind, [lin_RI*nS+l])
data = np.append(data, [op[k, l]])
m = nx_units*ny_units*nS
op_H = csr_matrix((data, (row_ind, col_ind)), [m, m],
dtype=np.complex128)
dim_op = [self.lattice_tensor_config, self.lattice_tensor_config]
return Qobj(op_H, dims=dim_op)
[docs] def operator_between_cells(self, op, row_cell, col_cell):
"""
A function that returns an operator matrix that applies op to specific
cells specified in the cells list
Parameters
----------
op : qutip.Qobj
Qobj representing the operator to be put between cells row_cell
and col_cell.
row_cell: int
The row index for cell for the operator op to be applied.
col_cell: int
The column index for cell for the operator op to be applied.
Returns
-------
oper_bet_cell : Qobj
Quantum object representing the operator with op applied between
the specified cells.
"""
if not isinstance(row_cell, int):
raise Exception("row_cell is required to be an int between 0 and\
num_cell - 1.")
if row_cell < 0 or row_cell > self.num_cell-1:
raise Exception("row_cell is required to be an int between 0\
and num_cell - 1.")
if not isinstance(col_cell, int):
raise Exception("row_cell is required to be an int between 0 and\
num_cell - 1.")
if col_cell < 0 or col_cell > self.num_cell-1:
raise Exception("row_cell is required to be an int between 0\
and num_cell - 1.")
nSb = self._H_intra.shape
if (not isinstance(op, Qobj)):
raise Exception("op in operator_between_cells need to be Qobj's.")
nSi = op.shape
if (nSb != nSi):
raise Exception("op in operstor_between_cells() is required to \
be dimensionally the same as Hamiltonian_of_cell.")
T = np.zeros((self.num_cell, self.num_cell), dtype=complex)
T[row_cell, col_cell] = 1
op_H = np.kron(T, op)
dim_op = [self.lattice_tensor_config, self.lattice_tensor_config]
return Qobj(op_H, dims=dim_op)
[docs] def plot_dispersion(self):
"""
Plots the dispersion relationship for the lattice with the specified
number of unit cells. The dispersion of the infinte crystal is also
plotted if num_cell is smaller than MAXc.
"""
MAXc = 20 # Cell numbers above which we do not plot the infinite
# crystal dispersion
if self.period_bnd_cond_x == 0:
raise Exception("The lattice is not periodic.")
if self.num_cell <= MAXc:
(kxA, val_ks) = self.get_dispersion(101)
(knxA, val_kns) = self.get_dispersion()
fig, ax = plt.subplots()
if self.num_cell <= MAXc:
for g in range(self._length_of_unit_cell):
ax.plot(kxA/np.pi, val_ks[g, :])
for g in range(self._length_of_unit_cell):
if self.num_cell % 2 == 0:
ax.plot(np.append(knxA, [np.pi])/np.pi,
np.append(val_kns[g, :], val_kns[g, 0]), 'ro')
else:
ax.plot(knxA/np.pi, val_kns[g, :], 'ro')
ax.set_ylabel('Energy')
ax.set_xlabel(r'$k_x(\pi/a)$')
plt.show(fig)
fig.savefig('./Dispersion.pdf')
[docs] def get_dispersion(self, knpoints=0):
"""
Returns dispersion relationship for the lattice with the specified
number of unit cells with a k array and a band energy array.
Returns
-------
knxa : np.array
knxA[j][0] is the jth good Quantum number k.
val_kns : np.array
val_kns[j][:] is the array of band energies of the jth band good at
all the good Quantum numbers of k.
