Fixed point stability and decay of correlations

Abstract

In the framework of the renormalization-group theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective ${\Phi }^4$ theories, having an N-component fundamental field ${\Phi }_i$ and containing up to fourth-order powers of the field components. Their renormalization-group flow is usually characterized by several fixed points. We give here strong arguments in favour of the following conjecture: the stable fixed point corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent $\eta$ describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the $\epsilon$-expansion. Then, we discuss its validity beyond the $\epsilon$-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.

16 pages

References in corpus (29)

• Critical Phenomena and Renormalization-Group Theory
• Critical Exponents of the N-vector model
• Quantum Field Theory in the Large N Limit: a review
• Critical exponents and equation of state of the three-dimensional Heisenberg universality class
• Universality of phase transitions of frustrated antiferromagnets
• The critical exponents of the superfluid transition in He4
• 25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice
• Multicritical phenomena in O(n_1)+O(n_2)-symmetric theories
• Critical exponents for 3D O(n)-symmetric model with n > 3
• The N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: a six-loop study
• Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems
• Eliminating leading corrections to scaling in the 3-dimensional O(N)-symmetric phi^4 model: N=3 and 4
• The critical behavior of frustrated spin models with noncollinear order
• Phase transitions in anisotropic superconducting and magnetic systems with vector order parameters: Three-loop renormalization-group analysis
• Critical behavior of O(2)xO(N) symmetric models
• Large-n Critical Behavior of O(n)xO(m) Spin Models
• A proof of the irreversibility of renormalization group flows in four dimensions
• Chiral phase transitions: focus driven critical behavior in systems with planar and vector ordering
• Anomalies, Unitarity and Quantum Irreversibility
• On the Trace Anomaly as a Measure of Degrees of Freedom
• Comment on "Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory"
• Exact Consequences of the Trace Anomaly in Four Dimensions
• Critical thermodynamics of three-dimensional chiral model for N > 3
• The normal-to-planar superfluid transition in Helium 3
• Randomly dilute spin models with cubic symmetry
• Instability of the O(5) multicritical behavior in the SO(5) theory of high-Tc superconductors
• The critical equation of state of the three-dimensional O(N) universality class: N>4
• Spin-density-wave order in cuprates
• Critical behavior of the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy

Cited by in corpus (18)

• Coherent and dissipative dynamics at quantum phase transitions
• Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• General Properties of Multiscalar RG Flows in $d=4-\epsilon$
• Strain-induced time-reversal odd superconductivity in graphene
• Long-range multi-scalar models at three loops
• Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• Itinerant quantum multi-criticality of two dimensional Dirac fermions
• Multicritical behavior of two-dimensional anisotropic antiferromagnets in a magnetic field
• Universal behavior of coupled order parameters below three dimensions
• The tri-fundamental quartic model
• Critical behaviour of the O(n)-${\varphi }^4$ model with an antisymmetric order parameter
• Multipartite entanglement and quantum Fisher information in conformal field theories
• Three-dimensional Abelian and non-Abelian gauge Higgs theories
• Disordered O(n) Loop Model and Coupled Conformal Field Theories
• Stability of fixed points and generalized critical behavior in multifield models
• On operator mixing in fermionic CFTs in non-integer dimensions
• Global superfluid phase diagram of three component fermions with magnetic ordering
• Conjecture on the lower bound of the length-scale critical exponent $\nu$ at continuous phase transitions