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In Bloch theory, using the reduced zone scheme, we can index the states of a crystal by either $k,G$ where $k$ is in the first Brillouin zone and $G$ is a reciprocal lattice vector, or $n,k$ where $k$ is in the first Brillouin zone and $n$ is the band index.

I would like to work in tight binding, for which the latter choice $n,k$ is natural. We might like to express operators in the $n,k$ basis, such as the density operator $\rho(r)=\psi^+(r)\psi(r)$ and its Fourier transform $\rho(q) = \int e^{-iqr} \rho(r)$.

This particular operator can be written as $$ \rho(q)=\sum\limits_{m,n,k}M_{m,n}(k,q)c^+_{m,k-q}c_{n,k} $$ where $$ M_{m,n}(k,q) = \int\limits_{\text{unit cell}} ds u^*_{m,k-q}(s)u_{n,k}(s) $$ is a matrix element of the periodic part of the Bloch wavefunctions.

Now, suppose I want to study a density fluctuation of wavevector $q$ which is outside the first Brillouin zone. Then I have some sort of ambiguity or redundancy which I do not yet understand, because the matrix element $M_{m,n}(k,q)$ has both the bands $m,n$ and a reciprocal lattice vector in $q$ (as it extends outside the first Brillouin zone), as if I'm both using the reduced zone scheme and not using it at the same time.

What is the proper way to do handle reciprocal lattice vectors outside the first Brillouin zone when working in the band basis?

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Working with the reduced-zone scheme (your band basis), all the physical properties are periodic function in the reciprocal lattice. Therefore, your matrix element is also a peroidic function in teh reciprocal lattice. For axmple, saying that $k-q$ falls into outside the first brillouin zone, what you have to do is folding it back to the first brillouin zone, by added or substracted integer multplcation of $G$:

\begin{align} M_{m,n}(k,q) &= \int\limits_{\text{unit cell}} ds u^*_{m,k-q}(s)u_{n,k}(s) \\ &= \int\limits_{\text{unit cell}} ds u^*_{m,k-q+G}(s)u_{n,k}(s) \end{align}

Since the periodic part of the Bloch function is also a periodic function in reciprocal lattice. $$ u^*_{m,k-q}(x) = u^*_{m,k-q+G}(x). $$

The effect of exntented band scheme had been replace by the band indices $m$ and $n$. What you worried (extended $k+q$) had been replaced by one of the inter-band matrix elements: The matrix element in exntended band scheme for $k-q$ outside 1st BZ: $m(k, k-q)$ had been replaced by $m_{n, n'}(k, k-g+G)$, been folded back onto the 1st BZ.

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