Energy Concentration in a Two-Dimensional Magnetic Skyrmion Model: Variational Analysis of Lattice and Continuum Theories

Low-lying eigenmodes of the Wilson–Dirac operator and correlations with topological objects

Recent progress in the treatment of Dirac fields in lattice gauge theory has allowed the chiral symmetry and associated anomaly on the lattice to be discussed in a manner similar to that in continuum theory. In particular, the index theorem on the lattice can be discussed. The analysis of the index theorem on the discrete lattice itself has certain subtle aspects, but lattice theory deals with completely regularized quantities, and thus some of the subtle aspects in continuum theory are now given a more rigorous basis. It is explained that all the results of chiral anomalies in continuum theory are reproduced in a suitable continuum limit of lattice gauge theory, providing a uniform and consistent treatment of both continuum and lattice theories.

Lateral phase drift of the topological charge density in stochastic optical fields

An infinite-dimensional family of analytic solutions in pure SU(2) Yang–Mills theory at finite density in (3+1) dimensions is constructed. It is labelled by two integeres (p and q) as well as by a two-dimensional free massless scalar field. The gauge field depends on all the 4 coordinates (to keep alive the topological charge) but in such a way to reduce the (3+1)-dimensional Yang–Mills field equations to the field equation of a 2D free massless scalar field. For each p and q, both the on-shell action and the energy-density reduce to the action and Hamiltonian of the corresponding 2D CFT. The topological charge density associated to the non-Abelian Chern–Simons current is non-zero. It is possible to define a non-linear composition within this family as if these configurations were “Lego blocks”. The non-linear effects of Yang–Mills theory manifest themselves since the topological charge density of the composition of two solutions is not the sum of the charge densities of the components. This leads to an upper bound on the amplitudes in order for the topological charge density to be well-defined. This suggests that if the temperature and/or the energy is/are high enough, the topological density of these configurations is not well-defined anymore. Semiclassically, one can show that (depending on whether the topological charge is even or odd) some of the operators appearing in the 2D CFT should be quantized as Fermions (despite the Bosonic nature of the classical field).

A physical interpretation of the eigenmodes of the Ginsparg-Wilson type Dirac operator

We apply a machine learning technique for identifying the topological charge of quantum gauge configurations in four-dimensional SU(3) Yang–Mills theory. The topological charge density measured on the original and smoothed gauge configurations with and without dimensional reduction is used as inputs for neural networks (NNs) with and without convolutional layers. The gradient flow is used for the smoothing of the gauge field. We find that the topological charge determined at a large flow time can be predicted with high accuracy from the data at small flow times by the trained NN; for example, the accuracy exceeds $99\%$ with the data at $t/a^2\le0.3$. High robustness against the change of simulation parameters is also confirmed with a fixed physical volume. We find that the best performance is obtained when the spatial coordinates of the topological charge density are fully integrated out in preprocessing, which implies that our convolutional NN does not find characteristic structures in multi-dimensional space relevant for the determination of the topological charge.

We discuss the connection between supersymmetric field theories and topological field theories and show how this connection may be used to construct local lattice field theories which maintain an exact supersymmetry. It is shown how metric independence of the continuum topological field theory allows us to derive the lattice theory by blocking out of the continuum in a deformed geometry. This, in turn allows us to prove the cut-off independence of certain supersymmetric Ward identities. 1.

We derive a generalized set of Ward identities that captures the effects of topological charge on Hall transport. The Ward identities follow from the (2+1)-dimensional momentum algebra, which includes a central extension proportional to the topological charge density. In the presence of topological objects like Skyrmions, we observe that the central term leads to a direct relation between the thermal Hall conductivity and the topological charge density. We extend this relation to incorporate the effects of a magnetic field and an electric current. The topological charge density produces a distinct signature in the electric Hall conductivity, which is identified in existing experimental data and yields further novel predictions. For insulating materials with translation invariance, the Hall viscosity can be directly determined from the Skyrmion density and the thermal Hall conductivity to be measured as a function of momentum.

