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arXiv:cond-mat/0504008 · Physical Review Letters

Comment on “Hausdorff Dimension of Critical Fluctuations in Abelian Gauge Theories”

Nikolay V. Prokof'ev, Boris V. Svistunov

Published Jun 1, 2006• 10 citations
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Abstract

Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection, $η+ D_H = 2$, between the anomalous scaling dimension $η$ of the U(1) universality class order parameter and the Hausdorff dimension $D_H$ of critical loops in loop representations of U(1) models. We show that the above relation is wrong and establish a correct relation that contains a new critical exponent.

DOI
10.1103/physrevlett.96.219701

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Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection, η+DH=2η+ D_H = 2η+DH​=2, between the anomalous scaling dimension ηηη of the U(1) universality class order parameter and the Hausdorff dimension DHD_HDH​ of critical loops in loop representations of U(1) models. We show that the above relation is wrong and establish a correct relation that contains a new critical exponent.
In a recent paper [Phys. Rev. E 92, 012123 (2015)], a modified ddd-dimensional ${\mathrm{\ensuremath{\Phi}}}^{4}$ model was investigated that differs from the standard one in that the ${\mathrm{\ensuremath{\Phi}}}^{4}$ term was replaced by a nonlocal one with a potential $u(\mathbit{x}\ensuremath{-}{\mathbit{x}}^{\ensuremath{'}})$ that depends on a parameter $\ensuremath{\sigma}$ and decays exponentially as $|\mathbit{x}\ensuremath{-}{\mathbit{x}}^{\ensuremath{'}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ on a scale ${|m|}^{\ensuremath{-}1}<\ensuremath{\infty}$. The authors claim the upper critical dimension of this model to be ${d}_{\ensuremath{\sigma}}=4+2\ensuremath{\sigma}$. Performing a one-loop calculation, they arrive at expansions in powers of ${\ensuremath{\epsilon}}_{\ensuremath{\sigma}}={d}_{\ensuremath{\sigma}}\ensuremath{-}d$ for critical exponents such as $\ensuremath{\eta}$ and related ones to $O({\ensuremath{\epsilon}}_{\ensuremath{\sigma}})$ whose $O({\ensuremath{\epsilon}}_{\ensuremath{\sigma}})$ coefficients depend on $\ensuremath{\sigma}$ and the ratio $w={m}^{2}/{\mathrm{\ensuremath{\Lambda}}}^{2}$, where $\mathrm{\ensuremath{\Lambda}}$ is the uv cutoff. It is shown that these claims are unfounded and based on misjudgments and an ill-conceived renormalization-group calculation.
In a recent paper, de Freitas et al. [Phys. Rev. E 61, 6330 (2000)] presented simulational results for the critical exponents of the two-species reaction-diffusion system $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A$ in dimension d=1.d=1.d=1. In particular, the correlation length exponent was found as $\ensuremath{\nu}=2.21(5)$ in contradiction to the well-known relation $\ensuremath{\nu}=2/d.$ In this Comment, the symmetry arguments leading to exact critical exponents for the universality class to which this reaction-diffusion system belongs are concisely reconsidered.
Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β\betaβ of the directed Ising universality class. The value of β\betaβ was conjectured to be 1314\frac{13}{14}1413​ and Monte Carlo simulation studies in the literature have repeatedly reproduced the consistent values to the conjecture. In this paper, however, we find that the conjectured value of β\betaβ is not correct. Rather, we obtain β\betaβ = 1.020(5) from different models belonging to the directed Ising class.
A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent ννν, which should hold for the large class of continuous transitions associated with ddd-dimensional Landau-Ginzburg-Wilson (LGW) Φ4Φ^4Φ4 theories with a multicomponent scalar field φφφ and a unique φ⋅φφ\cdot φφ⋅φ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If Δφ=(d−2+η)/2Δ_φ=(d-2+η)/2Δφ​=(d−2+η)/2 is the dimension of the order-parameter field φφφ, and Δε=d−1/νΔ_\varepsilon=d-1/νΔε​=d−1/ν is the RG dimension of the energy operator ε\varepsilonε, which can be identified with [φ⋅φ][φ\cdot φ][φ⋅φ] (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality Δε≥2ΔφΔ_\varepsilon \ge 2 Δ_φΔε​≥2Δφ​, which implies ν≥(2−η)−1ν\ge (2-η)^{-1}ν≥(2−η)−1 and γ=(2−η)ν≥1γ= (2-η)ν\ge 1γ=(2−η)ν≥1. These inequalities are supported by general arguments for ferromagnetic lattice models, by εεε-expansion results for generic LGW Φ4Φ^4Φ4 theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW Φ4Φ^4Φ4 theories. In particular, since unitarity requires η≥0η\ge 0η≥0, the above inequality implies ν≥1/2ν\ge 1/2ν≥1/2 for unitary theories. This lower bound is more restrictive than ν>1/dν> 1/dν>1/d, derived by noting that ν=1/dν=1/dν=1/d characterizes the singular finite-size behavior at first-order transitions.