sisl.physics.State

class sisl.physics.State(state, parent=None, **info)

Bases: ParentContainer

An object handling a set of vectors describing a given state

Parameters:
  • state (array_like) – state vectors state[i, :] containing the i’th state vector

  • parent (obj, optional) – a parent object that defines the origin of the state.

  • **info (dict, optional) – an info dictionary that turns into an attribute on the object. This info may contain anything that may be relevant for the state.

Notes

This class should be subclassed!

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, diag])

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([sum])

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 \(\sum_i|\psi_i\rangle\langle\psi'_i|\)

phase([method, ret_index])

Calculate the Euler angle (phase) for the elements of the state, in the range \(]-\pi;\pi]\)

remove(idx[, inplace])

Return a new state without the specified vectors

rotate([phi, individual])

Rotate all states (in-place) to rotate the largest component to be along the angle phi

sub(idx[, 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

state

dkind

The data-type of the state (in str)

dtype

Data-type for the state

info

parent

shape

Returns the shape of the state

__init__(state, parent=None, **info)[source]

Define a state container with a given set of states

align_norm(other, ret_index=False, inplace=False)[source]

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 (bool, optional) – also return indices for the swapped indices

  • inplace (bool, optional) – 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, ret_index=False, inplace=False)[source]

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 (bool, optional) – return which indices got swapped

  • inplace (bool, optional) – rotate the states in-place

See also

align_norm

re-order states such that site-norms have a smaller residual

change_gauge(gauge, offset=(0, 0, 0))[source]

In-place change of the gauge of the state coefficients

The two gauges are related through:

\[\tilde C_j = e^{i\mathbf k\mathbf r_j} C_j\]

where \(C_j\) and \(\tilde C_j\) belongs to the r and R gauge, respectively.

Parameters:
  • gauge ({'R', 'r'}) – specify the new gauge for the mode coefficients

  • offset (array_like, optional) – whether the coordinates should be offset by another phase-factor

copy()[source]

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

property dkind

The data-type of the state (in str)

property dtype

Data-type for the state

info
inner(ket=None, matrix=None, diag=True)[source]

Calculate the inner product as \(\mathbf A_{ij} = \langle\psi_i|\mathbf M|\psi'_j\rangle\)

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.

  • diag (bool, optional) – only return the diagonal matrix \(\mathbf A_{ii}\).

Notes

This does not take into account a possible overlap matrix when non-orthogonal basis sets are used.

Raises:

ValueError – if the number of state coefficients are different for the bra and ket

Returns:

a matrix with the sum of inner state products

Return type:

numpy.ndarray

ipr(q=2)[source]

Calculate the inverse participation ratio (IPR) for arbitrary q values

The inverse participation ratio is defined as

\[I_{q,i} = \frac{\sum_\nu |\psi_{i\nu}|^{2q}}{ \big[\sum_\nu |\psi_{i\nu}|^2\big]^q}\]

where \(i\) is the band index and \(\nu\) 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,i} = \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 [5]. Note that for eigen states the IPR reduces to:

\[I_{q,i} = \sum_\nu |\psi_{i\nu}|^{2q}\]

since the denominator is \(1^{q} = 1\).

Parameters:

q (int, optional) – order parameter for the IPR

iter(asarray=False)[source]

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()[source]

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

Returns:

the Euclidean norm for each state

Return type:

numpy.ndarray

norm2(sum=True)[source]

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

Parameters:

sum (bool, optional) – for true only a single number per state will be returned, otherwise the norm per basis element will be returned.

Returns:

the squared norm for each state

Return type:

numpy.ndarray

normalize()[source]

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:

a new state with all states normalized, otherwise equal to this

Return type:

State

outer(ket=None, matrix=None)[source]

Return the outer product by \(\sum_i|\psi_i\rangle\langle\psi'_i|\)

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 \(|\psi_i\rangle\)

  • 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:

a matrix with the sum of outer state products

Return type:

numpy.ndarray

parent
phase(method='max', ret_index=False)[source]

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 (bool, optional) – return indices for the elements used when method=='max'

remove(idx, inplace=False)[source]

Return a new state without the specified vectors

Parameters:
  • idx (int or array_like) – indices that are removed in the returned object

  • inplace (bool, optional) – whether the values will be removed inplace

Returns:

a new state without containing the requested elements, only if inplace is false

Return type:

State

rotate(phi=0.0, individual=False)[source]

Rotate all states (in-place) to rotate the largest component to be along the angle phi

The states will be rotated according to:

\[S' = S / S^\dagger_{\phi-\mathrm{max}} \exp (i \phi),\]

where \(S^\dagger_{\phi-\mathrm{max}}\) 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.

property shape

Returns the shape of the state

state
sub(idx, inplace=False)[source]

Return a new state with only the specified states

Parameters:
  • idx (int or array_like) – indices that are retained in the returned object

  • inplace (bool, optional) – whether the values will be retained inplace

Returns:

a new state only containing the requested elements, only if inplace is false

Return type:

State

tile(reps, axis, normalize=False, offset=0)[source]

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 (int) – number of repetitions along a specific lattice vector

  • axis (int) – lattice vector to tile along

  • normalize (bool, optional) – whether the states are normalized upon return, may be useful for eigenstates

  • offset (float, optional) – the offset for the phase factors

See also

Geometry.tile

translate(isc)[source]

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