sisl.physics.phonon.EigenmodePhonon

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

Bases: ModeCPhonon

Eigenmodes of phonons with eigenvectors and eigenvalues.

This holds routines that enable the calculation of (projected) density of states.

Methods

`DOS`(E[, distribution])	Calculate DOS for provided energies, E.
`PDOS`(E[, distribution])	Calculate PDOS for provided energies, E.
`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
`asCoefficient`()
`asState`()
`change_gauge`(gauge[, offset])	In-place change of the gauge of the state coefficients
`copy`()	Return a copy (only the coefficients and states are copied), `parent` and `info` are passed by reference
`degenerate`(atol)	Find degenerate coefficients with a specified precision
`derivative`([order, matrix, axes, operator])	Calculate the derivative with respect to \(\mathbf k\) for a set of states up to a given order
`displacement`([atol])	Calculate real-space displacements for a given mode (in units of the characteristic length)
`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 \(\|\psi\|^2=1\)
`occupation`([distribution])	Calculate the occupations for the states according to a distribution function
`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 indices
`rotate`([phi, individual, inplace])	Rotate all states to rotate the largest component to be along the angle phi
`sort`([ascending])	Sort and return a new `StateC` by sorting the coefficients (default to ascending)
`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
`velocity`(args, *kwargs)	Calculate velocity of the modes

Attributes

`c`
`dkind`	The data-type of the state (in str)
`dtype`	Data-type for the state
`hw`	Eigenmode values in units of \(\hbar \omega\) [eV]
`info`
`mode`	Eigenmodes (states)
`parent`
`shape`	Returns the shape of the state
`state`

DOS(E, distribution='gaussian')[source]

Calculate DOS for provided energies, E.

This routine calls sisl.physics.phonon.DOS with appropriate arguments and returns the DOS.

See DOS for argument details.

PDOS(E, distribution='gaussian')[source]

Calculate PDOS for provided energies, E.

This routine calls PDOS with appropriate arguments and returns the PDOS.

See PDOS for argument details.

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

asCoefficient()

asState()

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() → StateC: Return a copy (only the coefficients and states are copied), parent and info are passed by reference

degenerate(atol: float)

Find degenerate coefficients with a specified precision

Parameters:: atol – the precision above which coefficients are not considered degenerate
Returns:: list of numpy.ndarray – a list of indices

Calculate the derivative with respect to \(\mathbf k\) for a set of states up to a given order

These are calculated using the analytic expression (\(\alpha\) corresponding to the Cartesian directions), here only shown for the 1st order derivative:

\[\mathbf{d}_{\alpha ij} = \langle \psi_i | \frac{\partial}{\partial\mathbf k_\alpha} \mathbf H(\mathbf k) | \psi_j \rangle\]

In case of non-orthogonal basis the equations substitutes \(\mathbf H(\mathbf k)\) by \(\mathbf H(\mathbf k) - \epsilon_i\mathbf S(\mathbf k)\).

The 2nd order derivatives are calculated with the Berry curvature correction:

\[\mathbf d^2_{\alpha \beta ij} = \langle\psi_i| \frac{\partial^2}{\partial\mathbf k_\alpha\partial\mathbf k_\beta} \mathbf H(\mathbf k) | \psi_j\rangle - \frac12\frac{\mathbf{d}_{\alpha ij}\mathbf{d}_{\beta ij}} {\epsilon_i - \epsilon_j}\]

Notes

When requesting 2nd derivatives it will not be advisable to use a sub before calculating the derivatives since the 1st order perturbation uses the energy differences (Berry contribution) and the 1st derivative matrix for correcting the curvature.

For states at the \(\Gamma\) point you may get warnings about casting complex numbers to reals. In these cases you should force the state at the \(\Gamma\) point to be calculated in complex numbers to enable the correct decoupling.

Parameters:

order – an integer specifying which order of the derivative is being calculated.
matrix – whether the full matrix or only the diagonal components are returned
axes – NOTE: this argument may change in future versions. only calculate the derivative(s) along specified Cartesian directions. The axes argument will be sorted internally, so the order will always be xyz. For the higher order derivatives all those involving only the provided axes will be used.
operator – an operator that translates the \(\delta\) matrices to another operator. The same operator will be applied to both P and S matrices.

See also

SparseOrbitalBZ.dPk: function for generating the matrix derivatives
SparseOrbitalBZ.dSk: function for generating the matrix derivatives in non-orthogonal basis

Returns:

dv – the 1st derivative, has shape (3, state.shape[0]) for matrix=False, else has shape (3, state.shape[0], state.shape[0]) Also returned for order >= 2 since it is used in the higher order derivatives
ddv – the 2nd derivative, has shape (6, state.shape[0]) for matrix=False, else has shape (6, state.shape[0], state.shape[0]), the first dimension is in the Voigt representation Only returned for order >= 2

displacement(atol: float = 1e-9)[source]

Calculate real-space displacements for a given mode (in units of the characteristic length)

The displacements per mode may be written as:

\[\mathbf{u}_{I\alpha} = \epsilon_{I\alpha}\sqrt{\frac{\hbar}{m_I \omega}}\]

where \(I\) is the atomic index.

Even for negative frequencies the characteristic length is calculated for use of non-equilibrium modes.

Parameters:: atol – absolute tolerance for whether a phonon is 0 or not. Since the phonon energy is used in the calculation of the displacement vector we have to remove phonon modes with 0 energy. The displacements for phonon modes with an absolute energy below atol will be 0.
Returns:: numpy.ndarray – displacements per mode with final dimension (len(self), self.parent.na, 3), displacements are in Ang

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 = StateC(np.arange(10), 1)
>>> n = state.norm()
>>> norm_state = StateC(state.state / n.reshape(-1, 1), state.c.copy())
>>> norm_state.c[0] == 1

Returns:: State – a new state with all states normalized, otherwise equal to this

occupation(distribution='bose_einstein')[source]

Calculate the occupations for the states according to a distribution function

Parameters:: distribution (str or func, optional) – distribution used to find occupations
Returns:: numpy.ndarray – len(self) with occupation values

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) → StateC | None

Return a new state without the specified indices

Parameters:

index – indices that are removed in the returned object
inplace – whether the values will be removed inplace

Returns:

StateC – 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).

sort(ascending: bool = True)

Sort and return a new StateC by sorting the coefficients (default to ascending)

Parameters:: ascending – sort the contained elements ascending, else they will be sorted descending

sub(index: sisl.typing.SimpleIndex, inplace: bool = False) → StateC | 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:

StateC – a new object with a subset of the states, 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

velocity(*args, **kwargs)

Calculate velocity of the modes

This routine calls derivative with appropriate arguments (1st order derivative) and returns the velocity for the modes.

Note that the coefficients associated with the ModeCPhonon must correspond to the energies of the modes.

See derivative for details and possible arguments. One cannot pass the order argument as that is fixed to 1 in this call.

Notes

The states and energies for the modes may have changed after calling this routine. This is because of the velocity un-folding for degenerate modes. I.e. calling displacement and/or PDOS after this method may change the result.

See also

derivative: for details of the implementation

c

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

property dtype: Data-type for the state

property hw: Eigenmode values in units of \(\hbar \omega\) [eV]

info

property mode: Eigenmodes (states)

parent

property shape: Returns the shape of the state

state