Connection between the winding number and the Chern number
Han-Ting Chen, Chia-Hsun Chang, Hsien-chung Kao
备注：
13 pages, 39 figures, Major revision. Connection between the winding number and the Chern number has been shown. The part related to the modification of the Zak phase has been deleted since there may be some errors
Bulk-edge correspondence is one of the most distinct properties of topological insulators. In particular, the 1D winding number $\n$ has a one-to-one correspondence to the number of edge states in a chain of topological insulators with boundaries. By properly choosing the unit cells, we carry out numerical calculation to show explicitly in the extended SSH model that the winding numbers corresponding to the left and right unit cells may be used to predict the numbers of edge states on the two boundaries in a finite chain. Moreover, by drawing analogy between the SSH model and QWZ model, we show that the extended SSH model may be generalized to the extended QWZ model. By integrating the ``magnetic field'' over the momentum strip $0\le p_2 \le π, 0\le p_1 2π$ in the Brillouin zone, we show a identity relating the 2D Chern number and the difference between the 1D winding numbers at $p_2=0 $ and $p_2 =π$.
Chern numbers can be calculated within a frame of vortex fields related to phase conventions of a wave function. In a band protected by gaps the Chern number is equivalent to the total number of flux carrying vortices. In the presence of topological defects like Dirac cones this method becomes problematic, in particular if they lack a well-defined winding number. We develop a scheme to include topological defects into the vortex field frame. A winding number is determined by the behavior of the phase in reciprocal space when encircling the defect's contact point. To address the possible lack o…
Explicit equivalence between the spectral localizer and local Chern and winding markers
KTH Royal Institute of Technology（皇家理工学院）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基克科学基金会）；Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、内耳研究所）
Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles: the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties…
We construct useful sets of one-particle states in the quantum Hall system based on the von Neumann lattice. Using the set of momentum states, we develop a field-theoretical formalism and apply the formalism to the system subjected to a periodic potential. The topological formula of the Hall conductance written by the winding number of propagator is generalized to Bloch electrons. The relation between the winding number and the Chern number is clarified.
Micromotion area as proxy for anomalous Floquet topological systems
Kyoto University（京都大学）；University of Hamburg（汉堡大学）；The Hamburg Centre for Ultrafast Imaging（汉堡超快成像中心）；Technische Universität Berlin（柏林工业大学）；TU Dortmund University（多特蒙德工业大学）
Real topological phases featuring real Chern numbers and second-order boundary modes have been a focus of current research, but finding their material realization remains a challenge. Here, based on first-principles calculations and theoretical analysis, we reveal the already experimentally synthesized three-dimensional (3D) graphdiyne as the first realistic example of the recently proposed second-order real nodal-line semimetal. We show that the material hosts a pair of real nodal rings, each protected by two topological charges: a real Chern number and a 1D winding number. The two charges ge…
We propose a two-dimensional non-Hermitian Chern insulator with inversion symmetry, which is anisotropic and has staggered gain and loss in both x and y directions. In this system, conventional bulk-boundary correspondence holds. The Chern number is a topological invariant that accurately predicts the topological phase transition and the existence of helical edge states in the topologically nontrivial gapped phase. In the gapless phase, the band touching points are isolated and protected by the symmetry. The degenerate points alter the system topology, and the exceptional points can destroy th…
Optical fiber is a ubiquitous and indispensable component in communications, sensing, biomedicine and many other lightwave technologies and applications. Here we propose topological one-way fibers to remove two fundamental mechanisms that limit fiber performance: scattering and reflection. We design three-dimensional~(3D) photonic crystal fibers, inside which photons propagate only in one direction, that are completely immune to Rayleigh and Mie scatterings and significantly suppress the nonlinear Brillouin and Raman scatterings. A one-way fiber is also free from Fresnel reflection, naturally …
The winding number has been widely used as an invariant for diagnosing topological phases in one-dimensional chiral-symmetric systems. We put forward a real-space representation for the winding number. Remarkably, our method reproduces an exactly quantized winding number even in the presence of disorders that break translation symmetry but preserve chiral symmetry. We prove that our real-space representation of the winding number, the winding number defined through the twisted boundary condition, and the real-space winding number derived previously in [Phys. Rev. Lett. 113, 046802 (2014)], are…
We introduce a simple method to realize and detect photonic topological Chern insulators with one-dimensional circiut quantum electrodynamics arrays. By periodically modulating the couplings of the array, we show that this one-dimensional model can be mapped into a two-dimensional Chern insulator model. In addition to allowing the study of photonic Chern insulators, this approach also provides a natural platform to realise experimentally Laughlin's pumping argument. Based on scattering theory of topological insulators and input-output formalism, we show that the photonic edge state can be prob…
A magnetic helix can be wound into a classical Heisenberg chain by fixing one end while rotating the other one. We show that in quantum Heisenberg chains \new{of finite length}, the magnetization slips back to the trivial state beyond a finite turning angle. Avoided level crossings thus undermine classical topological protection. Yet, for special values of the axial Heisenberg anisotropy, stable spin helices form again, which are non-locally entangled. Away from these sweet spots, spin helices can be stabilized dynamically or by dissipation. For half-integer spin chains of odd length, a spin s…
The topology of electronic states in band insulators with mirror symmetry can be classified in two different ways. One is in terms of the mirror Chern number, an integer that counts the number of protected Dirac cones in the Brillouin zone of high-symmetry surfaces. The other is via a $\mathbb{Z}_2$ index that distinguishes between systems that have a nonzero quantized orbital magnetoelectric coupling ("axion-odd"), and those that do not ("axion-even"); this classification can also be induced by other symmetries in the magnetic point group, including time reversal and inversion. A systematic c…
