sisl.physics.phonon.EigenvectorPhonon

class sisl.physics.phonon.EigenvectorPhonon(state, parent=None, **info)[source]

Bases: ModePhonon

Eigenvectors of phonon modes, no eigenvalues retained

Methods

`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 \(\langle\psi\|\psi\rangle\)
`normalize`()	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
`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

Attributes

`dkind`	The data-type of the state (in str)
`dtype`	Data-type for the state
`info`
`mode`	Eigenmodes (states)
`parent`
`shape`	Returns the shape of the state
`state`

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 against
ret_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 False
index (array of int) – the indices that swaps self to be self_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 state
ret_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 and cell 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

copy() → State: Return a copy (only the state is copied). parent and info are passed by reference

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 product
- matrix a matrix with diagonals and the off-diagonals
- basis only do inner products for individual states, but return them basis-resolved
- atoms 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 false
state (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', 'atoms', 'basis'] = 'sum')

Return a vector with the norm of each state \(\langle\psi|\psi\rangle\)

Parameters:: projection – whether to compute the norm per state as a single number, atom-resolved or per basis dimension.

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 calculated
ret_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 axis
individual (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).

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

property dkind: The data-type of the state (in str)

property dtype: Data-type for the state

info

property mode: Eigenmodes (states)

parent

property shape: Returns the shape of the state

state