# Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**Example**

**1.**

**Example**

**2.**

**Definition**

**1.**

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 2. Local Conformal Invariance

**Definition**

**2.**

**Definition**

**3.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 3. Impossibility of a Local Meta-Conformal Invariance of the dle Process

- The deterministic part of the dle Langevin Equation (1) is distinct from the simple invariant equations $\mathcal{S}\phi =0$ of either ortho- or meta-1-conformal invariance.For analogy, consider briefly Schrödinger-invariant systems with a Langevin equation of the form $\mathcal{S}\phi ={\left(2\nu T\right)}^{1/2}\eta $, where $\eta =\frac{\mathrm{d}B}{\mathrm{d}t}$ is a white noise of unit variance, and such that the Schrödinger algebra is a dynamical symmetry of the noise-less equation (deterministic part) $\mathcal{S}{\phi}_{0}=0$. Then, the Bargman super-selection rules [69] which follow from the combination of spatial translation-invariance and Galilei-invariance with $z=2$, imply exact relations between averages of the full noisy theory and the averages calculated from its deterministic part [70]. In particular, the two-time response function of the full noisy equation $R(t,s;\mathit{r})={R}_{0}(t,s;\mathit{r})$, is identical to the response ${R}_{0}$ found when the noise is turned off and computed from the dynamical Schrödinger symmetry [16,70].We shall assume here that an analogous result can be derived also for the dle Langevin equation, although this has not yet been done. It seems plausible that such a result should exist, since in the example (5b) and (6b) of the dle process, the two-time response R is independent of T (which characterises the white noise), as it is the case for Schrödinger-invariance.
- The explicit response function (6b) of the dle process is distinct from the predictions (12) , (14) and (16), see also Table 2. The form of the meta-1 conformal two-point function (14), is clearly different for finite values of the scaling variable $v=({r}_{1}-{r}_{2})/({t}_{1}-{t}_{2})$, and similarly for the conformal Galilean case (16). The ortho-conformal two-point function (12) looks to be much closer, with the choice ${x}_{1}=\frac{1}{2}$ and the scale factor fixed to $\nu =1$, were it not for the extra factor $\nu (t-s)$.On the other hand, the two-time dle-correlator (6a) does not agree with (12) either, but might be similar to a two-point function computed in a semi-infinite space $t\ge 0$, $r\in \mathbb{R}$ with a boundary at $t=0$.

## 4. Riesz-Feller Fractional Derivative

**Definition**

**4.**

**Lemma**

**1.**

**Lemma**

**2.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**3.**

## 5. Non-Local Meta-Conformal Generators

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Corollary**

**2.**

**Corollary**

**3.**

**Proposition**

**4.**

**Proof.**

## 6. Ward Identities for Co-Variant Quasi-Primary n-Point Functions

## 7. Co-Variant Two-Time Correlators and Responses

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Case****A:**- $2\xi =1$. Then $\left({\epsilon}_{1}{\mathrm{\nabla}}_{v}+\mu v{\partial}_{v}+\mu \right)f(v)=0$ and ${x}_{1}\ne {x}_{2}$ is still possible.
**Case****B:**- ${x}_{1}={x}_{2}$. Then ${\xi}_{1}={\xi}_{2}$ and $\left({\epsilon}_{1}{\mathrm{\nabla}}_{v}+\mu v{\partial}_{v}+2\mu \xi \right)f(v)=0$.

