Phase transitions and order in two-dimensional generalized nonlinear σ models

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 stiff and 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, which is markedly weaker than the well-known lnL 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.