Unconventional quantum Hall effects in two-dimensional massive spin-1 fermion systems

Yong Xu and L.-M. Duan

Phys. Rev. B 96, 155301 – Published 2 October 2017

Abstract

Unconventional fermions with high degeneracies in three dimensions beyond Weyl and Dirac fermions have sparked tremendous interest in condensed matter physics. Here, we study quantum Hall effects (QHEs) in a two-dimensional unconventional fermion system with a pair of gapped spin-1 fermions. We find that the original unlimited number of zero-energy Landau levels in the gapless case develops into a series of bands, leading to a novel QHE phenomenon where the Hall conductance first decreases (or increases) to 0 and then revives as an infinite ladder of fine staircase when the Fermi surface is moved toward zero energy, and it suddenly reverses, with its sign being flipped, due to a Van Hove singularity when the Fermi surface is moved across 0. We further investigate the peculiar QHEs in a dice model with a pair of spin-1 fermions, which agree well with the results of the continuous model.

Received 8 May 2017

DOI:https://doi.org/10.1103/PhysRevB.96.155301

Physics Subject Headings (PhySH)

Integer quantum Hall effect Landau levels Semimetals

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yong Xu^* and L.-M. Duan

Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA and Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China

^*yongxuph@umich.edu

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Issue

Vol. 96, Iss. 15 — 15 October 2017

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Images

Figure 1
Hall conductance with respect to the chemical potential: (a) in a conventional 2D electron gas, where $B$ and $m$ denote the magnetic-field strength and the electron mass, respectively; (b) in single-layer graphene and (c) in bilayer graphene; and (d) in a system with two triply degenerate points, with a zoomed-in view in the inset. In (b)–(d), red and black lines represent the massless and massive cases with mass $m_{z} = 0$ and $m_{z} = 0.5 E_{B}$ , respectively. Here, $E_{B} = v \sqrt{e B ℏ}$ , with $v$ denoting the electron velocity.
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Figure 2
Eigenenergy of Hamiltonian (1) for (a) $m_{z} = 0$ and (b) $m_{z} > 0$ , with zero-energy states denoted by the green planes. Energy of LLs as a function of (c) $B$ for fixed $m_{z}$ (this $m_{z}$ is taken as the energy unit) and (d) $m_{z}$ for fixed $B$ (with the corresponding $E_{B}$ taken as the energy unit). In (c) and (d), black and red lines correspond to the results of two valleys with $τ_{z} = 1$ and $τ_{z} = - 1$ , respectively, and in the cyan region the Hall conductance vanishes. (e), (f) Zoomed-in view of (c) and (d), respectively. Here, $B_{0} = m_{z}^{2} / (e ℏ v^{2})$ .
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Figure 3
Hall conductance of a single valley for (a) massless electrons with $m_{z} = 0$ and (b) massive electrons with $m_{z} = 0.5 E_{B}$ . Black and red lines denote valleys with $τ_{z} = 1$ and $τ_{z} = - 1$ , respectively.
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Figure 4
(a) Schematics of a lattice structure of a tight-binding model with a pair of triply degenerate points. Each unit cell consists of three sites, A, B, and C, represented by cyan, black, and green filled circles, respectively. There is no hopping between A and C sublattices. The black line shows the zigzag boundary condition. The right panel in (a) displays the Brillouin zone, with $b_{1}$ and $b_{2}$ being the reciprocal vectors. Band structure of our tight-binding model subject to zigzag boundary conditions without magnetic fields in (b) with $m_{z} = 0$ and in (c) with $m_{z} = 0.5 t$ and with magnetic fields in (d) with $m_{z} = 0$ and in (e) with $m_{z} = 0.5 t$ , where red lines denote edge states. (f) Zoomed-in view of (e) around zero energy; (g) the corresponding Hall conductance. In (d)–(f), flat black lines represent LLs, and dashed blue lines are associated with states located at the same edge and hence not chiral. Here, we use 240 unit cells, with the magnetic flux per unit cell being $1 / 60 ϕ_{0}$ .
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Figure 5
(a) Schematic of a lattice structure, with armchair boundaries denoted by the black line. (b)–(f) Band structure obtained using the same parameters as in Figs. 4–4, but with armchair boundary conditions.
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