Phenomenology of local scale invariance: from conformal invariance to dynamical scaling
Phenomenology of local scale invariance: from conformal invariance to dynamical scaling
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DOI：
10.1016/s0550-3213(02)00540-0
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发表时间：
2002-05
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通讯作者：
M. Henkel
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文献类型：
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作者：
M. Henkel
表现出强各向异性或动态尺度行为的统计系统由各向异性指数θ或动态指数z来表征。对于给定的θ（或z）值，我们构建了局部尺度变换，可以将其视为具有时空相关膨胀因子的尺度变换。发现了两种不同类型的局部尺度变换。第一类可以描述具有给定θ值的静态系统的强各向异性标度，而第二类可以描述具有动态指数z的动态标度。局部标度变换作为某些非局部自由场理论的动态对称群。已知的局部尺度不变性的特殊情况是θ=1时的保形不变性和θ=2时的Schrödinger不变性。局部尺度不变性假设表明拟初等算子的两点函数满足由交换分数阶导数构造的一类线性分数阶微分方程。这些的显式解给出了平衡点上两点相关器和平衡点外两点响应函数的精确表达式。对于两次自动响应函数，可以找到一种特别简单和一般的形式。这些预测在ANNNI和ANNNS模型中的单轴Lifshitz点以及简单铁磁体的老化行为中得到了明确的证实，例如动力学Glauber-Ising模型和具有非守恒阶参量的动力学球面模型，它们经历了相序动力学或非平衡临界动力学。
Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2. The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.