Quantised Hall conductance in a perfect crystal

Dana, Itzhack; Avron, J. E.; Żak, J.

doi:10.1088/0022-3719/18/22/004

Cited by 146 publications

(160 citation statements)

References 20 publications

Supporting

Mentioning

154

Contrasting

Order By: Relevance

“…In a seminal paper by Thouless et al [25], it was shown that each gap in the spectrum of this model is associated with a quantized and nontrivial Chern number (Hall conductance). Later on, it was shown by Dana et al [26] that the nontriviality of the Chern numbers stems from the symmetry of this model with respect to the magnetic translation group. Choosing consistent boundary conditions [26], the Chern number ν r that is associated with a gap number r = 1, .…”

mentioning

confidence: 94%

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

2012

View full text Add to dashboard Cite

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e. a cut and project in any irrational angle.PACS numbers: 71.23. Ft, 73.43.Nq, 05.30.Rt Recent experimental developments in photonic crystals [1,2] and ultracold atoms [3][4][5] have made the study of the dynamics of particles in one-dimensional (1D) quasiperiodic systems experimentally accessible. These fascinating systems have long been the focus of extensive research. They have been studied mainly in the context of their transport and localization properties, showing a variety of interesting transitions between metallic, localized, and critical phases [6][7][8][9]. With their recently found nontrivial connection to topological phases of matter [10] there is a growing interest in their boundary phenomena [11,12].The behavior of particles in such systems is described by 1D tight-binding models with quasiperiodic modulations. There is an abundance of quasiperiodic modulations, among which the canonical types are the Harper model (also known as the Aubry-André model) [6,13] and the Fibonacci quasicrystal (QC) [14]. The quasiperiodicity of the Harper model enters in the form of a cosine modulation incommensurate with lattice spacing, whereas the Fibonacci QC has two discrete values that appear interchangeably according to the Fibonacci sequence. Moreover, the quasiperiodicity may appear in on-site terms (diagonal), in hopping terms (off diagonal), or in both (generalized). Each of these models describes different physical phenomena. Indeed, the Harper and the Fibonacci modulations have different localization phase diagrams, depending also on their appearance as diagonal or off-diagonal terms (see e.g., Refs. [7][8][9]15]). Notably, several attempts were made to gather these models under some general framework [9,16,17], but with only partial success.These 1D quasiperiodic models play a nontrivial role in the rapidly growing field of topological phases of matter [10][11][12]. This new paradigm classifies gapped systems such as band insulators and superconductors [18,19]. Each gap in these systems is attributed an index that characterizes topological properties of the wave functions in the bands below this gap. By definition, two gapped systems belong to the same topological phase if they can be deformed continuously from one into the other witho...

show abstract

mentioning

confidence: 94%

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

2012

View full text Add to dashboard Cite

show abstract

“…As was shown by Usov 14 , the Hall conductance of the µth subband may be presented as a sum of two terms σ xy = σ (1) xy + σ (2) xy , where…”

Section: Hall Conductancementioning

confidence: 91%

“…As was shown by Usov 14 , the Hall conductance of the th subband may be presented as a sum of two terms xy = (1) xy + (2) xy , where (1) xy = e 2 ph and (2) xy = i e 2 2 h I p X n=1 J n dk : (10) Here, the vector J n is determined by its components …”

Section: Hall Conductancementioning

confidence: 99%

Hall conductance of a two-dimensional electron gas in periodic lattice with triangular antidots

Demikhovskiǐ¹,

Перов²

2005

Physica E: Low-dimensional Systems and Nanostructures

View full text Add to dashboard Cite

The topic of this contribution is the investigation of quantum states and quantum Hall effect in electron gas subjected to a periodic potential of the lateral lattice. The potential is formed by triangular quantum antidos located on the sites of the square lattice. In a such system the inversion center and the four-fold rotation symmetry are absent. The topological invariants which characterize different magnetic subbands and their Hall conductances are calculated. It is shown that the details of the antidot geometry are crucial for the Hall conductance quantization rule. The critical values of lattice parameters defining the shape of triangular antidots at which the Hall conductance is changed drastically are determined. We demonstrate that the quantum states and Hall conductance quantization law for the triangular antidot lattice differ from the case of the square lattice with cylindrical antidots. As an example, the Hall conductances of magnetic subbands for different antidot geometries are calculated for the case when the number of magnetic flux quanta per unit cell is equal to three.

show abstract

“…Almost at the same time Zak [4] proposed an equivalent approach: projective representations of the translation group were considered as vector representations of its covering group, being in fact a central *e-mail: florek0amu.edu.pl (871) extension of the translation group and a group of factors (see also [5]). The problem investigated by these authors is strongly related to the Landau quantization [6] and, therefore, to the quantum Hall effect [7,8].…”

Section: Introductionmentioning

confidence: 99%

Product of Projective Representations in Description of Multi-Electron States in An External Magnetic Field

Florek

1999

Acta Phys. Pol. A

View full text Add to dashboard Cite

In this paper all inequivalent irreducible projective representations of the two-dimensional translation group for a given factor system are determined. A normalized, i.e. corresponding to the Landau gauge, factor system is considered. Obtained representations directly lead to concept of magnetic cells and to periodicity with respect to the charge of a moving particle. It is also shown that the quantization condition is imposed on the product qH of the charge q and the magnitude of magnetic field H. The Kronecker product of such representations is considered and it is proven that the multiplication of representations corresponds to the addition of charges of particles moving in a given external magnetic field. In general, coupling of d representations corresponds to d-particle states. Presented results can be applied in any problem related to two-dimensional electron gas in a magnetic field, for example in the fractional quantum Hall effect or high temperature superconductivity.

show abstract

Quantised Hall conductance in a perfect crystal

Cited by 146 publications

References 20 publications

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

Hall conductance of a two-dimensional electron gas in periodic lattice with triangular antidots

Product of Projective Representations in Description of Multi-Electron States in An External Magnetic Field

Contact Info

Product

Resources

About