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arXiv:hep-th/0103194 · Physical Review Letters

Anisotropic Scaling and Generalized Conformal Invariance at Lifshitz Points

Michel Pleimling, Malte Henkel

Published Aug 31, 2001• 49 citations
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Abstract

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.

DOI
10.1103/physrevlett.87.125702

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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 \ensuremathα=0.18(2)\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)\ensuremathα=0.18(2), \ensuremathβ=0.238(5)\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)\ensuremathβ=0.238(5), and \ensuremathγ=1.36(3)\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)\ensuremathγ=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.
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=1,2,3…N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}N=1,2,3…. Differential equations for the two-point function are derived and explicitly solved for all values of NNN. Known special cases are conformal invariance ( N=2N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N=2) and Schr\"odinger invariance ( N=1N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1N=1). For N=4N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4N=4 and N=6N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6N=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.
In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: ${\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}/\ensuremath{\nu}z}$, which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent zzz, evaluated from the behavior of the ratio ${F}_{2}(t)={\ensuremath{\langle}{M}^{2}\ensuremath{\rangle}}_{{m}_{0}=0}/{\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}^{2}\ensuremath{\sim}{t}^{3/z}$, along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent $\ensuremath{\theta}$ that characterizes the initial slip of magnetization and the global persistence exponent ${\ensuremath{\theta}}_{g}$ associated with the probability P(t)P(t)P(t) that magnetization keeps its signal up to time ttt. Our estimates for the dynamic critical exponents at the Lifshitz point are z=2.34(2)z=2.34(2)z=2.34(2) and ${\ensuremath{\theta}}_{g}=0.336(4)$, values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the zzz axis, i.e., J2=0{J}_{2}=0J2​=0), i.e., $z\ensuremath{\approx}2.07$ and ${\ensuremath{\theta}}_{g}\ensuremath{\approx}0.38$. We also present estimates for the static critical exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.
We investigate a critical Ising-like model in the curved geometry ${S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}$ obtained by a conformal mapping of the infinite 3D space R3{R}^{3}R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as Xh=0.5178(12){X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)Xh​=0.5178(12) and Xt=1.423(19){X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)Xt​=1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.
We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .
Field-theoretic renormalization-group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic $(m\ensuremath{\ne}d)$ and isotropic (m=d(m=d(m=d close to 8) situations. The general theory is illustrated for the N-vector ${\ensuremath{\varphi}}^{4}$ model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization-group transformations. The description of the isotropic behavior requires only one type of renormalization-group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization-group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class (N,d,m)(N,d,m)(N,d,m) reduces easily to the Ising-like (N,d)(N,d)(N,d) when m=0.m=0.m=0. We show that the isotropic universality class (N,m)(N,m)(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit $\stackrel{\ensuremath{\rightarrow}}{d}m$ near 8. The exponents for the uniaxial case d=3,N=m=1d=3,N=m=1d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.