non-Abelian Charges Research Articles

Non-Abelian symmetry-resolved entanglement entropy

We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2)SU(2) symmetry.

Interferometry of Non-Abelian Band Singularities and Euler Class Topology.

In systems with a real Bloch Hamiltonian band nodes can be characterized by a non-Abelian frame-rotation charge. The ability of these band nodes to annihilate pairwise is path dependent, since by braiding nodes in adjacent gaps the sign of their charges can be changed. Here, we theoretically construct and numerically confirm two concrete methods to experimentally probe these non-Abelian braiding processes and charges in ultracold atomic systems. We consider a coherent superposition of two bands that can be created by moving atoms through the band singularities at some angle in momentum space. Analyzing the dependency of excitations on the frame charges, we demonstrate an interferometry scheme passing through two band nodes, which reveals the relative frame charges and allows for measuring the multigap topological invariant. The second method relies on a single wave packet probing two nodes sequentially, where the frame charges can be determined from the band populations. Our results present a feasible avenue for measuring non-Abelian charges of band nodes and the direct experimental verification of braiding procedures, which can be applied in a variety of settings including the recently discovered anomalous non-Abelian phases arising under periodic driving.

Synthetic non-Abelian topological charges in ultracold atomic gases

Synthetic non-Abelian topological charges in ultracold atomic gases

Spin-charged point particle in a non-Abelian external field with the generalized uncertainty relation

The dynamics of a particle carrying a non-Abelian charge is studied in the presence of a minimal length. By choosing an appropriate non-Abelian gauge field, the system identifies with the Hamiltonian of the Jaynes-Cummings model whose solutions can be determined algebraically. The model has an underlying graded Lie algebra symmetry reminiscent of supersymmetric quantum mechanics. We calculate the energy levels and associated eigenstates using conservation of the number of excitations of the system. Then, we present the effect of the minimal length on the dynamics of the system and we are particularly interested in two particular cases, that of Rabi oscillations and that of the collapse-revival of the wave function. The results show that the higher the deformation parameter, the faster the oscillatory behavior of the atomic inversion.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Topological phase transition and surface states in a non-Abelian charged nodal line photonic crystal

Abstract Topological charges of nodal lines in a multigap system are represented by non-Abelian numbers, and the Euler class, a topological invariant, can be used to explain their topological phase transitions, such as pair-annihilation of nodal lines. Up until now, no discussion of phase transitions of nodal lines in photonic crystals using the Euler class has been reported, despite the fact that the Euler class and topological phase transition have recently been addressed in metallic or acoustic crystals. Here, we show how the deformation of a photonic crystal causes topological phase transitions in the nodal lines, and the Euler class can be used to theoretically predict the nodal lines’ stability based on the non-Abelian topological charge theory. Specifically, by manipulating the separation between the two single diamond structures and the extent of structural distortion, we numerically demonstrate the topological transition of nodal lines, e.g., from nodal lines to nodal rings. We then demonstrate that the range of surface states is strongly influenced by the topological phase transition of nodal lines. Moreover, the Zak phase was used to explain the surface states’ existence.

Non-Abelian physics in light and sound.

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Black holes with Abelian and Non-Abelian charges and their impact on matter accretion flows

In this paper, we study the black hole spacetime structure of a model consisting of the standard Maxwell theory and a p-power-Yang–Mills term. This non-linear contribution introduces a non-Abelian charge into the global solution, resulting in a modified structure of the standard Reissner–Nordström black hole. Specifically, we focus on the model with p=1/2, which gives rise to a new type of modified Reissner–Nordström black hole. For this class of black holes, we compute the event horizon, the innermost stable circular orbit, and the conditions to preserve the weak cosmic censorship conjecture. The latter condition sets a well-established relation between the electric and the Yang–Mills charges. As a first astrophysical implication, the accretion properties of spherical steady flows around this new modified Reissner–Nordström black hole are investigated in detail. Concretely, we compute the critical radius that establishes the condition for having stable transonic flow in terms of the local sound speed and the involved charges. Extensive numerical examples of how the Yang–Mills charge affects the accretion process of an isothermal fluid in comparison to the standard Reissner–Nordström and Schwarzschild black holes are displayed. Finally, analytical solutions in the fully relativistic regime, along with numerical computations, of the mass accretion rate for a polytropic fluid in terms of the electric and Yang–Mills charges are obtained. As a main result, the mass accretion rate efficiency is considerably improved, with respect to the standard Reissner–Nordström and Schwarzschild solutions, for negative values of the Yang–Mills charge.

Floquet non-Abelian topological insulator and multifold bulk-edge correspondence

Topological phases characterized by non-Abelian charges are beyond the scope of the paradigmatic tenfold way and have gained increasing attention recently. Here we investigate topological insulators with multiple tangled gaps in Floquet settings and identify uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. We demonstrate that the bulk-edge correspondence is multifold and follows the multiplication rule of the quaternion group Q8. The same quaternion charge corresponds to several distinct edge-state configurations that are fully determined by phase-band singularities of the time evolution. In the anomalous non-Abelian phase, edge states appear in all bandgaps despite trivial quaternion charge. Furthermore, we uncover an exotic swap effect—the emergence of interface modes with swapped driving, which is a signature of the non-Abelian dynamics and absent in Floquet Abelian systems. Our work, for the first time, presents Floquet topological insulators characterized by non-Abelian charges and opens up exciting possibilities for exploring the rich and uncharted territory of non-equilibrium topological phases.