**Proposition**

**7.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Corollary**

**4.**

**Proof.**

## Acknowledgments

## Conflicts of Interest

## References and Notes

- Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B. Infinite conformal symmetry in two-dimensional quantum field-theory. Nuclear Phys. B
**1984**, 241, 330–338. [Google Scholar] [CrossRef] - Krug, J.; Meakin, P. Kinetic roughening of laplacian fronts. Phys. Rev. Lett.
**1991**, 66, 703–706. [Google Scholar] [CrossRef] [PubMed] - Paulos, M.F.; Rychkov, S.; van Rees, B.C.; Zan, B. Conformal Invariance in the Long-Range Ising Model. Nuclear Phys. B
**2016**, 902, 249–291. [Google Scholar] [CrossRef] [Green Version] - Krug, J. Statistical physics of growth processes. In Scale invariance, Interfaces and Non-Equilibrium Dynamics; McKane, A., Droz, M., Vannimenus, J., Wolf, D., Eds.; NATO ASI Series; Plenum Press: London, UK, 1994; Volume B344, p. 1. [Google Scholar]
- Yoon, S.Y.; Kim, Y. Surface growth models with a random-walk-like nonlocality. Phys. Rev. E
**2003**, 68, 036121. [Google Scholar] [CrossRef] [PubMed] - Aarão Reis, F.D.A.; Stafiej, J. Crossover of interface growth dynamics during corrosion and passivation. J. Phys. Cond. Matt.
**2007**, 19, 065125. [Google Scholar] [CrossRef] - Zoia, A.; Rosso, A.; Kardar, M. Fractional Laplacian in Bounded Domains. Phys. Rev. E
**2007**, 76, 021116. [Google Scholar] [CrossRef] [PubMed] - Spohn, H. Bosonization, vicinal surfaces, and hydrodynamic fluctuation theory. Phys. Rev. E
**1999**, 60, 6411–6420. [Google Scholar] [CrossRef] - Popkov, V.; Schütz, G.M. Transition probabilities and dynamic structure factor in the ASEP conditioned on strong flux. J. Stat. Phys.
**2011**, 142, 627–639. [Google Scholar] [CrossRef] - Karevski, D.; Schütz, G.M. Conformal invariance in driven diffusive systems at high currents. arXiv
**2016**. [Google Scholar] - Spohn, H. Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys.
**2014**, 154, 1191–1227. [Google Scholar] [CrossRef] - Bertini, L.; De Sole, A.; Gabrielli, D.; Jona-Lasinio, G.; Landim, C. Macroscopic fluctuation theory. Rev. Mod. Phys.
**2015**, 87, 593–636. [Google Scholar] [CrossRef] - Barabási, A.-L.; Stanley, H.E. Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Halpin-Healy, T.; Zhang, Y.-C. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys. Rep.
**1995**, 254, 215–414. [Google Scholar] [CrossRef] - Krug, J. Origins of scale-invariance in growth processes. Adv. Phys.
**1997**, 46, 139–282. [Google Scholar] [CrossRef] - Henkel, M.; Pleimling, M. Non-Equilibrium Phase Transitions Volume 2: Ageing and Dynamical Scaling Far from Equilibrium; Springer: Heidelberg, Germany, 2010. [Google Scholar]
- Täuber, U.C. Critical Dynamics: A Field-Theory Approach to Equilibrium and Non-Equilibrium Scaling Behaviour; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Family, F.; Vicsek, T. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J. Phys. A Math. Gen.
