We study incompressible systems of motile particles with alignment interactions. Unlike their compressible counterparts, in which the order-disorder (i.e., moving to static) transition, tuned by either noise or number density, is discontinuous, in in
compressible systems this transition can be continuous, and belongs to a new universality class. We calculate the critical exponents to $O(epsilon)$in an $epsilon=4-d$ expansion, and derive two exact scaling relations. This is the first analytic treatment of a phase transition in a new universality class in an active system.
We present a hydrodynamic theory of polar active smectics, for systems both with and without number conservation. For the latter, we find quasi long-ranged smectic order in d=2 and long-ranged smectic order in d=3. In d=2 there is a Kosterlitz-Thoule
ss type phase transition from the smectic phase to the ordered fluid phase driven by increasing the noise strength. For the number conserving case, we find that giant number fluctuations are greatly suppressed by the smectic order; that smectic order is long-ranged in d=3; and that nonlinear effects become important in d=2.
We study theoretically the smectic A to C phase transition in isotropic disordered environments. Surprisingly, we find that, as in the clean smectic A to C phase transition, smectic layer fluctuations do not affect the nature of the transition, in sp
ite of the fact that they are much stronger in the presence of the disorder. As a result, we find that the universality class of the transition is that of the Random field XY model (RFXY).
We show that in suitable anisotropic ferromagnets, both stable and metastable ``tilted phases occur, in which the magnetization ${vec M}$ makes an angle between zero and $180$ degrees with the externally applied ${vec H}$. Tuning either the magnitude
of the external field or the temperature can lead to continuous transitions between these states. A unique feature is that one of these transitions is between two {it metastable} states. Near the transitions the longitudinal susceptibility becomes anomalous with an exponent which has an {it exact} scaling relation with the critical exponents.
Using a phenomenological Ginzburg-Landau theory for the magnetic conical cycloid state of a multiferroic, which has been recently reported in the cubic spinel CoCr$_{2}$O$_{4}$, we discuss its low-energy fluctuation spectrum. We identify the Goldston
e modes of the conical cycloidal order, and deduce their dispersion relations whose signature anisotropy in momentum space reflects the symmetries broken by the ordered state. We discuss the soft polarization fluctuations, the `electromagnons, associated with these magnetic modes and make several experimental predictions which can be tested in neutron scattering and optical experiments.
We show that the cycloidal magnetic order of a multiferroic can arise in the absence of spin and lattice anisotropies, for e.g., in a cubic material, and this explains the occurrence of such a state in CoCr$_2$O$_4$. We discuss the case when this ord
er coexists with ferromagnetism in a so called `conical cycloid state, and show that a direct transition to this state from the ferromagnet is necessarily first order. On quite general grounds, the reversal of the direction of the uniform magnetization in this state can lead to the reversal of the electric polarization as well, without the need to invoke `toroidal moment as the order parameter.
