{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T10:43:51Z","timestamp":1768733031752,"version":"3.49.0"},"reference-count":89,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,11,13]],"date-time":"2015-11-13T00:00:00Z","timestamp":1447372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant n-point functions. These are important for the physical identification of n-point functions as responses or correlators.<\/jats:p>","DOI":"10.3390\/sym7042108","type":"journal-article","created":{"date-parts":[[2015,11,16]],"date-time":"2015-11-16T05:40:47Z","timestamp":1447652447000},"page":"2108-2133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions"],"prefix":"10.3390","volume":"7","author":[{"given":"Malte","family":"Henkel","sequence":"first","affiliation":[{"name":"Groupe de Physique Statistique, Institut Jean Lamour (CNRS UMR 7198), Universit\u00e9 de Lorraine Nancy, B.P. 70239, F-54506 Vandoeuvre-l\u00e8s-Nancy Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Henkel, M., and Pleimling, M. (2010). Non-Equilibrium Phase Transitions Volume 2: Ageing and Dynamical Scaling Far from Equilibrium, Springer.","DOI":"10.1007\/978-90-481-2869-3"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"T\u00e4uber, U.C. (2014). Critical Dynamics: A Field Theory Apporach to Equilibrium and Non-Equilibrium Scaling Behaviour, Cambridge University Press.","DOI":"10.1017\/CBO9781139046213"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"93","DOI":"10.24033\/asens.603","article-title":"Les groupes de transformation continus, infinis, simples","volume":"26","author":"Cartan","year":"1909","journal-title":"Ann. Sci. Ecole Norm. S."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0550-3213(84)90052-X","article-title":"Infinite conformal symmetry in two-dimensional quantum field-theory","volume":"241","author":"Belavin","year":"1984","journal-title":"Nucl. Phys. B"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Di Francesco, P., Mathieu, P., and S\u00e9n\u00e9chal, D. (1997). Conformal Field-Theory, Springer.","DOI":"10.1007\/978-1-4612-2256-9"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Unterberger, J., and Roger, C. (2011). The Schr\u00f6dinger\u2013Virasoro Algebra, Springer.","DOI":"10.1007\/978-3-642-22717-2"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1007\/BF02186756","article-title":"Schr\u00f6dinger-invariance and strongly anisotropic critical systems","volume":"75","author":"Henkel","year":"1994","journal-title":"J. Stat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/S0550-3213(02)00540-0","article-title":"Phenomenology of local scale invariance: From conformal invariance to dynamical scaling","volume":"641","author":"Henkel","year":"2002","journal-title":"Nucl. Phys. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1007\/s00023-006-0289-1","article-title":"The Schr\u00f6dinger\u2013Virasoro Lie group and algebra: From geometry to representation theory","volume":"7","author":"Roger","year":"2006","journal-title":"Ann. Henri Poincare"},{"key":"ref_10","unstructured":"To see this explicitly, one should exponentiate these generators to create their corresponding finite transformations, see [11]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/S0550-3213(03)00252-9","article-title":"Schr\u00f6dinger invariance and space-time symmetries","volume":"660","author":"Henkel","year":"2003","journal-title":"Nucl. Phys. B"},{"key":"ref_12","first-page":"328","article-title":"\u00dcber die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen","volume":"6","author":"Lie","year":"1881","journal-title":"Arch. Math. 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Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Martelli, D., and Tachikawa, Y. (2010). Comments on Galiean conformal field-theories and their geometric realisation. J. High Energy Phys., 1005:091.","DOI":"10.1007\/JHEP05(2010)091"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3786","DOI":"10.1063\/1.532067","article-title":"Nonrelativistic conformal groups","volume":"38","author":"Negro","year":"1997","journal-title":"J. Math. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3810","DOI":"10.1063\/1.532068","article-title":"Nonrelativistic conformal groups II","volume":"38","author":"Negro","year":"1997","journal-title":"J. Math. Phys."},{"key":"ref_27","unstructured":"In the context of asymptotically flat 3D gravity, an isomorphic Lie algebra is known as BMS algebra, \n        \n          \n\t\t  \n            bms\n\t\t\t3\n           \n\t\t   \u2245\n\t\t   \n              CGA\n            \n            (\n            1\n            )\n          \n        \n       [28,29,30,31,32]."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bagchi, A., Detournay, S., and Grumiller, D. (2012). Flat-space chiral gravity. Phys. Rev. Lett., 109.","DOI":"10.1103\/PhysRevLett.109.151301"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"141302:1","DOI":"10.1103\/PhysRevLett.110.141302","article-title":"Holographies of 3D flat cosmological horizons","volume":"110","author":"Bagchi","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Barnich, G., and Comp\u00e8re, G. (2007). Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions. Class. Quantum Grav., 24.","DOI":"10.1088\/0264-9381\/24\/5\/F01"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3139","DOI":"10.1088\/0264-9381\/24\/11\/C01","article-title":"Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions (corrigendum)","volume":"24","author":"Barnich","year":"2007","journal-title":"Class. Quant. Grav."