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Recently Wang and Cheng proposed a self-consistent effective Hamiltoniantheory (SCEHT) for many-body fermionic systems (Wang & Cheng, 2019). This paperattempts to provide a mathematical foundation to the formulation of the SCEHTthat enables further study of excited states of the system in a more systematicand theoretical manner. Gauge fields are introduced and correct total energyfunctional in relations to the coupling gauge field is given. We also providesa Monte-Carlo numerical scheme for the search of the ground state that goesbeyond the SCEHT.

Mathematical Theory of Locally Coherent Quantum Many-Body Fermionic System