**1985**, 18, L75–L81. [Google Scholar] [CrossRef] - Yeung, C.; Rao, M.; Desai, R.C. Bounds on the decay of the auto-correlation in phase ordering dynamics. Phys. Rev. E
**1996**, 53, 3073–3077. [Google Scholar] [CrossRef] - Henkel, M.; Durang, X. Spherical model of interface growth. J. Stat. Mech.
**2015**, 2015, P05022. [Google Scholar] [CrossRef] - Henkel, M. Non-local meta-conformal invariance in diffusion-limited erosion. J. Phys. A Math. Theor.
**2016**, 49, 49LT02. [Google Scholar] [CrossRef] - Hase, M.O.; Salinas, S.R. Dynamics of a mean spherical model with competing interactions. J. Phys. A Math. Gen.
**2006**, 39, 4875–4899. [Google Scholar] [CrossRef] - Ebbinghaus, M.; Grandclaude, H.; Henkel, M. Absence of logarithmic scaling in the ageing behaviour of the 4D spherical model. Eur. Phys. J. B
**2008**, 63, 85–91. [Google Scholar] [CrossRef] - Edwards, S.F.; Wilkinson, D.R. The surface statistics of a granular aggregate. Proc. R. Soc. Lond. A
**1982**, 381, 17–31. [Google Scholar] [CrossRef] - Kardar, M.; Parisi, G.; Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett.
**1986**, 56, 889–892. [Google Scholar] [CrossRef] [PubMed] - Rodrigues, E.A.; Mello, B.A.; Oliveira, F.A. Growth exponents of the etching model in high dimensions. J. Phys. A Math. Theor.
**2015**, 48, 035001. [Google Scholar] [CrossRef] - Rodrigues, E.A.; Oliveira, F.A.; Mello, B.A. On the existence of an upper critical dimension for systems within the KPZ universality class. Acta. Phys. Pol. B
**2015**, 46, 1231–1237. [Google Scholar] [CrossRef] - Alves, W.S.; Rodrigues, E.A.; Fernades, H.A.; Mello, B.A.; Oliveira, F.A.; Costa, I.V.L. Analysis of etching at a solid-solid interface. Phys. Rev. E
**2016**, 94, 042119. [Google Scholar] [CrossRef] [PubMed] - Krech, M. Short-time scaling behaviour of growing interfaces. Phys. Rev. E
**1997**, 55, 668–679, Erratum in**1997**, 56, 1285. [Google Scholar] [CrossRef] - Henkel, M. On logarithmic extensions of local scale-invariance. Nuclear Phys. B
**2013**, 869, 282–302. [Google Scholar] [CrossRef] - Kelling, J.; Ódor, G.; Gemming, S. Local scale-invariance of the (2 + 1)-dimensional Kardar-Parisi-Zhang model. arXiv
**2016**. [Google Scholar] - Henkel, M.; Noh, J.D.; Pleimling, M. Phenomenology of ageing in the Kardar-Parisi-Zhang equation. Phys. Rev. E
**2012**, 85, 030102(R). [Google Scholar] [CrossRef] [PubMed] - Ódor, G.; Kelling, J.; Gemming, S. Ageing of the (2+1)-dimensional Kardar-Parisi-Zhang model. Phys. Rev. E
**2014**, 89, 032146. [Google Scholar] [CrossRef] [PubMed] - Kelling, J.; Ódor, G.; Gemming, S. Universality of (2 + 1)-dimensional restricted solid-on-solid models. Phys. Rev. E
**2016**, 94, 022107. [Google Scholar] [CrossRef] [PubMed] - Halpin-Healy, T.; Palansantzas, G. Universal correlators and distributions as experimental signatures of (2 + 1)-dimensional Kardar-Parisi-Zhang growth. Europhys. Lett.
**2014**, 105, 50001. [Google Scholar] - Kloss, T.; Canet, L.; Wschebor, N. Nonperturbative renormalization group for the stationary Kardar-Parisi- Zhang equation: Scaling functions and amplitude ratios in 1 + 1, 2 + 1 and 3 + 1 dimensions. Phys. Rev. E
**2012**, 86, 051124. [Google Scholar] [CrossRef] [PubMed] - Röthlein, A.; Baumann, F.; Pleimling, M. Symmetry-based determination of space-time functions in nonequilibrium growth processes. Phys. Rev. E
**2006**, 74, 061604, Erratum in**2007**, 76, 019901(E). [Google Scholar] [CrossRef] [PubMed] - Niederer, U. The maximal kinematical invariance group of the free Schrödinger equation. Helv. Phys. Acta
**1972**, 45, 802–810. [Google Scholar] - Lie, S. Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Mathematik og Naturvidenskab
**1882**, 6, 328–368. [Google Scholar] - Jacobi, C.G. Vorlesungen über Dynamik (1842/43), 4. Vorlesung. In “Gesammelte Werke”; Clebsch, A., Lottner, E., Eds.; Akademie der Wissenschaften: Berlin, Germany, 1866;
^{2}1884. [Google Scholar] - Henkel, M.; Stoimenov, S. Meta-conformal invariance and the boundedness of two-point correlation functions. J. Phys. A Math. Theor.
