Nontrivial Loop Research Articles

Generation of Homotopy Classes for Unconstrained 3D Wire Routing from Characteristic Loops

Routing of wire harnesses in complex machinery is a complicated problem. In this paper, we present an approach for computing all non-homotopic paths characteristic of all the homotopy classes associated with a pair of source–destination points embedded on the surface of the product. We introduce the notion of routing graphs, which are generated from specific non-trivial loops in the first homology group associated with the surface of the product. The routing graph, which is constructed only once, encodes the homotopy of the routing environment and is, therefore, used to find all the non-homotopic paths between multiple source–destination pairs. Based on the path embedding requirements, we classify the routing process into (1) On-Surface Routing, when the paths are constrained to the surface of the product, and (2) In-Air Routing, when the paths are allowed to venture into the product ambiance space. In the former case, the routing graph is defined by both the handle and tunnel loops in the first homology group, whereas in the latter case, it is defined by only the handle loops. We propose a linking number-based strategy for computing the handle and tunnel loops for a given closed surface. The methods are extended for routing in assemblies using the so-called shell structures associated with them. The results obtained demonstrate the efficacy of the proposed approach. The results show that the computation of the routing graph is very efficient, and the homotopy classes can be generated in very little time using the proposed approach.

Pair density wave and loop current promoted by Van Hove singularities in moiré systems

We theoretically show that in the presence of conventional or higher order Van Hove singularities (VHS), the bare finite momentum pairing, also known as the pair density wave (PDW), susceptibility can be promoted to the same order of the most divergent bare BCS susceptibility through a valley-contrasting flux $3\ensuremath{\phi}$ in each triangular plaquette at $\ensuremath{\phi}=\ensuremath{\pi}/3$ and $\ensuremath{\pi}/6$ in moir\'e systems. This makes the PDW order a possible leading instability for an electronic system with repulsive interactions. We confirm that it indeed wins over all other instabilities and becomes the ground state under certain conditions through the renormalization group calculation and a flux insertion argument. Moreover, we also find that a topological nontrivial loop current order becomes the leading instability if the Fermi surface with conventional VHS is perfectly nested at $\ensuremath{\phi}=\ensuremath{\pi}/3$. Similar to the Haldane model, this loop current state has the quantum anomalous Hall effect. If we dope this loop current state or introduce a finite next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, the chiral $d$-wave PDW becomes the dominant instability. Experimentally, the flux can be effectively tuned by an out-of-plane electric field in moir\'e systems based on graphene and transition metal dichalcogenides.

Coupled Dynamical Phase Transitions in Driven Disk Packings.

Under the influence of oscillatory shear, a monolayer of frictional granular disks exhibits two dynamical phase transitions: a transition from an initially disordered state to an ordered crystalline state and a dynamic active-absorbing phase transition. Although there is no reason apriori for these to be at the same critical point, they are. The transitions may also be characterized by the disk trajectories, which are nontrivial loops breaking time-reversal invariance.

Eigenvalue topology of non-Hermitian band structures in two and three dimensions

In the band theory for non-Hermitian systems, the energy eigenvalues, which are complex, can exhibit non-trivial topology which is not present in Hermitian systems. In one dimension, it was recently noted theoretically and demonstrated experimentally that the eigenvalue topology is classified by the braid group. The classification of eigenvalue topology in higher dimensions, however, remained an open question. Here, we give a complete description of eigenvalue topology in two and three dimensional systems, including the gapped and gapless cases. We reduce the topological classification problem to a purely computational problem in algebraic topology. In two dimensions, the Brillouin zone torus is punctured by exceptional points, and each nontrivial loop in the punctured torus acquires a braid group invariant. These braids satisfy the constraint that the composite of the braids around the exceptional points is equal to the commutator of the braids on the fundamental cycles of the torus. In three dimensions, there are exceptional knots and links, and the classification depends on how they are embedded in the Brillouin zone three-torus. When the exceptional link is contained in a contractible ball, the classification can be expressed in terms of the knot group of the link. Our results provide a comprehensive understanding of non-Hermitian eigenvalue topology in higher dimensional systems, and should be important for the further explorations of topologically robust open quantum and classical systems.

