Zero modes in the random hopping model

Brouwer, Piet W.; Racine, Étienne; Furusaki, Akira; Hatsugai, Yasuhiro; Morita, Yoshifumi; Mudry, Christopher

doi:10.1103/physrevb.66.014204

Cited by 44 publications

(40 citation statements)

References 61 publications

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“…It has the same spatial dependence as the quasi-localized solutions which are induced by radial potentials on 2D Dirac fermions [22]. The matching of localized states described above cannot be generalized to the case t ′ = 0, as the band of edge states is not degenerate in energy [23]. The localized state at the Fermi level becomes a resonance inside the continuum of extended states.…”

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confidence: 99%

Disorder Induced Localized States in Graphene

et al. 2006

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We consider the electronic structure near vacancies in the half-filled honeycomb lattice. It is shown that vacancies induce the formation of localized states. When particle-hole symmetry is broken, localized states become resonances close to the Fermi level. We also study the problem of a finite density of vacancies, obtaining the electronic density of states, and discussing the issue of electronic localization in these systems. Our results have also relevance for the problem of disorder in d-wave superconductors.PACS numbers: 81.05. Uw, 71.27.+a, Introduction. The problem of disorder in systems with Dirac fermions has been studied extensively in the last few years in the context of dirty d-wave superconductors [1]. Dirac fermions are also the elementary excitations of the honeycomb lattice at half-filling, equally known as graphene, which is realized in two-dimensional (2D) Carbon based materials with sp 2 bonding. It is well-known that disorder is ubiquitous in graphene and graphite (which is produced by stacking graphene sheets) and its effect on the electronic structure has been studied extensively [2,3,4,5,6,7,8,9,10,11,12,13]. It has been shown recently [14] that the interplay of disorder and electron-electron interactions is fundamental for the understanding of recent experiments in graphene devices [15]. Furthermore, experiments reveal that ferromagnetism is generated in heavily disordered graphite samples [16,17,18,19,20], but the understanding of the interplay of strong disorder and electron-electron interactions in these systems is still in its infancy. Different mechanisms for ferromagnetism in graphite have been proposed and they are either based on the nucleation of ferromagnetism around extended defects such as edges [2,8,12,13] or due to exchange interactions originating from unscreened Coulomb interactions [21]. Therefore, the understanding of the nature of the electronic states in Dirac fermion systems with strong disorder is of the utmost interest.In the following, we analyze in detail states near the Fermi energy induced by vacancies in a tight-binding model for the electronic states of graphene planes. We show that single vacancies in a graphene plane generate localized states which are sensitive to the presence of particle-hole symmetry breaking. Moreover, a finite density of such defects leads to strong changes in the local and averaged electronic Density Of States (DOS) with the creation of localized states at the Dirac point.The model. We consider a single band model described by the Hamiltonian:

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Disorder Induced Localized States in Graphene

et al. 2006

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“…If the distribution of vacant sites is uneven between the two sublattices, zero energy modes will necessarily appear. This follows from a theorem in linear algebra 34 and can be seen as follows. Assume, very generally, that we have a bipartite lattice, with sublattices A and B (It can be any bipartite lattice like the square or honeycomb lattices in 2D, cubic in 3D, etc.…”

Section: Vacancies and A Theoremmentioning

confidence: 99%

Modeling disorder in graphene

2008

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We present a study of different models of local disorder in graphene. Our focus is on the main effects that vacancies -random, compensated and uncompensated -, local impurities and substitutional impurities bring into the electronic structure of graphene. By exploring these types of disorder and their connections, we show that they introduce dramatic changes in the low energy spectrum of graphene, viz. localized zero modes, strong resonances, gap and pseudogap behavior, and non-dispersive midgap zero modes.

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“…We also impose a nearest-neighbor and next-nearest-neighbor exclusion constraint on the vacancies and do not allow them to interrupt the armchair edges. These restrictions, along with the compensated nature of the vacancy disorder, eliminate all previously studied and well-understood sources of vacancy-induced [9,28] zero modes in the spectrum of H.…”

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Vacancy-Induced Low-Energy States in Undoped Graphene

2016

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We demonstrate that a nonzero concentration n v of static, randomly placed vacancies in graphene leads to a density w of zero-energy quasiparticle states at the band center ϵ ¼ 0 within a tight-binding description with nearest-neighbor hopping t on the honeycomb lattice. We show that w remains generically nonzero in the compensated case (exactly equal number of vacancies on the two sublattices) even in the presence of hopping disorder and depends sensitively on n v and correlations between vacancy positions. For low, but not-too-low, jϵj=t in this compensated case, we show that the density of states ρðϵÞ exhibits a strong divergence of the form ρ Dyson ðϵÞ ∼ jϵj −1 =½logðt=jϵjÞ ðyþ1Þ , which crosses over to the universal low-energy asymptotic form (modified Gade-Wegner scaling) expected on symmetry grounds ρ GW ðϵÞ ∼ jϵj −1 e −b½logðt=jϵjÞ 2=3 below a crossover scale ϵ c ≪ t. ϵ c is found to decrease rapidly with decreasing n v , while y decreases much more slowly. DOI: 10.1103/PhysRevLett.117.116806 Static impurities, which give rise to random timeindependent terms in the single-particle Hamiltonian for quasiparticle excitations of a condensed matter system, can lead to the phenomenon of Anderson localization, whereby quasiparticle wave functions lose their plane-wave character and become localized [1]. Such localization transitions and universal low-energy properties of the localized phase have been successfully described in many cases using effective field theories [2,3] whose form depends on symmetry properties of the quasiparticle Hamiltonian in the presence of impurities. In some cases [4,5], it has also been possible to refine these field-theoretical predictions using real-space strong-disorder renormalization group ideas [6].In this Letter, we study the effects of a nonzero concentration n v of static, randomly located vacancies in graphene. We use a tight-binding description for electronic states of graphene, with hopping amplitude t between nearest-neighbor sites on a honeycomb lattice, and model vacancies by the deletion of the corresponding site in this tight-binding model [7][8][9][10][11]. We focus on the compensated case, i.e., exactly equal numbers of vacancies on the two sublattices of the honeycomb lattice, and demonstrate that vacancies generically lead to a nonuniversal density w of zero-energy quasiparticle states at the band center ϵ ¼ 0 even in this compensated case, including in the presence of hopping disorder. For low, but not-too-low, jϵj=t in this compensated case, the density of states (DOS) ρðϵÞ exhibits a strong divergence of the formfamiliar in the context of various random-hopping problems in one dimension [12][13][14][15][16][17][18][19][20]. At still lower energies, below a crossover scale ϵ c that is several orders of magnitude smaller than t even for moderately small values of n v (0.05-0.1), we show that the DOS crosses over to the low-energy asymptotic behavior [4-6,21] of the chiral orthogonal universality class (to which our tight-binding model belongs on symmetry grou...

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Zero modes in the random hopping model

Cited by 44 publications

References 61 publications

Disorder Induced Localized States in Graphene

Disorder Induced Localized States in Graphene

Modeling disorder in graphene

Vacancy-Induced Low-Energy States in Undoped Graphene

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