**2016**, 49, 47LT01. [Google Scholar] [CrossRef] - Cartan, É. Les groupes de transformation continus, infinis, simples. Annales Scientifiques de l’École Normale Supérieure (3e série)
**1909**, 26, 93–161. [Google Scholar] - Di Francesco, P.; Mathieu, P.; Sénéchal, D. Conformal Field-Theory; Springer: Heidelberg, Germany, 1997. [Google Scholar]
- Polyakov, A.M. Conformal symmetry of critical fluctuations. Sov. Phys. JETP Lett.
**1970**, 12, 381–383. [Google Scholar] - Henkel, M. Phenomenology of local scale invariance: From conformal invariance to dynamical scaling. Nuclear Phys. B
**2002**, 641, 405–486. [Google Scholar] [CrossRef] - Henkel, M. Dynamical symmetries and causality in non-equilibrium phase transitions. Symmetry
**2015**, 7, 2108–2133. [Google Scholar] [CrossRef] - Henkel, M.; Schott, R.; Stoimenov, S.; Unterberger, J. The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states. Conflu. Math.
**2012**, 4, 1250006. [Google Scholar] [CrossRef] - Stoimenov, S.; Henkel, M. From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation. Symmetry
**2015**, 7, 1595–1612. [Google Scholar] [CrossRef] - Havas, P.; Plebanski, J. Conformal extensions of the Galilei group and their relation to the Schrödinger group. J. Math. Phys.
**1978**, 19, 482–488. [Google Scholar] [CrossRef] - Henkel, M. Extended scale-invariance in strongly anisotropic equilibrium critical systems. Phys. Rev. Lett.
**1997**, 78, 1940–1943. [Google Scholar] [CrossRef] - Negro, J.; del Olmo, M.A.; Rodríguez-Marco, A. Nonrelativistic conformal groups. J. Math. Phys.
**1997**, 38, 3786–3809. [Google Scholar] [CrossRef] - Negro, J.; del Olmo, M.A.; Rodríguez-Marco, A. Nonrelativistic conformal groups II. J. Math. Phys.
**1997**, 38, 3810–3831. [Google Scholar] [CrossRef] - Henkel, M.; Unterberger, J. Schrödinger invariance and space-time symmetries. Nuclear Phys.
**2003**, B660, 407–435. [Google Scholar] [CrossRef] - Barnich, G.; Compère, G. Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions. Class. Quantum Gravity
**2007**, 24, F15–F23, Corrigendum in**2007**, 24, 3139. [Google Scholar] [CrossRef] - Bagchi, A.; Gopakumar, R.; Mandal, I.; Miwa, A. GCA in 2D. J. High Energy Phys.
**2010**, 8, 1–40. [Google Scholar] [CrossRef] - Duval, C.; Horváthy, P.A. Non-relativistic conformal symmetries and Newton-Cartan structures. J. Phys. A Math. Theor.
**2009**, 42, 465206. [Google Scholar] [CrossRef] - Cherniha, R.; Henkel, M. The exotic conformal Galilei algebra and non-linear partial differential equations. J. Math. Anal. Appl.
**2010**, 369, 120–132. [Google Scholar] [CrossRef] - Hosseiny, A.; Rouhani, S. Affine extension of galilean conformal algebra in 2 + 1 dimensions. J. Math. Phys.
**2010**, 51, 052307. [Google Scholar] [CrossRef] - Zhang, P.-M.; Horváthy, P.A. Non-relativistic conformal symmetries in fluid mechanics. Eur. Phys. J. C
**2010**, 65, 607–614. [Google Scholar] [CrossRef] - Barnich, G.; Gomberoff, A.; González, H.A. Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field-theories as the flat limit of Liouville theory. Phys. Rev.
**2007**, D87, 124032. [Google Scholar] [CrossRef] - It can be shown [57] that there are no cga-invariant scalar equations (in the classical Lie sense). However, if one considers the Newton-Hooke extension of the cga on a curved de Sitter/anti-de Sitter space (whose flat-space limit is not isomorphic to the cga), non-linear representations have been used to find non-linear invariant equations, related to the Pais-Uhlenbeck oscillator, see [62,63,64,65,66,67] and refs. therein.
- Chernyavsky, D. Coest spaces and Einstein manifolds with ℓ-conformal Galilei symmetry. Nuclear Phys. B
**2016**, 911, 471–479. [Google Scholar] [CrossRef] - Masterov, I. Remark on higher-derivative mechanics with ℓ-conformal Galilei symmetry. J. Math. Phys.
**2016**, 57, 092901. [Google Scholar] [CrossRef] - Krivonos, S.; Lechtenfeld, O.; Sorin, A. Minimal realization of ℓ-conformal Galilei algebra, Pais-Uhlenbeck oscillators and their deformation. J. High Energy Phys.
**2016**, 1610, 073. [Google Scholar] [CrossRef] - Chernyasky, D.; Galajinsky, A. Ricci-flat space-times with ℓ-conformal Galilei symmetry. Phys. Lett.
**2016**, 754, 249–253. [Google Scholar] [CrossRef] - Andrezejewski, K.; Galajinsky, A.; Gonera, J.; Masterov, I. Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator. Nuclear Phys. B
**2014**, 885, 150–162. [Google Scholar] [CrossRef] - Galajinsky, A.; Masterov, I. Dynamical realisation of ℓ-conformal Newton Hooke group. Phys. Lett. B
**2013**, 723, 190–195. [Google Scholar] [CrossRef] - Henkel, M.; Stoimenov, S. Physical ageing and Lie algebras of local scale-invariance. In Lie Theory and Its Applications in Physics; Dobrev, V., Ed.; Springer Proceedings in Mathematics & Statistics; Springer: Heidelberg, Germany, 2015; Volume 111, pp. 33–50. [Google Scholar]
- Bargman, V. Unitary ray representations of continuous groups. Ann. Math.
**1954**, 59, 1–46. [Google Scholar] [CrossRef] - Picone, A.; Henkel, M. Local scale-invariance and ageing in noisy systems. Nuclear Phys. B
**2004**, 688, 217–265. [Google Scholar] [CrossRef] - Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives; Gordon and Breach: Amsterdam, The Netherlands, 1993. [Google Scholar]
- Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math.
**2012**, 136, 521–573. [Google Scholar] [CrossRef] - Cinti, E.; Ferrari, F. Geometric inequalities for fractional Laplace operators and applications. Nonlinear Differ. Equ. Appl.
**2015**, 22, 1699–1714. [Google Scholar] [CrossRef] - Gel’fand, I.M.; Shilov, G.E. Generalized Functions, Volume 1: Properties and Operations; Academic Press: New York, NY, USA, 1964. [Google Scholar]
- Sethuraman, S. On microscopic derivation of a fractional stochastic Burgers equation. Commun. Math. Phys.
**2016**, 341, 625–665. [Google Scholar] [CrossRef] - Ovsienko, V.; Roger, C. Generalisations of Virasoro group and Virasoro algebras through extensions by modules of tensor-densities on S
^{1}. Indag. Math.**1998**, 9, 277–288. [Google Scholar] [CrossRef] - Henkel, M. Schrödinger-invariance and strongly anisotropic critical systems. J. Stat. Phys.
**1994**, 75, 1023–1061. [Google Scholar] [CrossRef] - Henkel, M. From dynamical scaling to local scale-invariance: A tutorial. Eur. Phys. J. Spec. Top.
**2017**. to be published. [Google Scholar]

