Ising model and auxiliary fields

Gaussian integration IIIn the new integration variablesI= ∏ m„∫ √Jmm d√ ˜φPm − ˜m2πφmJmm −1«φm ˜ √+ φ˜2 m 2Sm= det(J) ∏ m∫ dφm√2πe − P m,nφmJ −1 mn φn2 + √ 2 P m φmSmHere we used:1 det(U) = 12 ˜φT J −1 ˜φ = φ T J −1 φ3 ˜φT · S = φ · SWe are left with linear in S m terms⇒ trace out S m = ±1