Non-local meta-conformal invariance in diffusion-limited erosion

Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain

Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent $z=1$, none of the known variants of conformal invariance can act as its dynamical symmetry. In $d=1$ spatial dimensions, its infinite-dimensional dynamic symmetry is constructed and shown to be isomorphic to the direct sum of three loop-Virasoro algebras, with the maximal finite-dimensional sub-algebra $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$. The infinitesimal generators are spatially non-local and use the Riesz-Feller fr…

Meta-conformal invariance and the boundedness of two-point correlation functions

The covariant two-point functions, derived from Ward identities in direct space, can be affected by consistency problems and can become unbounded for large time- or space-separations. This difficulty arises for several extensions of dynamical scaling, for example Schrödinger-invariance, conformal Galilei invariance or meta-conformal invariance, but not for standard ortho-conformal invariance. For meta-conformal invariance in 1+1 dimensions, these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in…

Meta-Schrödinger invariance

The Meta-Schrödinger algebra arises as the dynamical symmetry in transport processes which are ballistic in a chosen `parallel' direction and diffusive and all other `transverse' directions. The time-space transformations of this Lie algebra and its infinite-dimensional extension, the meta-Schrödinger-Virasoro algebra, are constructed. We also find the representation suitable for non-stationary systems by proposing a generalised form of the generator of time-translations. Co-variant two-point functions of quasi-primary scaling operators are derived for both the stationary and the non-stationar…

Boundedness of meta-conformal two-point functions in one and two spatial dimensions

Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent $z=1$, and distinct from the standard ortho-conformal invariance. The meta-conformal Ward identities can be directly read off from the Lie algebra generators, but this procedure implicitly assumes that the co-variant correlators should depend holomorphically on time- and space coordinates. Furthermore, this assumption implies un-physical singularities in the co-variant correlators. A careful reformulation of the global meta-conformal Ward identities in a dualised space, combined with a regularity postul…

二维空间中安德森局域化跃迁的共形不变性、多重分形和有限尺寸标度

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

我们推广了多重分形指数α_0与典型波函数振幅标度之间的普遍关系。（Anderson）定域-离域跃迁和准一维（Q1 D）几何中典型定域长度的临界有限尺寸标度（FSS）振幅Λ c：（i）当在Q1 D样品的横向方向上施加开放边界条件时（条带几何形状），结果表明，对应的临界频率选择表面振幅Λ c^o与边界多重分形指数α_0^普遍相关s表示沿直线边界（表面）的典型波函数振幅。(ii)我们进一步提出了这些普遍的关系，这些对称类的状态密度在过渡消失的一般化。(iii)我们验证我们的广义关系[方程。(6)和（7）]数值计算了以下四种类型的二维Anderson跃迁：（a）自旋轨道（辛）对称类中的金属到（普通绝缘体）跃迁，（b）自旋轨道（辛）对称类中的金属到（Z_2拓扑绝缘体）跃迁，（c）整数量子Hall坪跃迁，和（d）自旋量子Hall坪跃迁。