By E. Krause
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Extra info for Advances in Fluid Mechanics
Transverse Ising Chain 21 from the first order perturbation theory. Near the critical point, however, all higher order contributions to the mass gap are expected to become relevant. 2]' using Raleigh-Schrodinger perturbation theory, estimated the higher order corrections in>' to the mass gap ~(>') and found that all the higher order contributions vanish identically in 1+1 dimension. In higher dimensions, of course, all the benificial properties of the perturbation scheme are lost, the perturbation series does not truncate in the first order.
Wannier, Phys. Rev. 60252 (1941). 2] J. B. Kogut, Rev. Mod. Phys. 51 659 (1979). 3] P. Pfeuty, Ann. Phys. 5779 (1970); E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16407 (1961). 4] S. Katsura, Phys. Rev. 1271508 (1962). 5] B. M. McCoy, Phys. Rev. 173531 (1968). 6) R. Shankar, in Current Topics in Condensed Matter and Particle Physics, Ed. J. Pati, Q. Shafi and Yu Lu, World Scientific (1993). 7) M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28 1516 (1972), see also, M. N. Barber in Phase Transition and Critical Phenomena, Ed.
Xt 2. Transverse Ising Chain 29 11" m = Go = (OIS: 10) = Z 7r dq 0 W;1 + -11" dq W;1 7r cos q, which gives an elliptic integral of the first kind, and is nonanalytic at 1. 3). 38) 0 -- ----. ~- >. 5 r/2J Fig. 3]. (SZ) with r /2J. Dashed The correlation functions given in App. 3 depend only on Gr. , G r is nonanalytic for>' = 1 and so are the correlation functions. The critical case>' = 1 can be studied more easily because of the simple form of G r (see App. 3). f. 2 r - 2 .... (r-l). C; and C: behave in the analogous way and all the correlations at >.