Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.

Journal of Physics C Solid State PhysicsSep 7, 197473 citations

Improved upper bounds on the critical temperature of the 1n2 ising spin chain

J. O. Vigfusson

Recent results on the upper-bound property of the magnetization in certain approximations to ferromagnetic spin systems are here used to derive a rigorous upper bound for the critical temperature of the one-dimensional ferromagnetic $\frac{1}{{n}^{2}}$ Ising spin system. We find that ${\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})$. The best previous bound was 0.360. A further improvement by the same technique is possible.Recent results on the upper-bound property of the magnetization in certain approximations to ferromagnetic spin systems are here used to derive a rigorous upper bound for the critical temperature of the one-dimensional ferromagnetic $\frac{1}{{n}^{2}}$ Ising spin system. We find that ${\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})$. The best previous bound was 0.360. A further improvement by the same technique is possible.

Physical review. B, Condensed matterSep 1, 19869 citations

\frac{1}{{n}^{2}}

Ising spin system. We find that ${\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})$. The best previous bound was 0.360. A further improvement by the same technique is possible.