The Haldane model and its localization dichotomy

Abstract

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden.Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [15]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [25], and finally we announce further generalizations to non-periodic models.Date: September 6, 2019. Extended version of the paper published in Rend. Mat. Appl. 39, 307-327 (2018). In comparison with the published version, we added some details and the whole Chapter 5.2010 Mathematics Subject Classification. 81Q70, 81V70, 47A56, 47A10.

Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators

“…Note that this generalizes the Chern number as for periodic systems the Chern number and the Chern marker agree [4,5]. Therefore, parallel to the periodic case, it is conjectured that the Chern marker characterizes the existence of localized Wannier basis for gapped systems [4,5]. Before continuing to state the conjecture more precisely and state the main result of this paper, which confirms the conjecture in one direction, let us start by making some definitions:…”

“…Note that this generalizes the Chern number as for periodic systems the Chern number and the Chern marker agree [4,5]. Therefore, parallel to the periodic case, it is conjectured that the Chern marker characterizes the existence of localized Wannier basis for gapped systems [4,5].…”

“…Following the reasoning at the beginning of the proof of Proposition 3.2, we see that χ a (P − P a+b )4 SWe now split the set {m ∈ Z 2 : |m| ∞ > a + b} into three parts and bound each part separatelyS 1 := m : |m 1 | > a + b and |m 2 | > a + b S 2 := m : |m 1 | > a + b and |m 2 | ≤ a + b S 3 := m : |m 1 | ≤ a + b and |m 2 | > a + b…”

“…Since the notion of the Chern number depends on the Bloch decomposition, the Chern number is no longer well defined for non-periodic systems. For generic systems, the Chern marker was proposed in [2,4] as an extension.…”

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

“…In this section, we first present basic features of the periodic bulk and edge Haldane models and their Bloch reductions in Sections 4.1 and 4.2. We do this only for the reader's convenience since excellent reviews already exist in the literature [47,76]. We then describe how we model defects and disorder in Section 4.3.…”