Extraordinary-log surface phase transition in the three-dimensional $XY$ model

TL;DR

This paper provides strong numerical evidence for a novel logarithmic universality in the surface phase transition of the three-dimensional XY model, revealing a new decay behavior of correlations involving logarithms.

Contribution

The study demonstrates the emergence of logarithmic universality in the 3D XY model and proposes a finite-size scaling form with a two-distance behavior, expanding understanding of critical phenomena.

Findings

Evidence for logarithmic decay of correlations in the 3D XY model.

Finite-size scaling shows a plateau with a logarithmic decay in system size.

The critical exponent relates to the RG parameter of the helicity modulus.

Abstract

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance $r$ as $g(r) \sim r^{2-d-\eta}$, with $d$ the spatial dimension and $\eta$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of O($N$) system. In this logarithmic universality, $g(r)$ decays in a power of logarithmic distance as $g(r) \sim ({\rm ln}r)^{-\hat{\eta}}$, dramatically different from the standard scenario. We explore the three-dimensional $XY$ model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with $L$ as $g(L) \sim ({\rm ln}L)^{-\hat{\eta}'}$ as well as the $r$-dependent term $g(r) \sim ({\rm ln}r)^{-\hat{\eta}}$, with ${\hat{\eta}'} \approx {\hat{\eta}}-1$. The critical exponent $\hat{\eta}'$, characterizing the height of the plateau, obeys the scaling relation $\hat{\eta}'=(N-1)/(2\pi \alpha)$ with the RG parameter $\alpha$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.