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Barnich, G., Gomberoff, A., and Gonz\u00e1lez, H.A. (2007). Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field-theories as the flat limit of Liouville theory. Phys. Rev. D, 87.","DOI":"10.1103\/PhysRevD.87.124032"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1250006:1","DOI":"10.1142\/S1793744212500065","article-title":"The Poincar\u00e9 algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states","volume":"4","author":"Henkel","year":"2012","journal-title":"Conflu. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physleta.2006.04.016","article-title":"Exotic galilean conformal symmetry and its dynamical realisations","volume":"357","author":"Lukierski","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.physletb.2007.04.058","article-title":"Acceleration-extended galilean symmetries with central charges and their dynamical realizations","volume":"650","author":"Lukierski","year":"2007","journal-title":"Phys. Lett. B"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.nuclphysb.2013.12.009","article-title":"Logarithmic exotic conformal galilean algebras","volume":"879","author":"Henkel","year":"2014","journal-title":"Nucl. Phys. B"},{"key":"ref_37","unstructured":"An infinite-dimensional extension of ecga does not appear to be possible."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"L589","DOI":"10.1088\/0305-4470\/39\/42\/L01","article-title":"On the identification of quasiprimary operators in local scale-invariance","volume":"39","author":"Henkel","year":"2006","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Stoimenov, S., and Henkel, M. (2013). Non-local representations of the ageing algebra in higher dimensions. J. Phys. A Math. Theor., 46.","DOI":"10.1088\/1751-8113\/46\/24\/245004"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Minic, D., Vaman, D., and Wu, C. (2012). Three-point function of aging dynamics and the AdS-CFT correspondence. Phys. Rev. Lett., 109.","DOI":"10.1103\/PhysRevLett.109.131601"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Knapp, A.W. (1986). Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press.","DOI":"10.1515\/9781400883974"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Duval, C., and Horv\u00e1thy, P.A. (2009). Non-relativistic conformal symmetries and Newton\u2013Cartan structures. J. Phys. A Math. Theor., 42.","DOI":"10.1088\/1751-8113\/42\/46\/465206"},{"key":"ref_43","unstructured":"Although it might appear that z = 2, the renormalisation of the interactions, required in interacting field-theories, can change this and produce non-trivial values of z, see e.g., [2]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.nuclphysb.2004.03.028","article-title":"Local scale-invariance and ageing in noisy systems","volume":"688","author":"Picone","year":"2004","journal-title":"Nucl. Phys. B"},{"key":"ref_45","first-page":"381","article-title":"Conformal symmetry of critical fluctuations","volume":"12","author":"Polyakov","year":"1970","journal-title":"Sov. Phys. JETP Lett."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Barab\u00e1si, A.-L., and Stanley, H.E. (1995). Fractal Concepts in Surface Growth, Cambridge University Press.","DOI":"10.1017\/CBO9780511599798"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Henkel, M., and Durang, X. (2015). Spherical model of interface growth. J. Stat. Mech.","DOI":"10.1088\/1742-5468\/2015\/05\/P05022"},{"key":"ref_48","unstructured":"For d = 1, the dynamics of the Arcetri model is identical [47] to the one of the spherical Sherrington\u2013Kirkpatrick model. The model is defined by the classical hamiltonian \n        \n          \n            H\n            =\n\t\t\t\u2212\n\t\t\t\n            1\n\t\t\t2\n\t\t\t\n\t\t\t\n              \u2211\n\t\t\t  \n\t\t\t  i\n\t\t\t  ,\n\t\t\t  j\n\t\t\t  =\n\t\t\t  1\n\t\t\t  \n              N\n            \n\t\t\t\n              J\n\t\t\t  \n\t\t\t  i\n\t\t\t  ,\n\t\t\t  j\n\t\t\t  \n            \n\t\t\t\n              s\n\t\t\t  i\n            \n\t\t\t\n              s\n\t\t\t  j\n            \n          \n        \n      , where the \n        \n          \n            \n              J\n\t\t\t  \n\t\t\t  i\n\t\t\t  ,\n\t\t\t  j\n\t\t\t  \n            \n          \n        \n       are independent centred gaussian variables, of variance \u223cO(1\/\ud835\udca9), and the si satisfy the spherical constraint \n        \n          \n            \n              \u2211\n\t\t\t  \n\t\t\t  i\n\t\t\t  =\n\t\t\t  1\n\t\t\t  \n              N\n            \n\t\t\t\n              s\n\t\t\t  i\n\t\t\t  2\n            \n\t\t\t=\n\t\t\tN\n          \n        \n      . As usual, the dynamics if given by a Langevin equation [49]. This problem is also equivalent to the statistics of the gap to the largest eigenvalue of a \ud835\udca9 \u00d7 \ud835\udca9 gaussian unitary matrix [50,51], for \ud835\udca9 \u2192 \u221e. Work is in progress on identifying interface growth models with \n        \n          \n            \u039e\n            (\n            t\n            )\n            \u2260\n            0\n          \n        \n      ."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Cugliandolo, L.F., and Dean, D. (1995). Full dynamical solution for a spherical spin-glass model. J. Phys. A Math. Gen., 28.","DOI":"10.1088\/0305-4470\/28\/15\/003"},{"key":"ref_50","unstructured":"Fyodorov, Y.V., Perret, A., and Schehr, G. Large-time zero-temperature dynamics of the spherical p = 2 spin model of finite size. Available online: http:\/\/arxiv.org\/pdf\/1507.08520.pdf."