The existence of quasi-long range order is demonstrated in nonequilibrium steady states in isotropic $XY$ spin chains including of two types of additional terms that each generate a gap in the energy spectrum. The system is driven out of equilibrium
by initializing a domain-wall magnetization profile through application of an external magnetic field and switching off the magnetic field at the same time the energy gap is activated. An energy gap is produced by either applying a staggered magnetic field in the $z$ direction or introducing a modulation to the $XY$ coupling. The magnetization, spin current, and spin-spin correlation functions are computed analytically in the thermodynamic limit at long times after the quench. For both types of systems, we find the persistence of power-law correlations despite the ground-state correlation functions exhibiting exponential decay.
An approximate solution is presented for simple harmonic motion in the presence of damping by a force which is a general power-law function of the velocity. The approximation is shown to be quite robust, allowing for a simple way to investigate ampli
tude decay in the presence of general types of weak, nonlinear damping.
Out-of-equilibrium behavior is explored in the one-dimensional anisotropic $XY$ model. Initially preparing the system in the isotropic $XX$ model with a linearly varying magnetic field to create a domain-wall magnetization profile, dynamics is genera
ted by rapidly changing the exchange interaction anisotropy and external magnetic field. Relaxation to a nonequilibrium steady state is studied analytically at the critical transverse Ising point, where correlation functions may be computed in closed form. For arbitrary values of anisotropy and external field, an effective generalized Gibbs ensemble is shown to accurately describe observables in the long-time limit. Additionally, we find spatial oscillations in the exponentially decaying, transverse spin-spin correlation functions with wavelength set by the magnetization jump across the initial domain wall. This wavelength depends only weakly on anisotropy and magnetic field in contrast to the current, which is highly dependent on these parameters.
The analogy of classical repulsive interactions emerging from the exchange of mediating particles is revisited with a quantitative approach. Simulations are presented for a particular toy model which are accessible to undergraduate students at any le
vel in the physics curriculum. Analytic treatment of the various regimes shows rigorously how effective force laws can emerge from an underlying microscopic model and should be accessible to advanced undergraduate physics majors. The analysis presented uses the concept of emergence as motivation for students to gain experience building and testing simplified models for complex physical processes.
Excitable membranes are an important type of nonlinear dynamical system and their study can be used to provide a connection between physical and biological circuits. We discuss two models of excitable membranes important in cardiac and neural tissues
. One model is based on the Fitzhugh-Nagumo equations and the other is based on a three-transistor excitable circuit. We construct a circuit that simulates reentrant tachycardia and its treatment by surgical ablation. This project is appropriate for advanced undergraduates as a laboratory capstone project, or as a senior thesis or honors project, and can also be a collaborative project, with one student responsible for the computational predictions and another for the circuit construction and measurements.
