[Paper Review] XY frustrated systems: continuous exponents in discontinuous phase transitions

Matthieu Tissier, Bertrand Delamotte+1 more|arXiv (Cornell University)|Jul 9, 2001

Theoretical and Computational PhysicsPhysics and Astronomy41 references31 citations

TL;DR

This paper resolves the paradox of XY frustrated magnets by showing they exhibit continuous scaling behavior with nonuniversal critical exponents and large correlation lengths despite undergoing weak first-order transitions. Using a nonperturbative functional renormalization group approach based on the effective average action, the authors demonstrate that slow renormalization group flow in coupling space—without fixed points or minima—generically produces scaling and pseudo-critical behavior, explaining experimental data from materials like CsNiCl3 and Ho without requiring universality.

ABSTRACT

XY frustrated magnets exhibit an unsual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first order phase transitions. We show that all perturbative approaches that have been used to investigate XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.

Motivation & Objective

• To resolve the longstanding controversy over the nature of phase transitions in three-dimensional XY frustrated magnets, particularly the coexistence of scaling laws with nonuniversal exponents and apparent first-order behavior.
• To explain why experimental and Monte Carlo data for systems like CsNiCl3 and Ho show scaling without evidence of universality, contradicting perturbative predictions of second-order transitions.
• To demonstrate that the absence of fixed points or minima in the nonperturbative RG flow does not preclude large correlation lengths and scaling behavior.
• To establish that the effective average action approach captures the physics of weak first-order transitions with pseudo-scaling, unlike perturbative methods.
• To show that the observed nonuniversal exponents and negative anomalous dimensions arise generically from slow RG flow in coupling space, not from true fixed points.

Proposed method

• Employing the nonperturbative functional renormalization group (FRG) via the effective average action (Wetterich equation), which allows for a systematic, nonperturbative treatment of the full coupling space.
• Using the Wilsonian exact renormalization group (ERG) framework to derive flow equations for the effective average action, including threshold functions that depend on the momentum scale and regulator.
• Analyzing the RG flow in the space of coupling constants, particularly focusing on the absence of fixed points or minima, yet observing slow flow over large domains.
• Applying the formalism to the O(N)×O(2)→O(N−2)×O(2)_{diag} symmetry breaking pattern characteristic of XY frustrated systems, with the order parameter represented as a matrix field Φ.
• Deriving and solving the flow equations for the effective action, including the threshold functions ld, md, nd that encode the momentum dependence and regulator dependence.
• Comparing results with perturbative calculations (e.g., six-loop expansions) and experimental data to validate the nonperturbative description.

Experimental results

Research questions

• RQ1 Why do XY frustrated magnets like CsNiCl3 and Ho exhibit scaling behavior with continuously varying critical exponents, contradicting the expectation of universality?
• RQ2 How can a system undergoing a weak first-order transition still display scaling laws and large correlation lengths?
• RQ3 Why do perturbative approaches predict a stable fixed point and second-order transition, while nonperturbative methods and experiments suggest a weak first-order transition?
• RQ4 What mechanism in the absence of fixed points or minima in the RG flow can still produce scaling and pseudo-critical behavior?
• RQ5 Can the effective average action method account for the observed nonuniversal exponents and negative anomalous dimensions in XY frustrated systems?

Key findings

• The nonperturbative functional renormalization group approach based on the effective average action successfully reproduces the nonuniversal critical exponents and negative anomalous dimension observed in XY frustrated magnets.
• Despite the absence of fixed points or minima in the RG flow, the flow remains slow over large domains in coupling constant space, leading to large correlation lengths and scaling behavior.
• The observed scaling behavior is not due to a true fixed point but arises generically from the slow evolution of the RG flow, explaining the pseudo-scaling and lack of universality.
• The method accounts for the experimental data from materials such as CsNiCl3 (β ≈ 0.24–0.25, ν ≈ 0.54) and Ho (β ≈ 0.39, ν ≈ 0.57) with good quantitative agreement.
• The results contrast sharply with perturbative approaches (e.g., Pelissetto et al.), which predict a stable fixed point and second-order transition, highlighting the failure of perturbation theory in this regime.
• The study suggests that similar behavior—scaling with continuously varying exponents—may occur generically in systems with a critical component number Nc separating second-order and first-order behavior.