Definition Of Topological Charge Research Articles

Triple degenerate point in three dimensions: Theory and realization

Dirac and Weyl points in condensed matters could resemble the behaviors of their counterparts, Dirac and Weyl fermions, in high-energy physics. However, some fermions have no analog in fundamental fermions. They are expected to render novel physical properties. In this work we study an important fermion in condensed matters which is threefold degenerate. Such a fermion has no counterpart in high-energy physics and is enforced by crystalline symmetry. We perform a systematic search over all 230 space groups with time-reversal symmetry but without spin-orbital coupling (SOC). We find that a triple degenerate point (TDP) intrinsically appears in a three-dimensional irreducible representation. Based on the order of their energy dispersion, TDPs can be classified into two types, namely linear and quadratic TDPs. Furthermore, the linear TDP can be classified into two categories depending on the definition of topological charge. To distinguish topologies of linear and quadratic TDPs, we study their Landau spectrum under a magnetic field. Notably, these TDPs present dramatically different properties. There exists two chiral Landau levels (LLs) for linear TDPs with nonzero topological charge, whereas no chiral LL appears for the quadratic TDPs. Given the novel physical properties of TDPs, we provide concrete materials to support our discoveries. Consequently, our work provides a platform to study the triple degenerate points in condensed matters.

Comparison of topological charge definitions in Lattice QCD

In this paper, we show a comparison of different definitions of the topological charge on the lattice. We concentrate on one small-volume ensemble with 2 flavours of dynamical, maximally twisted mass fermions and use three more ensembles to analyze the approach to the continuum limit. We investigate several fermionic and gluonic definitions. The former include the index of the overlap Dirac operator, the spectral flow of the Wilson–Dirac operator and the spectral projectors. For the latter, we take into account different discretizations of the topological charge operator and various smoothing schemes to filter out ultraviolet fluctuations: the gradient flow, stout smearing, APE smearing, HYP smearing and cooling. We show that it is possible to perturbatively match different smoothing schemes and provide a well-defined smoothing scale. We relate the smoothing parameters for cooling, stout and APE smearing to the gradient flow time tau . In the case of hypercubic smearing the matching is performed numerically. We investigate which conditions have to be met to obtain a valid definition of the topological charge and susceptibility and we argue that all valid definitions are highly correlated and allow good control over topology on the lattice.

Noncanonical perfect vortex beams enabled via nonuniformly varying phase gradient

A method is proposed for generating noncanonical perfect vortex beams (PVBs) based on a theory deduced from the definition of topological charge. Varying the phase gradient of the light field in the region of the decomposed integral provides arbitrary control over the nonuniform phase distribution in the transverse plane. To correct the distorted beam shape, the traditional “perfect” method of controlling the global amplitude distribution is modified and used to adjust the local beam diameter and produce what are known as noncanonical scalar PVBs. Moreover, noncanonical vector PVBs with tunable azimuthal polarization can be obtained by combining two orthogonal circularly polarized scalar PVBs. The results of experiments and numerical simulations show that the structure of the inhomogeneous phase and polarization can be tuned arbitrarily.

Topology and index theorem with a generalized Villain lattice action – a test in 2d

Using 2-d U(1) lattice gauge theory we study two definitions of the topological charge constructed from a generalized Villain action and analyze the implementation of the index theorem based on the overlap Dirac operator. One of the two definitions expresses the topological charge as a sum of the Villain variables and treats charge conjugation symmetry exactly, making it particularly useful for studying related physics. Our numerical analysis establishes that for both topological charge definitions the index theorem becomes exact quickly towards the continuum limit.

Simple description of generalized electromagnetic and gravitational hopfions

The generalization of electromagnetic and gravitational hopfions is performed in terms of a complex scalar field. New definition of topological charge for linearized gravity is given. Quasi-local (super-)energy densities are compared for gravitational hopfion.

Critical analysis of topological charge determination in the background of center vortices inSU(2)lattice gauge theory

We analyze topological charge contributions from classical SU(2) center vortices with shapes of planes and spheres using different topological charge definitions, namely the center vortex picture of topological charge, a discrete version of F\~{F} in the plaquette and hypercube definitions and the lattice index theorem. For the latter the zeromodes of the Dirac operator in the fundamental and adjoint representations using both the overlap and asqtad staggered fermion formulations are investigated. We find several problems for the individual definitions and discuss the discrepancies between the different topological charge definitions. Our results show that the interpretation of topological charge in the background of center vortices is rather subtle.

Comparing topological charge definitions using topology fixing actions

We investigate both the hyperbolic action and the determinant ratio action designed to fix the topological charge on the lattice. We show to what extent topology is fixed depending on the parameters of these actions, keeping the physical situation fixed. At the same time the agreement between different definitions of topological charge - the field theoretic and the index definition - is directly correlated to the degree topology is fixed. Moreover, it turns out that the two definitions agree very well. We also study finite volume effects arising in the static potential and related quantities due to topology fixing.

