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Discrete Holomorphicity at Two-Dimensional Critical Points · Around

2009

Discrete Holomorphicity at Two-Dimensional Critical Points

Cardy, John

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After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models.

These are objects whose correlation functions satisfy a discrete version of the Cauchy-Riemann relations.
Their existence appears to have a deep relation with the integrability of the model, and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory (CFT) describing the continuum limit.
This hypothesis sheds light on the connection between CFT and integrability, and, if verified, can also be used to prove that the scaling limit of certain discrete curves in these models is described by Schramm-Loewner evolution (SLE).

Built on

Fradkin, E., Kadanoff, L.P.: Disorder variables and para-fermions in two-dimensional statistical mechanics
1980
Earlier work this paper cites.
Fateev, V.A., Zamolodchikov, A.B.: Self-dual solutions of the star-triangle relations in Z
1982
Earlier work this paper cites.
Zamolodchikov, A.B, Fateev, V.A.:
1985
Earlier work this paper cites.
Nienhuis, B.: Critical and multicritical O
1990
Earlier work this paper cites.

Similar

Warnaar, S.O., Nienhuis, B.: Solvable lattice models labelled by Dynkin diagrams
1993
Cited alongside, same era.
For reviews for theoretical physicists, see Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains
2005
Cited alongside, same era.
Kenyon, R., Schlenker, J.M.:Rhombic embeddings of planar quad-graphs
2005
Cited alongside, same era.
Riva, V.,Cardy, J.: Holomorphic Parafermions in the Potts model and SLE
2006
Cited alongside, same era.

Then

Smirnov, S.: Towards conformal invariance of 2D lattice models
2006
Later among the works it cites.
Rajabpour, M.A., Cardy, J.: Discretely Holomorphic Parafermions in Lattice Z(N) Models
2007
Later among the works it cites.
Ikhlef, Y., Cardy, J.: Discretely Holomorphic Parafermions and Integrable Loop Models
2009
Closest in time.

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