Emil Prodan, Bryan Leung, Jean Bellissard
The theory of the higher Chern numbers in the presence of strong disorder is developed. Sharp quantization and homotopy invariance conditions are provided. The relevance of the result to the field of strongly disordered topological insulators is discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We establish a general local formula computing the topological anomaly of gauge theories in the framework of non-commutative geometry.
An odd index theorem for higher odd Chern characters of crossed product algebras is proved. It generalizes the Noether-Gohberg-Krein index theorem. Furthermore, a local formula for the associated cyclic cocycle is provided. When applied to the non-commutative Brillouin zone, this allows to define topological invariants for 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 region of Anderson localization.
REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the …
This review deals with strongly disordered topological insulators and covers some recent applications of a well established analytic theory based on the methods of Non-Commutative Geometry (NCG) and developed for the Integer Quantum Hall-Effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit 2-dimensional models, one for a Chern insulator and one for a Quantum spin-Hall insulator, we first gi…
5d-4d Correspondence in Twisted M-theory on a Conifold
本文在conifold背景下将扭量M理论描述为5维非对易Chern-Simons-物质理论，构造4维非对易规范手征WZW模型并证明其与体理论的规范不变耦合，从而建立并推广了5d-4d对应。
The problem of the consistent definition of gauge theories living on the non-commutative (NC) spaces with a non-constant NC parameter $Θ(x)$ is discussed. Working in the L$_\infty$ formalism we specify the undeformed theory, $3$d abelian Chern-Simons, by setting the initial $\ell_1$ brackets. The deformation is introduced by assigning the star commutator to the $\ell_2$ bracket. For this initial set up we construct the corresponding L$_\infty$ structure which defines both the NC deformation of the abelian gauge transformations and the field equations covariant under these transformations. To c…
The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quant…
Alain Connes' Non-Commutative Geometry program [Connes 1994] has been recently carried out [Prodan, Leung, Bellissard 2013, Prodan, Schulz-Baldes 2014] for the entire A- and AIII-symmetry classes of topological insulators, in the regime of strong disorder where the insulating gap is completely filled with dense localized spectrum. This is a short overview of these results, whose goal is to highlight the methods of Non-Commutative Geometry involved in these studies. The exposition proceeds gradually through the cyclic cohomology, quantized calculus with Fredholm-modules, local formulas for the …
We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $θ$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $θ$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reduc…
A non-commutative formula for the isotropic magneto-electric response of disordered insulators under magnetic fields is derived using the methods of non-commutative geometry. Our result follows from an explicit evaluation of the Ito derivative with respect to the magnetic field of the non-commutative formula for the electric polarization reported in Ref. 1. The quantization, topological invariance and connection to a second Chern number of the magneto-electric response are discussed in the context of 3-dimensional, disordered, time-reversal or inversion symmetric topological insulators.
Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus. The associated Chern numbers (topological charges) arise to be non-integer, in the commutative limit the well known integer Chern numbers of the complex line bundles over the 2-sphere are recovered.
Non-Commutative Gauge Theory at the Beach
本文证明非交换五维Chern-Simons理论可紧致化为KP方程的拉格朗日表述，其无色散极限由Poisson-Chern-Simons理论描述，且所有树级振幅消失，表面缺陷上的顶点代数在极限下收缩为$w_{1+\infty}$。
These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants (c,G,Λ,\hbar) enter as deformation parameters. The classical situation is summarised, where solutions can be characterised in terms of model spacetimes (which depend on c and Λ), together with global identifications via elements of the corresponding isometry groups. The quantum theory may be viewed as a deformation of this picture, with quantum groups replac…
Topological quantum numbers account for the precise quantization that occurs in the integer Hall effect. In this theory, Kubo's formula for the conductance acquires a topological interpretation in terms of Chern numbers and their non-commutative analog, the Fredholm Indices.
When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known. In this article we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation [A. H. Chamseddine, A. Connes, and V. Mukhanov, Journal of High Energy Physics, 98 (2014)] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on …
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 the general 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. In these geometric situations, the non- commutativity is introduced by a constant magnetic field and the theory splits into two pieces: commutative and non-commutative, both of which are governed by a $C^*$ geometry. In the n…
We discuss a new class of strong homotopy algebras constructed via inner deformations. Such deformations have a number of remarkable properties. In the simplest case, every one-parameter family of associative algebras leads to an $L_\infty$-algebra that can be used to construct a classical integrable model. Another application of this class of $L_\infty$-algebras is related with the three-dimensional bosonization duality in Chern--Simons vector models, where it implements the idea of the slightly-broken higher spin symmetry. One large class of associative algebras originates from Deformation Q…
Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L$_\infty$ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS$_5$ sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere ca…
Chern numbers in quantum graphs
通过将二维方格量子图映射为单顶点双环磁通图，解析推导能带简并条件，揭示拓扑相变并计算陈数相图。
Lattice models forming bands with higher Chern number offer an intriguing possibility for new phases of matter with no analogue in continuum Landau levels. Here, we establish the existence of a number of new bulk insulating states at fractional filling in flat bands with Chern number $C=N>1$, forming in a recently proposed pyrochlore model with strong spin-orbit coupling. In particular, we find compelling evidence for a series of stable states at $ν=1/(2N+1)$ for fermions as well as bosonic states at $ν=1/(N+1)$. By examining the topological ground state degeneracies and the excitation structu…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。