ADIABATIC PHASES FROM AN EFFECTIVE ACTION

Berry's phase is calculated as a term in the derivative xpansion of the effective action of a spin- system in an external field. A quantum mechanical system depending on an external parameter, taken slowly around a closed loop in parameter space exhibits in addition to the familiar dynamical phase variation ageometric phase factor [ 1]. In this short note we show that this phase factor occurs naturally in the derivative xpansion of the effective action [ 2]: Berry's phase can be calculated by straightforward and direct evaluation of a functional determinant. We shall concentrate on the ground state of a spin- system in an external magnetic field B with lagrangian [2,3] L = i (~"~"+ ~,,,.j~m~.Bj), (1) where the ~'s are real anticommuting variables. To obtain the effective action F(B) we have to evaluate the path integral exp [ i / " (B)] = f [D, ",exp ( i f dt[~i(,n~"+emnj,m,nBJ)]), (2) which is easily done, since L is only quadratic in the ~'s. We learn that exp [iF(B) ] =detl/2(OJ~mn +iemnjB j) , (3) or F(B) =- iTr In (¢O~mn "l-iEmnjB j) , (4) where to is the energy operator and the trace Tr is understood in a functional sense. Gerbert [3] arrives at the same expression for F(B), (3), only he goes on to evaluate F making strong use of symmetry principles. We shall instead evaluate (4) in a straightforward way. Since B (t) is time dependent, the operator ((.l)(~mn" ~ iEmnjB J) is not diagonal in the energy representation, and (4) cannot be evaluated exactly. Instead we may expand the density ofF(B) in powers of B [4,5]. Schematically, we may write F(B) = f dt [- V, rr(B) +A (B)II+ iBiMijBJ+...]. (5) Clearly V¢rr(B) is just the effective potential and A (B) Berry's connection. Vefr(B) can be evaluated by taking B (t) = Bo to be a constant [ 5,6]. Then (4) reduces to an integral in momentum space