This report reviews recent progress in computing Kubo formulas for general interacting Hamiltonians. The aim is to calculate electric and thermal magneto-conductivities in strong scattering regimes where Boltzmann equation and Hall conductivity proxies exceed their validity. Three primary approaches are explained.1. Degeneracy-projected polarization formulas for Hall-type conductivities, which substantially reduce the number of calculated current matrix elements. These expressions generalize the Berry curvature integral formulas to imperfect lattices.2. Continued fraction representation of dynamical longitudinal conductivities. The calculations produce a set of thermodynamic averages, which can be controllably extrapolated using their mathematical relations to low and high frequency conductivity asymptotics.3. Hall-type coefficients summation formulas, which are constructed from thermodynamic averages.The thermodynamic formulas are derived in the operator Hilbert space formalism, which avoids the opacity and high computational cost of the Hamiltonian eigenspectrum. The coefficients can be obtained by well established imaginary-time Monte Carlo sampling, high temperature expansion, traces of operator products, and variational wavefunctions at low temperatures.We demonstrate the power of approaches 1–3 by their application to well known models of lattice electrons and bosons. The calculations clarify the far-reaching influence of strong local interactions on the metallic transport near Mott insulators. Future directions for these approaches are discussed.

Recently, topolectrical circuits (TECs) boom in studying the topological states of matter. The resemblance between circuit Laplacians and tight-binding models in condensed matter physics allows for the exploration of exotic topological phases on the circuit platform. In this review, we begin by presenting the basic equations for the circuit elements and units, along with the fundamentals and experimental methods for TECs. Subsequently, we retrospect the main literature in this field, encompassing the circuit realization of (higher-order) topological insulators and semimetals. Due to the abundant electrical elements and flexible connections, many unconventional topological states like the non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states that are challenging to observe in conventional condensed matter physics, have been observed in circuits and summarized in this review. Furthermore, we show the capability of electrical circuits for exploring the physical phenomena in other systems, such as photonic and magnetic ones. Importantly, we highlight TEC systems are convenient for manufacture and miniaturization because of their compatibility with the traditional integrated circuits. Finally, we prospect the future directions in this exciting field, and connect the emerging TECs with the development of topology physics, (meta)material designs, and device applications.

This paper offers a review while also studying yet unexplored features of the area of chaotic systems subjected to parameter drift of non-negligible rate, an area where the methods of traditional chaos theory are not applicable. Notably, periodic orbit expansion cannot be applied since no periodic orbits exist, nor do long-time limits, since for drifting physical processes the observational time can only be finite. This means that traditional Lyapunov-exponents are also ill-defined. Furthermore, such systems are non-ergodic, time and ensemble averages are different, the ensemble approach being superior to the single-trajectory view. In general, attractors and phase portraits are time-dependent in a non-periodic fashion. We describe the use of general methods which remain nevertheless applicable in such systems. In the phase space, the analysis is based on stable and unstable foliations, their intersections defining a Smale horseshoe, and the intersection points can be identified with the chaotic set governing the core of the drifting chaotic dynamics. Because of the drift, foliations and chaotic sets are also time-dependent, snapshot objects. We give a formal description for the time-dependent natural measure, illustrated by numerical examples. As a quantitative indicator for the strength of chaos, the so-called ensemble-averaged pairwise distance (EAPD) can be evaluated at any time instant. The derivative of this function can be considered the instantaneous (largest) Lyapunov exponent. We show that snapshot chaotic saddles, the central concept of transient chaos, can be identified in drifting systems as the intersections of the foliations, possessing a time-dependent escape rate in general. In dissipative systems, we find that the snapshot attractor coincides with the unstable foliation, and can consist of more than one component. These are a chaotic one, an extended snapshot chaotic saddle, and multiple regular time-dependent attractor points. When constructing the time-dependent basins of attraction of the attractor points, we find that the basin boundaries are time-dependent and fractal-like, containing the stable foliation, and that they can even exhibit Wada properties. In the Hamiltonian case, we study the phenomenon of the break-up of tori due to the drift in terms of both foliations and EAPD functions. We find that time-dependent versions of chaotic seas are not always fully chaotic, they can contain non-chaotic regions. Within such regions we identify time-dependent non-hyperbolic regions, the analogs of sticky zones of classical Hamiltonian phase spaces. We provide approximate formulas for the information dimension of snapshot objects, based on time-dependent Lyapunov exponents and escape rates. Besides these results, we also give possible applications of our methods e.g. in climate science and in the area of Lagrangian Coherent Structures.

We present a comprehensive experimental study of the dynamics of electrostatic MEMS resonators under large excitations. We identified three frequency ranges where large oscillations occur; a non-resonant region driven by fast–slow dynamic interactions and two resonant regions. In these regions, we found a plethora of dynamic phenomena including cascades of period-doubling bifurcations, a bubble structure, homoclinic and cyclic-fold bifurcations, hysteresis, intermittencies, quasiperiodicity, chaotic attractors, odd-periodic windows within those attractors, Shilnikov orbits, and Shilnikov chaos.We encountered these complex nonlinear dynamics phenomena under relatively high dissipation levels, the quality factors of the resonators examined in this study were Q = 6.2 and 2.1. In the case of MEMS with higher quality factors (Q>100), it is quite reasonable to expect those phenomena to appear under relatively low excitation levels (compared to the static pull-in voltage). This calls for a new paradigm in the design of electrostatic MEMS that seeks to manage dynamic phenomena rather than attempt to avoid them and, thereby, overly restricting the design space. We believe this is feasible given the repeatable and predictable nature of those phenomena.