"""
# The _k_space_calculations() function is not used for get_dispersion
# because we calculate the infinite crystal dispersion in
# plot_dispersion using this coode and we do not want to calculate
# all the eigen-values, eigenvectors of the bulk Hamiltonian for too
# many points, as is done in the _k_space_calculations() function.
if self.period_bnd_cond_x == 0:
raise Exception("The lattice is not periodic.")
if knpoints == 0:
knpoints = self.num_cell
a = 1 # The unit cell length is always considered 1
kn_start = 0
kn_end = 2*np.pi/a
val_kns = np.zeros((self._length_of_unit_cell, knpoints), dtype=float)
knxA = np.zeros((knpoints, 1), dtype=float)
G0_H = self._H_intra
# knxA = np.roll(knxA, np.floor_divide(knpoints, 2))
for ks in range(knpoints):
knx = kn_start + (ks*(kn_end-kn_start)/knpoints)
if knx >= np.pi:
knxA[ks, 0] = knx - 2 * np.pi
else:
knxA[ks, 0] = knx
knxA = np.roll(knxA, np.floor_divide(knpoints, 2))
for ks in range(knpoints):
kx = knxA[ks, 0]
H_ka = G0_H
k_cos = [[kx]]
for m in range(len(self._H_inter_list)):
r_cos = self._inter_vec_list[m]
kr_dotted = np.dot(k_cos, r_cos)
H_int = self._H_inter_list[m]*np.exp(kr_dotted*1j)[0]
H_ka = H_ka + H_int + H_int.dag()
H_k = csr_matrix(H_ka)
qH_k = Qobj(H_k)
(vals, veks) = qH_k.eigenstates()
val_kns[:, ks] = vals[:]
return (knxA, val_kns)
[docs] def bloch_wave_functions(self):
r"""
Returns eigenvectors ($\psi_n(k)$) of the Hamiltonian in a
numpy.ndarray for translationally symmetric lattices with periodic
boundary condition.
.. math::
:nowrap:
\begin{eqnarray}
|\psi_n(k) \rangle = |k \rangle \otimes | u_{n}(k) \rangle \\
| u_{n}(k) \rangle = a_n(k)|a\rangle + b_n(k)|b\rangle \\
\end{eqnarray}
Please see section 1.2 of Asbóth, J. K., Oroszlány, L., & Pályi, A.
(2016). A short course on topological insulators. Lecture notes in
physics, 919 for a review.
Returns
-------
eigenstates : ordered np.array
eigenstates[j][0] is the jth eigenvalue.
eigenstates[j][1] is the corresponding eigenvector.
"""
if self.period_bnd_cond_x == 0:
raise Exception("The lattice is not periodic.")
(knxA, qH_ks, val_kns, vec_kns, vec_xs) = self._k_space_calculations()
dtype = [('eigen_value', np.longdouble), ('eigen_vector', Qobj)]
values = list()
for i in range(self.num_cell):
for j in range(self._length_of_unit_cell):
values.append((
val_kns[j][i], vec_xs[j+i*self._length_of_unit_cell]))
eigen_states = np.array(values, dtype=dtype)
# eigen_states = np.sort(eigen_states, order='eigen_value')
return eigen_states
[docs] def cell_periodic_parts(self):
r"""
Returns eigenvectors of the bulk Hamiltonian, i.e. the cell periodic
part($u_n(k)$) of the Bloch wavefunctios in a numpy.ndarray for
translationally symmetric lattices with periodic boundary condition.
.. math::
:nowrap:
\begin{eqnarray}
|\psi_n(k) \rangle = |k \rangle \otimes | u_{n}(k) \rangle \\
| u_{n}(k) \rangle = a_n(k)|a\rangle + b_n(k)|b\rangle \\
\end{eqnarray}
Please see section 1.2 of Asbóth, J. K., Oroszlány, L., & Pályi, A.
(2016). A short course on topological insulators. Lecture notes in
physics, 919 for a review.
Returns
-------
knxa : np.array
knxA[j][0] is the jth good Quantum number k.
vec_kns : np.ndarray of Qobj's
vec_kns[j] is the Oobj of type ket that holds an eigenvector of the
bulk Hamiltonian of the lattice.
"""
if self.period_bnd_cond_x == 0:
raise Exception("The lattice is not periodic.")
(knxA, qH_ks, val_kns, vec_kns, vec_xs) = self._k_space_calculations()
return (knxA, vec_kns)
[docs] def bulk_Hamiltonians(self):
"""
Returns the bulk momentum space Hamiltonian ($H(k)$) for the lattice at
the good quantum numbers of k in a numpy ndarray of Qobj's.
