Hidden Symmetry Research Articles

Is There Any Hidden Symmetry in the Stripe Structure of Perovskite High-Temperature Superconductors?

Local and fast structural probes using synchrotron radiation have shown nanoscale striped puddles and nanoscale phase separation in doped perovskites. It is known that the striped phases in doped perovskites are due to competing interactions involving charge, spin, and lattice degrees of freedom. In this work, we show that two different stripes can be represented as a superposition of a pair of stripes, U(θ n) or D(θ n), characterized by perovskite tilts where one of the pair is rotated in relation to the other partner by an angle Δθ n = π/2. The spatial distribution of the U and D stripes is reduced to all possible maps in the well-known mathematical four-color theorem. Both the periodic striped puddles and random structures can be represented by using planar graphs with a chromatic number χ ≤ 4. To observe the colors in mapping experiments, it is necessary to recover variously oriented tilting effects from the replica. It is established that there is an interplay between the annihilation/creation of new stripes and ordering/disordering tilts in relation to the θ n angle in the CuO2 plane, where the characteristic shape of the stripes coincides with the tilting-ordered regions.

Absolute Hidden Symmetry in Time and Absolute Asymmetry in Mass and Velocity between a Particle and Its Anti-Particle

In this paper we will the discuss possible hidden symmetry in time and the possible asymmetry in mass and velocity between a particle and its anti-particle. Possible hidden symmetry in time between a particle and its anti-particle manifested in their corresponding possible hidden variables in time, fr and -fr  was first discussed in Brodet (2017b). In this paper we will extend the discussion and discuss how the above possible hidden symmetry in time and its corresponding possible symmetry in absolute position, energy and momentum, may yield an asymmetry in mass and velocity of a particle and its corresponding anti-particle. We will deconstruct the particle’s and anti-particle’s absolute position, into three complex number describing the particle/anti-particle time, velocity and mass. The particle/anti-particle symmetry and asymmetry of the above complex numbers will be discussed in the context of parity violation and the known asymmetry in angular production of particles and anti-particles. Moreover, the possible mass asymmetry will be used to explain the CP violation in the neutral kaon system. Finally, experimental ways to investigate and test the above are presented.

A new variational principle, convexity, and supercritical Neumann problems

Utilizing a new variational principle that allows us to deal with problems beyond the usual locally compact structure, we study problems with a supercritical nonlinearity of the type (1) { − Δ u + u = a ( x ) f ( u ) a m p ; in Ω , u > 0 a m p ; in Ω , ∂ u ∂ ν = 0 a m p ; on ∂ Ω . \begin{equation}\tag {1} \begin {cases} -\Delta u + u = a(x) f(u) & \text {in $\Omega $}, \\ u>0 & \text {in $\Omega $}, \\ \frac {\partial u}{\partial \nu } = 0 & \text {on $\partial \Omega $}. \end{cases} \end{equation} To be more precise, Ω \Omega is a bounded domain in R N \mathbb {R}^N which satisfies certain symmetry assumptions, Ω \Omega is a domain of “ m m revolution" ( 1 ≤ m > N 1\leq m>N and the case of m = 1 m=1 corresponds to radial domains), and a > 0 a > 0 satisfies compatible symmetry assumptions along with monotonicity conditions. We find positive nontrivial solutions of (1) in the case of suitable supercritical nonlinearities f f by finding critical points of I I where \[ I ( u ) = ∫ Ω { a ( x ) F ∗ ( − Δ u + u a ( x ) ) − a ( x ) F ( u ) } d x I(u)=\int _\Omega \left \{ a(x) F^* \left ( \frac {-\Delta u + u}{a(x)} \right ) - a(x) F(u) \right \} dx \] over the closed convex cone K m K_m of nonnegative, symmetric, and monotonic functions in H 1 ( Ω ) H^1(\Omega ) where F ′ = f F’=f and where F ∗ F^* is the Fenchel dual of F F . We mention two important comments: First, there is a hidden symmetry in the functional I I due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with supercritical problems lacking the necessary compactness requirement. Second, the energy I I is not at all related to the classical Euler–Lagrange energy associated with (1). After we have proven the existence of critical points u u of I I on K m K_m , we then unitize a new abstract variational approach to show that these critical points in fact satisfy − Δ u + u = a ( x ) f ( u ) -\Delta u + u = a(x) f(u) . In the particular case of f ( u ) = | u | p − 2 u f(u)=|u|^{p-2} u we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.

Hidden symmetries in the HFB norm overlap functions

A brief consideration about hidden symmetries in the HFB norm overlap functions is presented, in particular, in association with the presence of pair-wise degeneracy of the eigenvalue spectrum of a product of two antisymmetric matrices.

