Nontrivial Topological Structures Research Articles

Topology of musical data

The musical realm is a promising area in which to expect to find nontrivial topological structures. This paper describes several kinds of metrics on musical data, and explores the implications of these metrics in two ways: via techniques of classical topology where the metric space of all-possible musical data can be described explicitly, and via modern data-driven ideas of persistent homology which calculates the Betti-number barcodes of individual musical works. Both analyses are able to recover three well-known topological structures in music: the circularity of octave-reduced musical scales, the circle of fifths, and the rhythmic repetition of timelines. Applications to a variety of musical works (for example, folk music in the form of standard MIDI files) are presented, and the barcodes show many interesting features. Examples show that individual pieces may span the complete space (in which case the classical and the data-driven analyses agree), or they may span only part of the space.

Nontrivial topological electronic structures in a single Bi(111) bilayer on different substrates: A first-principles study

Electronic structures, minimum energy configurations, and band topology of strained Bi(111) single bilayers placed on a variety of semiconducting and insulating substrates are investigated using first-principles calculations. A topological phase diagram of a free-standing Bi bilayer is presented to help guide the selection of suitable substrates. The insulating hexagonal-BN is identified as the best candidate substrate material for supporting nontrivial topological insulating phase of Bi bilayer thin films. A planar hexagonal Bi layer is predicted under tensile strain, which we show could be realized on a SiC substrate. The Bi bilayer becomes metallic under the compressive strain induced by Si and Ge substrates.

Symmetry-protected -quantization and quaternionic Berry connection with Kramers degeneracy

As for a generic parameter-dependent Hamiltonian with time reversal (TR) invariance, a non-Abelian Berry connection with Kramers (KR) degeneracy is introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many-body system with KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without KR degeneracy. Natural dimensions to have nontrivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without KR degeneracy, which describe the monopoles of Dirac and Yang. We have shown that the Yang monopole is literally a quaternionic Dirac monopole.The generic Berry phases with/without KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry-protected -quantization of these general Berry phases, a sufficient condition of the -quantization is given as the inversion/reflection equivalence.Topological charges of the SO(3) and SO(5) nonlinear σ-models are discussed in relation to the Chern numbers of the CP1 and HP1 models as well.

An investigation of quantum transport by the free-electron network model: Resonance and interference effects

Quantum coherent transport through a multiply-connected network is investigated by the free-electron network model (FENM). Within this model, we study π -conjugated molecules such as benzenedithiol (BDT) in order to understand the influence of nontrivial topological structures on the transport behavior. The analytical solutions for transmission functions and I– V characteristics of the simplest networked conjugated molecules are derived. Moreover, quantum effects such as resonance and interference are clearly revealed in this approach. We have also compared our FENM approach with the non-equilibrium Green’s function (NEGF) method within tight-binding calculation.

Localizations on complex networks

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.

Kinks in discrete light cone quantization

We investigate non-trivial topological structures in discrete light cone quantization (DLCQ) through the example of the broken symmetry phase of the two-dimensional φ4 theory using antiperiodic boundary condition (APBC). We present evidence for degenerate ground states which is both a signature of spontaneous symmetry breaking and mandatory for the existence of kinks. Guided by a constrained variational calculation with a coherent state ansatz, we then extract the vacuum energy and kink mass and compare with classical and semi-classical results. We compare the DLCQ results for the number density of bosons in the kink state and the Fourier transform of the form factor of the kink with corresponding observables in the coherent variational kink state.

Structural and electrical properties of TaSe 3 ring crystals

We have investigated structural and electrical properties of “ribbon-knot” crystals of TaSe 3 (superconductor) having nontrivial topological structures. They are shaped a ring, a Möbius strip and a 2 π-twisted loop, so they can be categorized by a ribbon knot topological invariant in the mathematics field. For investigation of structural property of the ring crystals, cutting operation was performed for 45 specimens of the TaSe 3 ring crystal using focused ion beam milling. The cutting operation transformed them to arc-shape crystals having homogeneous curvatures. At low temperature, thin ring crystals of TaSe 3 showed superconductivity, however, thick rings showed nonmetallic behavior. The experimental results suggest that the ring-shape structure affect the electrical behavior in the TaSe 3 ring crystal.

Topological isomerism : Should rotaxanes, endohedral fullerene complexes and in-out isomers of hydrogenated fullerenes be considered as such?

Rotaxanes are not considered to be topological isomers of their separated fragments. However, due to their similarity to catenanes, they are often discussed within the realm of topological chemistry. Similarly, “in” fullerene isomers (endohedral complexes, nested fullerenes, and the isomers of hydrogenated fullerenes) have not, until recently, been treated as topological isomers of the corresponding “out” isomers. In this paper the problem of topological isomerism is analysed departing from the standard topological notions of arbitrary continuous deformations imposed on the system under investigation as a sufficient condition for topological isomerism. Similarly to the isomer classification into conformational and configurational isomers, the energy barrier between the trivial (e. g., separated rings) and non-trivial (e. g., catenated) topological structures should be invoked into such classifications to limit allowed deformations of the systems under investigation. A precise mathematical model of the classification, based on topological invariants, is presented in an Appendix. Contrary to commonly accepted definitions, the problem of topological isomerism is analysed departing from the notions of arbitrary continuous deformations imposed on the system as a sufficient condition for the isomerism. The energy barrier between the trivial (e. g., separated fragments) and non-trivial (e. g., catenated) topological structures is invoked to limit allowed deformations and include the titles molecules into the realm of topological chemistry.

Decorrelating the topology in full QCD

We investigate the performance of the hybrid Monte Carlo algorithm in updating non-trivial global topological structures. We find that the hybrid Monte Carlo algorithm has serious problems decorrelating the global topological charge. This represents a warning which must be seriously considered when simulating full QCD, regardless of the number and type of fermions, with this or any similar algorithm. Simulated tempering is examined as a means of accelerating the decorrelation.

Hybrid Monte Carlo and topological modes of full QCD

We investigate the performance of the hybrid Monte Carlo algorithm, the standard algorithm used for lattice QCD simulations involving fermions, in updating non-trivial global topological structures. We find that the hybrid Monte Carlo algorithm has serious problems decorrelating the global topological charge. This represents a warning which must be seriously considered when simulating full QCD by hybrid Monte Carlo.