Proof of the Triviality of <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span> Field Theory and Some Mean-Field Features of Ising Models for <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>></mo><mn></mn><mn>4</mn><mn></mn></math></span>

Michael Aizenman

doi:10.1103/PhysRevLett.47.1

Proof of the Triviality of ${ϕ_{d}}^{4}$ Field Theory and Some Mean-Field Features of Ising Models for $d > 4$

Michael Aizenman

Phys. Rev. Lett. 47, 1 – Published 6 July 1981; Erratum Phys. Rev. Lett. 47, 886 (1981)

Abstract

It is rigorously proved that the continuum limits of Euclidean $ϕ_{d}^{4}$ lattice fields are free fields in $d > 4$ . An exact geometric characterization of criticality in Ising models is introduced, and used to prove other mean-field features for $d > 4$ and hyperscaling in $d = 2$ .