How to do inner products between different k

We have es.state[n, i] where n is a band index and i an orbital in the primitive cell. And H.tocsr() has indices [i, J] with i an orbital index in the primitive cell and J one in the supercell.

To compute general matrix elements we need <s1| A |s2> = s1.inner(matrix=A.tocsr(), ket=s2.state_sc) to compute general matrix elements between the Bloch states s1, s2.

What I have in mind is thus that s2.state_sc should have indices [m, J] such that the general inner product <s1(n)| A |s2(m)> takes the form [n, i] . [i,J] . ([m, J].T). (It is a separate discussion if we should transpose the ket internally or not)

_Originally posted by @tfrederiksen in https://github.com/zerothi/sisl/pull/355#discussion_r666131276_

[zerothi]

Maybe it is worth to understand how SIESTA is computing macroscopic polarization of dielectrics (via the geometric Berry phase) as implemented in the subroutine KSV_pol?

In siesta it is calculated using the matrix elements $phi_i(r-r_i)dkr*phi_j(r-r_j) + HC$ to get a phase on the orbital. Then subsequently it is the determinant of some overlap matrix times the wavefunction coefficients (not fully correct, but along these lines).

I would say this path is not feasible in sisl due to the matrix elements not being calculate-able in a fast way.

[zerothi]

<k1(n) | k2(m)> = k1.state[n, i].S[i, J].((k2.state_sc[m, J]).T)

But I think this is equivalent to <k1|S(k2)|k2>, right? S[i, J] * k2[m, J] == S[i, J] * exp(2j*pi * k2 @ isc(J)) * k2[m, j] or am I wrong here?

You are right, they are equivalent…

Here is a little script exploring this for the non-ortho case:

import sisl
import numpy as np

geom = sisl.geom.graphene()

H = sisl.Hamiltonian(geom, orthogonal=False)
H.construct([[0.1, 1.5], [[0, 1], [2.7, 0.3]]])

k = np.array([2/3, 1/3, 0])
es = H.eigenstate(k)
es_ext = np.concatenate([np.exp(2j * np.pi * k.dot(isc)) * es.state for _, isc in H.geom.sc], axis=1)

s = es.inner(es, diag=False)
s_ext = es.state.conj().dot(H.tocsr(-1).dot(es_ext.T))
assert np.allclose(s, s_ext)

ev = np.diag(es.state.conj().dot(H.tocsr().dot(es_ext.T)))
assert np.allclose(ev, es.eig)

k2 = np.array([0, 0, 0])
es2 = H.eigenstate(k2)
es2_ext = np.concatenate([np.exp(2j * np.pi * k2.dot(isc)) * es2.state for _, isc in H.geom.sc], axis=1)

s21 = es2.inner(es, diag=False) # using S(k2) from es2
s12_ext = es.state.conj().dot(H.tocsr(-1).dot(es2_ext.T))
assert np.allclose(s21.conj().T, s12_ext)

s12 = es.inner(es2, diag=False) # using S(k) from es
assert not np.allclose(s12, s12_ext)

A note, you shouldn’t use tocsr. When you start using NC/SOC what you want is Hk(format='sc'). I have just fixed so one can use format=‘sc’` for gamma-point.