Magnetic-Field-Induced Localization in 2D Topological Insulators

Abstract

Localization of the helical edge states in quantum spin Hall insulators requires breaking time reversal invariance. In experiments this is naturally implemented by applying a weak magnetic field B. We propse a model based on scattering theory that describes the localization of helical edge states due to coupling to random magnetic fluxes. We find that the localization length is proportional to B −2 when B is small, and saturates to a constant when B is sufficiently large. We estimate especially the localization length for the HgTe/CdTe quantum wells with known experimental parameters. PACS numbers: 73.20.Fz, The prediction and discovery of quantum spin Hall insulators (QSHIs) [1, 1, 2, 2] has opened a door to an unexpected category of topological phases in condensed matter [5][6][7][8], and revealed a new route to investigations of edge/boundarystate physics [9][10][11]. Although the prototypes of QSHIs [1, 1] are mainly based on two copies of quantum Hall insulators, which have been investigated for more than three decades [12,13], it was soon realized that the fundamental importance of time reversal invariance (TRI) distinguishes the two systems in a profound way [2]. Indeed, QSHIs, unlike the Zclassified quantum Hall insulators [14], belong to a class of two-dimensional time-reversal-invariant Z 2 topological insulators [2]. The defining feature of QSHIs, as its name suggests, is a pair of helical edge states that persist in the bulk insulating gap of the system [1, 1, 2, 2, 9].The topological power of QSHIs lies precisely in the robustness of the helical edge states against generic perturbations due to unavoidable disorder in every experimental setup, unless TRI is broken. In the presence of both TRI breaking and disorder, the helical edge states will be localized, and the general framework of Anderson's localization theory applies [15]. Nevertheless, the localization of the helical edge states distinguishes itself from conventional one-dimensional localization when the focus is placed on the crucial role TRI plays in the problem. This point becomes especially relevant as TRI can be broken continuously, for instance, by turning on a magnetic field gradually. Indeed, the sensibility of transport though helical edge states to weak magnetic field has been demonstrated experimentally in the measurement of magneto-conductance in topologically nontrivial HgTe/CdTe quantum wells [2, 3]. Related theoretical analyses have been carried out that include the interplay between TRI-breaking and disorder, but mainly consider magnetic impurities [17,18], or bulk random potential combined with magnetic field [19]. A transparent edge theory that focuses on the magnetic-field-dependent localization of the helical edge states, however, is still missing.In this paper we propose a model that explicitly addresses the question on how the localization of helical edge states occurs as a weak magnetic field is gradually turned on. Our model is based on the scattering theory of edge states in the presence of generic edge disorder. In...

Emission of time-bin entangled particles into helical edge states

“…Each level then crosses E F at a different time. Due to spin-orbit coupling, the particles have a finite probability of flipping their spin in the emission process 30,31 , which corresponds to a change in the propagation direction. The emitted particles are thus in a superposition of left-and rightmovers, leading to entanglement as discussed below.…”

“…The frozen (in time) scattering matrix of the SES can be calculated as in Ref. 31 and reads (up to a global phase)…”

Hyperfine interactions in two-dimensional HgTe topological insulators

“…66,67,69,78,92 Terms related to bulk inversion asymmetry, 8,22 Rashba spin-orbit coupling 13,76 and/or magnetic fields 19,21 can also be included here. The envelope function is given by…”

“…[13][14][15][16][17][18] The effect of external magnetic fields have also been considered. 6,8,[19][20][21][22][23][24] The TI state in HgTe QWs was predicted by Bernevig, Hughes, and Zhang (BHZ) 25 by using a simplified k · p model containing states with S-and P -like symmetries, respectively. They found that beyond a critical thickness of the HgTe QW, the TI state would appear as confirmed experimentally.…”

Effective Hamiltonian and Dynamics of Edge States in Two-Dimensional Topological Insulators under Magnetic Fields

“…However, the effect of perpendicular magnetic field in these conditions is much smaller and manifests itself at B z ∼ 1 T, similar to the case of in-plane field. To describe the effect of the perpendicular field, several theoretical approaches * Electronic address: raichev@isp.kiev.ua have been proposed, including numerical simulation of transport based on a 2D disordered site model [8], as well as analytical consideration of the possibility of enhanced backscattering due to edge spectrum nonlinearity induced by the magnetic field [9] and of interference effects at a disordered edge allowing for loops of the helical edge states [10]. The approach of Ref.…”