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 \(\langle\psi|\mathbf S|\psi\rangle\)

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

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

dkind

The data-type of the state (in str)

dtype

Data-type for the state

info

parent

shape

Returns the shape of the state

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 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', '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 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).

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