[Paper Review] Comment on ``Comparative study of Higgs transition in one-component and two-component lattice superconductor models"

Anatoly Kuklov, Munehisa Matsumoto+3 more|ArXiv.org|May 16, 2008

Particle physics theoretical and experimental studies6 citations

TL;DR

This comment challenges the claim by Motrunich and Vishwanath that the NCCP1 model exhibits a second-order Higgs transition line, arguing instead that rescaling their Monte Carlo data reveals a diverging flow inconsistent with scale invariance. Using the flowgram method with length-scale renormalization, the authors demonstrate perfect data collapse with non-scale-invariant behavior, indicating a generic first-order transition rather than a tricritical point at K ≈ 0.2.

ABSTRACT

We comment on the claim of continuous phase transition in the SU(2) symmetric Deconfined Critical Point action made by O.Motrunich and A.Vishwanath in ArXive:0805.1494

Motivation & Objective

• To critically assess the claim by Motrunich and Vishwanath of a second-order Higgs transition line in the NCCP1 model.
• To test whether the observed data in Ref. MV supports scale invariance at the transition point.
• To apply the flowgram method with length-scale renormalization to distinguish between second-order and weak first-order transitions.
• To determine whether the tricritical point at K ≈ 0.2 is supported by the data or contradicted by scaling behavior.

Proposed method

• Digitized flowgram data from Fig. 13 (lower panel) of Ref. MV for the NCCP1 model.
• Applied length-scale renormalization by rescaling system size as $ L \to C(g)L $, with $ g = 1/4K $, using a smooth, monotonically increasing function $ C(K) $.
• Performed two-parameter fit to determine $ C(K) = 0.0740/K + 0.00444(\exp(1.00/K) - 1) $, anchored at $ C(0.2) = 1 $.
• Evaluated the rescaled flowgrams for collapse and scale invariance across $ g \in [0, g_{\text{coll}}] $.
• Used the flowgram method’s established framework to assess whether divergent flow implies absence of scale invariance.
• Compared results to prior studies on U(1)×U(1) and SU(2)-symmetric NCCP1 models, where similar divergent flows indicated first-order transitions.

Experimental results

Research questions

• RQ1 Does the data from Motrunich and Vishwanath’s Monte Carlo simulations support scale invariance at the Higgs transition?
• RQ2 Is the claimed tricritical point at $ K \approx 0.2 $ consistent with a proper data collapse analysis using length-scale renormalization?
• RQ3 Can the flowgram method distinguish between second-order and weak first-order transitions in the NCCP1 model?
• RQ4 What is the scaling behavior of the flowgrams when rescaled using a physically motivated $ C(K) $ function?
• RQ5 Does the observed divergent flow in the rescaled data contradict the existence of a second-order transition line?

Key findings

• The rescaled flowgrams from Ref. MV’s data exhibit a perfect collapse with a diverging flow, indicating a breakdown of scale invariance.
• The scaling function $ C(K) $ was determined via two-parameter fit as $ C(K) = 0.0740/K + 0.00444(\exp(1.00/K) - 1) $, with $ C(0.2) = 1 $.
• The divergent flow behavior is inconsistent with the presence of a tricritical point at $ K \approx 0.2 $, contradicting the original claim.
• The observed scaling behavior closely resembles that found in prior studies of first-order transitions in U(1)×U(1) and SU(2)-symmetric NCCP1 models.
• The absence of scale invariance in the rescaled data strongly suggests a generic first-order transition rather than a second-order one.
• The authors conclude that the original interpretation of a second-order Higgs transition line is not supported by the data when analyzed with the flowgram method and data collapse.