Lack of symmetry restoration after a quantum quench: An entanglement asymmetry study

We consider the quantum quench in the XX spin chain starting from a tilted Néel state which explicitly breaks the U(1)U(1) symmetry of the post-quench Hamiltonian. Very surprisingly, the U(1)U(1) symmetry is not restored at large time because of the activation of a non-Abelian set of charges which all break it. The breaking of the symmetry can be effectively and quantitatively characterised by the recently introduced entanglement asymmetry. By a combination of exact calculations and quasi-particle picture arguments, we are able to exactly describe the behaviour of the asymmetry at any time after the quench. Furthermore we show that the stationary behaviour is completely captured by a non-Abelian generalised Gibbs ensemble. While our computations have been performed for a non-interacting spin chain, we expect similar results to hold for the integrable interacting case as well because of the presence of non-Abelian charges also in that case.

Non-Abelian Frame Charge Flow in Photonic Media

In PT-symmetric systems, the notion of non-Abelian frame charges enables multiband topological characterization of degeneracies by examining the eigenvector-frame rotations. Interestingly, some features of these frame charges can be viewed as an analog of electric charges confined in conducting wires, only that they flow in momentum space along nodal lines. But different from electric charges, these frame charges are not integers, and non-Abelian signatures emerge when braiding between adjacent band nodal lines occurs, which flips the direction of the flow. In photonic systems, the photonic Γ point serves as the source or sink of such a frame charge flow due to a hidden braiding induced by the often-ignored electrostatic mode at zero frequency. This source naturally generates non-Abelian topological signatures in PT-symmetric photonic systems that can be identified even in ordinary dielectric media. We use biaxial photonic crystals as examples to show how complex nodal line configurations can be explained as the topological consequences of the frame charge flow from the Γ point to the Brillouin zone boundaries. Our results reveal that non-Abelian band topology is intrinsically manifested in general photonic systems, making them easily accessible platforms for exploring non-Abelian physics.6 MoreReceived 11 September 2022Accepted 7 April 2023DOI:https://doi.org/10.1103/PhysRevX.13.021024Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMetamaterialsSymmetry protected topological statesTopological effects in photonic systemsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Minimal non-abelian nodal braiding in ideal metamaterials

Exploring new topological phases and phenomena has become a vital topic in condensed matter physics and materials sciences. Recent studies reveal that a braided colliding nodal pair can be stabilized in a multi-gap system with PT or {C}_{2z}T symmetry. This exemplifies non-abelian topological charges beyond the scope of conventional single-gap abelian band topology. Here, we construct ideal acoustic metamaterials to realize non-abelian braiding with the fewest band nodes. By emulating the time with a sequence of acoustic samples, we experimentally observe an elegant but nontrivial nodal braiding process, including nodes creation, braiding, collision, and repulsion (i.e., impossible to annihilate), and measure the mirror eigenvalues to elucidate the braiding consequence. The latter, at the level of wavefunctions, is of prime importance since essentially braiding physics aims to entangle multi-band wavefunctions. Furthermore, we experimentally unveil the highly intricate correlation between the multi-gap edge responses and the bulk non-abelian charges. Our findings pave the way for developing non-abelian topological physics that is still in its infancy.

Interference Measurements of Non-Abelian e/4 & Abelian e/2 Quasiparticle Braiding

The quantum Hall states at filling factors $\nu=5/2$ and $7/2$ are expected to have Abelian charge $e/2$ quasiparticles and non-Abelian charge $e/4$ quasiparticles. The non-Abelian statistics of the latter has been predicted to display a striking interferometric signature, the even-odd effect. By measuring resistance oscillations as a function of magnetic field in Fabry-P\'erot interferometers using new high purity heterostructures, we for the first time report experimental evidence for the non-Abelian nature of excitations at $\nu=7/2$. At both $\nu=5/2$ and $7/2$ we also examine, for the first time, the fermion parity, a topological quantum number of an even number of non-Abelian quasiparticles. The phase of observed $e/4$ oscillations is reproducible and stable over long times (hours) near both filling factors, indicating stability of the fermion parity. At both fractions, when phase fluctuations are observed, they are predominantly $\pi$ phase flips, consistent with either fermion parity change or change in the number of the enclosed $e/4$ quasiparticles. We also examine lower-frequency oscillations attributable to Abelian interference processes in both states. Taken together, these results constitute new evidence for the non-Abelian nature of $e/4$ quasiparticles; the observed life-time of their combined fermion parity further strengthens the case for their utility for topological quantum computation.

