Dirac Structure Research Articles

Localisation without supersymmetry: towards exact results from Dirac structures in 3D N = 0 gauge theory

We show, by introducing purely auxiliary gluinos and scalars, that the quantum path integral for a class of 3D interacting non-supersymmetric gauge theories localises. The theories in this class all admit a ‘Manin gauge theory’ formulation, that we introduce; it is obtained by enhancing the gauge algebra of the theory to a Dirac structure inside a Manin pair. This machinery allows us to do localisation computations for every theory in this class at once, including for 3D Yang-Mills theory, and for its Third Way deformation; the latter calculation casts the Third Way path integral into an almost 1-loop exact form.

Transition metal atom on the two-dimensional Dirac structure RhB4/OsB4: A novel multifunctional electrocatalyst

The design of low-cost multifunctional electrocatalysts with high carrier mobility, high surface free energy and base surface activity facilitates hydrogen evolution reactions (HER) and oxygen reduction reactions (ORR), which are essential for the development of clean energy. Recent studies have shown significant progress in developing HER/ORR bifunctional catalysts, such as transition metal alloys, oxides, sulfides, and carbon-based composites. These materials are highly efficient in catalysis and help reduce costs. In this work, density functional theory is used to predict the performance of single atom catalysis (SACs) embedded with transition metal (TM) using two-dimensional (2D) Dirac node-ring semi-metal RhB4/OsB4 monolayer as substrate. Systematic calculations show that the TM@RhB4/OsB4 structure with thermodynamic and kinetic stability presents multiple active sites on the basal surface. Notably, Fe@RhB4 is an excellent hydrogen evolution reaction HER catalyst because Dirac cones occur near the Fermi level of this structure, providing high carrier densities and accelerating charge transfer between the catalyst and reaction intermediates. When the substrate RhB4/OsB4 is modified by Sc and Cu atoms, the band gap of the structure becomes small, which increases the conductivity of the original structure. The ΔGH* values of Sc@RhB4 and Cu@OsB4 were 0.118, and 9.38 meV, respectively, outperforming the ideal catalyst Pt. In addition, it is found that Fe@RhB4, and Mn@OsB4 exhibit outstanding catalytic properties for the oxygen reduction reaction (ORR), with overpotentials of 0.41V, and 0.40V, respectively, surpassing the performance of the reference catalyst Pt. It is exciting that the Fe@RhB4 exhibits ideal HER/ORR bifunctional electrocatalytic properties. This study will provide a theoretical basis for the development of multifunctional electrocatalysts in the field of Dirac nodal ring semi-metal materials by single atom doping technology.

Renormalization of asymmetric staple-shaped Wilson-line operators in lattice and continuum perturbation theory

In this work, we study the renormalization of nonlocal quark bilinear operators containing an asymmetric staple-shaped Wilson line at the one-loop level in both lattice and continuum perturbation theory. These operators enter the first-principle calculation of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD using the formulation of large momentum effective theory. We provide appropriate RI′-type conditions that address the power and logarithmic divergences, as well as the mixing among staple operators of different Dirac structures, using a number of different possible projectors. A variant of RI′, including calculations of rectangular Wilson loops, which cancel the pinch-pole singularities of the staple operators at infinite length and reduce residual power divergences, is also employed. We calculate at one-loop order the conversion matrix, which relates the quasi-TMDPDFs in the RI′-type schemes to the reference scheme MS¯ for arbitrary values of the renormalization momentum scale and of the dimensions of the staple. Published by the American Physical Society 2024

Quasi-Lie Bialgebroids, Dirac Structures, and Deformations of Poisson Quasi-Nijenhuis Manifolds

We show how to deform a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Then we interpret this procedure in the context of quasi-Lie bialgebroids, as a particular case of the so called twisting of a quasi-Lie bialgebroid. Finally, we frame our result in the setting of Courant algebroids and Dirac structures.

