We study the non-equilibrium behavior of systems that are driven out of equilibrium from the interface. In the first part of this thesis, we study a model of a two-dimensional lattice gas that is in contact with two heat baths that are at different temperatures. Performing Monte Carlo simulations, we find that there are three possible types of non-equilibrium steady states, depending on the values of certain system parameters. They include a disordered phase, a fully phase separated state, and an interesting state with striped patterns in the half of the lattice where the temperature is lower. The last one is a novel non-equilibrium steady state that we study systematically by varying the system parameters. To obtain the non-equilibrium finite-size phase diagram, we perform a spectrum analysis to classify not only the three major states, but also the sub-states of the striped phase. In the second part of the thesis, we study magnetic friction that results when two Potts systems move with respect to each other. In this research, we first study a model that consists of two interacting Potts blocks, where one block moves on top of the other. As a result, the system is driven out of equilibrium constantly. In our research we find for weak interfacial couplings that the contacting surfaces behave rather similar to a free surface. If the interfacial coupling is strong, however, anisotropic spin patterns appear on the contacting surfaces. This study is extended to a three-dimensional Potts wedge with a tip sliding along the surface of a Potts block. It is found that the shape of the Potts lattice influences the surface behavior of the system. / Ph. D.

<div>We propose an order parameter theory of the quantum Hall nematic in high fractional Landau levels in terms of an Ising description. This new model solves a couple of extant problems in the literature: (1) The low-temperature behavior of the measured resistivity anisotropy is captured better by our model than previous theoretical treatments based on the electron nematic having XY symmetry. (2) Our model allows for the development of true long-range order at low temperature, consistent with the observation of anisotropic low-temperature transport.<br></div><div><div> We furthermore propose new experimental tests based on hysteresis that can distinguish whether any two-dimensional electron nematic is in the XY universality class (as previously proposed in high fractional Landau levels), or in the Ising universality class (as we propose). Given the growing interest in electron nematics in many materials, we expect our proposed test of universality class to be of broad interest.</div><div> </div><div> Whereas the XY model in two dimensions does not have a long-range ordered phase, the addition of uniaxial random field disorder induces a long-range ordered phase in which the spontaneous magnetization points perpendicular to the random field direction, via an order-by-disorder transition. We have shown that this spontaneous magnetization is robust against a rotating driving field, up to a critical driving field amplitude. Thus we have found evidence for a new non-equilibrium phase transition that was unknown before in this model. Moreover, we have discovered an incredible anomaly at this nonequilibrium phase transition: the critical region is accompanied by a cascade of period multiplication events. This physics is reminiscent of the period bifurcation cascade signaling the transition to chaos in nonlinear systems, and of the approach to the irreversibility transition in models of yield in amorphous solids~\cite{reichhardt-dahmen,leishangthem_yielding_2017}. This period multiplication cascade is surprising to be present in a statistical mechanics model, and suggests that the non-equilibrium transition as a function of driving field amplitude is part of a larger class of transitions in dynamical systems.</div><div> Moreover, we show that this multi-period behavior represents a new emergent classical discrete time-crystal, since the new period is robust against changes to initial conditions and low-temperature fluctuations over hundreds of driving period cycles.</div><div><br></div><div> We expect this work to be of broad interest, further encouraging cross-fertilization between the rapidly growing field of time-crystals with the well-established fields of nonequilibrium phase transitions and dynamical systems.</div><div> </div><div> Geometrical configurations gave us a better understanding of the multi-period behavior of the limit-cycles.</div><div> Moreover, surface probes are continually evolving and generating vast amounts of spatially resolved data of quantum materials, which reveal a lot of detail about the microscopic and macroscopic properties of the system. <br></div><div> Materials undergoing a transition between two distinct states, phase separate.</div><div> These phase-separated regions form intricate patterns on the observable surface, which can encode model-specific information, including interaction, dimensionality, and disorder. </div><div> While there are rigorous methods for understanding these patterns, they turn out to be time-consuming as well as requiring expertise. </div><div> We show that a well-tuned machine learning framework can decipher this information with minimal effort from the user.</div><div> We expect this to be widely used by the scientific community to fast-track comprehension of the underlying physics in these materials.</div><div><br></div></div>