Nontrivial Symmetry Research Articles

Localized states with nontrivial symmetries: Localized labyrinthine patterns.

The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with nontrivial symmetries such as labyrinthine patterns are observed in different physical contexts. Here we report stable localized disordered patterns in spatially extended dissipative systems. These two- and three-dimensional localized structures consist of an isolated labyrinth embedded in a homogeneous steady state. Their partial bifurcation diagram allows us to explain this phenomenon as a manifestation of a pinning-depinning transition. We illustrate our findings on the Swift-Hohenberg-type of equationsand other well-established models for plant ecology, nonlinear optics, and reaction-diffusion systems.

Symmetries and Solvability of a Class of Higher Order Systems of Ordinary Difference Equations

We perform Lie analysis for a system of higher order difference equations with variable coefficients and derive non-trivial symmetries. We use these symmetries to find exact formulas for the solutions in terms of k. Furthermore, a detailed study for a specific value of k is presented. Our findings generalize some results in the literature.

Symplectic modular symmetry in heterotic string vacua: flavor, CP, and R-symmetries

We examine a common origin of four-dimensional flavor, CP, and U(1)R symmetries in the context of heterotic string theory with standard embedding. We find that flavor and U(1)R symmetries are unified into the Sp(2h + 2, ℂ) modular symmetries of Calabi-Yau threefolds with h being the number of moduli fields. Together with the {mathbb{Z}}_2^{mathrm{CP}} CP symmetry, they are enhanced to GSp(2h + 2, ℂ) ≃ Sp(2h + 2, ℂ) ⋊ {mathbb{Z}}_2^{mathrm{CP}} generalized symplectic modular symmetry. We exemplify the S3, S4, T′, S9 non-Abelian flavor symmetries on explicit toroidal orbifolds with and without resolutions and ℤ2, S4 flavor symmetries on three-parameter examples of Calabi-Yau threefolds. Thus, non-trivial flavor symmetries appear in not only the exact orbifold limit but also a certain class of Calabi-Yau three-folds. These flavor symmetries are further enlarged to non-Abelian discrete groups by the CP symmetry.

Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field

We classify simple symmetries for an Ornstein-Uhlenbeck process, describing a particle in an external force field $f(x)$. It turns out that for sufficiently regular (in a sense to be defined) forces there are nontrivial symmetries only if $f(x)$ is at most linear. We fully discuss the isotropic case, while for the non-isotropic we only deal with a generic situation (defined in detail in the text).

Addendum: Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics (2018 J. Phys. Commun. 2 115018)

An extension is proposed to the internal symmetry transformations associated with mass, entropy and other Clebsch-related conservation in geophysical fluid dynamics. Those symmetry transformations were previously parameterized with an arbitrary function of materially conserved Clebsch potentials. The extension consists in adding potential vorticity q to the list of fields on which a new arbitrary function depends. If , where is an arbitrary function of specific entropy s, then the symmetry is trivial and gives rise to a trivial conservation law. Otherwise, the symmetry is non-trivial and an associated non-trivial conservation law exists. Moreover, the notions of trivial and non-trivial Casimir invariants are defined. All non-trivial symmetries that become hidden following a reduction of phase space are associated with non-trivial Casimir invariants of a non-canonical Hamiltonian formulation for fluids, while all trivial conservation laws are associated with trivial Casimir invariants.

Symmetry classification of uniform states in spin-2 Bose-Einstein condensates and neutron P23 superfluids

We clarify a relation between the Gross-Pitaevskii energy functional for spin-2 spinor Bose-Einstein condensates and the Ginzburg-Landau theory for neutron $^3P_2$ superfluidis (spin-triplet $P$-wave pairing with total angular momentum two). We then classify all uniform states with nontrivial unbroken symmetries, with the help of geometric invariant theory.

Holographic symmetry algebras for gauge theory and gravity

All 4D gauge and gravitational theories in asymptotically flat spacetimes contain an infinite number of non-trivial symmetries. They can be succinctly characterized by generalized 2D currents acting on the celestial sphere. A complete classification of these symmetries and their algebras is an open problem. Here we construct two towers of such 2D currents from positive-helicity photons, gluons, or gravitons with integer conformal weights. These generate the symmetries associated to an infinite tower of conformally soft theorems. The current algebra commutators are explicitly derived from the poles in the OPE coefficients, and found to comprise a rich closed subalgebra of the complete symmetry algebra.

Classical symmetries and the Quantum Approximate Optimization Algorithm

We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular, we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques toward predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.

Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

Abstract We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.

Reconstructing a minimal topological dynamical system from a set of return times

AbstractWe investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations, this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption, two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric closure, then that Kronecker system must be a factor. We also study similar problems involving nilsystems and polynomial return times. We state a number of questions on whether these results extend to other homogeneous spaces and transitive group actions, some of which are already interesting for finite groups.

Observation of Unconventional Charge Density Wave without Acoustic Phonon Anomaly in Kagome Superconductors AV3Sb5 ( A=Rb , Cs)

The combination of non-trivial band topology and symmetry breaking phases gives rise to novel quantum states and phenomena such as topological superconductivity, quantum anomalous Hall effect and axion electrodynamics. Evidence of intertwined charge density wave (CDW) and superconducting order parameters has recently been observed in a novel kagome material AV3Sb5 (A=K,Rb,Cs) that features a Z2 topological invariant in the electronic structure. However, the origin of the CDW and its intricate interplay with topological state has yet to be determined. Here, using hard x-ray scattering, we demonstrate a three-dimensional (3D) CDW with 2*2*2 superstructure in (Rb,Cs)V3Sb5. Unexpectedly, we find that the CDW fails to induce acoustic phonon anomalies at the CDW wavevector but yields a novel Raman mode which quickly damps into a broad continuum below the CDW transition temperature. Our observations exclude strong electron-phonon coupling driven CDW in AV3Sb5 and point to an unconventional and electronic-driven mechanism that couples the CDW and the topological band structure.