**Figure 1.**Schematics of the genesis of an eroding surface through the dle process. (

**a**) Initial state: a diffusing particle (red path) arrives on a flat surface (full black line) and erodes a small part of it; (

**b**) Analogous process at a later time, when the surface has been partially eroded.

**Figure 2.**(

**a**) Schematic illustration of a vicinal surface, formed by terraces. Fluctuations between terraces are described by steps and kinks; (

**b**) Reinterpretation of the steps of a vicinal surface as non-intersecting world lines in $1+1$ dimensions of an ensemble of fermionic particles.

**Figure 3.**Scaling function $f(u)$ of the covariant two-point correlator $\mathcal{C}(t,r)={t}^{-2{x}_{1}}f(r/t)$, over against the scaling variable $u=r/t$, for ortho-, meta-1- and Galilean-conformal invariance, Equations (12), (14) and (16) respectively, where ${x}_{1}={\gamma}_{1}=\frac{1}{2}$ and $\mu =1$.

**Figure 4.**Scaling function $f(u)$ of the covariant meta-2-conformal two-point response $\mathcal{R}(t,r)={t}^{-{x}_{1}-{x}_{2}}f(r/t)$, over against the scaling variable $u=r/t$, for $\xi ={\xi}_{1}=[0.6,1.0,1.4]$ in the left, middle and right panels, respectively, and several values of the amplitude ratio ρ.

**Table 1.**Exponents of growing interfaces in the Kardar-Parisi-Zhang (kpz), Edwards-Wilkinson (ew), Arcetri (for both $T={T}_{c}$ and $T<{T}_{c}$) and dle universality classes. The numbers in bracket give the estimated error in the last digit(s).

Model | d | z | β | a | b | ${\mathit{\lambda}}_{\mathit{C}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | References |
---|---|---|---|---|---|---|---|---|

kpz | 1 | $3/2$ | $1/3$ | $-1/3$ | $-2/3$ | 1 | 1 | [25,29,32] |

2 | $1.61(2)$ | $0.2415(15)$ | $0.30(1)$ | $-0.483(3)$ | $1.97(3)$ | $2.04(3)$ | [33,34] | |

2 | $1.61(2)$ | $0.241(1)$ | - | $-0.483$ | $1.91(6)$ | - | [35] | |

2 | $1.61(5)$ | $0.244(2)$ | - | - | - | - | [26] | |

2 | $1.627(4)$ | $0.229(6)$ | - | - | - | - | [36] | |

2 | $1.61(2)$ | $0.2415(15)$ | $0.24(2)$ | $-0.483(3)$ | $1.97(3)$ | $2.00(6)$ | [31,33] | |

ew | $<2$ | 2 | $(2-d)/4$ | $d/2-1$ | $d/2-1$ | d | d | |

2 | 2 | 0(log)${\phantom{\rule{3.33333pt}{0ex}}}^{\#}$ | 0 | 0 | 2 | 2 | [24,37] | |

$>2$ | 2 | 0 | $d/2-1$ | $d/2-1$ | d | d | ||

Arcetri $T={T}_{c}$ | $<2$ | 2 | $(2-d)/4$ | $d/2-1$ | $d/2-1$ | $3d/2-1$ | $3d/2-1$ | |

2 | 2 | 0(log)${\phantom{\rule{3.33333pt}{0ex}}}^{\#}$ | 0 | 0 | 2 | 2 | [20] | |

$>2$ | 2 | 0 | $d/2-1$ | $d/2-1$ | d | d | ||

$T<{T}_{c}$ | d | 2 | $1/2$ | $d/2-1$ | $-1$ | $d/2-1$ | $d/2-1$ | |

dle | $<1$ | 1 | $(1-d)/2$ | $d-1$ | $d-1$ | d | d | |

1 | 1 | 0(log)${\phantom{\rule{3.33333pt}{0ex}}}^{\#}$ | 0 | 0 | 1 | 1 | [4,21] | |

$>1$ | 1 | 0 | $d-1$ | $d-1$ | d | d |

**Table 2.**Comparison of local ortho-conformal, conformal Galilean and meta-1 conformal invariance, in $(1+1)D$. The non-vanishing Lie algebra commutators, the defining equation of the generators, the invariant differential operator $\mathcal{S}$ and the covariant two-point function is indicated, where applicable. Physically, the co-variant quasiprimary two-point function ${\mathcal{C}}_{12}=\langle {\phi}_{1}(t,r){\phi}_{2}(0,0)\rangle $ is a correlator, with the constraints ${x}_{1}={x}_{2}$ and ${\gamma}_{1}={\gamma}_{2}$.

Ortho | Galilean | Meta-1 | |
---|---|---|---|

Lie | $\left[{X}_{n},{X}_{m}\right]=(n-m){X}_{n+m}$ | $\left[{X}_{n},{X}_{m}\right]=(n-m){X}_{n+m}$ | $\left[{X}_{n},{X}_{m}\right]=(n-m){X}_{n+m}$ |

algebra | $\left[{X}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}=(n-m){Y}_{n+m}$ | $\left[{X}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}=(n-m){Y}_{n+m}$ | $\left[{X}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}=(n-m){Y}_{n+m}$ |

