Hidden Symmetry Research Articles

Hidden symmetry of the 16D oscillator and its 9D coulomb analogue

We present the quadratic Hahn algebra QH(3) as an algebra of the hidden symmetry for a certain class of exactly solvable potentials, generalizing the 16D oscillator and its 9D coulomb analogue to the generalized version of the Hurwitz transformation based on SU (1,1)⊕ SU (1,1) . The solvability of the Schrodinger equation of these problems by the variables separation method are discussed in spherical and parabolic (cylindrical) coordinates. The overlap coefficients between wave functions in these coordinates are shown to coincide with the Clebsch – Gordan coefficients for the SU(1,1) algebra.

Condensed matter physics in time crystals

Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.

Topological domain wall states in a nonsymmorphic chiral chain

The Su-Schrieffer-Heeger (SSH) model, containing dimerized hopping and a constant onsite energy, has become a paradigmatic model for one-dimensional topological phases, soliton excitations and fractionalized charge in the presence of chiral symmetry. Motivated by the recent developments in engineering artificial lattices, we study an alternative model where hopping is constant but the onsite energy is dimerized. We find that it has a non-symmorphic chiral symmetry and supports topologically distinct phases described by a $\mathbb{Z}_{2}$ invariant $\nu$. In the case of multimode ribbon we also find topological phases protected by hidden symmetries and we uncover the corresponding $\mathbb{Z}_{2}$ invariants $\nu_{n}$. We show that, in contrast to the SSH case, zero-energy states do not necessarily appear at the boundary between topologically distinct phases, but instead these systems support a new kind of bulk-boundary correspondence: The energy of the topological domain wall states typically scales to zero as $1/w$, where $w$ is the width of the domain wall separating phases with different topology. Moreover, under specific circumstances we also find a faster scaling $e^{-w/\xi}$, where $\xi$ is an intrinsic length scale. We show that the spectral flow of these states and the charge of the domain walls are different than in the case of the SSH model.

Searching in the dark: the hunt for the dark photon

The existence of Dark Matter (DM) is a well established fact since many decades, thanks to the observation of the effects of its gravitational interaction with the ordinary matter in the Universe. However, our knowledge of the Dark Matter features is still rather scarce. Indeed, one of the biggest quests in fundamental science today is the investigation of Dark Matter nature, from its origin to its composition, and the way its constituents interact with the ordinary matter, apart from gravity. Huge and ambitious efforts have been spent in the last years into its identification, concentrating especially on the search of viable Weakly Interacting Massive Particle candidates. However, no positive results have been achieved so far along this direction. On the other hand, many fascinating new ideas and models for its interpretation have been blooming: among them, an intriguing hypothesis is that the Dark Matter constituents could be neutral under Standard Model interactions, but they could interact through a new, still unknown, force under a "hidden" charge. This new hidden symmetry would be mediated by a massive gauge boson, the dark photon, which is expected to couple to the Standard Model via a kinetic mixing. The search for such a massive mediator has been pursued with large enthusiasm and dedication in the latest years, as its observation could be within the reach of many already existing experimental facilities, both based on accelerators

The Symmetries and the General Harmonic Solution to Equations of Maxwell Electrodynamics with an Axion

The effective Lagrangian is derived and a group of hidden symmetries is determined for Maxwell’s theory with an axion in the stationary case. The general harmonic solution is constructed, and it is shown that a central source can have only a Coulomb-type magnetic field. The electric field of such a source has a complex potential, which can be finite everywhere with a special choice of parameters. The differential scattering cross section of the test particle on a dione with an axionic charge is calculated within the first order of the perturbation theory.

Massive vector fields in Kerr-Newman and Kerr-Sen black hole spacetimes

The superradiant instability modes of ultralight massive vector bosons are studied for weakly charged rotating black holes in Einstein-Maxwell gravity (the Kerr- Newman solution) and low-energy heterotic string theory (the Kerr-Sen black hole). We show that in both these cases, the corresponding massive vector (Proca) equations can be fully separated, exploiting the hidden symmetry present in these spacetimes. The resultant ordinary differential equations are solved numerically to find the most unstable modes of the Proca field in the two backgrounds and compared to the vacuum (Kerr black hole) case.

