Terms Of Topological Invariants Research Articles

Maximally non-integrable almost complex structures: an h-principle and cohomological properties

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2nge 10 (respectively ge 6) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).

The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields

AbstractWe revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an explicit description of the cohomology groups of generalised (‘harmonic’) Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet–Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [42] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.

Intrinsic sign problem in fermionic and bosonic chiral topological matter

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of gapped, chiral, topological phases of matter. Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or 'qudits'), and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge, and the topological spins of anyons. We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and non-chiral topological matter.

Topological invariants, zero mode edge states and finite size effect for a generalized non-reciprocal Su-Schrieffer-Heeger model

Intriguing issues in one-dimensional non-reciprocal topological systems include the breakdown of usual bulk-edge correspondence and the occurrence of half-integer topological invariants. In order to understand these unusual topological properties, we investigate the topological phase diagrams and the zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger model, based on some analytical results. Meanwhile, we provide a concise geometrical interpretation of the bulk topological invariants in terms of two independent winding numbers and also give an alternative interpretation related to the linking properties of curves in three-dimensional space. For the system under the open boundary condition, we construct analytically the wavefunctions of zero-mode edge states by properly considering a hidden symmetry of the system and the normalization condition with the use of biorthogonal eigenvectors. Our analytical results directly give the phase boundary for the existence of zero-mode edge states and unveil clearly the evolution behavior of edge states. In comparison with results via exact diagonalization of finite-size systems, we find our analytical results agree with the numerical results very well.

Persistent Cohomology for Data With Multicomponent Heterogeneous Information.

Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and reveal certain interaction patterns in terms of topological invariants. However, in addition to the geometric information, there is a wide variety of nongeometric information of molecular structures, such as element types, atomic partial charges, atomic pairwise interactions, and electrostatic potential functions, that is not described by persistent homology. Although element-specific homology and electrostatic persistent homology can encode some nongeometric information into geometry based topological invariants, it is desirable to have a mathematical paradigm to systematically embed both geometric and nongeometric information, i.e., multicomponent heterogeneous information, into unified topological representations. To this end, we propose a persistent cohomology based framework for the enriched representation of data. In our framework, nongeometric information can either be distributed globally or reside locally on the datasets in the geometric sense and can be properly defined on topological spaces, i.e., simplicial complexes. Using the proposed persistent cohomology based framework, enriched barcodes are extracted from datasets to represent heterogeneous information. We consider a variety of datasets to validate the present formulation and illustrate the usefulness of the proposed method based on persistent cohomology. It is found that the proposed framework outperforms or at least matches the state-of-the-art methods in the protein-ligand binding affinity prediction from massive biomolecular datasets without resorting to any deep learning formulation.

Topological Invariants in Terms of Green’s Function for the Interacting Kitaev Chain**Supported by the National Natural Science Foundation of China under Grant No 11274379, and the Research Funds of Renmin University of China under Grant No 14XNLQ07.

A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green’s function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green’s functions can distinguish more topological nonequivalent phases than the fermion parity.

Schwarzschild instanton in emergent gravity

In the bottom-up approach of emergent gravity, we attempt to find symplectic gauge fields emerging from Euclidean Schwarzschild instanton, which is studied as electromagnetism defined on the symplectic space [Formula: see text]. Geometrical engineering with the emergent metric sets up the Seiberg–Witten map between commutative and non-commutative gauge fields, preparing the ground for the evaluation of topological invariants in terms of the underlying gauge theory quantities.

Persistent Homology Analysis of RNA

Abstract Topological data analysis has been recently used to extract meaningful information frombiomolecules. Here we introduce the application of persistent homology, a topological data analysis tool, for computing persistent features (loops) of the RNA folding space. The scaffold of the RNA folding space is a complex graph from which the global features are extracted by completing the graph to a simplicial complex via the notion of clique and Vietoris-Rips complexes. The resulting simplicial complexes are characterised in terms of topological invariants, such as the number of holes in any dimension, i.e. Betti numbers. Our approach discovers persistent structural features, which are the set of smallest components to which the RNA folding space can be reduced. Thanks to this discovery, which in terms of data mining can be considered as a space dimension reduction, it is possible to extract a new insight that is crucial for understanding the mechanism of the RNA folding towards the optimal secondary structure. This structure is composed by the components discovered during the reduction step of the RNA folding space and is characterized by minimum free energy.

Arrangements of spheres and projective spaces

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections and the topology of the tangent bundle complement. In the present paper, we find a closed form formula for the homotopy type of the complement and express some of its topological invariants in terms of the associated combinatorial information.

Stability of theX−Yphase of the two-dimensionalC4point group insulator

Noninteracting insulating electronic states of matter can be classified according to their symmetries in terms of topological invariants which can be related to effective surface theories. These effective surface theories are in turn topologically protected against the effects of disorder. Topological crystalline insulators are, on the other hand, trivial in the sense of the above classification but still possess surface modes. In this work we consider an extension of the Bernevig-Hughes-Zhang model that describes a topological crystalline insulator. We explicitly show that the surface properties of this state can be as robust as in topologically nontrivial insulators, but only if the $S_z$-component of the spin is conserved. However, in the presence of Rashba spin-orbit coupling this protection vanishes and the surface states localize, even if the crystalline symmetries are intact on average.