Adiabatic cycles of quantum spin systems

Motivated by the $\mathrm{\ensuremath{\Omega}}$-spectrum proposal of unique gapped ground states by Kitaev, we study adiabatic cycles in gapped quantum spin systems from various perspectives. We give a few exactly solvable models in one and two spatial dimensions and discuss how nontrivial adiabatic cycles are detected. For one spatial dimension, we study the adiabatic cycle in detail with the matrix product state and show that the symmetry charge can act on the space of matrices without changing the physical states, which leads to nontrivial loops with symmetry charges. For generic spatial dimensions, based on the Bockstein isomorphism ${H}^{d}(G,U(1))\ensuremath{\cong}{H}^{d+1}(G,\mathbb{Z})$, we study a group cohomology model of the adiabatic cycle that pumps a symmetry-protected topological phase on the boundary by one period. It is shown that the spatial texture of the adiabatic Hamiltonian traps a symmetry-protected topological phase in one dimension lower.

Chains of rotating boson stars

Boson stars are stationary, axially symmetric solutions of a complex scalar field theory coupled to gravity. Recently, multisolitonic configurations interpreted as static chains of multiple boson stars bound by gravity and carrying no angular momentum were reported. We propose to generalize those solutions to the stationary case by constructing chains of rotating boson stars and analyze their properties. The nonlinear elliptic field equations are solved using the finite element method. We find that chains with an even number of constituents exhibit the same spiral-like frequency dependence of their mass, angular momentum and Noether charge as single boson stars. In contrast, sequences of chains with an odd number of constituents show nontrivial loops starting and ending at the flat vacuum. As a consequence, such solutions cannot be uniquely parametrized by a single parameter. We conjecture that all rotating chains correspond to excitations of single boson stars or pairs of them. We also analyze their flat space limit and find that they reduce to chains of $Q$-balls.

Distinguishing phases via non-Markovian dynamics of entanglement in topological quantum codes under parallel magnetic field

We investigate the static and the dynamical behavior of localizable entanglement and its lower bounds on nontrivial loops of topological quantum codes with parallel magnetic field. Exploiting the connection between the stabilizer states and graph states in the absence of the parallel field and external noise, we identify a specific measurement basis, referred to as the canonical measurement basis, that optimizes localizable entanglement when measurement is restricted to single-qubit Pauli measurements only, thereby providing a lower bound. We also propose an approximation of the lower bound that can be computed for larger systems according to the computational resource in hand. Additionally, we compute a lower bound of the localizable entanglement that can be computed by determining the expectation value of an appropriately designed witness operator. We study the behavior of these lower bounds in the vicinity of the topological to nontopological quantum phase transition of the system, and perform a finite-size scaling analysis. We also investigate the dynamical features of these lower bounds when the system is subjected to Markovian or non-Markovian single-qubit dephasing noise. We find that in the case of the non-Markovian dephasing noise, at large time, the canonical measurement-based lower bound oscillates with a larger amplitude when the initial state of the system undergoing dephasing dynamics is chosen from the nontopological phase, compared to the same for an initial state from the topological phase. These features can be utilized to distinguish the topological phase of the system from the nontopological phase in the presence of dephasing noise.

Berry phase in the rigid rotor: Emergent physics of odd antiferromagnets

The rigid rotor is a classic problem in quantum mechanics, describing the dynamics of a rigid body with its center of mass held fixed. It is well known to describe the rotational spectra of molecules. The configuration space of this problem is SO(3), the space of all rotations in three dimensions. This is a topological space with two types of closed loops: trivial loops that can be adiabatically shrunk to a point and nontrivial loops that cannot. In the traditional formulation of the problem, stationary states are periodic over both types of closed loops. However, periodicity conditions may change if Berry phases are introduced. We argue that time-reversal symmetry allows for only one new possibility---a Berry phase of $\ensuremath{\pi}$ attached to all nontrivial loops. We derive the corresponding stationary states by exploiting the connection between SO(3) and SU(2) spaces. The solutions are antiperiodic over any nontrivial loop; i.e., stationary states reverse sign under a $2\ensuremath{\pi}$ rotation about any axis. Remarkably, this framework is realized in the low-energy physics of certain quantum magnets. The magnets must satisfy the following conditions: (a) the classical ground states are unpolarized, carrying no net magnetization, (b) the set of classical ground states is indexed by SO(3), and (c) the product $N\ifmmode\times\else\texttimes\fi{}S$ is a half-integer, where $N$ is the number of spins and $S$ is the spin quantum number. We demonstrate this result in a family of Heisenberg antiferromagnets defined on polygons with an odd number of vertices. At each vertex, we have a spin-$S$ moment that is coupled to its nearest neighbors. In the classical limit, these magnets have coplanar ground states. Their quantum spectra, at low energies, correspond to ``spherical top'' and ``symmetric top'' rigid rotors. For integer values of $S$, we recover traditional rigid-rotor spectra. With half-integer $S$, we obtain rotor spectra with a Berry phase of $\ensuremath{\pi}$.

