It is well known that the 2d XY model exhibits an unusual infinite order phase transition belonging to the Kosterlitz-Thouless (KT) universality class. Introduction of a nematic coupling into the XY Hamiltonian leads to an additional phase transition in the Ising universality class [1]. In this paper, using a combination of extensive Monte Carlo simulations and finite size scaling, we show that the higher order harmonics lead to a qualitatively different phase diagram, with additional ordered phases originating from the competition between the ferromagnetic and pseudo-nematic couplings. The new phase transitions belong to the 2d Potts, Ising, or KT universality classes.The low temperature behavior of two dimensional (2d) systems with continuous symmetries is controlled by topological defects, such as vortices and domain walls. Although massless Goldstone excitations, such as spin waves, destroy the long-range order of these systems, a pseudo-long-range order with algebraically decaying correlation functions still remains possible. At low temperatures the topological defects which undermine the pseudo-long-range order are all paired up, while above the critical temperature these defects unbind, leading to exponentially decaying correlation functions and a loss of the pseudo-long-range order. A classical example of such system is the XY model. At low temperature, the topological defects, in the form of integer valued vortices, are all joined in vortex-antivortex pairs, resulting in algebraically decaying spin-spin correlation functions. Above the Kosterlitz-Thouless (KT) critical temperature [2,3], these pairs unbind and the correlation functions decay exponentially.Unlike in 3d, for 2d systems the arguments based purely on symmetry considerations are not sufficient to fix the universality class of possible phase transitions [4][5][6][7], and even in 3d a second order phase transition can be preempted by a first order one [8]. Violations of strong universality are even more common in 2d. Thus, it is possible for systems with the same underlying symmetries and the same coarse-grained Landau-GinzburgWilson Hamiltonian not to belong to the same universality class. It is, therefore, interesting to ask what phase transitions are possible for 2d Hamiltonians invariant under the transformation θ → θ + 2π. In this paper, we will study using extensive Monte Carlo simulations and finite size scaling (FSS) analysis, a large class of generalized XY models which, while preserving the same θ → θ+2π symmetry, have very complex phase diagrams, with phase transitions belonging to the Ising and Potts universality classes, in addition to the usual KT phase transition. In some of these models, transitions can be understood in terms of new topological defects, such as fractional vortices and domain walls [1,9]. Apart from the fundamental considerations regarding the connection between symmetry and universality, our purpose is to describe new, previously unnoticed, ordered phases which occur in 2d systems with continuous symmetry.The m...

Model ecosystems with quenched, symmetric interspecies interactions have been extensively studied using the replica method of the statistical mechanics of disordered systems. Here, we consider a more general scenario in which both the species abundances and the interspecies interactions change with time, albeit in widely separated timescales. The equilibrium of the coupled dynamics is studied analytically within the partial annealing framework, in which the number of replicas n takes on positive as well as negative values. In the case n > 0, which describes ecosystems characterized by the cooperative interspecies interactions, we find a discontinuous transition to a regime of zero diversity, whereas in the case where competition prevails, n < 0, we find that the species diversity is maximum.

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Within the framework of the random replicator model of ecosystems, we use equilibrium statistical mechanics tools to study the effect of manipulating the ecosystem so as to guarantee that a fixed fraction of the surviving species at equilibrium display a predefined set of characters (e.g., characters of economic value). Provided that the intraspecies competition is not too weak, we find that the consequence of such intervention on the ecosystem composition is a significant increase on the number of species that become extinct, and so the impoverishment of the ecosystem.