Rigorous Inequalities for Critical-Point Correlation Exponents

Simple rigorous proofs are given for the inequalities $\gamma \le (2-\eta )\nu$, $2-\eta \le \frac{d(\delta -1)}{(\delta +1)}$, and $2-\eta \le \frac{d{\gamma }^{\prime }}{(2_{\beta }+{\gamma }^{\prime })}\le \frac{d{\gamma }^{\prime }}{(2-{\alpha }^{\prime })}$ satisfied by the exponents $\nu$ and $\eta$ describing the decay of the spin-spin correlation function in a $d$-dimensional ferromagnet near its critical point. The notation is standard, but new, refined definitions of $\nu$ and $\eta$ are utilized in the proofs. The exponent ${\eta }_E$ describing the decay of the energy-energy correlation function in an Ising ferromagnet is proved to satisfy $2-{\eta }_E\le d{\alpha }^{\prime }$, $2-{\eta }_E\le \frac{d{\alpha }_c}{(1+\zeta +{\alpha }_c)}$, where the specific heat $C_M$ at $T=T_c$ diverges with magnetization $M$ as $M^{-{\alpha }_c}$, while the energy derivative ${\left|\frac{\partial U}{\partial M}\right|}_{T_c}$ varies as $M^{\zeta }$. (The mean-field or classical values are ${\alpha }_c=0\mathrm{}$, $\zeta =1\mathrm{}$.) The proofs are based on general and "intuitively obvious" positivity and monotonicity properties of ferromagnetic correlation functions. The necessary properties (and certain supplementary lemmas) can be established rigorously for Ising models of arbitrary spin, lattice structure, and ferromagnetic coupling ($J_{\mathrm{ij}}\ge 0$).