Towards the fractional quantum Hall effect: a noncommutative geometry perspective

Marcolli, Matilde; Mathai, Varghese

doi:10.1007/978-3-8348-0352-8_12

Cited by 29 publications

(45 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can obtain this way a model of the fractional quantum Hall effect via noncommutative geometry (cf. [152], [153], [154]), where one uses hyperbolic geometry to simulate the interactions. The noncommutative geometry approach to the quantum Hall effect described above was extended to hyperbolic geometry in [37].…”

Section: Brillouin Zone and The Quantum Hall Effectmentioning

confidence: 99%

A Walk in the Noncommutative Garden

Connes

Marcolli

2008

An Invitation to Noncommutative Geometry

Self Cite

188

331

View full text Add to dashboard Cite

Section: Brillouin Zone and The Quantum Hall Effectmentioning

confidence: 99%

A Walk in the Noncommutative Garden

Connes

Marcolli

2008

An Invitation to Noncommutative Geometry

Self Cite

188

331

View full text Add to dashboard Cite

“…If the Hamiltonian H has a spectrum bounded from below, then each gap in the spectrum gives rise to a projector P <E onto the Eigenspaces with Eigenvalues less than any fixed value E in the gap, see e.g. [4,30]. The gap labeling then associates the K-theory class of P <E to the gap.…”

Section: Spanning Treesmentioning

confidence: 99%

Singular Geometry of the Momentum Space: From wire networks to quivers and monopoles

Kaufmann

Khlebnikov²,

Wehefritz-Kaufmann

2016

jsing

View full text Add to dashboard Cite

Abstract.A new nano-material in the form of a double gyroid has motivated us to study (non)-commutative C * geometry of periodic wire networks and the associated graph Hamiltonians.Here we present a general more abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. The resulting effective C * -geometry is that of the momentum space, which parameterizes the quasi-momenta.This geometry is usually singular, where the singularities describe so-called band intersections in physics. We give geometric and algebraic methods to study these intersections; their origin being singularity theory and representation theory. A technique we newly apply to this situation is the use of topological invariants, which we formalize and explain in the paper. This uses K-theory and Chern classes as well as "slicing methods" for their computation. In this method the invariants can be computed using Berry's connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.Adding a constant magnetic field or allowing projective representations makes the C * geometry non-commutative. In this case, we can also use K-theory, albeit in a different way, to make statements about the band structure using gap labeling.

show abstract

“…We recall briefly the definition and properties of twisted group C * -algebras, as this will be useful in the following. For a similar overview and applications to the case of Fuchsian groups see [20].…”

Section: Twisted Group Algebras and The Magnetic Laplacianmentioning

confidence: 99%

“…We recall here briefly the relation between spectral theory of Harper operators and K-theory of twisted group C * -algebras (cf. [6], [20] §3). We then proceed in the following section to analyze the specific case of C * (S(Λ, V )).…”

Section: 7mentioning

confidence: 99%

Solvmanifolds and noncommutative tori with real multiplication

Marcolli

2008

Communications in Number Theory and Physics

Self Cite

View full text Add to dashboard Cite

Abstract. We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained by deforming the fiber tori to noncommutative spaces. The 3-dimensional solvmanifold is the homotopy quotient in the sense of Baum-Connes of the noncommutative space obtained as the crossed product of the noncommutative torus by the action of the units of the real quadratic field. This noncommutative space is identified with the twisted group C * -algebra of the fundamental group of the 3-manifold. The twisting can be interpreted as the cocycle arising from a magnetic field, as in the theory of the quantum Hall effect. We prove a twisted index theorem that computes the range of the trace on the K-theory of this noncommutative space and gives an estimate on the gaps in the spectrum of the associated Harper operator.

show abstract

Towards the fractional quantum Hall effect: a noncommutative geometry perspective

Cited by 29 publications

References 35 publications

A Walk in the Noncommutative Garden

A Walk in the Noncommutative Garden

Singular Geometry of the Momentum Space: From wire networks to quivers and monopoles

Solvmanifolds and noncommutative tori with real multiplication

Contact Info

Product

Resources

About