Hi, > Thank you Joseph for the highlight. It make sens since the tests on a > square lattice seemed working fine. > > I checked the 'plot_2d_bands' function but could not identify the > transformation you were mentioning. Could you help me in this please?
It requires a bit of reading but we can extract the following snippets
from 'wraparound' (I have reworded some comments to make sense without
the surrounding context of the 'wraparound' module):
# calculate the reciprocal symmetry vectors: these
# are the columns of 'A'
B = np.array(symmetry.periods).T
A = B.dot(np.linalg.inv(B.T.dot(B)))
...
# transforms a momentum in the basis (kx, ky) to the basis of
# reciprocal symmetry vectors
def momentum_to_lattice(k):
k, residuals = scipy.linalg.lstsq(A, k)[:2]
if np.any(abs(residuals) > 1e-7):
raise RuntimeError("Requested momentum doesn't correspond"
" to any lattice momentum.")
return k
We could then use 'momentum_to_lattice' like so:
syst = make_wrapped_around_syst()
H_k(kx, ky):
k = (kx, ky)
k_prime = momentum_to_lattice(k)
return syst.hamiltonian_submatrix(args=k_prime)
Does this help?
> Just to check my understanding for this module, could you tell me if I am
> right:
> 1) one can plot the system sys (before wraping)
> 2) check the extra-cell hoppings
> 3) take them off and add terms of the form -t exp(i k .Vj) for the
> corresponding intra-cell hopping (Vj is the vector of the eliminated
> hopping)
> with this, we obtain the Hamiltonian of the wrapped system
>
> am I correct ?
Yes, in addition to the caveat that inter-cell hoppings between a site
and its image under the symmetry will correspond to *onsite* terms in
the wrapped around system (imagine a simple chain of sites with
nearest-neighbor hoppings).
Happy Kwanting,
Joe
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