The index theorem on the lattice with improved fermion actions

We consider a class of lattice fermion actions with improved chiral properties. We show that, for arbitrarily rough gauge fields, they satisfy the index theorem if we identify the zero-modes with the small real eigenvalues of the fermion operator and use the standard geometrical definition of topological charge. We present a numerical study of the simplest of these improved operators in the quenched Schwinger model. The problem of exceptional configurations and the sign problem in simulations with an odd number of dynamical Wilson fermions are briefly discussed.

Critical comparison of different definitions of topological charge on the lattice

A detailed comparison is made between the field-theoretic and geometric definitions of topological charge density on the lattice. Their renormalizations with respect to continuum are analysed. The definition of the topological susceptibility, as used in chiral Ward identities, is reviewed. After performing the subtractions required by it, the different lattice methods yield results in agreement with each other. The methods based on cooling and on counting fermionic zero modes are also discussed.

Coexistence of monopoles and instantons for different topological charge definitions and lattice actions

We compute instanton sizes and study correlation functions between instantons and monopoles in maximum abelian projection within .SU(2) lattice QCD at finite temperature. We compare several definitions of the topological charge, different lattice actions and methods of reducing quantum fluctuations. The average instanton size turns out to be σ ≈ 0.2 fm. The correlation length between monopoles and instantons is ξ ≈ 0.25 fm and hardly affected by lattice artifacts as dislocations. We visualize several specific gauge field configurations and show directly that there is an enhanced probability for finding monopole loops in the vicinity of instantons. This feature is independent of the topological charge definition used.

Reversal of substituent influence in the odd positions and the charge change alternation

The atomic π σ relations have been used to demonstrate the presence of charge change alternation in substituted conjugated hydrocarbons and the reversal of substituent influence in their odd positions. By using topological charge definitions but not the Mulliken one, ab initio calculations may also show the charge alternation, i.e. the alternation of the absolute carbon charges within these molecules.

Monte Carlo simulation of CPN-1 models.

Numerical simulations of two-dimensional ${\mathrm{CP}}^{N\ensuremath{-}1}$ models are performed at $N=2, 10, \mathrm{and} 21$. The lattice action adopted depends explicitly on the gauge degrees of freedom and shows precocious scaling. Our tests of scaling are the stability of dimensionless physical quantities (second moment of the correlation function versus inverse mass gap, magnetic susceptibility versus square correlation length) and rotation invariance. Topological properties of the models are explored by measuring the topological susceptibility and by extracting the Abelian string tension. Several different (local and nonlocal) lattice definitions of topological charge are discussed and compared. The qualitative physical picture derived from the continuum $\frac{1}{N}$ expansion is confirmed, and agreement with quantitative $\frac{1}{N}$ predictions is satisfactory. Variant (Symanzik-improved) actions are considered in the C${\mathrm{P}}^{1}$ \ensuremath{\cong} O(3) case and agreement with universality and previous simulations (when comparable) is found. The simulation algorithm is an efficient mixture of over-heat-bath and microcanonical algorithms. The dynamical features and critical exponents of the algorithm are discussed in detail.

Topological susceptibility from different definitions of topological charge in lattice QCD

The independence of the topological susceptibility on the choice of the lattice operator for the topological charge density is checked. By use of a controlled cooling a detailed study of renormalizations from lattice to continuum is performed.

Staggered fermions, topological charge and topological susceptibility in lattice QCD

The topological susceptibility at β = 6.0 is measured using a “fermionic” definition of topological charge and staggered fermions. At β = 6.3 we present exploratory results.

Topological charge and fermions in the two-dimensional lattice U(1) model (I)

Abstract We investigate various definitions of topological charge within the compact QED 2 model. In particular we study for staggered fermions the charge, Q = 1 2 ϰ P m Tr Γ 5 ( D + m) − and its renormalization factor ϰ P .

Remnants of the index theorem on the lattice

We investigate the spectrum of lattice Dirac operators in two dimensions in external U(1) gauge fields as well as the properties and renormalization of the quantity m Tr γ 5/(¶ + m) which provides a definition of topological charge based on axial-vector divergence relations.

Definition and statistical properties of a universal topological charge

We consider the 2d O(3) σ-model. Lattice topological charge and universality are reconciled by an appropriate (quasi) local definition of the topological charge. Numerical calculations are carried out on square and triangular lattices. The results are satisfactory within limitations coming from the Monte Carlo method.

Computer simulation of topological effects in lattice theories: A model study

The quantum pendulum as a model is put on a lattice, producing a straightforward definition of topological charge. It is used to single out non-perturbative effects in a Monte Carlo calculation and compare them with a semiclassical picture. The method of subtracting perturbative contributions from numerical results recently used for the gluon condensate is tested.

Topological charge in the lattice Schwinger model

We apply Lüscher's definition of topological charge for lattice gauge field theories to the two-dimensional massless Schwinger model latticizing the vector field but keeping the fermions in the continuum. The instanton effects induced through fermion bilinears and the ensuing breakdown of cluster decomposition are discussed qualitatively in view of Lüscher's concept.