Approximations for the Mori Continued Fraction in Paramagnets

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Abstract
We point out that an approximation previously used 3 to describe the 1 dimensional Heisenberg paramagnet contains an inconsistency at low temperature. We show that using the form for K1(t) used in reference 3, one finds K 3 [ s =δ 1 1/2 ]/ K 3 [0]α T +1/2 at low temperatures, so that the approximation K 3 [ s ] ≈ K 3 [0] fails in the region of the spin wave resonance away from k =0 at low temperatures. Here the Laplace transform R[s] of a response function related to the structure factor is given by R [ s ] = R (0)/( s + δ 1 /( s + δ 2 /( s + K 3 [ s ]))) = R (0)/( s + K 1 [ s ]) and δ 1 = 〈ω 2 〉 ; δ 2 = 〈ω 4 〉/ 〈ω 2 〉 − 〈ω 2 〉 . We have constructed a fully selfconsistent version of the approximation and explored its consequences for the classical Heisenberg chain and the xy model. We conclude that there are serious consistency problems whenever δ 1 ≧ δ 2 .

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