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arXiv:2309.15307 · Physical review. D/Physical review. D.

Functional renormalization group approach to dipolar fixed point which is scale invariant but nonconformal

Yu Nakayama

Published Jul 23, 2024• 4 citations
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Abstract

A dipolar fixed point introduced by Aharony and Fisher is a physical example of interacting scale-invariant but nonconformal field theories. We find that the perturbative critical exponents computed in ε expansions violate the conformal bootstrap bound. We formulate the functional renormalization group equations Wetterich and Polchinski to study the fixed point. We present some results in three dimensions within (uncontrolled) local potential approximations (with or without perturbative anomalous dimensions). Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.025020

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A dipolar fixed point introduced by Aharony and Fisher is a physical example of interacting scale-invariant but nonconformal field theories. We find that the perturbative critical exponents computed in ε expansions violate the conformal bootstrap bound. We formulate the functional renormalization group equations Wetterich and Polchinski to study the fixed point. We present some results in three dimensions within (uncontrolled) local potential approximations (with or without perturbative anomalous dimensions). Published by the American Physical Society 2024
Aharony and Fisher showed that non-local dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point, called the dipolar fixed point. The non-perturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale-invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 40340^3403 by introducing the local cost function parameterized by a parameter λλλ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit λ=∞λ= \inftyλ=∞. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant λ=8λ=8λ=8 (e.g. ν=0.601(2)(−2+0)ν=0.601(2)(^{+0}_{-2})ν=0.601(2)(−2+0​) in the local Heisenberg-dipolar model while ν=0.712(1)(−0+3)ν=0.712(1)(^{+3}_{-0})ν=0.712(1)(−0+3​) in the Heisenberg model), and the spin correlation function has a feature that it becomes divergence-free, implying the lack of conformal invariance.
We compute the critical exponents of three-dimensional magnets with strong dipole-dipole interactions using the functional renormalization group (FRG) within the local potential approximation including the wave function renormalization (LPA′). The system is governed by the Aharony fixed point, which is scale-invariant but lacks conformal invariance. Our nonperturbative FRG analysis identifies this fixed point and determines its scaling behavior. The resulting critical exponents are found to be close to those of the Heisenberg O (3) universality class, as computed within the same FRG/LPA′ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.
We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.
Aharony and Fisher showed that nonlocal dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point called the dipolar fixed point. The nonperturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 403{40}^{3}403 by introducing the local cost function parameterized by a parameter $\ensuremath{\lambda}$ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit $\ensuremath{\lambda}=\ensuremath{\infty}$. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant $\ensuremath{\lambda}=8$ [e.g., $\ensuremath{\nu}=0.601(2){(}_{\ensuremath{-}2}^{+0}){{(}_{\ensuremath{-}4}^{+5})}_{{\ensuremath{\beta}}_{c}}$ in the local Heisenberg-dipolar model while $\ensuremath{\nu}=0.712(1){(}_{\ensuremath{-}0}^{+3}){{(}_{\ensuremath{-}1}^{+1})}_{{\ensuremath{\beta}}_{c}}$ in the Heisenberg model], and the spin correlation function has a feature that it becomes divergence free, implying the lack of conformal invariance.
Equations related to the Polchinski version of the exact renormalization group (RG) equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterization invariance, are postulated. Reparameterization invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O (ε 2 ) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O ( N ) symmetry, for various N and for ranges of η, are obtained within the local potential approximation. The associated η, which follows from the derivative approximation described here, is found for various N . The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.