Please see section 1.2 of Asbóth, J. K., Oroszlány, L., & Pályi, A.
(2016). A short course on topological insulators. Lecture notes in
physics, 919 for a review.
Returns
-------
knxa : np.array
knxA[j][0] is the jth good Quantum number k.
qH_ks : np.ndarray of Qobj's
qH_ks[j] is the Oobj of type oper that holds a bulk Hamiltonian
for a good quantum number k.
"""
if self.period_bnd_cond_x == 0:
raise Exception("The lattice is not periodic.")
(knxA, qH_ks, val_kns, vec_kns, vec_xs) = self._k_space_calculations()
return (knxA, qH_ks)
def _k_space_calculations(self, knpoints=0):
"""
Returns bulk Hamiltonian, its eigenvectors and eigenvectors of the
space Hamiltonian at all the good quantum numbers of a periodic
translationally invariant lattice.
Returns
-------
knxa : np.array
knxA[j][0] is the jth good Quantum number k.
qH_ks : np.ndarray of Qobj's
qH_ks[j] is the Oobj of type oper that holds a bulk Hamiltonian
for a good quantum number k.
vec_xs : np.ndarray of Qobj's
vec_xs[j] is the Oobj of type ket that holds an eigenvector of the
Hamiltonian of the lattice.
vec_kns : np.ndarray of Qobj's
vec_kns[j] is the Oobj of type ket that holds an eigenvector of the
bulk Hamiltonian of the lattice.
"""
if knpoints == 0:
knpoints = self.num_cell
a = 1 # The unit cell length is always considered 1
kn_start = 0
kn_end = 2*np.pi/a
val_kns = np.zeros((self._length_of_unit_cell, knpoints), dtype=float)
knxA = np.zeros((knpoints, 1), dtype=float)
G0_H = self._H_intra
vec_kns = np.zeros((knpoints, self._length_of_unit_cell,
self._length_of_unit_cell), dtype=complex)
vec_xs = np.array([None for i in range(
knpoints * self._length_of_unit_cell)])
qH_ks = np.array([None for i in range(knpoints)])
for ks in range(knpoints):
knx = kn_start + (ks*(kn_end-kn_start)/knpoints)
if knx >= np.pi:
knxA[ks, 0] = knx - 2 * np.pi
else:
knxA[ks, 0] = knx
knxA = np.roll(knxA, np.floor_divide(knpoints, 2))
dim_hk = [self.cell_tensor_config, self.cell_tensor_config]
for ks in range(knpoints):
kx = knxA[ks, 0]
H_ka = G0_H
k_cos = [[kx]]
for m in range(len(self._H_inter_list)):
r_cos = self._inter_vec_list[m]
kr_dotted = np.dot(k_cos, r_cos)
H_int = self._H_inter_list[m]*np.exp(kr_dotted*1j)[0]
H_ka = H_ka + H_int + H_int.dag()
H_k = csr_matrix(H_ka)
qH_k = Qobj(H_k, dims=dim_hk)
qH_ks[ks] = qH_k
(vals, veks) = qH_k.eigenstates()
plane_waves = np.kron(np.exp(-1j * (kx * range(self.num_cell))),
np.ones(self._length_of_unit_cell))
for eig_no in range(self._length_of_unit_cell):
unit_cell_periodic = np.kron(
np.ones(self.num_cell), veks[eig_no].dag())
vec_x = np.multiply(plane_waves, unit_cell_periodic)
dim_H = [list(np.ones(len(self.lattice_tensor_config),
dtype=int)), self.lattice_tensor_config]
if self.is_real:
if np.count_nonzero(vec_x) > 0:
vec_x = np.real(vec_x)
length_vec_x = np.sqrt((Qobj(vec_x) * Qobj(vec_x).dag())[0][0])
vec_x = vec_x / length_vec_x
vec_x = Qobj(vec_x, dims=dim_H)
vec_xs[ks*self._length_of_unit_cell+eig_no] = vec_x.dag()
for i in range(self._length_of_unit_cell):
v0 = np.squeeze(veks[i].full(), axis=1)
vec_kns[ks, i, :] = v0
val_kns[:, ks] = vals[:]
return (knxA, qH_ks, val_kns, vec_kns, vec_xs)
[docs] def winding_number(self):
"""
Returns the winding number for a lattice that has chiral symmetry and
also plots the trajectory of (dx,dy)(dx,dy are the coefficients of
sigmax and sigmay in the Hamiltonian respectively) on a plane.
