sisl.physics.electron.StateElectron
- class sisl.physics.electron.StateElectron(state, parent=None, **info)[source]
Bases:
_electron_State
,State
A state describing a physical quantity related to electrons
Methods
Sk
([format])Retrieve the overlap matrix corresponding to the originating parent structure.
align_norm
(other[, ret_index, inplace])Align self with the site-norms of other, a copy may optionally be returned
align_phase
(other[, ret_index, inplace])Align self with the phases for other, a copy may be returned
change_gauge
(gauge[, offset])In-place change of the gauge of the state coefficients
copy
()Return a copy (only the state is copied).
inner
([ket, matrix, projection])Calculate the inner product as \(\mathbf A_{ij} = \langle\psi_i|\mathbf M|\psi'_j\rangle\)
ipr
([q])Calculate the inverse participation ratio (IPR) for arbitrary q values
iter
([asarray])An iterator looping over the states in this system
norm
()Return a vector with the Euclidean norm of each state \(\sqrt{\langle\psi|\psi\rangle}\)
norm2
([projection])Return a vector with the norm of each state \(\langle\psi|\mathbf S|\psi\rangle\)
Return a normalized state where each state has \(|\psi|^2=1\)
outer
([ket, matrix])Return the outer product by \(\sum_\alpha|\psi_\alpha\rangle\langle\psi'_\alpha|\)
phase
([method, ret_index])Calculate the Euler angle (phase) for the elements of the state, in the range \(]-\pi;\pi]\)
remove
(index[, inplace])Return a new state without the specified vectors
rotate
([phi, individual, inplace])Rotate all states to rotate the largest component to be along the angle phi
spin_moment
([project])Calculate spin moment from the states
sub
(index[, inplace])Return a new state with only the specified states
tile
(reps, axis[, normalize, offset])Tile the state vectors for a new supercell
translate
(isc)Translate the vectors to a new unit-cell position
wavefunction
(grid[, spinor, eta])Expand the coefficients as the wavefunction on grid as-is
Attributes
The data-type of the state (in str)
Data-type for the state
Returns the shape of the state
- Sk(format=None)
Retrieve the overlap matrix corresponding to the originating parent structure.
When
self.parent
is a Hamiltonian this will return \(\mathbf S(\mathbf k)\) for the \(\mathbf k\)-point these eigenstates originate from.- Parameters:
format (
str
, optional) – the returned format of the overlap matrix. This only takes effect for non-orthogonal parents.
- align_norm(other: State, ret_index: bool = False, inplace: bool = False)
Align self with the site-norms of other, a copy may optionally be returned
To determine the new ordering of self first calculate the residual norm of the site-norms.
\[\delta N_{\alpha\beta} = \sum_i \big(\langle \psi^\alpha_i | \psi^\alpha_i\rangle - \langle \psi^\beta_i | \psi^\beta_i\rangle\big)^2\]where \(\alpha\) and \(\beta\) correspond to state indices in self and other, respectively. The new states (from self) returned is then ordered such that the index \(\alpha \equiv \beta'\) where \(\delta N_{\alpha\beta}\) is smallest.
- Parameters:
other (
State
) – the other state to align againstret_index – also return indices for the swapped indices
inplace – swap states in-place
- Returns:
self_swap (
State
) – A swapped instance of self, only if inplace is Falseindex (
array
ofint
) – the indices that swaps self to beself_swap
, i.e.self_swap = self.sub(index)
Only if inplace is False and ret_index is True
Notes
The input state and output state have the same number of states, but their ordering is not necessarily the same.
See also
align_phase
rotate states such that their phases align
- align_phase(other: State, ret_index: bool = False, inplace: bool = False)
Align self with the phases for other, a copy may be returned
States will be rotated by \(\pi\) provided the phase difference between the states are above \(|\Delta\theta| > \pi/2\).