Hidden Symmetries of Complex Analysis

We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.

Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics

From a manifestly invariant Lagrangian density based on Clebsch fields and suitable for geophysical fluid dynamics, non-trivial conservation laws and their associated symmetries are described in arbitrary coordinates via Noether’s first theorem. Potential vorticity conservation is however shown to be a trivial law of the second kind with no relevance to Noether’s first theorem. A canonical Hamiltonian formulation is obtained in which Dirac constraints explicitly include the possibly time-dependent metric tensor. It is shown that all Dirac constraints are primary and of the second class, which implies that no infinite-dimensional symmetry transformations of Clebsch fields exist and that Noether’s second theorem does not apply to the governing equations. Therefore, the considered Lagrangian density does not admit a symmetry associated with potential vorticity conservation via Noether’s two theorems. Finally, the corresponding non-canonical Hamiltonian structure with time-dependent strong constraints is derived using tensor components for arbitrary coordinates. The existence of Casimir invariants is linked to trivial conservation laws of the second kind and to symmetries that become hidden after a transformation away from canonical dynamical fields.

Multiple Coulomb phase in the fluoride pyrochlore CsNiCrF6

The Coulomb phase is an idealized state of matter whose properties are determined by factors beyond conventional considerations of symmetry, including global topology, conservation laws and emergent order. Theoretically, Coulomb phases occur in ice-type systems such as water ice and spin ice; in dimer models; and in certain spin liquids. However, apart from ice-type systems, more general experimental examples are very scarce. Here we study the partly disordered material CsNiCrF6 and show that this material is a multiple Coulomb phase with signature correlations in three degrees of freedom: charge configurations, atom displacements and spin configurations. We use neutron and X-ray scattering to separate these correlations and to determine the magnetic excitation spectrum. Our results show how the structural and magnetic properties of apparently disordered materials may inherit, and be dictated by, a hidden symmetry—the local gauge symmetry of an underlying Coulomb phase. Neutron and X-ray scattering experiments show that the partly disordered material CsNiCrF6 supports multiple Coulomb phases with structural and magnetic properties dictated by underlying local gauge symmetry.

Hidden symmetries in real and theoretical networks

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called latent symmetry, which is an extension of the standard notion of symmetry on networks, which can be directed, weighted or both. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a size, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erdős–Rényi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that latent symmetries are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

Cosmologies with Scalar Fields from Higher Dimensions Applied to Bianchi Type VIh=-1 Model: Classical and Quantum Solutions

We construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalar dilaton field on a time-dependent torus. The corresponding action in four dimensions is similar to the action of K-essence theories. This approach is applied to anisotropic cosmological Bianchi type VI(h=-1) model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. The classical Einstein field equations give us a hidden symmetry, corresponding to the equality between two radii B=C, which permits us to solve exactly the equations of motion. One relation between the scale factors (A,C) via the solutions is found. With this hidden symmetry, then we solve the FRW model, finding that the scale factor goes to B radii. Also the corresponding Wheeler-DeWitt (WDW) equation in the context of Standard Quantum Cosmology is solved, building a wavepacket when the scalar fields have a hyperbolic behavior, obtaining some qualitative results when we analyze the projection plane to the wall formed by the probability density. Bohm’s formalism for this cosmological model is revisited too.

Explicit Solutions for the (2 + 1)-Dimensional Jaulent–Miodek Equation Using the Integrating Factors Method in an Unbounded Domain

In this work, we prove that the integrating factors can be used as a reduction method. Analytical solutions of the Jaulent–Miodek (JM) equation are obtained using integrating factors as an extension of a recent work where, through hidden symmetries, the JM was reduced to ordinary differential equations (ODEs). Some of these ODEs had no quadrature. We here derive several new solutions for these non-solvable ODEs.

Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

ABSTRACTThis paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove similar results for the wave equation on the exterior of a very slowly rotating Kerr spacetime with recent work applying the vector-field method to the relativistic Vlasov equation.

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.

Asymptotic behavior of observables in the asymmetric quantum Rabi model

The asymmetric quantum Rabi model with broken parity invariance shows spectral degeneracies in the integer case, that is when the asymmetry parameter equals an integer multiple of half the oscillator frequency, thus hinting at a hidden symmetry and accompanying integrability of the model. We study the expectation values of spin observables for each eigenstate and observe characteristic differences between the integer and noninteger cases for the asymptotics in the deep strong coupling regime, which can be understood from a perturbative expansion in the qubit splitting. We also construct a parent Hamiltonian whose exact eigenstates possess the same symmetries as the perturbative eigenstates of the asymmetric quantum Rabi model in the integer case.

