Universality for Critical Lines in Ising, Vertex and Dimer Models

V. Mastropietro

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

ABSTRACT

In planar lattice statistical mechanics models like coupled
Ising with quartic interactions, vertex and dimer models, the exponents depen
d
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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