sisl.physics.electron.EigenstateElectron
- class sisl.physics.electron.EigenstateElectron(state, c, parent=None, **info)[source]
Bases:
StateCElectron
Eigen states of electrons with eigenvectors and eigenvalues.
This holds routines that enable the calculation of (projected) density of states, spin moments (spin texture).
Plotting
Plotting functions for the
EigenstateElectron
class.plot.wavefunction
(*[, i, ...])Creates a
EigenstateData
object and then plots aWavefunctionPlot
from it.Methods
COHP
(E, *args, **kwargs)Calculate COHP for provided energies, E.
COOP
(E, *args, **kwargs)Calculate COOP for provided energies, E.
COP
(E, M, *args, **kwargs)Calculate COP for provided energies, E using matrix M
DOS
(E[, distribution])Calculate DOS for provided energies, E.
PDOS
(E[, distribution])Calculate PDOS for provided energies, E.
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
asState
()berry_curvature
([sum, distribution, ...])Calculate the Berry curvature matrix for a set of states (Kubo)
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
andinfo
are passed by referencedegenerate
(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
effective_mass
(*args, **kwargs)Calculate effective mass tensor for the states, units are (ps/Ang)^2
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 \(|\psi|^2=1\)
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 \(]-\pi;\pi]\)
position
(*args, **kwargs)Calculate position for the states
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)spin_berry_curvature
([sigma, sum, ...])Calculate the spin Berry curvature
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
velocity
(*args, **kwargs)Calculate velocity for the states
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
Eigenvalues for each state
Returns the shape of the state
- COHP(E, *args, **kwargs)[source]
Calculate COHP for provided energies, E.
This routine calls
COP
with appropriate arguments.
- COOP(E, *args, **kwargs)[source]
Calculate COOP for provided energies, E.
This routine calls
COP
with appropriate arguments.
- COP(E, M, *args, **kwargs)[source]
Calculate COP for provided energies, E using matrix M
This routine calls
COP
with appropriate arguments.
- DOS(E, distribution='gaussian')[source]
Calculate DOS for provided energies, E.
This routine calls
sisl.physics.electron.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.
- 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
- asCoefficient()
- asState()
- berry_curvature(sum: bool = True, *, distribution: str | Callable[[Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]], ndarray] | None = None, derivative_kwargs: dict = {}, operator: Callable[[Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], Literal['x', 'y', 'z', 'xx', 'yy', 'zz', 'yz', 'xz', 'xy'] | None], Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]] | tuple[Callable[[Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], Literal['x', 'y', 'z', 'xx', 'yy', 'zz', 'yz', 'xz', 'xy'] | None], Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]], Callable[[Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], Literal['x', 'y', 'z', 'xx', 'yy', 'zz', 'yz', 'xz', 'xy'] | None], Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]]] = lambda M, d: ..., eta: float = 0.0) ndarray
Calculate the Berry curvature matrix for a set of states (Kubo)
The Berry curvature is calculated using the following expression (\(\alpha\), \(\beta\) corresponding to Cartesian directions):
\[\boldsymbol\Omega_{i,\alpha\beta} = 2i\hbar^2\sum_{j\neq i} \frac{v^{\alpha}_{ij} v^\beta_{ji}} {[\epsilon_i - \epsilon_j]^2 + i\eta^2}\]For details on the Berry curvature, see Eq. (11) in [14] or Eq. (2.59) in [1].
The
operator
argument can be used to define the Berry curvature in other quantities. E.g. the spin Berry curvature is defined by replacing \(v^\alpha\) by the spin current operator. seespin_berry_curvature
for details.For additional details on the spin Berry curvature, see Eq. (1) in [9] and Eq. (2) in [4].
Notes
There exists reports on some terms missing in the above formula, for details see [3].
- Parameters:
state – the state describing the electronic states we wish to calculate the Berry curvature of.
sum – only return the summed Berry curvature (over all states).
distribution – An optional distribution enabling one to automatically sum states across occupied/unoccupied states. This is useful when calculating AHC/SHC contributions since it can improve numerical accuracy. If this is None, it will do the above equation exactly.
derivative_kwargs – arguments passed to
derivative
. Sinceoperator
is defined here, one cannot haveoperator
in derivative_kwargs.operator – the operator to use for changing the dPk matrices. Note, that this may change the resulting units, and it will be up to the user to adapt the units accordingly.
eta – direct imaginary part broadening of the Lorentzian.
See also
derivative
method for calculating the exact derivatives
spin_berry_curvature
calculate the spin Berry curvature
ahc
anomalous Hall conductivity
shc
spin Hall conductivity
- Returns:
bc (
numpy.ndarray
) – If sum is False, it will be at least a 3D array with the 3rd dimension having the contribution from state i. If one passes axes to the derivative_kwargs argument one will get dimensions according to the number of axes requested, by default all axes will be used (even if they are non-periodic). The dtype will be imaginary. The unit is \(\mathrm{Ang}^2\).
