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Register a new user2D topological matter from a boundary Greens functions perspective: Faddeev-LeVerrier algorithm implementation
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Since the breakthrough of twistronics a plethora of topological phenomena in two dimensions has appeared, specially relating topology and electronic correlations. These systems can be typically analyzed in terms of lattice models of increasing complexity using Greens function techniques. In this work we introduce a general method to obtain the boundary Greens function of such models taking advantage of the numerical Faddeev-LeVerrier algorithm to circumvent some analytical constraints of previous works. As an illustration we apply our formalism to analyze the edge features of Chern insulators, topological superconductors as the Kitaev square lattice and the Checkerboard lattice in the flat band topological regime. The efficiency of the method is demonstrated by comparison to standard recursive Greens function calculations.
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In this note we present the Greens functions and density of states for the most frequently encountered 2D lattices: square, triangular, honeycomb, kagome, and Lieb lattice. Though the results are well know, we hope that their derivation performed in a uniform way is of some pedagogical value.
The introduction of topological invariants, ranging from insulators to metals, has provided new insights into the traditional classification of electronic states in condensed matter physics. A sudden change in the topological invariant at the boundary of a topological nontrivial system leads to the formation of exotic surface states that are dramatically different from its bulk. In recent years, significant advancements in the exploration of the physical properties of these topological systems and regarding device research related to spintronics and quantum computation have been made. Here, we review the progress of the characterization and manipulation of topological phases from the electron transport perspective and also the intriguing chiral/Majorana states that stem from them. We then discuss the future directions of research into these topological states and their potential applications.
Topological insulators and superconductors are characterized by their gapless boundary modes. In this paper, we develop a recursive approach to the boundary Green function which encodes this nontrivial boundary physics. Our approach describes the various topologically trivial and nontrivial phases as fixed points of a recursion and provides direct access to the phase diagram, the localization properties of the edge modes, as well as topological indices. We illustrate our approach in the context of various familiar models such as the Su-Schrieffer-Heeger model, the Kitaev chain, and a model for a Chern insulator. We also show that the method provides an intuitive approach to understand recently introduced topological phases which exhibit gapless corner states.
The Greens function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Greens functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Greens functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Greens function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Greens function method, which can be of interest when simulating simultaneously the properties of surface and bulk.
A systematic study of the microscopic and thermodynamical properties of pure neutron matter at finite temperature within the Self-Consistent Greens Function approach is performed. The model dependence of these results is analyzed by both comparing the results obtained with two different microscopic interactions, the CD-BONN and the Argonne V18 potentials, and by analyzing the results obtained with other approaches, such as the Brueckner--Hartree--Fock approximation, the variational approach and the virial expansion.
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