Returns
-------
winding_number : int or str
knxA[j][0] is the jth good Quantum number k.
"""
winding_number = 'defined'
if (self._length_of_unit_cell != 2):
raise Exception('H(k) is not a 2by2 matrix.')
if (self._H_intra[0, 0] != 0 or self._H_intra[1, 1] != 0):
raise Exception("Hamiltonian_of_cell has nonzero diagonals!")
for i in range(len(self._H_inter_list)):
H_I_00 = self._H_inter_list[i][0, 0]
H_I_11 = self._H_inter_list[i][1, 1]
if (H_I_00 != 0 or H_I_11 != 0):
raise Exception("inter_hop has nonzero diagonal elements!")
chiral_op = self.distribute_operator(sigmaz())
Hamt = self.Hamiltonian()
anti_commutator_chi_H = chiral_op * Hamt + Hamt * chiral_op
is_null = (np.abs(anti_commutator_chi_H.full()) < 1E-10).all()
if not is_null:
raise Exception("The Hamiltonian does not have chiral symmetry!")
knpoints = 100 # choose even
kn_start = 0
kn_end = 2*np.pi
knxA = np.zeros((knpoints+1, 1), dtype=float)
G0_H = self._H_intra
mx_k = np.array([None for i in range(knpoints+1)])
my_k = np.array([None for i in range(knpoints+1)])
Phi_m_k = np.array([None for i in range(knpoints+1)])
for ks in range(knpoints+1):
knx = kn_start + (ks*(kn_end-kn_start)/knpoints)
knxA[ks, 0] = knx
for ks in range(knpoints+1):
kx = knxA[ks, 0]
H_ka = G0_H
k_cos = [[kx]]
for m in range(len(self._H_inter_list)):
r_cos = self._inter_vec_list[m]
kr_dotted = np.dot(k_cos, r_cos)
H_int = self._H_inter_list[m]*np.exp(kr_dotted*1j)[0]
H_ka = H_ka + H_int + H_int.dag()
H_k = csr_matrix(H_ka)
qH_k = Qobj(H_k)
mx_k[ks] = 0.5*(qH_k*sigmax()).tr()
my_k[ks] = 0.5*(qH_k*sigmay()).tr()
if np.abs(mx_k[ks]) < 1E-10 and np.abs(my_k[ks]) < 1E-10:
winding_number = 'undefined'
if np.angle(mx_k[ks]+1j*my_k[ks]) >= 0:
Phi_m_k[ks] = np.angle(mx_k[ks]+1j*my_k[ks])
else:
Phi_m_k[ks] = 2*np.pi + np.angle(mx_k[ks]+1j*my_k[ks])
if winding_number == 'defined':
ddk_Phi_m_k = np.roll(Phi_m_k, -1) - Phi_m_k
intg_over_k = -np.sum(ddk_Phi_m_k[0:knpoints//2])+np.sum(
ddk_Phi_m_k[knpoints//2:knpoints])
winding_number = intg_over_k/(2*np.pi)
X_lim = 1.125 * np.abs(self._H_intra.full()[1, 0])
for i in range(len(self._H_inter_list)):
X_lim = X_lim + np.abs(self._H_inter_list[i].full()[1, 0])
plt.figure(figsize=(3*X_lim, 3*X_lim))
plt.plot(mx_k, my_k)
plt.plot(0, 0, 'ro')
plt.ylabel('$h_y$')
plt.xlabel('$h_x$')
plt.xlim(-X_lim, X_lim)
plt.ylim(-X_lim, X_lim)
plt.grid()
plt.show()
plt.savefig('./Winding.pdf')
plt.close()
return winding_number
[docs] def display_unit_cell(self, label_on=False):
"""
Produces a graphic displaying the unit cell features with labels on if
defined by user. Also returns a dict of Qobj's corresponding to the
labeled elements on the display.
