The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.

arXiv:hep-th/0103194Physical Review LettersAug 31, 200149 citations

Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems

Malte Henkel

A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}$ . Differential equations for the two-point function are derived and explicitly solved for all values of $N$ . Known special cases are conformal invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ ) and Schr\"odinger invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ ). For $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6$ , the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}$. Differential equations for the two-point function are derived and explicitly solved for all values of $N$. Known special cases are conformal invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$) and Schr\"odinger invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$). For $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6$, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.

arXiv:cond-mat/9610174Physical Review LettersMar 10, 1997118 citations

Conformal Invariance and Critical Phenomena

Malte Henkel

Springer eBooksJan 1, 1999387 citations

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}

. Differential equations for the two-point function are derived and explicitly solved for all values of

N

. Known special cases are conformal invariance (

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

) and Schr\"odinger invariance (

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

). For

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4

and

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6

, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.