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

Upper bounds for the transition temperatures of generalized Ising models

A. Brooks Harris

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

\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.