},{"key":"ref_51","unstructured":"Perret, A. (2015). Statistique D\u2019extr\u00eames de Variables Al\u00e9atoires Fortement Corr\u00e9\u00e9es. [Ph.D. Thesis, Universit\u00e9 Paris Sud]."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1063\/1.1703954","article-title":"Time-dependent statistics of the Ising model","volume":"4","author":"Glauber","year":"1963","journal-title":"J. Math. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Godr\u00e8che, C., and Luck, J.-M. (2000). Response of non-equilibrium systems at criticality: Exact results for the Glauber-Ising chain. J. Phys. A Math. Gen., 33.","DOI":"10.1088\/0305-4470\/33\/6\/305"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1088\/0305-4470\/37\/3\/004","article-title":"On the universality of the fluctuation-dissipation ratio in non-equilibrium critical dynamics","volume":"37","author":"Henkel","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3369","DOI":"10.1103\/PhysRevE.61.3369","article-title":"Fluctuation-dissipation ratio in the one-dimensional kinetic Ising model","volume":"61","author":"Lippiello","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_56","unstructured":"A historical comment: We have been aware of this since the very beginning of our investigations, in the early 1990s. The exact result Equation (33) looked strange, since the time-space response of the Glauber\u2013Ising model does have the nice form \n        \n          \n                  R\n\t\t\t\t  \n                  (\n                  t\n                  ,\n                  s\n                  ;\n                  r\n                  )\n                \n\t\t\t\t=\n\t\t\t\tR\n\t\t\t\t  \n                  (\n                  t\n                  ,\n                  s\n                  )\n                \n\t\t\t\texp\n\t\t\t\t\n                  [\n\t\t\t\t  \u2212\n\t\t\t\t  \n\t\t\t\t  1\n\t\t\t\t  2\n\t\t\t\t  \n\t\t\t\t  M\n\t\t\t\t  \n              r\n\t\t\t  2\n            \n                        \/\n                        \n                  (\n                  t\n                  \u2212\n                  s\n                  )\n                \n\t\t\t\t  ]\n\t\t\t\t  \n          \n        \n      , as expected from Galilei-invariance. Only several years later, we saw how the representations of the Schr\u00f6dinger algebra had to be generalised, which was only possible by giving up explicitly time-translation-invariance [38,44]."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.nuclphysb.2012.12.007","article-title":"On logarithmic extensions of local scale-invariance","volume":"869","author":"Henkel","year":"2013","journal-title":"Nucl. Phys. B"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Henkel, M., Noh, J.D., and Pleimling, M. (2012). Phenomenology of ageing in the Kardar\u2013Parisi\u2013Zhang equation. Phys. Rev. E, 85.","DOI":"10.1103\/PhysRevE.85.030102"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Henkel, M., and Rouhani, S. (2013). Logarithmic correlators or responses in non-relativistic analogues of conformal invariance. J. Phys. A Math. Theor., 46.","DOI":"10.1088\/1751-8113\/46\/49\/494004"},{"key":"ref_60","unstructured":"The specific structure of the dynamical functional \n        \n          \n\t\t  J\n\t\t  \n                    [\n                    \u03d5\n                    ,\n                    \n                      \u03d5\n                      \u02dc\n                    \n                    ]\n\t\t\t\t\t\n\t\t\t\t\t\n        \n      , see Equation (22), of the Arcetri model (and, more generally, of the kinetic spherical model [44]) leads to \n        \n          \n\t\t  \u03be\n                    +\n                    \n                      \u03be\n                      \u02dc\n                    \n\t\t\t\t\t=\n\t\t\t\t\t0\n\t\t\t\t\t\n        \n      , such that time-translation-invariance appears to be formally satisfied, in contrast to the 1D Glauber\u2013Ising model, where \n        \n          \n\t\t  \u03be\n                    +\n                    \n                      \u03be\n                      \u02dc\n                    \n\t\t\t\t\t=\n\t\t\t\t\t\n\t\t\t\t\t1\n\t\t\t\t\t4\n\t\t\t\t\t\n\t\t\t\t\t\n        \n      ."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"3487","DOI":"10.1142\/S0217979294001470","article-title":"Schr\u00f6dinger invariance in discrete stochastic systems","volume":"8","author":"Henkel","year":"1994","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_62","unstructured":"The scaling from Equation (39) is indeed recovered in several simple lattice models, see [61] for more details."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1016\/j.nuclphysb.2011.02.008","article-title":"On non-local representations of the ageing algebra","volume":"847","author":"Stoimenov","year":"2011","journal-title":"Nucl. Phys. B"},{"key":"ref_64","unstructured":"See [39] for an application to the kinetics of the phase-separating (model-B dynamics) spherical model."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0550-3213(93)90528-W","article-title":"Logarithmic operators in conformal field theory","volume":"410","author":"Gurarie","year":"1993","journal-title":"Nucl. Phys. 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