Topological Analysis of Syntactic Structures

We use the persistent homology method of topological data analysis and dimensional analysis techniques to study data of syntactic structures of world languages. We analyze relations between syntactic parameters in terms of dimensionality, of hierarchical clustering structures, and of non-trivial loops. We show there are relations that hold across language families and additional relations that are family-specific. We then analyze the trees describing the merging structure of persistent connected components for languages in different language families and we show that they partly correlate to historical phylogenetic trees but with significant differences. We also show the existence of interesting non-trivial persistent first homology groups in various language families. We give examples where explicit generators for the persistent first homology can be identified, some of which appear to correspond to homoplasy phenomena, while others may have an explanation in terms of historical linguistics, corresponding to known cases of syntactic borrowing across different language subfamilies.

Topological phenomena demonstrated in photorefractive photonic lattices [Invited

Topological photonics has attracted widespread research attention in the past decade due to its fundamental interest and unique manner in controlling light propagation for advanced applications. Paradigmatic approaches have been proposed to achieve topological phases including topological insulators in a variety of photonic systems. In particular, photonic lattices composed of evanescently coupled waveguide arrays have been employed conveniently to explore and investigate topological physics. In this article, we review our recent work on the demonstration of topological phenomena in reconfigurable photonic lattices established by site-to-site cw-laser-writing or multiple-beam optical induction in photorefractive nonlinear crystals. We focus on the study of topological states realized in the celebrated one-dimensional Su-Schrieffer-Heeger lattices, including nonlinear topological edge states and gap solitons, nonlinearity-induced coupling to topological edge states, and nonlinear control of non-Hermitian topological states. In the two-dimensional case, we discuss two typical examples: universal mapping of momentum-space topological singularities through Dirac-like photonic lattices and realization of real-space nontrivial loop states in flatband photonic lattices. Our work illustrates how photorefractive materials can be employed conveniently to build up various synthetic photonic microstructures for topological studies, which may prove relevant and inspiring for the exploration of fundamental phenomena in topological systems beyond photonics.

Topological order of the Rys F-model and its breakdown in realistic square spin ice: Topological sectors of Faraday loops

Both the Rys F-model and antiferromagnetic square ice possess the same ordered, antiferromagnetic ground state, but the ordering transition is of second order in the latter, and of infinite order in the former. To tie this difference to topological properties and their breakdown, we introduce a Faraday lines representation where loops carry the energy and magnetization of the system. Because the F-model does not admit monopoles, its Faraday loops have distinct topological properties, absent in square ice, and which allow for a natural partition of its phase space into topological sectors. Then, the Néel temperature corresponds to a transition from topologically trivial to non-trivial Faraday loops. Because magnetization is a homotopy invariant of the Faraday loops, and it is zero for topologically trivial ones, the susceptibility is zero below a critical field. In square spin ice, instead, monopoles destroy the homotopy invariance and the parity distinction among loops, thus erasing this rich topological structure. Consequently, even trivial loops can be magnetized in square ice, and their susceptibility is never zero.

On Small n-Hawaiian Loops

We define small n-Hawaiian loop as a special case of small pointed map, and study the group consisting of classes of small n-Hawaiian loops, $${\mathcal {H}}_n^s(X, x_0)$$ . As an example of spaces with non-trivial small 1-Hawaiian loops, we present 1st Hawaiian group of harmonic archipelago space as a special quotient group of 1st Hawaiian group of 1-dimensional Hawaiian earring. We define whisker topology on nth Hawaiian group which is Hausdorff if and only if $${\mathcal {H}}_n^s(X, x)$$ is trivial. We also prove that on metric spaces, null-homotopies can become small enough if and only if natural homomorphism maps $${\mathcal {H}}_n(X, x_0)$$ isomorphically on $$L_n(X, x_0)$$ if and only if $${\mathcal {H}}_n^s(X, x_0)$$ is trivial. Thus in spaces without non-trivial small n-Hawaiian loop, n-SLT paths transfer $${\mathcal {H}}_n$$ isomorphically along the points. We show that harmonic archipelago is an example with non-trivial small 1-Hawaiian loop and non-isomorphic 1st Hawaiian groups at two different points. Then we define n-SHLT paths which transfers nth Hawaiian group isomorphically along the points.

Phonon manipulation with valley-Hall junctions

In this work, we investigate experimentally the existence of in-plane valley Hall edge states in hexagonal lattices with relaxed space inversion symmetry and we exploit them to realize non-trivial backscattering-free interfaces. We propose special tessellations of lattice domains in which multiple non-trivial domain walls coalesce to form junctions, and we study how guided waves behave when they impinge on such junctions. We show that, through a proper selection of the interface types and orientations, it is possible to achieve junctions with different and exotic wave manipulation capabilities. Specifically, we discuss two applications. The first is a direction-selective splitter, which endows its host lattice with asymmetric wave transport characteristics. The second is a signal delayer that is realized by embedding in the lattice a non-trivial waveguide loop, along which the energy can be temporarily trapped and periodically released. These junctions can serve as building blocks for a new structural logic paradigm enabled by topological mechanics.