The importance of instanton (1) and monopole (2) solutions of the Yang-Mills (YM) equations seems to be out of question nowadays (3). While monopoles may exist as physical objects (as already speculated by DIRAC}, the physical meaning of instanton solutions is not so clear, although they seem to be very relevant in the problem of quark confinement. Nevertheless, using only regular solutions of the YM equations, one cannot get a sufficiently convincing explanation of the experimental fact that quarks are confined. This has brought many people to the search for singular solutions as merons (4) and singular monopoles, located at the zeros of the fourth component of the gauge field A~ (which in this formulation replaces the Higgs field)(5). ADLER has recently obtained monopolelike solutions of the YM equations in the presence of quark sources hut ignoring spin currents. He has derived configurations which display various geometries for the zero set, namely points, strings, spheres and 3-dimensional sets. Of greatest importance are finite-string configurations carrying the topological charge at the extremes, where quark insertion is possible {remember that topological charge density is maximum in the vicinity of quarks). To be noticed is the fact that, in 2and 3-dimensional extended zero sets, a topological density can appear on surfaces. But in strings {including Dirac's monopole string) the topological charge is always calculated as a local integral over the string ends, the contribution of the string itself being null or unphysical (5.6). Moreover, explicit string solutions of the YM equations have been always obtained in the limit of no t ime dependence (static limit). In the present paper we construct exact, string-singular solutions of the YM equations without external sources, but in the full 4-dimensional Euclidean space E 4.

Inherently global nature of topological charge fluctuations in QCD

We study the autocorrelations of observables constructed from the topological charge density, such as the topological charge on a time slice or in a subvolume, using a series of hybrid Monte Carlo simulations of pure SU(3) gauge theory with both periodic and open boundary conditions. We show that the autocorrelation functions of these observables obey a simple diffusion equation and we measure the diffusion coefficient, finding that it scales like the square of the lattice spacing. We use this result and measurements of the rate of tunneling between topological charge sectors to calculate the scaling behavior of the autocorrelation times of these observables on periodic and open lattices. There is a characteristic lattice spacing at which open boundary conditions become worthwhile for reducing autocorrelations and we show how this lattice spacing is related to the diffusion coefficient, the tunneling rate, and the lattice Euclidean time extent.

Optical vortex density limitation

Magnetic skyrmions are spatially localized whirls of spin moments in two dimensions, featuring a nontrivial topological charge and a well-defined topological charge density. We demonstrate that the quantum dynamics of magnetic skyrmions is governed by a dipole conservation law associated with the topological charge, akin to that in fracton theories of excitations with constrained mobility. The dipole conservation law enables a natural definition of the collective coordinate to specify the skyrmion’s position, which ultimately leads to a greatly simplified equation of motion in the form of the Thiele equation. In this formulation, the skyrmion mass, whose existence is often debated, actually vanishes. As a result, an isolated skyrmion is intrinsically pinned to be immobile and cannot move at a constant velocity. In a spin-wave theory, we show that such dynamics corresponds to a precise cancellation between a highly nontrivial motion of the quasiclassical skyrmion spin texture and a cloud of quantum fluctuations in the form of spin waves. Given this quenched kinetic energy of quantum skyrmions, we identify close analogies to the bosonic quantum Hall problem. In particular, the topological charge density is shown to obey the Girvin-MacDonald-Platzman algebra that describes neutral modes of the lowest Landau level in the fractional quantum Hall problem. Consequently, the conservation of the topological dipole suggests that magnetic skyrmion materials offer a promising platform for exploring fractonic phenomena with close analogies to fractional quantum Hall states.

Low-dimensional long-range topological structure in the QCD vacuum

The first analytic topologically non-trivial solutions in the (3 + 1)-dimensional gauged non-linear sigma model representing multi-solitons at finite volume with manifest ordered structures generating their own electromagnetic field are presented. The complete set of seven coupled non-linear field equations of the gauged non-linear sigma model together with the corresponding Maxwell equations are reduced in a self-consistent way to just one linear Schrodinger-like equation in two dimensions. The corresponding two dimensional periodic potential can be computed explicitly in terms of the solitons profile. The present construction keeps alive the topological charge of the gauged solitons. Both the energy density and the topological charge density are periodic and the positions of their peaks show a crystalline order. These solitons describe configurations in which (most of) the topological charge and total energy are concentrated within three-dimensional tube-shaped regions. The electric and magnetic fields vanish in the center of the tubes and take their maximum values on their surface while the electromagnetic current is contained within these tube-shaped regions. Electromagnetic perturbations of these families of gauged solitons are shortly discussed.