- 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
- copy() StateC
Return a copy (only the coefficients and states are copied),
parent
andinfo
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
ofnumpy.ndarray
– a list of indices
- derivative(order: Literal[1, 2] = 1, matrix: bool = False, axes: sisl.typing.CartesianAxes = 'xyz', operator: Callable[[Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], Literal['x', 'y', 'z', 'xx', 'yy', 'zz', 'yz', 'xz', 'xy'] | None], Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes]] = lambda M, d=None: ...)
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
andS
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])
formatrix=False
, else has shape(3, state.shape[0], state.shape[0])
Also returned fororder >= 2
since it is used in the higher order derivativesddv
– the 2nd derivative, has shape(6, state.shape[0])
formatrix=False
, else has shape(6, state.shape[0], state.shape[0])
, the first dimension is in the Voigt representation Only returned fororder >= 2
- effective_mass(*args, **kwargs)
Calculate effective mass tensor for the states, units are (ps/Ang)^2
This routine calls
derivative(2, *args, **kwargs)
and returns the effective mass for all states.Note that the coefficients associated with the
StateCElectron
must correspond to the energies of the states.Notes
Since some directions may not be periodic there will be zeros. This routine will invert elements where the values are different from 0.
It is not advisable to use a
sub
before calculating the effective mass since the 1st order perturbation uses the energy differences and the 1st derivative matrix for correcting the curvature.The returned effective mass is given in the Voigt notation.
For \(\Gamma\) point calculations it may be beneficial to pass dtype=np.complex128 to the eigenstate argument to ensure their complex values. This is necessary for the degeneracy decoupling.
See also
derivative
for details of the implementation
- 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 = 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='fermi_dirac')[source]
Calculate the occupations for the states according to a distribution function
- Parameters:
distribution (
str
orfunc
, 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 calculatedret_index – return indices for the elements used when
method=='max'
- plot.wavefunction(*, i: int = 0, geometry: Geometry | None = None, grid_prec: float = 0.2, grid: Grid | None = None, axes: Axes = ['z'], represent: Literal['real', 'imag', 'mod', 'phase', 'deg_phase', 'rad_phase'] = 'real', transforms: Sequence[str | Callable] = (), reduce_method: Literal['average', 'sum'] = 'average', boundary_mode: str = 'grid-wrap', nsc: tuple[int, int, int] = (1, 1, 1), interp: tuple[int, int, int] = (1, 1, 1), isos: Sequence[dict] = [], smooth: bool = False, colorscale: Colorscale | None = None, crange: tuple[float, float] | None = None, cmid: float | None = None, show_cell: Literal['box', 'axes', False] = 'box', cell_style: dict = {}, x_range: Sequence[float] | None = None, y_range: Sequence[float] | None = None, z_range: Sequence[float] | None = None, plot_geom: bool = False, geom_kwargs: dict = {}, backend: str = 'plotly') WavefunctionPlot
Creates a
EigenstateData
object and then plots aWavefunctionPlot
from it.- Parameters:
i – The index of the eigenstate to plot.
geometry – Geometry to use to project the eigenstate to real space. If None, the geometry associated with the eigenstate is used.
grid_prec – The precision of the grid where the wavefunction is projected.
grid – The grid to plot.
axes – The axes to project the grid to.
represent – The representation of the grid to plot.
transforms – List of transforms to apply to the grid before plotting.
reduce_method – The method used to reduce the grid axes that are not displayed.
boundary_mode – The method used to deal with the boundary conditions. Only used if the grid is to be orthogonalized. See scipy docs for more info on the possible values.
nsc – The number of unit cells to display in each direction.
interp – The interpolation factor to use for each axis to make the grid smoother.
isos – List of isosurfaces or isocontours to plot. See the showcase notebooks for examples.
smooth – Whether to ask the plotting backend to make an attempt at smoothing the grid display.
colorscale – Colorscale to use for the grid display in the 2D representation. If None, the default colorscale is used for each backend.
crange – Min and max values for the colorscale.
cmid – The value at which the colorscale is centered.
show_cell – Method used to display the unit cell. If False, the cell is not displayed.
cell_style – Style specification for the cell. See the showcase notebooks for examples.
x_range – The range of the x axis to take into account. Even if the X axis is not displayed! This is important because the reducing operation will only be applied on this range.
y_range – The range of the y axis to take into account. Even if the Y axis is not displayed! This is important because the reducing operation will only be applied on this range.
z_range – The range of the z axis to take into account. Even if the Z axis is not displayed! This is important because the reducing operation will only be applied on this range.
plot_geom – Whether to plot the associated geometry (if any).
geom_kwargs – Keyword arguments to pass to the geometry plot of the associated geometry.
backend – The backend to use to generate the figure.