Returns
-------
Hcell : dict
Hcell[i][j] is the Hamiltonian segment for $H_{i,j}$ labeled on the
graphic.
"""
CNS = self.cell_num_site
Hcell = [[{} for i in range(CNS)] for j in range(CNS)]
for i0 in range(CNS):
for j0 in range(CNS):
Qin = np.zeros((self._length_for_site, self._length_for_site),
dtype=complex)
for i in range(self._length_for_site):
for j in range(self._length_for_site):
Qin[i, j] = self._H_intra[
i0*self._length_for_site+i,
j0*self._length_for_site+j]
dim_site = list(np.delete(self.cell_tensor_config, [0], None))
dims_site = [dim_site, dim_site]
Hcell[i0][j0] = Qobj(Qin, dims=dims_site)
fig = plt.figure(figsize=[CNS*2, CNS*2.5])
ax = fig.add_subplot(111, aspect='equal')
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
if (CNS == 1):
ax.plot([self.positions_of_sites[0]], [0], "o", c="b", mec="w",
mew=0.0, zorder=10, ms=8.0)
if label_on is True:
plt.text(x=self.positions_of_sites[0]+0.2, y=0.0,
s='H'+str(i)+str(i), horizontalalignment='center',
verticalalignment='center')
x2 = (1+self.positions_of_sites[CNS-1])/2
x1 = x2-1
h = 1-x2
ax.plot([x1, x1], [-h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x2, x2], [-h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x1, x2], [h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x1, x2], [-h, -h], "-", c="k", lw=1.5, zorder=7)
plt.axis('off')
plt.show()
plt.close()
else:
for i in range(CNS):
ax.plot([self.positions_of_sites[i]], [0], "o", c="b", mec="w",
mew=0.0, zorder=10, ms=8.0)
if label_on is True:
x_b = self.positions_of_sites[i]+1/CNS/6
plt.text(x=x_b, y=0.0, s='H'+str(i)+str(i),
horizontalalignment='center',
verticalalignment='center')
if i == CNS-1:
continue
for j in range(i+1, CNS):
if (Hcell[i][j].full() == 0).all():
continue
c_cen = (self.positions_of_sites[
i]+self.positions_of_sites[j])/2
c_radius = (self.positions_of_sites[
j]-self.positions_of_sites[i])/2
circle1 = plt.Circle((c_cen, 0), c_radius, color='g',
fill=False)
ax.add_artist(circle1)
if label_on is True:
x_b = c_cen
y_b = c_radius - 0.025
plt.text(x=x_b, y=y_b, s='H'+str(i)+str(j),
horizontalalignment='center',
verticalalignment='center')
x2 = (1+self.positions_of_sites[CNS-1])/2
x1 = x2-1
h = (self.positions_of_sites[
CNS-1]-self.positions_of_sites[0])*8/15
ax.plot([x1, x1], [-h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x2, x2], [-h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x1, x2], [h, h], "-", c="k", lw=1.5, zorder=7)
ax.plot([x1, x2], [-h, -h], "-", c="k", lw=1.5, zorder=7)
plt.axis('off')
plt.show()
plt.close()
return Hcell
[docs] def display_lattice(self):
r"""
Produces a graphic portraying the lattice symbolically with a unit cell
marked in it.
Returns
-------
inter_T : Qobj
The coefficient of $\psi_{i,N}^{\dagger}\psi_{0,i+1}$, i.e. the
coupling between the two boundary sites of the two unit cells i and
i+1.
