non-Abelian Systems Research Articles

Improved HDRG decoders for qudit and non-Abelian quantum error correction

Hard-decision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and non-Abelian systems. In this work, we develop a method of performing HDRG decoding which combines strengths of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size L from to for any We apply our algorithm to decoding quantum double models and a non-Abelian anyon model with Fibonacci-like fusion rules, and show that it indeed significantly outperforms previous HDRG decoders. Furthermore, we provide the first study of continuous error correction with imperfect syndrome measurements for the quantum double models. The parallelized runtime of our algorithm is for the perfect measurement case. In the continuous case with imperfect syndrome measurements, the averaged runtime is for Abelian systems, while continuous error correction for non-Abelian anyons stays an open problem.

Measures of entanglement in non-Abelian anyonic systems

Bipartite entanglement entropies, calculated from the reduced density matrix of a subsystem, provide a description of the resources available within a system for performing quantum information processing. However, these quantities are not uniquely defined on a system of non-Abelian anyons. This paper describes how reduced density matrices and bipartite entanglement entropies (such as the von Neumann and Renyi entropies) may be constructed for non-Abelian anyonic systems, in ways which reduce to the conventional definitions for systems with only local degrees of freedom.

Non-Abelian color fields from relativistic color charge configurations in the classical limit

We study the dynamics of color fields as generated by simple configurations of relativistic particles with Abelian and non-Abelian [SU(2)] charges in the classical limit. We find that chromodynamic (non-Abelian) systems generally show Coulomb-like features by analogy with electrodynamics. A peculiar feature in the non-Abelian case is the additional strength of the chromoelectric and chromomagnetic fields caused by the contribution of changing the color charge. This change of color SU(2) charges results in a rotation of the color vector, which is getting very fast at close partonic distances. The presence of this non-Abelian additional term in the chromoelectric and chromomagnetic fields creates a color charge glow, which is manifested as a distinct color wave disturbance arising due to the finite distance at which the color interaction becomes active. This situation may be relevant to the hadronization phase in ultrarelativistic heavy-ion collisions, where the partonic state is governed by strong local color fluctuations.

Supersymmetric non-abelian multiwaves in D = 3 AdS superspace

We present a covariant, supersymmetric and kappa-symmetric action for non-Abelian multiwave system (nAmW) in D=3 AdS superspace. Its flat superspace limit provides a simplest counterpart of the recently proposed action for 11 dimensional system of N nearly coincident M-waves (multiple M0-branes), which is presently known for the case of flat target superspace only.

Obtaining gauge invariant actions via symplectic embedding formalism

AbstractThe concept of gauge invariance is one of the most subtle and useful concepts in modern theoretical physics. It is one of the Standard Model cornerstones. The main benefit due to the gauge invariance is that it can permit the comprehension of difficult systems in physics with an arbitrary choice of a reference frame at every instant of time. It is the objective of this work to show a path of obtaining gauge invariant theories from non‐invariant ones. Both are named also as first‐ and second‐class theories respectively, obeying Dirac's formalism. Namely, it is very important to understand why it is always desirable to have a bridge between gauge invariant and non‐invariant theories. Once established, this kind of mapping between first‐class (gauge invariant) and second‐class systems, in Dirac's formalism can be considered as a sort of equivalence. This work describe this kind of equivalence obtaining a gauge invariant theory starting with a non‐invariant one using the symplectic embedding formalism developed by some of us some years back. To illustrate the procedure it was analyzed both Abelian and non‐Abelian theories. It was demonstrated that this method is more convenient than others. For example, it was shown exactly that this embedding method used here does not require any special modification to handle with non‐Abelian systems.

Strong-disorder renormalization for interacting non-Abelian anyon systems in two dimensions

We consider the effect of quenched spatial disorder on systems of interacting, pinned non-Abelian anyons as might arise in disordered Hall samples at filling fractions \nu=5/2 or \nu=12/5. In one spatial dimension, such disordered anyon models have previously been shown to exhibit a hierarchy of infinite randomness phases. Here, we address systems in two spatial dimensions and report on the behavior of Ising and Fibonacci anyons under the numerical strong-disorder renormalization group (SDRG). In order to manage the topology-dependent interactions generated during the flow, we introduce a planar approximation to the SDRG treatment. We characterize this planar approximation by studying the flow of disordered hard-core bosons and the transverse field Ising model, where it successfully reproduces the known infinite randomness critical point with exponent \psi ~ 0.43. Our main conclusion for disordered anyon models in two spatial dimensions is that systems of Ising anyons as well as systems of Fibonacci anyons do not realize infinite randomness phases, but flow back to weaker disorder under the numerical SDRG treatment.

