I am currently trying to solve the path integral of fermion field on a two-dimensional square lattice. The Hamiltonian is given by tijai*aj with i and j summed over. Here "a" are creation and annihilation operators. How do I get the ground state for this fermionic field? How could I get this in path integral approach and particularly what cut in path integral means? Also, suggest me some materials to read upon?
The answer to your question can be found in Sec. 2.5 (Feynman Path Integral, p.123) of the book Quantum Many-Particle Systems, by JW Negele, and H Orland (Westview Press, Boulder, Colorado, 1998). In essence, once you have determined the expression for the Feynman propagator associated to your problem. you expand the action around the classical path, following which you determine the path that makes the action stationary, subject to the relevant initial and boundary conditions. Generally, when one has to do with an interacting problem, one can only determine the relevant field configuration in an asymptotic sense (say, to leading order in the constant of Planck), however since your problem is quadratic in the creation and annihilation operators, you will be able to solve your problem exactly.
For fermions there are so-called Grassmannian path integrals. see the attached lnk
https://doi.org/10.1209/0295-5075/15/6/009
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