Source code for qutip.lattice

# This file is part of QuTiP: Quantum Toolbox in Python.
#
#    Copyright (c) 2011 and later, The QuTiP Project.
#    All rights reserved.
#
#    Redistribution and use in source and binary forms, with or without
#    modification, are permitted provided that the following conditions are
#    met:
#
#    1. Redistributions of source code must retain the above copyright notice,
#       this list of conditions and the following disclaimer.
#
#    2. Redistributions in binary form must reproduce the above copyright
#       notice, this list of conditions and the following disclaimer in the
#       documentation and/or other materials provided with the distribution.
#
#    3. Neither the name of the QuTiP: Quantum Toolbox in Python nor the names
#       of its contributors may be used to endorse or promote products derived
#       from this software without specific prior written permission.
#
#    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
#    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
#    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
#    PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
#    HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
#    SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
#    LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
#    DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
#    THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
#    (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
#    OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
###############################################################################


__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. Methods ------- Hamiltonian() Hamiltonian of the crystal. basis() basis with the particle localized at a certain cell, site with specified degree of freedom. distribute_operator() Distributes an input operator over all the cells. x() Position operator for the crystal. k() Crystal momentum operator for the crystal. operator_at_cells() Distributes an input operator over user specified cells . operator_between_cells() A function that returns an operator matrix that applies an operator between a two specified cells. plot_dispersion() Plots dispersion relation of the crystal. get_dispersion() Returns the dispersion relation of the crystal. bloch_wave_functions() Returns the eigenstates of the Hamiltonian (which are Bloch wavefunctions) for a translationally symmetric periodic lattice. array_of_unk() Returns eigenvectors of the bulk Hamiltonian, i.e. the cell periodic part of the Bloch wavefunctios in a numpy.ndarray for translationally symmetric lattices with periodic boundary condition. bulk_Hamiltonians() Returns the bulk Hamiltonian for the lattice at the good quantum numbers of lattice momentum, k in a numpy ndarray of Qobj's. """ 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).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]).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).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) = np.shape(op) 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.complex) 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('$k_x(\pi/a)$') 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): """ 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', '<f16'), ('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): """ 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) < 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 is '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[1, 0]) for i in range(len(self._H_inter_list)): X_lim = X_lim + np.abs(self._H_inter_list[i][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): """ 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)