"""
dim_I = [self.cell_tensor_config, self.cell_tensor_config]
H_inter = Qobj(np.zeros((self._length_of_unit_cell,
self._length_of_unit_cell)), dims=dim_I)
for no, inter_hop_no in enumerate(self._H_inter_list):
H_inter = H_inter + inter_hop_no
H_inter = np.array(H_inter)
csn = self.cell_num_site
Hcell = [[{} for i in range(csn)] for j in range(csn)]
for i0 in range(csn):
for j0 in range(csn):
Qin = np.zeros((self._length_for_site, self._length_for_site),
dtype=complex)
for i in range(self._length_for_site):
for j in range(self._length_for_site):
Qin[i, j] = self._H_intra[
i0*self._length_for_site+i,
j0*self._length_for_site+j]
dim_site = list(np.delete(self.cell_tensor_config, [0], None))
dims_site = [dim_site, dim_site]
Hcell[i0][j0] = Qobj(Qin, dims=dims_site)
j0 = 0
i0 = csn-1
Qin = np.zeros((self._length_for_site, self._length_for_site),
dtype=complex)
for i in range(self._length_for_site):
for j in range(self._length_for_site):
Qin[i, j] = H_inter[i0*self._length_for_site+i,
j0*self._length_for_site+j]
inter_T = Qin
fig = plt.figure(figsize=[self.num_cell*3, self.num_cell*3])
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
ax = fig.add_subplot(111, aspect='equal')
for nc in range(self.num_cell):
x_cell = nc
for i in range(csn):
ax.plot([x_cell + self.positions_of_sites[i]], [0], "o",
c="b", mec="w", mew=0.0, zorder=10, ms=8.0)
if nc > 0:
# plot inter_cell_hop
ax.plot([x_cell-1+self.positions_of_sites[csn-1],
x_cell+self.positions_of_sites[0]], [0.0, 0.0],
"-", c="r", lw=1.5, zorder=7)
x_b = (x_cell-1+self.positions_of_sites[
csn-1] + x_cell + self.positions_of_sites[0])/2
plt.text(x=x_b, y=0.1, s='T',
horizontalalignment='center',
verticalalignment='center')
if i == csn-1:
continue
for j in range(i+1, csn):
if (Hcell[i][j].full() == 0).all():
continue
c_cen = self.positions_of_sites[i]
c_cen = (c_cen+self.positions_of_sites[j])/2
c_cen = c_cen + x_cell
c_radius = self.positions_of_sites[j]
c_radius = (c_radius-self.positions_of_sites[i])/2
circle1 = plt.Circle((c_cen, 0),
c_radius, color='g', fill=False)
ax.add_artist(circle1)
if (self.period_bnd_cond_x == 1):
x_cell = 0
x_b = 2*x_cell-1+self.positions_of_sites[csn-1]
x_b = (x_b+self.positions_of_sites[0])/2
plt.text(x=x_b, y=0.1, s='T', horizontalalignment='center',
verticalalignment='center')
ax.plot([x_cell-1+self.positions_of_sites[csn-1],
x_cell+self.positions_of_sites[0]], [0.0, 0.0],
"-", c="r", lw=1.5, zorder=7)
x_cell = self.num_cell
x_b = 2*x_cell-1+self.positions_of_sites[csn-1]
x_b = (x_b+self.positions_of_sites[0])/2
plt.text(x=x_b, y=0.1, s='T', horizontalalignment='center',
verticalalignment='center')
ax.plot([x_cell-1+self.positions_of_sites[csn-1],
x_cell+self.positions_of_sites[0]], [0.0, 0.0],
"-", c="r", lw=1.5, zorder=7)
x2 = (1+self.positions_of_sites[csn-1])/2
x1 = x2-1
h = 0.5
if self.num_cell > 2:
xu = 1 # The index of cell over which the black box is drawn
x1 = x1+xu
x2 = x2+xu
ax.plot([x1, x1], [-h, h], "-", c="k", lw=1.5, zorder=7, alpha=0.3)
ax.plot([x2, x2], [-h, h], "-", c="k", lw=1.5, zorder=7, alpha=0.3)
ax.plot([x1, x2], [h, h], "-", c="k", lw=1.5, zorder=7, alpha=0.3)
ax.plot([x1, x2], [-h, -h], "-", c="k", lw=1.5, zorder=7, alpha=0.3)
plt.axis('off')
plt.show()
plt.close()
dim_site = list(np.delete(self.cell_tensor_config, [0], None))
dims_site = [dim_site, dim_site]
return Qobj(inter_T, dims=dims_site)