See also
WavefunctionPlot
The plot class used to generate the plot.
EigenstateData
The class to which data is converted.
- position(*args, **kwargs)
Calculate position for the states
This routine calls
derivative(1, *args, **kwargs)
and returns the velocity for the states.Note that the coefficients associated with the
StateCElectron
must correspond to the energies of the states.The unit is Ang/ps.
Notes
The velocities are calculated without the Berry curvature contribution see Eq. (2) in [14]. It is thus typically denoted as the effective velocity operater (see Ref. 21 in [14]. The missing contribution may be added in later editions, for completeness sake, it is:
\[\delta \mathbf v = - \mathbf k\times \Omega_i(\mathbf k)\]where \(\Omega_i\) is the Berry curvature for state \(i\).
- Parameters:
*args, **kwargs – arguments passed directly to
derivative
, see that method for argument details.
See also
derivative
for details of the implementation
- 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 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).
- 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
- spin_berry_curvature(sigma: CartesianAxisStrLiteral | npt.ArrayLike = 'z', sum: bool = True, *, distribution: _TDist | None = None, J_axes: CartesianAxisStrLiteral | Sequence[CartesianAxisStrLiteral] = 'xyz', **berry_kwargs) np.ndarray
Calculate the spin Berry curvature
This is equivalent to calling
berry_curvature
with the spin current operator and the regular velocity operator instead of \(v^\alpha\):def noop(M, d): return M def Jz(M, d): if d in J_axes: return (M @ sigma_z + sigma_z @ M) * 0.5 return M state.berry_curvature(..., operator=(Jz, noop))
I.e. the left velocity operator being swapped with the spin current operator:
\[J^{\gamma\alpha} = \frac12 \{ v^\alpha, \hat{\sigma}^\gamma \}\]where \(\{\}\) means the anticommutator.
When calling it like this, the spin Berry curvature is found in the index corresponding to the axes the spin operator is acting on. The regular spin Berry curvature is found in all indices (J_axes).
E.g. if
J_axes = 'xy', sigma = 'z'
, thenshc[[0, 1]]
will be the spin Berry curvature using the Pauli matrix \(\hat{\sigma}^z\) (not the spin-operator \(\hat{s}^z = \dfrac\hbar2\hat{\sigma}^z\)), andshc[2]
will be the normal Berry curvature (since only the left velocity operator will be changed for J_axes).Notes
For performance reasons, it can be very benificial to extract the above methods and call
berry_curvature
directly. This is because the \(\sigma\) operator gets created on every call of this method.This, repeated matrix creation, might change in the future.
- Parameters:
sigma – which Pauli matrix is used, alternatively one can pass a custom spin matrix, or the full sigma.
J_axes – the direction(s) where the \(J^\sigma\) operator will be applied, defaults to all.
**kwargs – see
berry_curvature
for the remaining arguments.
See also
berry_curvature
calculate the Berry curvature (internally called)
derivative
method for calculating the exact derivatives
ahc
anomalous Hall conductivity
shc
spin Hall conductivity
- Returns:
bc (
numpy.ndarray
) – Spin Berry curvature + (possibly Berry curvature) returned in certain dimensions. If one passes axes to the derivative_kwargs argument one will get dimensions according to the number of axes requested, by default all axes will be used (even if they are non-periodic). The dtype will be imaginary. The unit is \(\mathrm{Ang}^2\).
- 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) 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 for the states
This routine calls
derivative(1, *args, **kwargs)
and returns the velocity for the states.Note that the coefficients associated with the
StateCElectron
must correspond to the energies of the states.The unit is Ang/ps.
Notes
The velocities are calculated without the Berry curvature contribution see Eq. (2) in [14]. It is thus typically denoted as the effective velocity operater (see Ref. 21 in [14]. The missing contribution may be added in later editions, for completeness sake, it is:
\[\delta \mathbf v = - \mathbf k\times \Omega_i(\mathbf k)\]where \(\Omega_i\) is the Berry curvature for state \(i\).
- Parameters:
*args, **kwargs – arguments passed directly to
derivative
, see that method for argument details.
See also
derivative
for details of the implementation
- 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.
- c
- property dkind
The data-type of the state (in str)
- property dtype
Data-type for the state
- property eig
Eigenvalues for each state
- info
- parent
- plot
Plotting functions for the
EigenstateElectron
class.
- property shape
Returns the shape of the state
- state