Abstract
We propose and construct a numerical algorithm to calculate the Berry conductivity in topological band insulators. The method is applicable to cold atom systems as well as solid state setups, both for the insulating case where the Fermi energy lies in the gap between two bulk bands as well as in the metallic regime. This method interpolates smoothly between both regimes. The algorithm is gauge-invariant by construction, efficient, and yields the Berry conductivity with known and controllable statistical error bars. We apply the algorithm to several paradigmatic models in the field of topological insulators, including Haldaneʼs model on the honeycomb lattice, the multi-band Hofstadter model, and the BHZ model, which describes the 2D spin Hall effect observed in CdTe/HgTe/CdTe quantum well heterostructures.
Highlights
Topological insulators (TI) are a topological state of quantum matter that constitutes a new paradigm in condensed matter physics [1,2,3,4]
Before presenting our numerical algorithm to calculate the Berry conductivity, we briefly review the expressions for the intrinsic Hall conductivity for the insulating case where the value of the Fermi energy lies in the gap between two bands
The algorithm works for the insulating case where the Fermi energy lies in the gap between two bulk bands; it works for the situation in which it lies within a band
Summary
Topological insulators (TI) are a topological state of quantum matter that constitutes a new paradigm in condensed matter physics [1,2,3,4]. In transport properties and measurements, bulk carriers often dominate over the contribution stemming from surface or edge states This question plays a fundamental role in the physics of the anomalous quantum Hall effect (AHE) [35, 36], which precedes the upsurge of topological insulators as a prominent field in condensed matter. This intrinsic contribution, which is dominant in metallic ferromagnets with moderate conductivity, depends only on band structure properties and is largely independent of the scattering that affects other AHE mechanisms Understanding this intrinsic and anomalous contribution has become possible with the seminal work by Haldane [37], who uncovered by a fully quantum-mechanical treatment— unlike precedent work based on semiclassical methods [38]—the topological origin of this contribution and its relation to the physics taking place at the Fermi surface. We conclude with a short summary and a discussion of possible future extensions of the presented method (section 4)
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