Topics in Condensed Matter Theory: Berry Curvature Effects in Transport and Numerical Analytic Continuation.

다국어 초록 (Multilingual Abstract)

This thesis is composed of two unrelated pieces of work: the first develops a semiclassical theory of transport in topological magnets, and the second presents a machine learning-based method of numerical analytic continuation.In Part I, we calculate the electric and thermal currents carried by electrons in the presence of general magnetic textures in three-dimensional crystals, including three-dimensional topological spin textures. We show, within a controlled, semiclassical approach that includes all phase space Berry curvatures, that the transverse electric, thermoelectric, and thermal Hall conductivities have two contributions in addition to the usual effect proportional to a magnetic field. These are an anomalous contribution governed by the momentum-space Berry curvature arising from the average magnetization, and a topological contribution determined by the real-space Berry curvature and proportional to the topological charge density, which is non-zero in skyrmion phases. This justifies the phenomenological analysis of transport signals employed in a wide range of materials as the sum of these three parts. We prove that the Wiedemann-Franz and Mott relations hold, even in the presence of topological spin textures, and justifying their use in analyzing the transport signals in these materials. This theory also predicts the existence of the in-plane anomalous and topological Hall effects in three-dimensional, low symmetry materials. We present a symmetry analysis which predicts when the in-plane Hall effect (IPHE) is forbidden, and predict which crystal structures could harbor an IPHE which is larger than the usual out-of-plane Hall effect.In Part II, we present a method of numerical analytic continuation of determinantal Quantum Monte Carlo (DQMC) Green's functions, utilizing an unsupervised neural network (NN). Many have used supervised machine learning methods to attack this problem, but the most interesting applications of DQMC are on systems in which the physical state is totally unknown, so it is advantageous to use an unsupervised NN, which requires no prior knowledge of the system's physical state. This autoencoder-type NN trains on real DQMC data, training the NN to produce spectral functions which reproduce the DQMC data as closely as possible. Regularization is provided by 1) a pretraining step, which trains the NN on general characteristics of spectral functions, and 2) the imposition of limited physical assumptions, such as sum rules. Ultimately, this NN is shown to outperform the standard maximum entropy method on the analytic continuation of noisy data, thus saving computational effort.