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arXiv:1706.06463 · Physical review. B./Physical review. B

Analytical results for the Green's functions of lattice fermions

A. Komnik, Stefan G. Heinze

Published Oct 3, 2017• 11 citations
arXivHTMLPDF

Abstract

We present a further development of methods for analytical calculations of Green's functions of lattice fermions based on recurrence relations. Applying it to tight-binding systems and topological superconductors in different dimensions, we obtain a number of noteworthy results. In particular, we derive an explicit expression for an arbitrary Green's function of an open Kitaev chain, and we discover nonlocal fermionic corner states in a two-dimensional $p$-wave superconductor.

DOI
10.1103/physrevb.96.155103

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We present a further development of methods for analytical calculations of Green's functions of lattice fermions based on recurrence relations. Applying it to tight-binding systems and topological superconductors in different dimensions, we obtain a number of noteworthy results. In particular, we derive an explicit expression for an arbitrary Green's function of an open Kitaev chain, and we discover nonlocal fermionic corner states in a two-dimensional ppp-wave superconductor.
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.
Since the breakthrough of twistronics a plethora of topological phenomena in correlated systems has appeared. These devices can be typically analyzed in terms of lattice models using Green's function techniques. In this work we introduce a general method to obtain the boundary Green's function of such models taking advantage of the numerical Faddeev-LeVerrier algorithm to circumvent some analytical constraints of previous works. We illustrate our formalism analyzing the edge features of a Chern insulator, the Kitaev square lattice model for a topological superconductor and the Checkerboard lattice hosting topological flat bands. The efficiency and accuracy of the method is demonstrated by comparison to recursive Green's function calculations.
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We study the many-body physics of different quantum systems using a hierarchy of correlations, which corresponds to a generalization of the 1/Z1/\mathcal{Z}1/Z hierarchy. The decoupling scheme obtained from this hierarchy is adapted to calculate double-time Green's functions and due to its nonperturbative nature, we describe quantum phase transition and topological features characteristic of strongly correlated phases. As concrete examples we consider spinless fermions in a dimerized chain and in a honeycomb lattice. We present analytical results which are valid for any dimension and can be generalized to different types of interactions (e.g., long-range interactions), which allows us to shed light on the effect of quantum correlations in a very systematic way. Furthermore, we show that this approach provides an efficient framework for the calculation of topological invariants in interacting systems.
We develop a general numerical method to study the zero-temperature properties of strongly correlated electron models on large lattices. The technique, which resembles Green's-function Monte Carlo, projects the ground-state component from a trial wave function with no approximations. We use this method to determine the phase diagram of the two-dimensional $t\ensuremath{-}J$ model, using the Maxwell construction to investigate electronic phase separation. The shell effects of fermions on finite-sized periodic lattices are minimized by keeping the number of electrons fixed at a closed-shell configuration and varying the size of the lattice. Results obtained for various electron numbers corresponding to different closed shells indicate that the finite-size effects in our calculation are small. For any value of interaction strength, we find that there is always a value of the electron density above which the system can lower its energy by forming a two-component phase separated state. Our results are compared with other calculations on the $t\ensuremath{-}J$ model. We find that the most accurate results are consistent with phase separation at all interaction strengths.