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arXiv:2301.01021

A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O($N$) models at large $N$

Shunsuke Yabunaka, Bertrand Delamotte

Published Jan 3, 2023• 0 citations
arXivHTMLPDF

Abstract

We show that at $N=\infty$ and below its upper critical dimension, $d

DOI
10.48550/arxiv.2301.01021

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A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O(NNN) models at large NNN
We show that at N=∞N=\inftyN=∞ and below its upper critical dimension, $d
We show that at $N=\ensuremath{\infty}$ and below its upper critical dimension, d<dupd<{d}_{\mathrm{up}}d<dup​, the critical and tetracritical behaviors of the O(N)\mathrm{O}(N)O(N) models are associated with the same renormalization group fixed point (FP) potential. Only their derivatives make them different with the subtleties that taking their $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the $\ensuremath{\epsilon}$---and the 1/N1/N1/N---expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at $N=\ensuremath{\infty}$ and d=dupd={d}_{\mathrm{up}}d=dup​ can be understood from a finite-NNN analysis.
We employ the nonperturbative functional renormalization group to study models with an O(N(1) ⊕O(N)(2)) symmetry. Here different fixed points exist in three dimensions, corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters. We discuss the critical behavior of the symmetry-enhanced isotropic, the decoupled and the biconical fixed point, and analyze their stability in the N(1),N(2) plane. We study the fate of nontrivial fixed points during the transition from three to four dimensions, finding evidence for a triviality problem for coupled two-scalar models in high-energy physics. We also point out the possibility of noncanonical critical exponents at semi-Gaussian fixed points and show the emergence of Goldstone modes from discrete symmetries.
We explore universal critical behavior in models with two competing order parameters, and an O(N) ⊕O(M) symmetry for dimensions d≤3. In d=3, there is always exactly one stable renormalization group fixed point, corresponding to bicritical or tetracritical behavior. Employing pseudospectral techniques to solve functional renormalization group equations in a two-dimensional field space, we uncover a more intricate structure of fixed points in d<3, where two additional bicritical fixed points play a role. Towards d=2, we discover ranges of N=M with several simultaneously stable fixed points, indicating the coexistence of several universality classes.