Universality for Critical Lines in Ising, Vertex and Dimer Models

V. Mastropietro

2021, v.27, Issue 4, 687-706


In planar lattice statistical mechanics models like coupled
Ising with quartic interactions, vertex and dimer models, the exponents depen
on all the Hamiltonian details. This corresponds, in the Renormalization Grou
p language,
to a line of fixed points. A form of universality is expected to hold, implyi
ng that
all the exponents can be expressed by exact ``Kadanoff'' relations in terms o
f a single one of them.
This conjecture has been recently established and we review
here the key steps of the proof,
obtained by rigorous Renormalization Group methods and valid irrespectively o
n the
solvability of the model. The exponents are expressed by
convergent series in the coupling and, thanks to a set of cancellations due t
o emerging chiral symmetries, the extended scaling relations are proven to b
e true.
%Anomalies cancel out in the Standatd model at a perturbative level,,
%provided that the cut-off are removed. The Standard Model is presumably emer
ging from %some more fundamental theory, so it is natural to ask if the cance
llation is valid at a non %pertyrvatie level and
%when some symmetry is vioated,
but is this camcellation
%present at a non-perturbative level and with finite cut-off,
%The anomaly cancellation, which almost determines the values of the charges in the Standard Model,
%is a one loop condition and its validity in presence of radiative corrections follows at a perturbative level by the Alder-Bardeen %theorem. We consider an effective electroweak theory for massless fermions
%with quartic
%Fermi interactions. Non-perturbative effects are rigorously excluded by the absolute convergence of the perturbative expansion %and the domain of anaiticity has a radius
%proportional to the ratio of tbe cut-off and the gause mass.
%The presence of cut-offs breaks local phse symmetry and produce extra terms in the Ward Identities.Nevertheless, using peculiar identities for such extra terms and regularity properties
%we prove that anomaly cancellation condition still hold at the dominnant level, while
%corrections produed by cut-offs are subdominant andcan be rigorously bounded.

Keywords: Universlity, Renormalization Group, Critical exponents


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