On the solutions of generalized Bogomolny equations

Generalized Bogomolny equations are encountered in the localization of the topological N=4 SYM theory. The boundary conditions for 't Hooft and surface operators are formulated by giving a model solution with some special singularity. In this note we consider the generalized Bogomolny equations on a half space and construct model solutions for the boundary 't Hooft and surface operators. It is shown that for the 't Hooft operator the equations reduce to the open Toda chain for arbitrary simple gauge group. For the surface operators the solutions of interest are rational solutions of a periodic non-abelian Toda system.

Disorder-induced Majorana metal in interacting non-Abelian anyon systems

We demonstrate that a thermal metal of Majorana fermions forms in a two-dimensional system of interacting non-Abelian (Ising) anyons in the presence of moderate disorder. This bulk metallic phase arises in the $\ensuremath{\nu}=5/2$ quantum Hall state when disorder pins the anyonic quasiparticles. More generally, it naturally occurs for various proposed systems supporting Majorana fermion zero modes when disorder induces the random pinning of a finite density of vortices. This includes all two-dimensional topological superconductors in so-called symmetry class D. A distinct experimental signature of the thermal metal phase is the presence of bulk heat transport down to zero temperature.

On Integrability of the Kontsevich Non-Abelian ODE System

We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries.

Inner structure of QED and its gauge invariant angular momenta: A new viewpoint to the final resolution of the nucleon spin crisis

We discover an inner structure of QED while the gauge potential is decomposed into two orthogonal components. Based on this, the Lagrangian of the electron-photon system is expanded to a new form and by the general method of field theory, the gauge invariant spin and orbital angular momentum operators of the electron and photon are naturally obtained from Noether's theorem. Our method, which can be generalized to the non-Abelian systems to investigate the inner structure of QCD, provides a new perspective to look on the nucleon spin crisis and opens a window into a strict and systematic resolution of this long-standing problem.

Nonabelian (2,0) tensor multiplets and 3-algebras

Using 3-algebras we obtain a nonabelian system of equations that furnish a representation of the (2, 0)-supersymmetric tensor multiplet. The on-shell conditions are quite restrictive so that the system can be reduced to five-dimensional gauge theory along with six-dimensional abelian (2, 0) tensor multiplets. We briefly discuss possible applications to D4-branes using a spacelike reduction and M5-branes using a null reduction.

Physics of a two-dimensional electron gas with cold atoms in non-Abelian gauge potentials

Motivated by the possibility of creating non-Abelian fields using cold atoms in optical lattices, we explore the richness and complexity of noninteracting two-dimensional electron gases (2DEGs) in a lattice, subjected to such fields. In the continuum limit, a non-Abelian system characterized by a two-component ``magnetic flux'' describes a harmonic oscillator existing in two different charge states (mimicking a particle-hole pair) where the coupling between the states is determined by the non-Abelian parameter, namely, the difference between the two components of the ``magnetic flux.'' A key feature of the non-Abelian system is a splitting of the Landau energy levels, which broaden into bands, as the spectrum depends explicitly on the transverse momentum. These Landau bands result in a coarse-grained ``moth,'' a continuum version of the generalized Hofstadter butterfly. Furthermore, the bands overlap, leading to effective relativistic effects. Importantly, similar features also characterize the corresponding two-dimensional lattice problem when at least one of the components of the magnetic flux is an irrational number. The lattice system with two competing ``magnetic fluxes'' penetrating the unit cell provides a rich environment in which to study localization phenomena. Some unique aspects of the transport properties of the non-Abelian system are the possibility of inducing localization by varying the quasimomentum, and the absence of localization of certain zero-energy states exhibiting a linear energy-momentum relation. Furthermore, non-Abelian systems provide an interesting localization scenario where the localization transition is accompanied by a transition from relativistic to nonrelativistic theory.

Interacting D2 branes in ten dimensions and non-Abelian Born–Infeld theory

In this paper we extend the bosonic D-brane action in D = 10 obtained by duality from the D = 11 membrane wrapped on S1 to an SU(2) non-Abelian system. This system presents only first class constraints, whose algebra closes off-shell and generalizes the algebra of diffeomorphisms of the D2 brane to include non-Abelian symmetry generators. From the SU(2)D-brane action, we also obtain the SU(2) Born–Infeld theory by performing a covariant reduction to a flat background. This calculation agrees up to fourth order with the result obtained from the superstring amplitudes and gives an alternative approach to analyse non-Abelian Born–Infeld theories.