$\left[{Y}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}=(n-m){X}_{n+m}$ | $\left[{Y}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}=0$ | $\left[{Y}_{n},{Y}_{m}\right]\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}=\mu (n-m){Y}_{n+m}$ | |

generators | (9) | (15) | (13) |

$\mathcal{S}$ | ${\partial}_{t}^{2}+{\partial}_{r}^{2}$ | - | $-\mu {\partial}_{t}+{\partial}_{r}$ |

${\mathcal{C}}_{12}$ | ${t}^{-2{x}_{1}}{\left(1+{\left(\frac{r}{t}\right)}^{2}\right)}^{-{x}_{1}}$ | ${t}^{-2{x}_{1}}exp\left(-2\left|\frac{{\gamma}_{1}r}{t}\right|\right)$ | ${t}^{-2{x}_{1}}{\left(1+\frac{\mu}{{\gamma}_{1}}\left|\frac{{\gamma}_{1}r}{t}\right|\right)}^{-2{\gamma}_{1}/\mu}$ |

**Table 3.**Comparison of non-local meta-2 conformal invariance, and meta-conformal galilei invariance in $(1+1)D$. The non-vanishing Lie algebra commutators, the defining equation of the generators and the invariant differential operator $\mathcal{S}$ are indicated. The usual generators are ${X}_{n}={A}_{n}+{B}_{n}^{+}+{B}_{n}^{-}$, ${Y}_{n}={B}_{n}^{+}+{B}_{n}^{-}$ and ${Z}_{n}={B}_{n}^{+}-{B}_{n}^{-}$, see also Table 2. Physically, the co-variant quasiprimary two-point function ${\mathcal{R}}_{12}=\langle {\phi}_{1}(t,r){\tilde{\phi}}_{2}(0,0)\rangle $ is a response function. In case B, one has $\psi =2{\xi}_{1}-1$.

Meta-2 Conformal | Meta-Conformal Galilean | Constraints | |
---|---|---|---|

Lie | $\left[{A}_{n},{A}_{m}\right]\phantom{\rule{3.33333pt}{0ex}}=(n-m){A}_{n+m}$ | $\left[{X}_{n},{X}_{m}\right]=(n-m){X}_{n+m}$ | |

algebra | $\left[{B}_{n}^{\pm},{B}_{m}^{\pm}\right]\phantom{\rule{0.222222em}{0ex}}=(n-m){B}_{n+m}^{\pm}$ | $\left[{X}_{n},{B}_{m}^{\pm}\right]\phantom{\rule{0.222222em}{0ex}}=(n-m){B}_{n+m}^{\pm}$ | |

generators | (29) | (43) | |

$\mathcal{S}$ | $-\mu {\partial}_{t}+{\mathrm{\nabla}}_{r}$ | - | |

${\mathcal{R}}_{12}$ | ${t}^{1-{x}_{1}-{x}_{2}}\xb7\nu t{\left({\nu}^{2}{t}^{2}+{r}^{2}\right)}^{-1}$ | - | ${\xi}_{1}+{\xi}_{2}=1$ (A)${x}_{1}-{\xi}_{1}={x}_{2}-{\xi}_{2}$ |

${t}^{2{\xi}_{1}-2{x}_{1}}{\left({\nu}^{2}{t}^{2}+{r}^{2}\right)}^{-{\xi}_{1}}$ $\xb7sin\left[\pi {\xi}_{1}-2{\xi}_{1}arctan\left(\frac{r}{\nu t}\right)\right]$ | ${t}^{-2{x}_{1}}exp\left(-2\left|{\gamma}_{1}r/t\right|\right)$ | ${x}_{1}={x}_{2}$ (B)${\xi}_{1}={\xi}_{2}$, or ${\gamma}_{1}={\gamma}_{2}$ |

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Henkel, M.
Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain. *Symmetry* **2017**, *9*, 2.
https://doi.org/10.3390/sym9010002

**AMA Style**

Henkel M.
Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain. *Symmetry*. 2017; 9(1):2.
https://doi.org/10.3390/sym9010002

**Chicago/Turabian Style**

Henkel, Malte.
2017. "Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain" *Symmetry* 9, no. 1: 2.
https://doi.org/10.3390/sym9010002