Possible hidden symmetry in time

Possible hidden symmetry in time

Hidden symmetry-protected Z2 topological insulator in a cubic lattice

Usually ${Z}_{2}$ topological insulators are protected by time-reversal symmetry. Here we present a new type of ${Z}_{2}$ topological insulators in a cubic lattice which is protected by a novel hidden symmetry, while time-reversal symmetry is broken. The hidden symmetry has a composite antiunitary operator consisting of fractional translation, complex conjugation, sublattice exchange, and local gauge transformation. Based on the hidden symmetry, we define the hidden symmetry polarization and ${Z}_{2}$ topological invariant to characterize the topological insulators. The surface states have band structures with an odd number of Dirac cones, where pseudospin-momentum locking occurs. When the hidden symmetry-breaking perturbations are added on the boundaries, a gap opens in the surface band structure, which confirms that the topological insulator and the surface states are protected by the hidden symmetry. We also discuss the realization and detection of this new kind of ${Z}_{2}$ topological insulator in optical lattices with ultracold atom techniques.

Photon motion in Kerr\u2013de\xa0Sitter\xa0 spacetimes

We study the general motion of photons in the Kerr–de Sitter black-hole and naked singularity spacetimes. The motion is governed by the impact parameters X, related to the axial symmetry of the spacetime, and q, related to its hidden symmetry. Appropriate ‘effective potentials’ governing the latitudinal and radial motion are introduced and their behavior is examined by the ‘Chinese boxes’ technique giving regions allowed for the motion in terms of the impact parameters. Restrictions on the impact parameters X and q are established in dependence on the spacetime parameters M, Lambda , a. The motion can be of orbital type (crossing the equatorial plane, q>0) and vortical type (tied above or below the equatorial plane, q<0). It is shown that for negative values of q, the reality conditions imposed on the latitudinal motion yield stronger constraints on the parameter X than that following from the reality condition of the radial motion, excluding the existence of vortical motion of constant radius. The properties of the spherical photon orbits of the orbital type are determined and used along with the properties of the effective potentials as criteria of classification of the KdS spacetimes according to the properties of the motion of the photon.

Hidden symmetries in N-layer dielectric stacks

The optical properties of a multilayer system with arbitrary N layers of dielectric media are investigated. Each layer is one of two dielectric media, with a thickness one-quarter the wavelength of light in that medium, corresponding to a central frequency f0. Using the transfer matrix method, the transmittance T is calculated for all possible 2N sequences for small N. Unexpectedly, it is found that instead of 2N different values of T at f0 (T0), there are only discrete values of T0, for even N, and (N + 1) for odd N. We explain this high degeneracy in T0 values by finding symmetry operations on the sequences that do not change T0. Analytical formulae were derived for the T0 values and their degeneracies as functions of N and an integer parameter for each sequence we call ‘charge’. Additionally, the bandwidth at f0 and filter response of the transmission spectra are investigated, revealing asymptotic behavior at large N.

Extended symmetry analysis of generalized Burgers equations

Using enhanced classification techniques, we carry out the extended symmetry analysis of the class of generalized Burgers equations of the form ut + uux + f(t, x)uxx = 0. This enhances all the previous results on symmetries of these equations and includes the description of admissible transformations, Lie symmetries, Lie and nonclassical reductions, hidden symmetries, conservation laws, potential admissible transformations, and potential symmetries. The study is based on the fact that the class is normalized, and its equivalence group is finite-dimensional.

Exploiting the hidden symmetry of spinning black holes: conservation laws and numerical tests

The Kerr black hole is stationary and axisymmetric, which leads to conservation of energy and azimuthal angular momentum along the orbits of free test particles in its vicinity, but also to conservation laws for the evolution of continuum matter fields. However, the Kerr space-time possesses an additional "hidden symmetry" which exhibits itself in an unexpected conserved quantity along geodesics known as the Carter constant. We investigate the possibility of using this hidden symmetry to obtain conservation laws and other identities which could be used to test astrophysical simulations of the evolution of matter fields near spinning black holes. After deriving such identities, we set up a simple numerical toy model on which we demonstrate how they can detect the violations of evolution equations in a numerical simulation. Even though one of the expressions we derive is in the form of a conservation law, we end up recommending an equivalent but simpler expression that is not in the form of a conservation law for practical implementation.

Hidden antiunitary symmetry behind “accidental” degeneracy and its protection of degeneracy

In quantum mechanics, accidental degeneracy refers to energy degeneracy that occurs coincidentally, without any protection by symmetry. Here, we prove a theorem stating that any two-fold degeneracy (accidental or not) in a quantum system is protected by a novel hidden symmetry, which can be expressed by an antiunitary operator with its square being -1. In this sense, the so-called accidental degeneracy is not really accidental, and this actually implies a hidden antiunitary symmetry.