I've explored the entanglement of modes by expanding the ground-state solution of a many-body problem as an infinite sum of Slater determinants of one-particle Hermite functions. The one-particle basis of Hermite functions is however defined over all space.
I'm intrigued to see how certain correlation measures relate to geometric features of the system but I would need to expand the ground state in terms of a basis that is defined over a particular spatial region for a particular sub-Hilbert space.
For instance, the 2d radial Hermite functions could be piecewise-defined and divided into two spatial regions; a circle with radius a, for the spatial region $(0,a)$ and a "donut" over the region $(a,\infty)$. This however seems rather hand-wavy and as far as I can tell doesn't seem to give rise to an orthonormal basis for a particular sub-Hilbert space. This should at least serve as an illustration of what I am trying to do.
My question is the following; does anyone have any suggestions as to how I can partition the Hilbert space such that the sub-Hilbert spaces are defined over certain spatial regions which can be easily intuited? and particularly what sort of orthonormal bases would span these Hilbert spaces?
I realize this may not actually be possible given; Can we always find a Hilbert space corresponding to a region of spacetime? (or at least very challenging!). I'm not well versed in QFT but many thanks in advance for any help or suggestions.