Phase transitions and order in two-dimensional generalized nonlinear $\sigma$-models

TL;DR

This paper investigates phase transitions and order in generalized 2D nonlinear sigma models coupled with additional degrees of freedom, revealing unusual transitions and scaling behaviors that differ from pure models, with implications for low-dimensional systems.

Contribution

It introduces a class of generalized 2D nonlinear sigma models with added variables, showing novel phase transitions and weak divergence of fluctuations, expanding understanding of order in low-dimensional systems.

Findings

Unusual phase transitions between stiff and soft phases in 2D models.

Variance of Goldstone mode fluctuations scales as ln(ln L), weaker than ln L.

Long-range noise can induce true long-range order in 2D.

Abstract

We study phase transitions and the nature of order in a class of classical generalized $O(N)$ nonlinear $\sigma$-models (NLS) constructed by minimally coupling pure NLS with additional degrees of freedom in the form of (i) Ising ferromagnetic spins, (ii) an advective Stokesian velocity and (iii) multiplicative noises. In examples (i) and (ii), and also (iii) with the associated multiplicative noise being not sufficiently long-ranged, we show that the models may display a class of unusual phase transitions between {\em stiff} and {\em soft phases}, where the effective spin stiffness, respectively, diverges and vanishes in the long wavelength limit at two dimensions ($2d$), unlike in pure NLS. In the stiff phase, in the thermodynamic limit the variance of the transverse spin (or, the Goldstone mode) fluctuations are found to scale with the system size $L$ in $2d$ as $\ln\ln L$ with a model-dependent amplitude, that is markedly weaker than the well-known $\ln L$-dependence of the variance of the broken symmetry modes in models that display quasi-long range order in $2d$. Equivalently, for $N=2$ at $2d$ the equal-time spin-spin correlations decay in powers of inverse logarithm of the spatial separation with model-dependent exponents. These transitions are controlled by the model parameters those couple the $O(N)$ spins with the additional variables. In the presence of long-range noises in example (iii), true long-range order may set in $2d$, depending upon the specific details of the underlying dynamics. Our results should be useful in understanding phase transitions in equilibrium and nonequilibrium low-dimensional systems with continuous symmetries in general.