Short-Range Spin Interaction and Critical Exponents

Yoshitake Yamazaki

E. Brézin, Giorgio Parisi A, Federico Ricci‐Tersenghi

E. Nogueira, S. Coutinho, Fernando Dantas Nobre +1 more

Josef Sak

Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$.

Masuo Suzuki

Y. Yamazaki, Masuo Suzuki

Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.