Question

What's the right Fourier transformation definition of creation or annihilation operators in lattice model?

Answer 1

Very often one can view the solid as lattice model and then use the the language of second quantization,namely taking the occupation number representation, to express the Hamiltonian of the system. In this representation the creation and annihilation operators play the fundamental role. But in recently I just found some discrepancies for the definition of these operators for lattice model:  Vijay Shenoy Version (starting at 13:10):  For 3D lattice model the Fourier transformation of the creation operator between K space and real space will be:  c†iσ=1N−−√∑k∈BZe−ik⃗ ⋅r⃗ c†σ(k⃗ )(1) here the subindices σ and i are the spin index and site index, respectively and N stands for all K points in first Brillouin zone. And Vijay didn't show the inverse transformation.  KaiSun's Version (at page 78):  He stated that we have a continuous k-space and it is periodic (the Brillouin zone), but the real space is discrete for the lattice model.The Fourier transoformation relations are (for 1D system): ck=a−−√2π−−√∑icie−ikx(2) ci=a−−√2π−−√∫BZdkckeikx(3) I think the argument of professor KaiSun is very convincible. But what's the difference between (1) and (3) and are there something wrong for the coefficients in Sun's definition? I mean a−−√2π−−√→12πa−−−√. Thanks in advance.

Answer 2

(1) is essentially the discrete version of (3), it applies to systems of finite size. The coefficients simply ensure that applying the Fourier transform and then the inverse Fourier transform get you back where you started. Different choices are possible.