- Parameters:
other (
State
) – the other state to align onto this stateret_index – return which indices got swapped
inplace – rotate the states in-place
See also
align_norm
re-order states such that site-norms have a smaller residual
- change_gauge(gauge: sisl.typing.GaugeType, offset=(0, 0, 0))
In-place change of the gauge of the state coefficients
The two gauges are related through:
\[\tilde C_\alpha = e^{i\mathbf k\mathbf r_\alpha} C_\alpha\]where \(C_\alpha\) and \(\tilde C_\alpha\) belongs to the
atom
andcell
gauge, respectively.- Parameters:
gauge – specify the new gauge for the mode coefficients
offset (
array_like
, optional) – whether the coordinates should be offset by another phase-factor
- inner(ket=None, matrix=None, projection: Literal['diag', 'atoms', 'basis', 'matrix'] = 'diag')
Calculate the inner product as \(\mathbf A_{ij} = \langle\psi_i|\mathbf M|\psi'_j\rangle\)
Inner product calculation allows for a variety of things.
for
matrix
it will compute off-diagonal elements as well
\[\mathbf A_{\alpha\beta} = \langle\psi_\alpha|\mathbf M|\psi'_\beta\rangle\]for
diag
only the diagonal components will be returned
\[\mathbf a_\alpha = \langle\psi_\alpha|\mathbf M|\psi_\alpha\rangle\]for
basis
, only do inner products for individual states, but return them basis-resolved
\[\mathbf A_{\alpha\beta} = \psi^*_{\alpha,\beta} \mathbf M|\psi_\alpha\rangle_\beta\]for
atoms
, only do inner products for individual states, but return them atom-resolved
- Parameters:
ket (
State
, optional) – the ket object to calculate the inner product with, if not passed it will do the inner product with itself. The object itself will always be the bra \(\langle\psi_i|\)matrix (
array_like
, optional) – whether a matrix is sandwiched between the bra and ket, defaults to the identity matrix. 1D arrays will be treated as a diagonal matrix.projection – how to perform the final projection. This can be used to sum specific sub-elements, return the diagonal, or the full matrix.
diag
only return the diagonal of the inner productmatrix
a matrix with diagonals and the off-diagonalsbasis
only do inner products for individual states, but return them basis-resolvedatoms
only do inner products for individual states, but return them atom-resolved
Notes
This does not take into account a possible overlap matrix when non-orthogonal basis sets are used. One have to add the overlap matrix in the matrix argument, if needed.
- Raises:
ValueError – if the number of state coefficients are different for the bra and ket
RuntimeError – if the matrix shapes are incompatible with an atomic resolution conversion
- Returns:
numpy.ndarray
– a matrix with the sum of inner state products
- ipr(q: int = 2)
Calculate the inverse participation ratio (IPR) for arbitrary q values
The inverse participation ratio is defined as
\[I_{q,\alpha} = \frac{\sum_i |\psi_{\alpha,i}|^{2q}}{ \big[\sum_i |\psi_{\alpha,i}|^2\big]^q}\]where \(\alpha\) is the band index and \(i\) is the orbital. The order of the IPR is defaulted to \(q=2\), see (1) for details. The IPR may be used to distinguish Anderson localization and extended states:
\begin{align} \lim_{L\to\infty} I_{2,\alpha} = \left\{ \begin{aligned} 1/L^d &\quad \text{extended state} \\ \text{const.} &\quad \text{localized state} \end{aligned}\right. \end{align}For further details see [7]. Note that for eigen states the IPR reduces to:
\[I_{q,\alpha} = \sum_i |\psi_{\alpha,i}|^{2q}\]since the denominator is \(1^{q} = 1\).
- Parameters:
q – order parameter for the IPR
- iter(asarray: bool = False)
An iterator looping over the states in this system
- Parameters:
asarray (
bool
, optional) – if true the yielded values are the state vectors, i.e. a numpy array. Otherwise an equivalent object is yielded.- Yields:
state (
State
) – a state only containing individual elements, if asarray is falsestate (
numpy.ndarray
) – a state only containing individual elements, if asarray is true
- norm()
Return a vector with the Euclidean norm of each state \(\sqrt{\langle\psi|\psi\rangle}\)
- Returns:
numpy.ndarray
– the Euclidean norm for each state
- norm2(projection: Literal['sum', 'orbitals', 'basis', 'atoms'] = 'sum')
Return a vector with the norm of each state \(\langle\psi|\mathbf S|\psi\rangle\)
\(\mathbf S\) is the overlap matrix (or basis), for orthogonal basis \(\mathbf S \equiv \mathbf I\).
- Parameters:
projection – whether to compute the norm per state as a single number or as orbital-/atom-resolved quantity
See also
inner
used method for calculating the squared norm.
- Returns:
numpy.ndarray
– the squared norm for each state
- normalize()
Return a normalized state where each state has \(|\psi|^2=1\)
This is roughly equivalent to:
>>> state = State(np.arange(10)) >>> n = state.norm() >>> norm_state = State(state.state / n.reshape(-1, 1))
Notes
This does not take into account a possible overlap matrix when non-orthogonal basis sets are used.
