Local scale invariance, conformal invariance and dynamical scaling

Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space-time symmetries of certain non-local free field theories. Physical applications include uniaxial Lifshitz points and ageing in simple ferromagnets. Scale invariance is a central notion of modern theories of critical and collective phenomena. We are interested in systems with strongly anisotropic or dynamical criticality. In these systems, two-point functions satisfy the scaling form G(t, r) = bG(bt, br) = tΦ ( rt ) = rΩ ( tr ) (1) where t stands for ‘temporal’ and r for ‘spatial’ coordinates, x is a scaling dimension, θ the anisotropy exponent (when t corresponds to physical time, θ = z is called the dynamical exponent) and Φ,Ω are scaling functions. Physical realizations of this are numerous, see [1] and references therein. For isotropic critical systems, θ = 1 and the ‘temporal’ variable t becomes just another coordinate. It is well-known that in this case, scale invariance (1) with a constant rescaling factor b can be replaced by the larger group of conformal transformations b = b(t, r) such that angles are preserved. It turns out that in the case of one space and one time dimensions, conformal invariance becomes an important dynamical symmetry from which many physically relevant conclusions can be drawn [2]. Given the remarkable success of conformal invariance descriptions of equilibrium phase transitions, one may wonder whether similar extensions of scale invariance also exist when θ 6= 1. Indeed, for θ = 2 the analogue of the conformal group is known to be the Schrödinger group [3, 4] (and apparently already known to Lie). While applications of the Schrödinger group as dynamical space-time symmetry are known [5], we are interested here in the more general case when θ 6= 1, 2. We shall first describe the construction of these local scale transformations, show that they act as a dynamical symmetry, then derive the functions Φ,Ω and finally comment upon some physical applications. For details we refer the reader to [6]. The defining axioms of our notion of local scale invariance from which our results will be derived, are as follows (for simplicity, in d = 1 space dimensions). (i) We seek space-time transformations with infinitesimal generators Xn, such that time undergoes a Möbius transformation t→ t′ = αt+ β γt+ δ ; αδ − βγ = 1 (2) and we require that even after the action on the space coordinates is included, the commutation relations [Xn, Xm] = (n−m)Xn+m (3) remain valid. This is motivated from the fact that this condition is satisfied for both conformal and Schrödinger invariance. Local scale invariance, conformal invariance and dynamical scaling 2 (ii) The generator X0 of scale transformations is X0 = −t∂t − 1 θ r∂r − x θ (4) with a scaling dimension x. Similarly, the generator of time translations is X−1 = −∂t. (iii) Spatial translation invariance is required. (iv) Since the Schrödinger group acts on wave functions through a projective representation, generalizations thereof should be expected to occur in the general case. Such extra terms will be called mass terms. Similarly, extra terms coming from the scaling dimensions should be present. (v) The generators when applied to a two-point function should yield a finite number of independent conditions, i.e. of the form XnG = 0. Proposition 1: Consider the generators