Langevin simulation of the chirally decomposed sine-Gordon model

A large class of quantum and statistical field theoretical models, encompassing relevant condensed matter and non-Abelian gauge systems, are defined in terms of complex actions. As the ordinary Monte Carlo methods are useless in dealing with these models, alternative computational strategies have been proposed along the years. The Langevin technique, in particular, is known to be frequently plagued with difficulties such as strong numerical instabilities or subtle ergodic behavior. Regarding the chirally decomposed version of the sine-Gordon model as a prototypical case for the failure of the Langevin approach, we devise a truncation prescription in the stochastic differential equations which yields numerical stability and is assumed not to spoil the Berezinskii-Kosterlitz-Thouless transition. This conjecture is supported by a finite size scaling analysis, whereby a massive phase ending at a line of critical points is clearly observed for the truncated stochastic model.

The Hamilton-Jacobi treatment for non-abelian Chern-Simons system

The non-abelian Chern-Simons field interacting with N component complex field is treated as a constrained system using the Hamilton-Jacobi approach. The reduced phase space Hamiltonian density is obtained with out introducing Lagrange multipliers and with out any additional gauge fixing condition. The quantization of this system is also discussed.

Noncommutative branes from M theory

The analysis of the worldvolume effective actions of the M-theory Kaluza-Klein monopole and 9-brane suggests that it should be possible to describe non-abelian configurations of M2-branes or M5-branes if the M2-branes are transverse to the eleventh direction and the M5-branes are wrapped on it. This is determined by the fact that the Kaluza-Klein monopole and the M9-brane are constrained to move in particular isometric spacetimes. We show that the same kind of situation is implied by the analysis of the brane descent relations in M-theory. We compute some of the non-commutative couplings of the worldvolume effective actions of these non-abelian systems of M2 and M5 branes and show that they indicate the existence of configurations corresponding to N branes expanding into a higher dimensional M-brane. The reduction to Type II brings up new descriptions of coincident D-branes at strong coupling. We show that these systems have the right non-commutative charges to describe certain expanded configurations playing a role in the framework of the AdS/CFT correspondence. Finally, we discuss the realization of non-commutative brane configurations as topological solitons in non-abelian brane-antibrane systems.

Some New Non-Abelian 2D Dualities

Starting from certain 3D non-abelian dual systems, we discuss a number of related dual systems in 2D, some of which are obtained by dimensional reduction. The dualities relate massive scalar and vector fields, and may be relevant for string theory in the context of massive type IIA supergravity. Supersymmetric extensions of the models are also presented.

Vortex strings and nonabelian sine-Gordon theories

We generalize the Lund-Regge model for vortex string dynamics in 4-dimensional Minkowski space to the arbitrary n-dimensional case. The n-dimensional vortex equation is identified with a nonabelian sine-Gordon equation and its integrability is proven by finding the associated linear equations of the inverse scattering. An explicit expression of vortex coordinates in terms of the variables of the nonabelian sine-Gordon system is derived. In particular, we obtain the n-dimensional vortex soliton solution of the Hasimoto-type from the one soliton solution of the nonabelian sine-Gordon equation.

Mass gap, Abelian dominance and vortex dynamics in SU(2) spin model

We discuss a new approach to the investigation of the nature of the mass gap in spin systems with continuous global symmetries which is much analogous to the method of abelian projection in the gauge theories. We suggest that the abelian degrees of freedom, in particular, abelian vortices are responsible for the mass gap generation phenomena in the non-abelian spin systems. To check our hypothesis we study numerically the three-dimensional SU(2) spin model in the Maximal Abelian projection. We find that the abelian mass gap in the projected theory coincides with the full non-abelian mass gap within numerical errors. The study of the percolation properties of the abelian vortex trajectories shows that the phase transition and the mass gap generation in the 3D SU(2) spin model are driven by the abelian vortex condensation.

Worldline approach to Sudakov-type form factors in non-Abelian gauge theories

We calculate Sudakov-type form factors for isolated spin-1/2 particles (fermions) entering non-Abelian gauge-field systems. We consider both the on- and the off-mass-shell case using a methodology which rests on a worldline casting of field theories. The simplicity and utility of our approach derives from the fact that we are in a position to make, a priori, a more transparent separation (factorization), with respect to a given scale, between short- and long-distance physics than diagramatic methods.