Quantum Transport in Disordered Systems Under Magnetic Fields: A Study Based on Operator Algebras

Prodan, Emil

doi:10.1093/amrx/abs017

Cited by 42 publications

(61 citation statements)

References 74 publications

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“…Using Hölder's inequality (17), the multi-linear functional is seen to be continuous in the standard Sobolev norm · d+1,1 . Due to (21), the functional is also continuous w.r.t. · ′ d+1,1 .…”

Section: The Index Theorem In the Strong Disorder Regimementioning

confidence: 99%

“…In [17], the position-space formula (3) was proposed as a phase label for disordered systems in the chiral unitary class. Furthermore, for explicit 1 and 3-dimensional topological models from the chiral unitary class, the invariant was evaluated numerically in the presence of strong disorder [17,27], by implementing techniques from [21]. It was found that Ch d (U ) remains quantized and non-fluctuating as the disorder strength is increased, up to a critical disorder strength where the localization length of the system diverges and the invariant sharply changes its value.…”

Section: Introductionmentioning

confidence: 99%

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Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Prodan

Schulz-Baldes

2016

Journal of Functional Analysis

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Abstract. An index theorem for higher Chern characters of odd Fredholm modules over crossed product algebras is proved, together with a local formula for the associated cyclic cocycle. The result generalizes the classic NoetherGohberg-Krein index theorem, which in its simplest form states that the winding number of a complex-valued function over the circle is equal to the index of the associated Toeplitz operator. When applied to the non-commutative Brillouin zone, this generalization allows to define topological invariants for all condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields, whenever the Fermi level lies in a region of Anderson localized spectrum.

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Section: The Index Theorem In the Strong Disorder Regimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Prodan

Schulz-Baldes

2016

Journal of Functional Analysis

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“…Our method is inspired by previous mathematical works on disordered tight-binding models, which can be classified into two distinct concepts. First, an algebraic treatment of electronic transport in disordered systems 14,15 that allows for a rigorous definition of quantum-mechanical operators in a disordered material. Second, the fact that local tight-binding models create exponentially localized observables, that is, they make it possible to controllably remove finite-size and edge effects from calculations 16 .…”

Section: Introductionmentioning

confidence: 99%

Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle

et al. 2017

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The ability in experiments to control the relative twist angle between successive layers in twodimensional (2D) materials offers a new approach to manipulating their electronic properties; we refer to this approach as "twistronics". A major challenge to theory is that, for arbitrary twist angles, the resulting structure involves incommensurate (aperiodic) 2D lattices. Here, we present a general method for the calculation of the electronic density of states of aperiodic 2D layered materials, using parameter-free hamiltonians derived from ab initio density-functional theory. We use graphene, a semimetal, and MoS2, a representative of the transition metal dichalcogenide (TMDC) family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multi-layered stacks in the presence of local disorder and external fields. We comment on the interesting features of their Density of States (DoS) as a function of twist-angle and local configuration and on how these features can be experimentally observed.

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“…[5,[28][29][30][31]. It is more convenient to work with the homotopically equivalent flatband Hamiltonian: Q ¼ P þ − P − , where P AE are the projectors onto the positive or negative energy spectrum.…”

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confidence: 99%

Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder

et al. 2014

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The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

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Quantum Transport in Disordered Systems Under Magnetic Fields: A Study Based on Operator Algebras

Cited by 42 publications

References 74 publications

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle

Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder

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