Infinite-dimensional fermionic symmetry in supersymmetric gauge theories

We establish the existence of an infinite-dimensional fermionic symmetry in four-dimensional supersymmetric gauge theories by analyzing semiclassical photino dynamics in abelian mathcal{N} = 1 theories with charged matter. The symmetry is parametrized by a spinor-valued function on an asymptotic S2 at null infinity. It is not manifest at the level of the Lagrangian, but acts non-trivially on physical states, and its Ward identity is the soft photino theorem. The infinite-dimensional fermionic symmetry resides in the same mathcal{N} = 1 supermultiplet as the physically non-trivial large gauge symmetries associated with the soft photon theorem.

Aspects of symmetry and topology in the charge density wave phase of 1T–TiSe2

The charge density wave (CDW) in 1T–TiSe2 harbors a nontrivial symmetry configuration. It is important to understand this underlying symmetry both for gaining a handle on the mechanism of CDW formation and for probing the CDW experimentally. Here, based on first-principles computations within the framework of the density functional theory, we unravel the connection between the symmetries of the normal and CDW states and the electronic structure of 1T–TiSe2. Our analysis highlights the key role of irreducible representations of the electronic states and the occurrence of band gaps in the system in driving the CDW. By showing how symmetry-related topology can be obtained directly from the electronic structure, our study provides a practical pathway in search of topological CDW insulators.

Existence and structure of symmetric Beltrami flows on compact 3-manifolds

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian 3-manifold (M¯,g) with (possibly empty) boundary and a smooth flow of isometries ϕt:M¯→M¯ we show that, if M¯ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field X, tangent to the boundary, which is an eigenfield of curl and satisfies (ϕt)⁎X=X, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and (M¯,g) has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable 3-manifolds with and without boundary.

A reaction–diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modelling

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

Nucleation of protein mesocrystals via oriented attachment

Self-assembly of proteins holds great promise for the bottom-up design and production of synthetic biomaterials. In conventional approaches, designer proteins are pre-programmed with specific recognition sites that drive the association process towards a desired organized state. Although proven effective, this approach poses restrictions on the complexity and material properties of the end-state. An alternative, hierarchical approach that has found wide adoption for inorganic systems, relies on the production of crystalline nanoparticles that become the building blocks of a next-level assembly process driven by oriented attachment (OA). As it stands, OA has not yet been observed for protein systems. Here we employ cryo-transmission electron microscopy (cryoEM) in the high nucleation rate limit of protein crystals and map the self-assembly route at molecular resolution. We observe the initial formation of facetted nanocrystals that merge lattices by means of OA alignment well before contact is made, satisfying non-trivial symmetry rules in the process. As these nanocrystalline assemblies grow larger we witness imperfect docking events leading to oriented aggregation into mesocrystalline assemblies. These observations highlight the underappreciated role of the interaction between crystalline nuclei, and the impact of OA on the crystallization process of proteins.

NONLINEAR DIFFUSION EQUATION WITH A PERTURBED CONVECTIONTERM: POTENTIAL SYMMETRIES WITH RESPECT TO THE SECOND CONSERVATION LAW

We consider the nonlinear diffusion equation with a perturbed convection term. The potential symmetries for the exact equation with respect to the second conservation law are classified. It is found that these exist only in the linear case. It is further shown that no nontrivial approximate potential symmetries of order one exists for the perturbed equation with respect to the other conservation law.

Invariance or equivalence: a tale of two principles

The presence of symmetries in physical theories implies a pernicious form of underdetermination. In order to avoid this theoretical vice, philosophers often espouse a principle called Leibniz Equivalence, which states that symmetry-related models represent the same state of affairs. Moreover, philosophers have claimed that the existence of non-trivial symmetries motivates us to accept the Invariance Principle, which states that quantities that vary under a theory’s symmetries aren’t physically real. Leibniz Equivalence and the Invariance Principle are often seen as part of the same package. I argue that this is a mistake: Leibniz Equivalence and the Invariance Principle are orthogonal to each other. This means that it is possible to hold that symmetry-related models represent the same state of affairs whilst having a realist attitude towards variant quantities. Various arguments have been presented in favour of the Invariance Principle: a rejection of the Invariance Principle is inter alia supposed to cause indeterminism, undetectability or failure of reference. I respond that these arguments at best support Leibniz Equivalence.

Symmetry indicators for inversion-symmetric non-Hermitian topological band structures

We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the associated Chern-Simons form, whose evaluation usually requires a computationally expensive choice of smooth gauge. In each case, we discuss the non-Hermitian surface states associated with nontrivial symmetry indicators.

Physics of symmetry protected topological phases involving higher symmetries and its applications

We discuss physical constructions, and the boundary properties of various symmetry protected topological phases that involve 1-form symmetries, from one spatial dimension (1d) to four spatial dimensions (4d). For example, the prototype 3d boundary state of 4d SPT states involving 1-form symmetries can be either a gapless photon phase (quantum electrodynamics) or gapped topological order enriched by 1-form symmetries, namely the loop excitations of these topological orders carry nontrivial 1-form symmetry charges. This study also serves the purpose of diagnosing anomaly of 3d states of matter. Connection between SPT states with 1-form symmetries and condensed matter systems such as quantum dimer models at one lower dimension will also be discussed. Whether a quantum dimer model can have a trivial gapped phase or not depends on the nature of its corresponding bulk state in one higher dimension.