Generalized Dirac monopoles in momentum space are constructed in even d+1 dimensions from the Weyl Hamiltonian in terms of Green's functions. In 3+1 spacetime dimensions, the (unit) charge of the monopole is equal to both the winding number and the Chern number, expressed as the integral of the Berry curvature. Based on the equivalence of the Chern and winding numbers, a chirally coupled field theory action is proposed for the Weyl semimetal phase. At the one loop order, the effective action yields both the chiral magnetic effect and the anomalous Hall effect. The Chern number appears as a coe…
A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group $π_{3}(S^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number $C_{i}\in π_{2}(S^{2})=\mathbb{Z}$ to another insulator with the Chern number $C_{f} $, the preimage of the Hamiltonian vector forms links having the Hopf number $C_{f}-C_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second-Chern number $N\in π_{4}(S^…
Topology has appeared in different physical contexts. The most prominent application is topologically protected edge transport in condensed matter physics. The Chern number, the topological invariant of gapped Bloch Hamiltonians, is an important quantity in this field. Another example of topology, in polarization physics, are polarization singularities, called L lines and C points. By establishing a connection between these two theories, we develop a novel technique to visualize and potentially measure the Chern number: it can be expressed either as the winding of the polarization azimuth alon…
We propose a method of measuring topological invariants of a photonic crystal through phase spectroscopy. We show how the Chern numbers can be deduced from the winding numbers of the reflection coefficient phase. An explicit proof of existence of edge states in system with nonzero reflection phase winding number is given. The method is illustrated for one- and two-dimensional photonic crystals of nontrivial topology.
Topological quantum phases of matter have been a topic of intense interest in contemporary condensed matter physics. Extensive efforts are devoted to investigate various exotic properties of topological matters including topological insulators, topological superconductors, and topological semimetals. For topological insulators, the dissipationless transport via gapless helical edge or surface states is supposed to play a defining role, which unfortunately has proved difficult to realize in experiments due to inevitable backscattering induced in the sample boundary. Motivated by the fundamental…
The winding numbers for the even d+1 spacetime dimensional Weyl Hamiltonians are calculated in terms of the related Green's functions. It is shown that these winding numbers result in the divergence of the Dirac monopole fields, hence they are equal to the unit topological charge. It is demonstrated that the winding numbers are also equal to the Chern numbers which are expressed as the integral of the Berry field strength. Explicit calculations are presented for the 3+1 and 5+1 dimensional cases. Relevance of these topological invariants for the physical systems like the semiclassical chiral k…
Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including non-minimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by …
We report on a new topological vortex solution in U(1)$\times$U(1) Maxwell-Chern-Simons theory. The existence of the vortex is envisaged by analytical means, and a numerical solution is obtained by integrating the equations of motion. These vortices have a long-range force because one of the U(1)s remains unbroken in the infrared, which is guarded by the Coleman-Hill theorem. The sum of the winding numbers of an ensemble of vortices has to vanish; otherwise the system would have a logarithmically divergent energy. In turn, these vortices exhibit classical confinement. We investigate the rich p…
The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well defined global section. This property implies that the Uhlmann connection is topologically trivial and as a consequence, the corresponding Chern character vanishes. We propose modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related …
Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices, that can also be viewed as representing non-equilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet (CF) winding number. This begs the question: How can a model without Chern number describe IQH systems? We resolve this apparent paradox by showing that non-zero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, de…
This work unveils how geometric features of two-band non-Hermitian Hamiltonians can completely classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, where a singularity ring (SR) captures the degeneracy structure of generic exceptional points. The SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our …
Anomalous topological phases in a Chern insulator connected to leads in a cylindrical geometry
International Center for Theoretical Sciences (ICTS)（国际理论科学中心）；Indian Institute of Science Education and Research, Pune（印度科学教育研究所浦那分校）；Advanced Institute for Materials Research (WPI-AIMR), Tohoku University（东北大学先进材料研究机构）；University of Kashmir（克什米尔大学）；Indian Institute of Science（印度理工学院）；The Institute of Mathematical Sciences（数学科学研究院）
The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the quasienergy gaps. We propose a scheme for measuring these winding numbers in a system of fermionic cold atoms in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of experimentally feasible dri…
A hallmark feature of topological physics is the presence of one-way propagating chiral modes at the system boundary. The chirality of edge modes is a consequence of the topological character of the bulk. For example, in a non-interacting quantum Hall model, edge modes manifest as mid-gap states between two topologically distinct bulk bands. The bulk-boundary correspondence dictates that the number of chiral edge modes, a topological invariant called the winding number, is completely determined by the bulk topological invariant, the Chern number. Here, for the first time, we measure the windin…
The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time-evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this winding number in terms of local data associated to the the eigenvalue crossings of such a map over a three dimensional manifold, based on an idea from Nathan and Rudner, New Journal of Physics, 17(12) 125014, 2015. We show that, up to homotopy, the crossings are a finite set of points and non degene…
