The Brillouin zone objects are all special classes enabling easy manipulation of an underlying physical quantity.
Quite often a physical quantity will be required to be averaged, or calculated individually over a number of k-points. In this regard the Brillouin zone objects can help.
The BrillouinZone object allows direct looping of contained k-points while invoking particular methods from the contained object. This is best shown with an example:
>>> H = Hamiltonian(...) >>> bz = BrillouinZone(H) >>> bz.apply.array.eigh()
This will calculate eigenvalues for all k-points associated with the
return everything as an array. The dispatch property of
BrillouinZone object has several use cases (here
array is shown).
This may be extremely convenient when calculating band-structures:
>>> H = Hamiltonian(...) >>> bs = BandStructure(H, [[0, 0, 0], [0.5, 0, 0]], 100) >>> bs_eig = bs.apply.array.eigh() >>> plt.plot(bs.lineark(), bs_eig)
and then you have all eigenvalues for all the k-points along the path.
Sometimes one may want to post-process the data for each k-point. As an example lets post-process the DOS on a per k-point basis while calculating the average:
>>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> E = np.linspace(-2, 2, 100) >>> def wrap_DOS(eigenstate): ... # Calculate the DOS for the eigenstates ... DOS = eigenstate.DOS(E) ... # Calculate the velocity for the eigenstates ... v = eigenstate.velocity() ... V = (v ** 2).sum(1) ... return DOS.reshape(-1, 1) * v ** 2 / V.reshape(-1, 1) >>> DOS = mp.apply.average.eigenstate(wrap=wrap_DOS, eta=True)
This will, calculate the Monkhorst pack k-averaged DOS split into 3 Cartesian
directions based on the eigenstates velocity direction. This method of manipulating
the result can be extremely powerful to calculate many quantities while running an
BrillouinZone average. The eta flag will print, to stdout, a progress-bar.
The usage of the
wrap method are also passed optional arguments,
parent which is
H in the above example.
weight are the current k-point and weight of the
corresponding k-point. An example could be to manipulate the DOS depending on the k-point and
>>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> E = np.linspace(-2, 2, 100) >>> def wrap_DOS(eigenstate, k, weight): ... # Calculate the DOS for the eigenstates and weight by k_x and weight ... return eigenstate.DOS(E) * k * weight >>> DOS = mp.apply.sum.eigenstate(wrap=wrap_DOS, eta=True)
When using wrap to calculate more than one quantity per eigenstate it may be advantageous
oplist to handle cases of BrillouinZone.apply.average and BrillouinZone.apply.sum.
>>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> E = np.linspace(-2, 2, 100) >>> def wrap_multiple(eigenstate): ... # Calculate DOS/PDOS for eigenstates ... DOS = eigenstate.DOS(E) ... PDOS = eigenstate.PDOS(E) ... # Calculate velocity for the eigenstates ... v = eigenstate.velocity() ... return oplist([DOS, PDOS, v]) >>> DOS, PDOS, v = mp.apply.average.eigenstate(wrap=wrap_multiple, eta=True)
Which does mathematical operations (averaging/summing) using
In some cases quantities are needed for all \(k\) points and in such cases
it may not always be that the returned quantities are commensurate.
Lets re-use the previous
wrap_multiple function and try and return the
>>> DOS_PDOS_v = mp.apply.eigenstate(wrap=wrap_multiple, eta=True)
This will raise an error since
wrap_multiple returns an oplist (same as a list)
and thus is unable to convert this into an equivalent
this can not be merged together in a single
numpy.ndarray since the shapes of the returned
quantities are not commensurate. One cannot concatenate the 3 different quantities.
To accomblish this one may use an
zip flag where the two lines are equivalent:
>>> DOS, PDOS, v = mp.apply.array.renew(zip=True).eigenstate(wrap=wrap_multiple, eta=True) >>> DOS, PDOS, v = mp.apply(zip=True).array.eigenstate(wrap=wrap_multiple, eta=True)
and the data is unpacked as wanted.
apply method looping k-points may be explicitly parallelized.
To run parallel do:
>>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> with mp.apply.renew(pool=True) as par: ... par.array.eigh()
This requires you also have the package
The above will run in parallel using a default number of processors
Return value of
Note that this may interfere with BLAS implementation which defaults
to use all CPU’s for threading. The total processors/threads that will
be created is
SISL_NUM_PROCS * OMP_NUM_THREADS. Try and ensure this is below
the actual core-count of your machine (or the number of requested cores in a
Alternatively one can control the number of processors locally by doing:
>>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> with mp.apply.renew(pool=2) as par: ... par.eigh()
which will request 2 processors (regardless of core-count).
As a last resort you can pass your own
Pool of workers that
will be used for the parallel processing.
>>> from multiprocessing import Pool >>> pool = Pool(4) >>> H = Hamiltonian(...) >>> mp = MonkhorstPack(H, [10, 10, 10]) >>> with mp.apply.renew(pool=pool) as par: ... par.array.eigh()
Pool should implement some standard methods that are
existing in the
pathos enviroment such as
uimap methods. See the
pathos documentation for detalis.
A class to construct Brillouin zone related quantities
Create a Monkhorst-Pack grid for the Brillouin zone
Create a path in the Brillouin zone for plotting band-structures etc.