Title:

Problems in the quantum theory of manyparticle systems

A finite temperature perturbation theory for the Heisenberg model of ferromagnetism is presented. A modified Wick's theorem proved for S=½ and finite temperatures allows the terms in the expansion for the temperature dependent Green's function to be expressed by linked Feynmanlike diagrams. The diagrams differ from the usual boson and fermion diagrams in the appearance of time (temperature) dependent dotted lines between propagators reflecting the effect of spin statistics. For temperatures such that ^{T}⁄_{TC} is finite, the diagrams for the irreducible selfenergies are classified with respect to a high density parameter ^{1}⁄_{Z} where Z is the number of spins interacting with a given spin. The results of molecular field theory are found in first order. The next two orders in the selfenergy graphs yield at low T spin wave theory and Dyson's results in Born approximation respectively. The high density classification is also verified directly at low T, and in an appendix, higher order graphs are found to yield the expressions for the lifetimes of the excitations of the system given by TahirKheli and ter Haar. The magnetization expression obtained from the graphs of the first three orders yields a Curie point since the first order graphs are renormalized to include in their internal lines all graphs of all other orders considered. The Curie point found is that of molecular field theory. The expressions for the magnetization at low temperatures and near the Curie point are then used as a guide in approximating the magnetization from graphs of the first three orders at all temperatures. This approximate expression is calculated as a function of the temperature using a long range potential, and the curve is compared to the experimental results for the magnetization of Cu K_{2} Cl_{4}.2H_{2}O, a spin ½ ferromagnetic insulator with next nearest neighbour interactions. The results from the random phase approximation of Tyablikov and Englert are also compared with the experimental results, and it is seen that both curves fit within the experimental errors of the points up to ^{T}⁄_{TC}~.6, but both fall below the experimental points as the Curie temperature is approached. The difference between the two theoretical curves is negligible.
