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 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 show that at $N=\ensuremath{\infty}$ and below its upper critical dimension, d<dup, the critical and tetracritical behaviors of the 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=dup can be understood from a finite-N analysis.
Mass generation at a fixed point: A Functional Renormalization Group Study of the tricritical O(N) model in d=3 and N=∞
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) model in three dimensions in the limit N→∞. 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 ν which takes the value ν=1/2 along the regular part of the BMB line, that is, for 0≤λ<λBMB, jumps to ν=1/3 on the singular part of this line with the BMB FP, corresponding to λ=λBMB, being the pivotal point between these two regimes. We also show how its singular potential emerges dynamically along the renormalization flow.
Nonequilibrium dynamics play an important role in all contexts of physics, both classical and quantum as well as living and nonliving, so it is crucial to develop a foundational understanding of nonequilibrium phase transitions. In this work we investigate an important class of nonequilibrium dynamics in the form of nonreciprocal interactions. In particular we study how nonreciprocal coupling between two O(n_{i}) order parameters (with i=1,2) affects the universality at a multicritical point, extending the analysis of J. T. Young et al. [Phys. Rev. X 10, 011039 (2020)2160-330810.1103/PhysRevX.10.011039], which considered the case n_{1}=n_{2}=1, i.e., a Z_{2}×Z_{2} model. We show that nonequilibrium fixed points (NEFPs) emerge for a broad range of n_{1},n_{2} and exhibit intrinsically nonequilibrium critical phenomena, namely a violation of fluctuation-dissipation relations at all scales and underdamped oscillations near criticality in contrast to the overdamped relaxational dynamics of the corresponding equilibrium models. Furthermore, the NEFPs exhibit an emergent discrete scale invariance in certain physically relevant regimes of n_{1},n_{2}, but not others, depending on whether the critical exponent ν is real or complex. The boundary between these two regions is described by an exceptional point in the renormalization group (RG) flow, leading to distinctive features in correlation functions and the phase diagram. Another contrast with the previous work is the number and stability of the NEFPs as well as the underlying topology of the RG flow. Finally, we investigate an extreme form of nonreciprocity where one order parameter is independent of the other order parameter but not vice versa. Unlike the Z_{2}×Z_{2} model, which becomes nonperturbative in this case, we identify a distinct nonequilibrium universality class whose dependent field similarly violates fluctuation-dissipation relations but does not exhibit discrete scale invariance or underdamped oscillations near criticality.
The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed.Three models are considered: the O(N ) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice.We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents.We first show that, for the O(N ) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs).Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied.The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters.On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters.For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters.It appears to be spurious, as expected.As for the frustrated models, there are two cases depending on the value of the number of spin components.When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP.On the contrary, for N < Nc, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N ) and cubic models.We conclude from this analysis that the stable FP found for N < Nc in frustrated models is spurious.Since Nc > 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.
Analyticity of critical exponents of the O(N) models from nonperturbative renormalization
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