CubicN-vector model and randomly dilute Ising model in general dimensions

Abstract

Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, $N$. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}Nl\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}Nl\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of $N$ and $d$ are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic $N$-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of $d$, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At $d=3$, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the $N$-component cubic model, the spin dimensionality ${N}_{c}$, at which the isotropic and cubic fixed points change stability, is determined as a function of $d$. The results support ${N}_{c}g3$ for three dimensions.