We present a theoretical study of finite momentum states in the context of degenerate gases and iron-based magnet. The unifying theme of these seemingly disparate states of condensed matter is the finite momentum of their respective grounds states and the associated enhanced fluctuations.;For the degenerate atomic gases, we study in the first part of the thesis a system of two species of bosonic atoms interacting through a p-wave Feshbach resonance as realized in Rubidium-85/Rubidium-87 mixture. In mapping out the phase diagram, we show that the system exhibits atomic (ASF), molecular (MSF) and atomic-molecular (AMSF) superfluid phases, where atoms, molecules, and atoms and molecules Bose condense, respectively. The ASF and MSF states are respectively characterized by a nonzero s-wave atomic and p-wave (orbital) spinor molecular condensates. The AMSF is distinguished by the presence of both of these condensates, with the s-wave atomic condensate component necessarily periodically modulated at a wavevector that is tunable with a magnetic field; that is, generically AMSF is a robust supersolid, that simultaneously breaks spatial translational and gauge symmetries. We explore the rich phenomenology of these phases and phase transitions between them, that we find to be strongly influenced by the quantum and thermal fluctuations.;In the second part of the thesis, we study magnetism in Fe1+yTe, a parent compound of the iron-based high-temperature superconductors. Motivated by earlier studies that have provided evidences of finite momentum spiral states in these materials, we show that a spin-1 exchange model, supplemented by a single-ion anisotropy accounts well for the experimentally observed magnetic phase diagram, that prominently exhibits commensurate bi-collinear and incommensurate spin-spiral orders with the associated low-energy spin-wave spectra. We derive the low energy hydrodynamic models for these magnetic states and use it to describe the magneto-structural and commensurate-incommensurate transitions, and the static and dynamic structure functions across temperature - Fe doping phase diagram.
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