- Returns:
State
– a new state with all states normalized, otherwise equal to this
- outer(ket=None, matrix=None)
Return the outer product by \(\sum_\alpha|\psi_\alpha\rangle\langle\psi'_\alpha|\)
- Parameters:
ket (
State
, optional) – the ket object to calculate the outer product of, if not passed it will do the outer product with itself. The object itself will always be the bra \(|\psi_\alpha\rangle\)matrix (
array_like
, optional) – whether a matrix is sandwiched between the ket and bra, defaults to the identity matrix. 1D arrays will be treated as a diagonal matrix.
Notes
This does not take into account a possible overlap matrix when non-orthogonal basis sets are used.
- Returns:
numpy.ndarray
– a matrix with the sum of outer state products
- phase(method: Literal['max', 'all'] = 'max', ret_index: bool = False)
Calculate the Euler angle (phase) for the elements of the state, in the range \(]-\pi;\pi]\)
- Parameters:
method (
{'max', 'all'}
) – for max, the phase for the element which has the largest absolute magnitude is returned, for all, all phases are calculatedret_index – return indices for the elements used when
method=='max'
- remove(index: sisl.typing.SimpleIndex, inplace: bool = False) State | None
Return a new state without the specified vectors
- Parameters:
index – indices that are removed in the returned object
inplace – whether the values will be removed inplace
- Returns:
State
– a new state without containing the requested elements, only if inplace is false
- rotate(phi: float = 0.0, individual: bool = False, inplace: bool = False) State | None
Rotate all states to rotate the largest component to be along the angle phi
The states will be rotated according to:
\[\mathbf S' = \mathbf S / \mathbf S^\dagger_{\phi-\mathrm{max}} \exp (i \phi),\]where \(\mathbf S^\dagger_{\phi-\mathrm{max}}\) is the phase of the component with the largest amplitude and \(\phi\) is the angle to align on.
- Parameters:
phi (
float
, optional) – angle to align the state at (in radians), 0 is the positive real axisindividual (
bool
, optional) – whether the rotation is per state, or a single maximum component is chosen.inplace – whether to do the rotation on the object it-self (True), or return a copy with the rotated states (False).
- spin_moment(project=False)
Calculate spin moment from the states
This routine calls
spin_moment
with appropriate arguments and returns the spin moment for the states.See
spin_moment
for details.- Parameters:
project (
bool
, optional) – whether the moments are orbitally resolved or not
- sub(index: sisl.typing.SimpleIndex, inplace: bool = False) State | None
Return a new state with only the specified states
- Parameters:
index – indices that are retained in the returned object
inplace – whether the values will be retained inplace
- Returns:
State
– a new state only containing the requested elements, only if inplace is false
- tile(reps: int, axis: int, normalize: bool = False, offset: float = 0) State
Tile the state vectors for a new supercell
Tiling a state vector makes use of the Bloch factors for a state by utilizing
\[\psi_{\mathbf k}(\mathbf r + \mathbf T) \propto e^{i\mathbf k\cdot \mathbf T}\]where \(\mathbf T = i\mathbf a_0 + j\mathbf a_1 + l\mathbf a_2\). Note that axis selects which of the \(\mathbf a_i\) vectors that are translated and reps corresponds to the \(i\), \(j\) and \(l\) variables. The offset moves the individual states by said amount, i.e. \(i\to i+\mathrm{offset}\).
- Parameters:
reps – number of repetitions along a specific lattice vector
axis – lattice vector to tile along
normalize – whether the states are normalized upon return, may be useful for eigenstates, equivalent to
state.tile().normalize()
offset – the offset for the phase factors
See also
Geometry.tile
,Grid.tile
,Lattice.tile
- translate(isc)
Translate the vectors to a new unit-cell position
The method is thoroughly explained in
tile
while this one only selects the corresponding state vector- Parameters:
isc (
(3,)
) – number of offsets for the statevector
See also
tile
equivalent method for generating more cells simultaneously
- wavefunction(grid, spinor=0, eta=None)
Expand the coefficients as the wavefunction on grid as-is
See
wavefunction
for argument details, the arguments not present in this method are automatically passed from this object.
- property dkind
The data-type of the state (in str)
- property dtype
Data-type for the state
- info
- parent
- property shape
Returns the shape of the state
- state