This approximation becomes exact above the upper critical dimension. For 10 points each:
[10m] Name this approximation often used in statistical physics, which reduces an interacting Hamiltonian to a sum of one-body Hamiltonians. For example, in Pierre Weiss’ solution to the Ising model, this approximation assumes each site “feels” a potential given by h plus Jzm.
ANSWER: mean-field approximation [accept mean-field theory; accept self-consistent field approximation, accept molecular field approximation; prompt on effective field]
[10e] In the canonical ensemble, assuming a mean field minimizes this quantity for the system by the Bogoliubov inequality. This thermodynamic potential named after a German is equal to U minus TS.
ANSWER: Helmholtz free energy [prompt on F or A; prompt on free energy]
[10h] The mean field is the maximizer in the saddle point approximation of the integral obtained from this exact transformation. This transformation decouples two-particle interactions, and is used to bosonize a model named for one of its two namesakes.
ANSWER: Hubbard-Stratonovich transformation (the model is the Hubbard model.)
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