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Physical Review Letters

One Fixed Point Can Hide Another One: Nonperturbative Behavior of the Tetracritical Fixed Point of O ( N ) Models at Large N

Shunsuke Yabunaka, Bertrand Delamotte

Published Jun 30, 2023• 2 citations
Publisher

Abstract

We show that at $N=\ensuremath{\infty}$ and below its upper critical dimension, $d<{d}_{\mathrm{up}}$, the critical and tetracritical behaviors of the $\mathrm{O}(N)$ 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 $1/N$---expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at $N=\ensuremath{\infty}$ and $d={d}_{\mathrm{up}}$ can be understood from a finite-$N$ analysis.

DOI
10.1103/physrevlett.130.261602

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We show that at N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞ 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→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ---and the 1/N1/N1/N---expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞ and d=dupd={d}_{\mathrm{up}}d=dup​ can be understood from a finite-NNN analysis.
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 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.
Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O(NNN) model in d=3d=3d=3 and N=∞N=\inftyN=∞
Renormalization group (RG) fixed points are commonly associated with scale invariance and a divergent correlation length. We show that this connection can fail in the tricritical O(N)O(N)O(N) model in three dimensions in the limit N→∞N\to\inftyN→∞. Revisiting the line of fixed points identified by Bardeen, Moshe, and Bander, we use the functional renormalization group to clarify the mechanism leading to mass generation at its singular endpoint (the BMB fixed point). We demonstrate that the generated mass is nonuniversal and originates from the nonanalytic structure of the effective potential. We show that the critical exponent ννν which takes the value ν=1/2ν= 1/2ν=1/2 along the regular part of the BMB line, that is, for 0≤λ<λBMB0 \leq λ< λ_{\rm BMB}0≤λ<λBMB​, jumps to ν=1/3ν= 1/3ν=1/3 on the singular part of this line with the BMB FP, corresponding to λ=λBMBλ= λ_{\rm BMB}λ=λBMB​, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.
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We employ the functional renormalization group framework at the second order in the derivative expansion to study the O(N) O ( N ) models continuously varying the number of field components N N and the spatial dimensionality d d . We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents \nu ν and \eta η across a line in the (d,N) ( d , N ) plane, which passes through the point (2,2) ( 2 , 2 ) . By direct numerical evaluation of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the (d,N) ( d , N ) plane, however no evidence of discontinuous or singular first and second derivatives of these functions for d>2 d > 2 . The computed derivatives of \eta(d,N) η ( d , N ) and \nu^{-1}(d,N) ν − 1 ( d , N ) become increasingly large for d\to 2
Recently developed renormalization-group techniques are summarized and exploited to yield a renormalization-group analysis of bicritical and tetracritical points (which arise in antiferromagnets and boson systems). For $n\ensuremath{\lesssim}3$ an isotropic or Heisenberg fixed point dominates and gives bicritical behavior; but for $n\ensuremath{\gtrsim}4$ a new "biconical" fixed point with irrational $\ensuremath{\epsilon}$-expansion coefficients appears. This describes a tetracritical point and may be relevant to displacive phase transitions.
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.