Enhanced higher harmonic generation from nodal topology

Among topological materials, nodal loop semimetals (NLSMs) are arguably the most topologically sophisticated, with their valence and conduction bands intersecting along arbitrarily intertwined nodes. But unlike the well-known topological band insulators with quantized edge conductivities, nodal loop materials possess topologically nontrivial Fermi surfaces, not bands. Hence an important question arises: Are there also directly measurable or even technologically useful physical properties characterizing nontrivial nodal loop topology? In this work, we provide an affirmative answer by showing, for the first time, that nodal linkages protect the higher harmonic generation (HHG) of electromagnetic signals. Specifically, nodal linkages enforce non-monotonicity in the intra-band semi-classical response of nodal materials, which will be robust against perturbations preserving the nodal topology. These nonlinearities distort incident radiation and produce higher frequency peaks in the teraHertz (THz) regime, as we quantitatively demonstrate for a few known nodal materials. Since THz sources are not yet ubiquitous, our new mechanism for HHG will greatly aid applications like material characterization and non-ionizing imaging of object interiors.

The fibre of the degree 3 map, Anick spaces and the double suspension

AbstractLet S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Automatic block decomposition based on dual surfaces

A high-quality block structure of a 3D model can support many important applications; however, the automated generation of a high-quality block structure is still a challenging problem. In this paper, a dual surface-based approach to automated and valid block decomposition of 3D models is proposed. First, dual loops for block decomposition are generated with the help of a computed frame field whose three types of degenerated singularities are corrected. Then, a required dual surfaces set is constructed to suitably separate all the boundary elements of the 3D model and singularities of the frame field with the help of the basis of the non-trivial loops on the boundary surfaces. Finally, a valid block structure is obtained by performing dual operations along the dual surfaces on the hex mesh generated by splitting the tetrahedral mesh of the 3D model. Experimental results showed the effectiveness of the proposed approach.

Peculiarities of magnetic ordering in the S=5/2 two-dimensional square-lattice antimonate NaMnSbO4

An orthorhombic compound, NaMnSbO4, represents a square net of magnetic Mn2+ ions residing in vertex-shared oxygen octahedra. Its static and dynamic magnetic properties were studied using magnetic susceptibility, specific heat, magnetization, electron spin resonance (ESR), nuclear magnetic resonance (NMR) and density functional calculations. Thermodynamic data indicate an establishment of the long-range magnetic order with TN about 44 K, which is preceded by a short-range one at about 55 K. In addition, a non-trivial wasp-waisted hysteresis loop of the magnetization was observed, indicating that the ground state is most probably canted antiferromagnetic. Temperature dependence of the magnetic susceptibility is described reasonably well in the framework of 2D square lattice model with the main exchange parameter J = -5.3 K, which is in good agreement with density functional analysis, NMR and ESR data.

Localization, Topology, and Quantized Transport in Disordered Floquet Systems.

We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.

Heavy quark diffusion in a Polyakov loop plasma

We calculate the transport coefficients, drag and momentum diffusion, of a heavy quark in a thermalized plasma of light quarks in the background of Polyakov loop. Quark thermal mass and the gluon Debye mass are calculated in a non-trivial Polyakov loop background. The constituent quark masses and the Polyakov loop is estimated within a Polyakov loop quark meson (PQM) model. The relevant scattering amplitudes for heavy quark and light partons in the background of Polyakov loop has been estimated within the matrix model. We have also compared the results with the Polyakov loop parameter estimated from lattice QCD simulations. We have studied the temperature and momentum dependence of heavy quark drag and diffusion coefficients. It is observed that the temperature dependence of the drag coefficient is quite weak which may play a key role to understand heavy quark observables at RHIC and LHC energies.

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

We study various statistical properties of the double-dimer model, a generalization of the dimer model, on rectangular domains of the square lattice. We take advantage of the Grassmannian representation of the dimer model, first to calculate the probability distribution of the number of nontrivial loops around a cylinder, which is consistent with the previously known result, and then to calculate the expectation value of the number of loops surrounding two faces and the left-passage probability, both in the discrete and the continuum cases. We also briefly explain the calculation of some related observables. As a by-product, we obtain the partition function of the dimer model in the presence of two and four monomers, and a single monomer on the boundary.