From the adiabatic theorem of quantum mechanics to topological states of matter

Abstractmagnified imageOwing to the enormous interest the rapidly growing field of topological states of matter (TSM) has attracted in recent years, the main focus of this review is on the theoretical foundations of TSM. Starting from the adiabatic theorem of quantum mechanics which we present from a geometrical perspective, the concept of TSM is introduced to distinguish gapped many body ground states that have representatives within the class of non‐interacting systems and mean field superconductors, respectively, regarding their global geometrical features. These classifying features are topological invariants defined in terms of the adiabatic curvature of these bulk insulating systems. We review the general classification of TSM in all symmetry classes in the framework of K‐Theory. Furthermore, we outline how interactions and disorder can be included into the theoretical framework of TSM by reformulating the relevant topological invariants in terms of the single particle Green's function and by introducing twisted boundary conditions, respectively. We finally integrate the field of TSM into a broader context by distinguishing TSM from the concept of topological order which has been introduced to study fractional quantum Hall systems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Magnetic spectrum of trigonally warped bilayer graphene: Semiclassical analysis, zero modes, and topological winding numbers

We investigate the fine structure in the energy spectrum of bilayer graphene in the presence of various stacking defaults, such as a translational or rotational mismatch. This fine structure consists of four Dirac points that move away from their original positions as a consequence of the mismatch and eventually merge in various manners. The different types of merging are described in terms of topological invariants (winding numbers) that determine the Landau-level spectrum in the presence of a magnetic field as well as the degeneracy of the levels. The Landau-level spectrum is, within a wide parameter range, well described by a semiclassical treatment that makes use of topological winding numbers. However, the latter need to be redefined at zero energy in the high-magnetic-field limit as well as in the vicinity of saddle points in the zero-field dispersion relation.

Quantization and fractional quantization of currents in periodically driven stochastic systems. I. Average currents

This article studies Markovian stochastic motion of a particle on a graph with finite number of nodes and periodically time-dependent transition rates that satisfy the detailed balance condition at any time. We show that under general conditions, the currents in the system on average become quantized or fractionally quantized for adiabatic driving at sufficiently low temperature. We develop the quantitative theory of this quantization and interpret it in terms of topological invariants. By implementing the celebrated Kirchhoff theorem we derive a general and explicit formula for the average generated current that plays a role of an efficient tool for treating the current quantization effects.

Soft topological objects in topological media

Topological invariants in terms of the Green's function in momentum and real space determine properties of smooth textures within topological media. In space dimension D=1 the topological invariant N_3 in terms of the Green's function G(\omega,k_x,x) determines the fermion number of the 1D soliton, while in space dimension D=3 the topological invariant N_5 in terms of the Green's function G(\omega,k_x,k_y,k_z,z) determines quantization of Hall conductivity in the soliton plane within the topological insulators.

Single-particle Green’s functions and interacting topological insulators

We study topological insulators characterized by the integer topological invariant $\mathbb{Z}$, in even and odd spatial dimensions. These are well understood in the case when there are no interactions. We extend the earlier work on this subject to construct their topological invariants in terms of their Green's functions. In this form, they can be used even if there are interactions. Specializing to one and two spatial dimensions, we further show that if two topologically distinct topological insulators border each other, the difference of their topological invariants is equal to the difference between the number of zero-energy boundary excitations and the number of zeros of the Green's function at the boundary. In the absence of interactions Green's functions have no zeros; thus, there are always edge states at the boundary, as is well known. In the presence of interactions, in principle, Green's functions could have zeros. In that case, there could be no edge states at the boundary of two topological insulators with different topological invariants. This may provide an alternative explanation to the recent results on one-dimensional interacting topological insulators.

Inequalities and homotopy relations in reaction topology

Global properties of potential energy hypersurfaces are analyzed using a topological model of reaction mechanisms. Homotopies (continuous deformations) of hypersurfaces are used to describe relations between reaction mechanisms in terms of topological invariants. Path-homotopy relations in open sets of a topological space (M, TC), where these open sets of manifold M represent chemical structures, are suggested for the analysis of topologically significant details of reaction mechanisms. In multiply connected subsets of M path-homotopy classes imply that the contribution of a single chemical structure to an overall reaction may lead to a variety of topologically nonequivalent reaction mechanisms, even if the reaction involves a fixed sequence of chemical structures.

On divisorial filtrations on sheaves

In this paper, we generalize the notion of Poincare series of a multi-index divisorial filtration corresponding to a collection of sigma-processes to the case of an arbitrary locally-free sheaf on the space of blow-ups of the complex plane ℂ2. For an arbitrary sheaf, we establish a representation of the series in terms of topological invariants of the sheaf. In particular, for the sheaf of functions, this representation coincides with the Poincare series obtained by Gusein—Zade and Delgado.

Elastic energy of liquid crystals in convex polyhedra

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler-Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratio is varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.

The topological approach to perceptual organization

To address the fundamental question of “what are the primitives of visual perception”, a theory of topological structure and functional hierarchy in visual perception has been proposed. This holds that the global nature of perceptual organization can be described in terms of topological invariants, global topological perception is prior to the perception of other featural properties, and the primitives of visual form perception are geometric invariants at different levels of structural stability. In Part I of this paper, I will illustrate why and how the topological approach to perceptual organization has been advanced. In Part II, I will provide empirical evidence supporting the early topological perception, while answering some commonly considered counteraccounts. In Part III, to complete the theory, I will apply the mathematics of tolerance spaces to describe global properties in discrete sets. In Part IV, I will further present experimental data to demonstrate the global-to-local functional hierarchy in form perception, which is stratified with respect to structural stability defined by Klein's Erlangen Program. Finally, in Part V, I will discuss relations of the global-to-local topological model to other theories: The topological approach reformulates both classical Gestalt holism and Gibson's direct perception of invariance, while providing a challenge to computational approaches to vision based on the local-to-global assumption.

Elastic energy of liquid crystals in convex polyhedra

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler–Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratios are varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.