In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the topological
space. Among the possibly many, optimal persistent cycles bring forth further information due to having guaranteed quality. However, topological features usually go through variations in the lifecycle of a bar which a single persistent cycle may not capture. Hence, for persistent homology induced from PL functions, we propose levelset persistent cycles consisting of a sequence of cycles that depict the evolution of homological features from birth to death. Our definition is based on levelset zigzag persistence which involves four types of persistence intervals as opposed to the two types in standard persistence. For each of the four types, we present a polynomial-time algorithm computing an optimal sequence of levelset persistent $p$-cycles for the so-called weak $(p+1)$-pseudomanifolds. Given that optimal cycle problems for homology are NP-hard in general, our results are useful in practice because weak pseudomanifolds do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that we exploited earlier for standard persistent cycles. Note that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case.
Tomography is a widely used tool for analyzing microstructures in three dimensions (3D). The analysis, however, faces difficulty because the constituent materials produce similar grey-scale values. Sometimes, this prompts the image segmentation proce
ss to assign a pixel/voxel to the wrong phase (active material or pore). Consequently, errors are introduced in the microstructure characteristics calculation. In this work we develop a filtering algorithm based on topological persistence, a technique used in topological data analysis. One problem faced when evaluating filtering algorithms is that real image data in general are not equipped with the ground truth information about the microstructure characteristics. For this study, we construct synthetic images for which the ground truth values are known. Specifically, we compare interconnected pore tortuosity and phase fraction. Experimental results show that our filtering algorithm provides a significant improvement in reproducing tortuosity close to the ground truth, even when the grey-scale values of the phases are similar.
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one in
strument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^omega)$ time complexity are not known, where $omega< 2.37286$ is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with $m$ additions and deletions on a graph with $n$ vertices and edges, the algorithm for $0$-dimension runs in $O(mlog^2 n+mlog m)$ time and the algorithm for 1-dimension runs in $O(mlog^4 n)$ time. The algorithm for $0$-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on $2$-manifolds. The algorithm for $1$-dimension pairs a negative edge with the earliest positive edge so that a $1$-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for $0$-dimension to compute the $(p-1)$-dimensional zigzag persistence for $mathbb{R}^p$-embedded complexes in $O(mlog^2 n+mlog m+nlog n)$ time.
We study the electronic transport properties at the intersection of three topological zero-lines as the elementary current partition node that arises in minimally twisted bilayer graphene. Unlike the partition laws of two intersecting zero-lines, we
find that (i) the incoming current can be partitioned into both left-right adjacent topological channels and that (ii) the forward-propagating current is nonzero. By tuning the Fermi energy from the charge-neutrality point to a band edge, the currents partitioned into the three outgoing channels become nearly equal. Moreover, we find that current partition node can be designed as a perfect valley filter and energy splitter controlled by electric gating. By changing the relative electric field magnitude, the intersection of three topological zero-lines can transform smoothly into a single zero line, and the current partition can be controlled precisely. We explore the available methods for modulating this device systematically by changing the Fermi energy, the energy gap size, and the size of central gapless region. The current partition is also influenced by magnetic fields and the system size. Our results provide a microscopic depiction of the electronic transport properties around a unit cell of minimally twisted bilayer graphene and have far-reaching implications in the design of electron-beam splitters and interferometer devices.
Magnetic hopfion is three-dimensional (3D) topological soliton with novel spin structure that would enable exotic dynamics. Here we study the current driven 3D dynamics of a magnetic hopfion with unit Hopf index in a frustrated magnet. Attributed to
spin Berry phase and symmetry of the hopfion, the phase space entangles multiple collective coordinates, thus the hopfion exhibits rich dynamics including longitudinal motion along the current direction, transverse motion perpendicular to the current direction, rotational motion and dilation. Furthermore, the characteristics of hopfion dynamics is determined by the ratio between the non-adiabatic spin transfer torque parameter and the damping parameter. Such peculiar 3D dynamics of magnetic hopfion could shed light on understanding the universal physics of hopfions in different systems and boost the prosperous development of 3D spintronics.
We theoretically study the finite-size effects in the dynamical response of a quantum anomalous Hall insulator in the disk geometry. Semi-analytic and numerical results are obtained for the wavefunctions and energies of the disk within a continuum Di
rac Hamiltonian description subject to a topological infinite mass boundary condition. Using the Kubo formula, we obtain the frequency-dependent longitudinal and Hall conductivities and find that optical transitions between edge states contribute dominantly to the real part of the dynamic Hall conductivity for frequency values both within and beyond the bulk band gap. We also find that the topological infinite mass boundary condition changes the low-frequency Hall conductivity to $ e^2/h $ in a finite-size system from the well-known value $ e^2/2h $ in an extended system. The magneto-optical Faraday rotation is then studied as a function of frequency for the setup of a quantum anomalous Hall insulator mounted on a dielectric substrate, showing both finite-size effects of the disk and Fabry-Perot resonances due to the substrate. Our work demonstrates the important role played by the boundary condition in the topological properties of finite-size systems through its effects on the electronic wavefunctions.
We theoretically investigate a folded bilayer graphene structure as an experimentally realizable platform to produce the one-dimensional topological zero-line modes. We demonstrate that the folded bilayer graphene under an external gate potential ena
bles tunable topologically conducting channels to be formed in the folded region, and that a perpendicular magnetic field can be used to enhance the conducting when external impurities are present. We also show experimentally that our proposed folded bilayer graphene structure can be fabricated in a controllable manner. Our proposed system greatly simplifies the technical difficulty in the original proposal by considering a planar bilayer graphene (i.e., precisely manipulating the alignment between vertical and lateral gates on bilayer graphene), laying out a new strategy in designing practical low-power electronics by utilizing the gate induced topological conducting channels.
Chiral magnets give rise to the anti-symmetric Dzyaloshinskii-Moriya (DM) interaction, which induces topological nontrivial textures such as magnetic skyrmions. The topology is characterized by integer values of the topological charge. In this work,
we performed the Monte-Carlo calculation of a two-dimensional model of the chiral magnet. A surprising upturn of the topological charge is identified at high fields and high temperatures. This upturn is closely related to thermal fluctuations at the atomic scale, and is explained by a simple physical picture based on triangulation of the lattice. This emergent topology is also explained by a field-theoretic analysis using $CP^{1}$ formalism.
Earlier Monte-Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY t
ransition, i.e. one with a dynamical critical exponent $z=1$. The transition changes from $z=1$ to the class of criticality with $z to infty$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.
The experimental study of the modulation of the envelope of spin-echo signals due to internal and external fields is an important spectroscopic tool to detect very small internal magnetic fields. We derive the free induction decay and the frequency s
pectrum and amplitude of spin-echo signals for arbitrary orientation of fields with respect to crystalline axis for nuclei in a crystal of orthorhombic symmetry. Results reproduce the results that no modulation should be observed in tetragonal crystals for fields either along the c-axis or any direction in the basal plane and give details of the signal as a function of the orthorhombicity parameter. They are used to discuss recent experimental results and provide guidelines for future experiments.
