The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1/r^{d+s}, admits a second-order phase transition with continuously varying critical exponents. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1/r^{d+s}, admits a second-order phase transition with continuously varying critical exponents. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.

arXiv:1703.03430Physical Review LettersJun 15, 2017107 citations

Validity of the long-range expansion in the n -vector model

M. A. Gusmão, W. K. Theumann

The critical behavior of an $n$ -component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.The critical behavior of an $n$-component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.

Physical review. B, Condensed matterDec 1, 198320 citations

Wilson-type expansions of critical exponents for long-range interactions

Masuo Suzuki, Y. Yamazaki, George Igarashi

Physics Letters ADec 1, 197244 citations

Critical scaling function for long-range interactions to order 1/n

W. K. Theumann

Physics Letters AJul 1, 19752 citations

The critical behavior of an

n

-component system in

d

dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.