Non-Abelian symmetry can increase entanglement entropy

The pillars of quantum theory include entanglement and operators' failure to commute. The Page curve quantifies the bipartite entanglement of a many-body system in a random pure state. This entanglement is known to decrease if one constrains extensive observables that commute with each other (Abelian ``charges''). Non-Abelian charges, which fail to commute with each other, are of current interest in quantum thermodynamics. For example, noncommuting charges were shown to reduce entropy-production rates and may enhance finite-size deviations from eigenstate thermalization. Bridging quantum thermodynamics to many-body physics, we quantify the effects of charges' noncommutation -- of a symmetry's non-Abelian nature -- on Page curves. First, we construct two models that are closely analogous but differ in whether their charges commute. We show analytically and numerically that the noncommuting-charge case has more entanglement. Hence charges' noncommutation can promote entanglement.

Observation of Non-Abelian Charged Nodes Linking Nonadjacent Gaps.

Nodal links are special configurations of band degeneracies in the momentum space, where nodal line branches encircle each other. In PT symmetric systems, nodal lines can be topologically characterized using the eigenvector frame rotations along an encircling loop and the linking structure can be described with non-Abelian frame charges involving adjacent bands. While the commutation rules between the frame charges are well established, the underlying relationship between distant band gap closing nodes remains to be explored. In this Letter, we present a photonic multiple nodal links system, where the nodal lines of nonadjacent bands are investigated with symmetry constraints on frame charges. Through an orthogonal nodal chain, the nodal line from the lower two bands predicts the existence of nodal lines formed between the higher bands. We designed and fabricated a metamaterial, with which the multiple nodal links and the topological connection between nonadjacent nodal lines are experimentally demonstrated.

Viscosity solutions for the relativistic inhomogeneous Vlasov equation in Schwarzschild outer space-time in the presence of the Yang-Mills field

This paper provide evidence of the existence and uniqueness result for the viscosity solutions of inhomogeneous Vlasov equation. we consider the Cauchy-Dirichlet problem for the relativistic Vlasov equation with near vacuum initial data where the distribution function depends on the time, the position, the momentum and the non-Abelian charge of particles. We consider this equation on aSchwarzschild outer space-time with Yang-Mills field.

Simulation of Lindblad equations for quarkonium in the quark-gluon plasma

We study the properties of the Lindbladian quantum mechanical evolution of quarkonia with non-Abelian charges (color-singlet and octet) in the quark-gluon plasma. We confirm that heavy quark recoils in the Lindblad equation correctly thermalize quarkonium colorful states within statistical errors from the simulation method. We also demonstrate that the Lindblad equation in the dipole limit can provide an efficient alternative method, which is applicable to a finite time evolution before thermalization and dramatically reduces the numerical cost. Our findings will serve as a foundation for large-scale simulation of quarkonium dynamics in the relativistic heavy-ion collisions.

Experimental Observation of Non-Abelian Earring Nodal Links in Phononic Crystals.

Nodal lines are symmetry-protected one-dimensional band degeneracies in momentum space, which can appear in numerous topological configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian topological physics have been proposed in space-time inversion (PT) symmetric systems. One of the most special configurations in such systems is the earring nodal link, composing of a nodal chain linking with an isolated nodal line. Such earring nodal links have not been observed in real systems. We designed phononic crystals with earring nodal links, and experimentally observed two different kinds of earring nodal links by measuring the band structures. We found that the order of the nodal chain and line can be switched after band inversion but their link cannot be severed. Our Letter provides experimental evidence for phenomena unique to non-Abelian band topology and our acoustic system provides a convenient platform for studying the new materials carrying non-Abelian charges.

Sign freedom of non-abelian topological charges in phononic and photonic topological semimetals

The topological nature of nodal lines in three-band systems can be described by non-abelian topological charges called quaternion numbers. Due to the gauge freedom of the eigenstates, the sign of quaternion numbers can be flipped by performing a gauge transformation, i.e., choosing a different basis of eigenstates. However, the sign flipping has not been explicitly shown in realistic systems such as phononic and photonic topological semimetals. Here, we elaborate on the sign freedom of non-abelian topological charges by visualizing numerically calculated topological charges in phononic and photonic topological semimetals. For this, we employ a common reference point method for multiple nodal lines and thus confirm that the sign flipping does not cause any inconsistency in building the quaternion group.

Multigap topology and non-Abelian braiding of phonons from first principles

Non-Abelian states of matter, in which the final state depends on the order of the interchanges of two quasiparticles, can encode information immune from environmental noise with the potential to provide a robust platform for topological quantum computation. We demonstrate that phonons can carry non-Abelian frame charges at the band-crossing points of their frequency spectrum, and that external stimuli can drive their braiding. We present a general framework to understand the topological configurations of phonons from first-principles calculations using a topological invariant called Euler class, and provide a complete analysis of phonon braiding by combining different topological configurations. Taking a well-known dielectric material ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ as a representative example, we demonstrate that electrostatic doping gives rise to phonon band inversions that can induce redistribution of the frame charges, leading to non-Abelian braiding of phonons. Our work provides a quasiparticle platform for realizable non-Abelian braiding in reciprocal space, and expands the tool set for studying braiding processes.