Fermion self-energy and damping rate in a hot magnetized plasma

We derive a general expression for the fermion self-energy in a hot magnetized plasma by using the Landau-level representation. In the one-loop approximation, the Dirac structure of the self-energy is characterized by five different functions that depend on the Landau-level index n and the longitudinal momentum pz. We derive general expressions for all five functions and obtain closed-form expressions for their imaginary parts. The latter receive contributions from three types of on shell processes, which are interpreted in terms of Landau-level transitions, accompanied by a single photon (gluon) emission or absorption. By making use of the imaginary parts of the self-energy functions, we also derive the Landau-level dependent fermion damping rates Γn(pz) and study them numerically in a wide range of model parameters. We also demonstrate that the two-spin degeneracy of the Landau levels is lifted by the one-loop self-energy corrections. While the spin splitting of the damping rates is small, it may be important for some spin and chiral effects. We argue that the general method and the numerical results for the rates can have interesting applications in heavy-ion physics, astrophysics, and cosmology, where strongly magnetized QED or QCD plasmas are ubiquitous. Published by the American Physical Society 2024

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

Stability of 2D Crystals from Tessellation

AbstractReticular chemistry has been a cornerstone in the design of novel 2D materials. Despite numerous possibilities for topological arrangements, only a few with high symmetry can form stable networks. Here, starting from 2D carbons, four types of highly stable tessellations are discovered, which consist of chains of non‐hexagonal rings separated by hexagonal ribbons. A modified Read–Shockley model is established to perfectly describe the stability of these highly stable frameworks, which is based on the interaction between non‐hexagonal rings. Moreover, these four types of tessellations and the modified Read–Shockley model are found to be of general validity in designing highly stable 2D materials, which is verified by the calculations on polymorphs of boron nitride and molybdenum disulfide. Besides, among the studied 2D carbon allotropes, two semi‐metallic structures with highly anisotropic Dirac cones and one semimetal with a Dirac nodal line at the Fermi level are discovered, as protected by their D2h symmetry. Spin‐orbital coupling is further found to open small bandgaps for these three Dirac structures, making them nontrivial topological insulators. The in‐depth understanding of the stability of 2D crystals in this study provides a new way for rational design of 2D crystals that may show peculiar electronic structures.

Dirac Structures for a Class of Port-Hamiltonian Systems in Discrete Time

Dirac Structures for a Class of Port-Hamiltonian Systems in Discrete Time

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

Coregular submanifolds and Poisson submersions

In this paper, we analyze submersions with Poisson fibres . These are submersions whose total space carries a Poisson structure, on which the ambient Poisson structure pulls back, as a Dirac structure, to Poisson structures on each individual fibre. Our “Poisson–Dirac viewpoint” is prompted by natural examples of Poisson submersions with Poisson fibres –in toric geometry and in Poisson–Lie groups– whose analysis was not possible using the existing tools in the Poisson literature. The first part of the paper studies the Poisson–Dirac perspective of inducing Poisson structures on submanifolds. This is a rich landscape, in which subtle behaviours abound, as illustrated by a surprising “jumping phenomenon” concerning the complex relation between the induced and the ambient symplectic foliations, which we discovered here. These pathologies, however, are absent from the well-behaved and abundant class of coregular submanifolds, with which we are mostly concerned here. The second part of the paper studies Poisson submersions with Poisson fibres –the natural Poisson generalization of flat symplectic bundles. These Poisson submersions have coregular Poisson–Dirac fibres, and behave functorially with respect to such submanifolds. We discuss the subtle collective behavior of the Poisson fibres of such Poisson fibrations, and explain their relation to pencils of Poisson structures. The third and final part applies the theory developed to Poisson submersions with Poisson fibres which arise in Lie theory. We also show that such submersions are a convenient setting for the associated bundle construction, and we illustrate this by producing new Poisson structures with a finite number of symplectic leaves. Some of the points in the paper being fairly new, we illustrate the many fine issues that appear with an abundance of (counter-)examples.

