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arXiv:1707.04383 · Physical Review Letters

Surprises in O ( N ) Models: Nonperturbative Fixed Points, Large N Limits, and Multicriticality

Shunsuke Yabunaka, Bertrand Delamotte

Published Nov 7, 2017• 51 citations
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Abstract

We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=∞. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N)⊗O(2) models relevant to frustrated magnetic systems.

DOI
10.1103/physrevlett.119.191602

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We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=∞. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N)⊗O(2) models relevant to frustrated magnetic systems.
We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.
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 study models with three coupled vector fields characterized by O(N_{1})⊕O(N_{2})⊕O(N_{3}) symmetry. Using the nonperturbative functional renormalization group, we derive β functions for the couplings and anomalous dimensions in d dimensions. Specializing to the case of three dimensions, we explore interacting fixed points that generalize the O(N) Wilson-Fisher fixed point. We find a symmetry-enhanced isotropic fixed point, a large class of fixed points with partial symmetry enhancement, as well as partially and fully decoupled fixed-point solutions. We discuss their stability properties for all values of N_{1},N_{2}, and N_{3}, emphasizing important differences to the related two-field models. For small numbers of field components, we find no stable fixed-point solutions, and we argue that this can be attributed to the presence of a large class of possible (mixed) couplings in the three-field and multifield models. Furthermore, we contrast different mechanisms for stability interchange between fixed points in the case of the two- and three-field models, which generically proceed through fixed-point collisions.
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 compute critical exponents of O(N)O(N)O(N) models in fractional dimensions between d=2d=2d=2 and 4, and for continuous values of the number of field components NNN, in this way completing the RG classification of universality classes for these models. These curves represent nonperturbative approximation to the exact results, they respect all the qualitative features expected from such quantities conciliating previously known perturbative results in three dimensions with exact results in two dimensions and giving a strong indication of what could be the exact behavior of such curves. We also report critical exponents for some multicritical universality classes in the cases $N\ensuremath{\ge}2$ and N=0N=0N=0. Finally, in the large-NNN limit our critical exponents correctly approach those of the spherical model, allowing us to set $N\ensuremath{\sim}100$ as the threshold for the quantitative validity of leading order large-NNN estimates.
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