Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization

Abstract

We employ the functional renormalization group framework at the second order in the derivative expansion to study the $O(N)$ models continuously varying the number of field components $N$ and the spatial dimensionality $d$. We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents $\nu$ and $\eta$ across a line in the $(d,N)$ plane, which passes through the point $(2,2)$. By direct numerical evaluation of $\eta (d,N)$ and ${\nu }^{-1}(d,N)$ 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 $(d,N)$ plane, however no evidence of discontinuous or singular first and second derivatives of these functions for $d>2$. The computed derivatives of $\eta (d,N)$ and ${\nu }^{-1}(d,N)$ become increasingly large for $d\to 2$ and $N\to 2$ and it is only in this limit that $\eta (d,N)$ and ${\nu }^{-1}(d,N)$ as obtained by us are evidently nonanalytical. By scanning the dependence of the subleading eigenvalue of the RG transformation on $N$ for $d>2$ we find no indication of its vanishing as anticipated by the Cardy-Hamber scenario. For dimensionality $d$ approaching 3 there are no signatures of the Cardy-Hamber line even as a crossover and its existence in the form of a nonanalyticity of the anticipated form is excluded.

arXiv commentSubmission to SciPost, 24 pages, 12 figures; Reviewed version: strengthened conclusions, added sec. 6, subsec. 5.3 and Appendix, improved plots' clarity

References in corpus (32)

• Exact evolution equation for the effective potential
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• Aspects of the Functional Renormalisation Group
• Functional renormalization group approach to correlated fermion systems
• The nonperturbative functional renormalization group and its applications
• Quantum Field Theory in the Large N Limit: a review
• Quantum simulation of an extra dimension
• Emergent Photons and New Transitions in the O(3) Sigma Model with Hedgehog Suppression
• Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order ${\partial }^4$
• Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models
• Nonperturbative Renormalization Flow and Essential Scaling for the Kosterlitz-Thouless Transition
• Convergence of Non-Perturbative Approximations to the Renormalization Group
• Precision calculation of critical exponents in the $O(N)$ universality classes with the nonperturbative renormalization group
• Kosterlitz-Thouless Phase Transition In The Two Dimensional Linear Sigma Model
• A unified picture of ferromagnetism, quasi-long range order and criticality in random field models
• O(N)-Universality Classes and the Mermin-Wagner Theorem
• Critical exponents of O(N) models in fractional dimensions
• Quantum simulation of non-trivial topology
• Functional renormalization group approach to the sine-Gordon model
• Higgs amplitude mode in the vicinity of a $(2+1)$-dimensional quantum critical point
• Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• Surprises in the $O(N)$ models: nonperturbative fixed points, large $N$ limit and multi-criticality
• Critical Phenomena in Continuous Dimension
• Nonperturbative renormalization group treatment of amplitude fluctuations for $|\phi {|}^4$ topological phase transitions
• Conformal invariance in the nonperturbative renormalization group: a rationale for choosing the regulator
• Longitudinal fluctuations in the Berezinskii-Kosterlitz-Thouless phase
• Interplay of spin waves and vortices in the 2D XY model at small vortex-core energy
• Dual lattice functional renormalization group for the Berezinskii-Kosterlitz-Thouless transition: irrelevance of amplitude and out-of-plane fluctuations
• Thermodynamics of the two-dimensional XY model from functional renormalization
• Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices
• Kosterlitz-Thouless signatures in the low-temperature phase of layered three-dimensional systems

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