Dynamical renormalization group analysis of $O(n)$ model in steady shear flow

TL;DR

This paper applies a dynamical renormalization group approach to the $O(n)$ model under steady shear flow, revealing new fixed points and critical behavior that differ from equilibrium systems, especially in low dimensions.

Contribution

It introduces a novel anisotropic scaling analysis under shear flow, identifying a stable Gaussian fixed point and revising critical dimension estimates for the $O(n)$ model.

Findings

Reveals a new stable Gaussian fixed point with anisotropic scaling.

Shows mean-field critical exponents are valid in 2D and 3D under shear.

Indicates shear flow stabilizes long-range order even in 2D, contrary to equilibrium expectations.

Abstract

We study the critical behavior of the $O(n)$ model under steady shear flow using a dynamical renormalization group (RG) method. Incorporating the strong anisotropy in scaling ansatz, which has been neglected in earlier RG analyses, we identify a new stable Gaussian fixed point. This fixed point reproduces the anisotropic scaling of static and dynamical critical exponents for both non-conserved (Model A) and conserved (Model B) order parameters. Notably, the upper critical dimensions are $d_{\text{up}} = 2$ for the non-conserved order parameter (Model A) and $d_{\text{up}} = 0$ for the conserved order parameter (Model B), implying that the mean-field critical exponents are observed even in both $d=2$ and $3$ dimensions. Furthermore, the scaling exponent of the order parameter is negative for all dimensions $d \geq 2$, indicating that shear flow stabilizes the long-range order associated with continuous symmetry breaking even in $d = 2$. In other words, the lower critical dimensions are $d_{\rm low} < 2$ for both types of order parameters. This contrasts with equilibrium systems, where the Hohenberg -- Mermin -- Wagner theorem prohibits continuous symmetry breaking in $d = 2$.