Eli Fonseca, Jacob Shapiro, Ahmed Sheta, Angela Wang, Kohtaro Yamakawa
We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.
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We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We …
This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. A Fedosov-Hörmander formula implementing in Euclidean spaces an Atiyah-Singer index theorem allows for an explicit computation of the index in terms of the symbol of the Fredholm operator. For Hamiltonians admitting an appropri…
The edge spectrum of Chern insulators with rough boundaries
陈氏绝缘体是周期性带状绝缘体，其性质是它们在占据带上的投影具有非零的陈氏数。具有均匀边界的Chern绝缘体显示出充满整个绝缘间隙的连续谱。对应于这部分谱的局域态密度位于边界附近，因此称为边缘谱。一个有趣的问题出现了，即，如果一个粗糙的边界，它可以被看作是一个强随机的潜力，这些准一维状态，将破坏连续边缘谱。本文展示了如何通过一个新制定的抽象框架，其中的期望值的电流的一般观察连接到一个特定的Fredholm运营商的指数可以回答这样的问题。对于本申请，我们将连接的期望值的电荷边缘电流与指标的Fredholm算子，保持不变的任意变形下的边界。
Absolutely continuous edge spectrum of topological insulators with an odd time-reversal symmetry
该研究证明受奇时间反演对称性保护的非平凡二维拓扑绝缘体具有绝对连续边谱，借助Wold分解的时间反演对称版本分离出其弹道边缘模，为相关拓扑绝缘体的边缘态性质提供了理论证明。
Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators
研究通过分解投影值映射揭示了时间反演对称拓扑绝缘体中Z₂不变量与陈数的关联。
We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1- and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to $\mathbb{Z}_2$ topological phases. We review several propo…
The Feynman amplitudes with the two-dimensional Wess-Zumino action functional have a geometric interpretation as bundle gerbe holonomy. We present details of the construction of a distinguished square root of such holonomy and of a related 3d-index and briefly recall the application of those to the building of topological invariants for time-reversal-symmetric two- and three-dimensional crystals, both static and periodically forced.
We study disordered topological insulators with time reversal symmetry. Relying on the noncommutative index theorem which relates the Chern number to the projection onto the Fermi sea and the magnetic flux operator, we give a precise definition of the $Z_2$ index which is a noncommutative analogue of the Atiyah-Singer $Z_2$ index. We prove that the noncommutative $Z_2$ index is robust against any time-reversal symmetric perturbation including disorder potentials as long as the spectral gap at the Fermi level does not close.
Spin transport and lack of quantisation for time-reversal symmetric insulators on the honeycomb structure
研究了蜂窝结构上时间反演对称绝缘体的自旋输运，发现其自旋电导率不依赖于自旋电流选择，并非普遍量子化，与Fu-Kane-Mele指数无直接关联。
A time-reversal invariant topological insulator occupying a Euclidean half-space determines a 'Quaternionic' self-adjoint Fredholm family. We show that the discrete spectrum data for such a family is geometrically encoded in a non-trivial 'Real' gerbe. The gerbe invariant, rather than a naïve counting of Dirac points, precisely captures how edge states completely fill up the bulk spectral gap in a topologically protected manner.
Chern character and Fermi point
本文通过费米点提出拓扑K理论的Chern特征标，并证明了四维拓扑绝缘体的边缘指数偶数性和体-边对应关系。
This work concerns with the description of the topological phases of band insulators of class DIII by using the equivariant cohomology. The main result is the definition of a cohomology class for general systems of class DIII which generalizes the well known $\mathbb{Z}_2$-invariant given by the Teo-Kane formula in the one-dimension case. In the two-dimensional case this cohomology invariant allows a complete description of the strong and weak phases. The relation with the KR-theory, the Noether-Fredholm index and the classification of "Real" gerbes are also discussed.
We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to -1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a $\mathbb{Z}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four $\mathbb{Z}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indice…
We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invari…
A bounded operator $T$ on a separable, complex Hilbert space is said to be odd symmetric if $I^*T^tI=T$ where $I$ is a real unitary satisfying $I^2=-1$ and $T^t$ denotes the transpose of $T$. It is proved that such an operator can always be factorized as $T=I^*A^tIA$ with some operator $A$. This generalizes a result of Hua and Siegel for matrices. As application it is proved that the set of odd symmetric Fredholm operators has two connected components labelled by a $Z_2$-index given by the parity of the dimension of the kernel of $T$. This recovers a result of Atiyah and Singer. Two examples o…
Continuous bulk and interface description of topological insulators
该研究分析狄拉克方程组形式的拓扑绝缘体连续模型，利用Fredholm算子指标描述体块与界面拓扑性质并证明其对空间异质性扰动的稳定性，先针对二维材料提出后推广到任意维度，为拓扑与空间涨落相互作用分析提供定量工具。
We introduce a web of strongly correlated interacting 3+1D topological superconductors/insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their new SU(N) generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of field theories of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism …
Topological Anderson Insulators by homogenization theory
本文提出微扰 Dirac 方程模型，证明大幅高频微扰能诱导产生与未微扰状态不同的拓扑相，并在两相交界面处实现稳健的非对称输运，为拓扑 Anderson 绝缘体提供了基于均匀化理论与 Fredholm 算子指数表征的数学理论支撑。
When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin's Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic $Z_2$ signature, while for particle-hole symmetric systems it leads to a criterion for the exis…
Employing the Foldy-Wouthuysen transformation it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2+1 dimensions is discussed and by means of a dimensional reduction approach the 1+1 dimensional descendant of the 2+1 dimensional Chern-Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derive…
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 …
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, the fractional $S$-duality. Such $\text{QED}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($θ$) can take a nontrivial but still time-reversal invariant value $π/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversa…
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