Port-Hamiltonian formulations of the incompressible Euler equations with a free surface

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v,Σ), (η,ϕ∂,Σ) and (ω,ϕ∂,Σ), with v the velocity, η the solenoidal velocity, ϕ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity.

Nijenhuis operator and Lie like bracket on generalised tangent bundle

The evolution of Nijenhuis operators has started during nineties. The Nijenhuis operator is formulated using the deformation on Poisson Nijenhuis manifolds and Lie algebras. In this paper, the Nijenhuis operator is suggested on the generalised tangent bundle TM + T*M followed by the deformation on Lie algebra. A new geometric structure is formulated in association with Nijenhuis relation. It is proved that if the Nijenhuis operator, N: G(TM+ T*M)->G(TM + T*M) is restricted to the Dirac structure of the generalised tangent bundle TM + T*M, then the deformation is a trivial. However, on the whole space G(TM + T*M), it is only a deformation weaker than the trivial deformation. A bracket like a Lie bracket is defined on G(TM + T*M)+G(T*M). The bracket is skew symmetric and does not satisfy Jacobi identity property. A structure like Dirac structure is also defined on that space.

A Lagrangian variational modular approach to thermodynamic systems

The diakoptical method developed by Kron (1963) has been known as a powerful tool in modeling and analyzing large-scale physical systems. The principal idea lies in the process of tearing and interconnection for a physical system, namely, one can reticulate the system into several subsystems, each of which is called a primitive system when it once torn apart from the original system. The original system can be regarded as an interconnected system via the interconnection structure, which is known to be modeled as a Dirac structure. This paper illustrates the idea of a Lagrangian variational modular approach by extending the case in mechanics done by Jacobs and Yoshimura (2014) to the case of thermodynamic systems, by following Gay-Balmaz and Yoshimura (2023), together with examples of non-simple thermodynamic systems with internal heat transfer and mechanical interactions.

Variational and Dirac structures for interconnected distributed-discrete systems

We present how the interconnection of distributed and discrete systems can be realized within the setting of variational principles and Dirac structures on the Lagrangian side. For the distributed system we focus on the case when the configuration space is an infinite-dimensional vector space and the system is subject to boundary energy flow. A key property of our approach is that it systematically extends the canonical variational and symplectic structures of finite and infinite-dimensional Lagrangian and Hamiltonian mechanics.

Multi-scale models of physical systems originating from model reduction and interconnection

In this paper we consider model reduction by clustering, and show how this leads to spatial layered network and energy storage structures. Furthermore, we consider the Dirac structures that arise from multi-scale power-conserving interconnections in physical network systems and boundary control distributed-parameter systems.

Structure-preserving spatial discretization of a coupled Heat-Wave system formulated as an irreversible port-Hamiltonian system.

The port-Hamiltonian (pH) framework allows one to properly model, interconnect, simulate and control various types of systems. Yet, properly modeling irreversibility remains a challenge as one has to include a nonlinear relation between flows and efforts, leading to a nonlinear Dirac structure. In this work, we will focus on the modelling of a distributed coupling of the heat and wave equations as pH systems. In particular, the representation of the coupled dynamics as an irreversible pH system is presented and its properties discussed. A discretization in space which preserves both the second and first principles of thermodynamics is detailed. Finally some numerical results are presented.

On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems

Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system composed of a stochastic vibrating string actuated at the boundary by a mass–spring system with external input and output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the development of stabilizing methods.

Complex Dirac structures with constant real index on flag manifolds

Complex Dirac structures with constant real index on flag manifolds

On Dirac structures admitting a variational approach

On Dirac structures admitting a variational approach

A geometric framework for discrete time port‐Hamiltonian systems

AbstractPort‐Hamiltonian systems provide an energy‐based formulation with a model class that is closed under structure preserving interconnection. For continuous‐time systems, these interconnections are constructed by geometric objects called Dirac structures. In this paper, we derive this geometric formulation and the interconnection properties for scattering passive discrete‐time port‐Hamiltonian systems.