Ehlers symmetry in four dimensions

Starting with the light-cone Hamiltonian for gravity, we perform a field redefinition that reveals a hidden symmetry in four dimensions, namely the Ehlers $SL(2,R)$ symmetry. The field redefinition, which is non-local in space but local in time, acts as a canonical transformation in the Hamiltonian formulation keeping the Poisson bracket relations unaltered. We discuss the electro-magnetic duality symmetry of gravity in the light-cone formalism, which forms the $SO(2)$ subgroup of the Ehlers symmetry. The helicity states in the original Hamiltonian are not in a representation of the enhanced symmetry group. In order to make the symmetry manifest, we make a change of variables in the path integral from the helicity states to new fields that transform linearly under the $SO(2)$ duality symmetry.

Quiver Representations and Dimension Reduction in Dynamical Systems

Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.

Investigation of breaking dynamics for Riemann waves in shallow water

Abstract Breaking dynamics of Riemann waves, has a profound impact on studying the dynamics of shock-breaking systems. Calogero– Bogoyavlenskii– Schiff (CBS) equation describes the interaction between Riemann propagating wave along y-axis with long propagating wave along x-axis. Two extensions of the CBS (2+1)-dimensional equation are investigated. Optimization of the commutative product for the nonlocal symmetry is constructed and two successive symmetry reductions reduced the equations to ordinary differential equations (ODEs). Hidden symmetries are used in the second reduction step. The singular manifold method (SMM) is then applied to the ordinary differential equations. This results in a Schwarzian derivative whose solution leads to Cuspon and Kink waves.

Symmetries of Relativistic Hydrogen Atom

The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

We focus on the method recently proposed by Lunin and Frolov–Krtouš–Kubizňák to solve the Maxwell equation on the Kerr-NUT-(A)dS spacetime by separation of variables. In their method, it is crucial that the background spacetime has hidden symmetries because they generate commuting symmetry operators with which the separation of variables can be achieved. In this work we reproduce these commuting symmetry operators in a covariant fashion. We first review the procedure known as the Eisenhart–Duval lift to construct commuting symmetry operators for given equations of motion. Then we apply this procedure to the Lunin–Frolov–Krtouš–Kubizňák (LFKK) equation. It is shown that the commuting symmetry operators obtained for the LFKK equation coincide with the ones previously obtained by Frolov–Krtouš–Kubizňák, up to first-order symmetry operators corresponding to Killing vector fields. We also address the Teukolsky equation on the Kerr-NUT-(A)dS spacetime and its symmetry operator is constructed.

Kitaev honeycomb models in magnetic fields: Dynamical response and dual models

Motivated by recent reports of a field-induced intermediate phase (IP) in the antiferromagnetic honeycomb Kitaev model that may be a spin liquid whose nature is distinct from the Kitaev Z2 phase, we present a detailed numerical study on the nature and dynamical response (such as dynamical spin-structure factors and resonant inelastic x-ray scattering intensities) of this field-induced IP and neighboring phases in a family of Kitaev-based models related by hidden symmetries and duality transformations. We further show that the same field-induced IP can appear in models relevant for $\alpha$-RuCl${_3}$, which exhibit a ferromagnetic Kitaev coupling and additional interactions. In $\alpha$-RuCl${_3}$, the IP represents a new phase, that is likely independent from the putative field-induced (spin-liquid) phase recently reported from thermal Hall conductivity measurements.

Hidden Symmetry and Invariance in Optical Forces

Like any physical quantity whose symmetry properties mimic its source, the optical force acting on a neutral spherical particle has a symmetry that mimics the incident field. The optical force cons...

Seesaw mechanism and pseudo C-symmetry

It is shown that the specific “charge conjugation” transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely (nu _{R})^{C}=Coverline{nu _{R}}^{T} for a chiral fermion nu _{R} (and similarly for nu _{L}), is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. When carefully examined, it also fails as a classical transformation law in a Lorentz invariant field theory. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. The pseudo C-symmetry is effective to identify Majorana neutrinos analogously to the classical Majorana condition. The analysis of CP breaking in weak interactions is performed using the conventional CP transformation, which is defined independently of the pseudo C-transformation, in the seesaw model after mass diagonalization. A way to ensure an operatorially consistent formulation of C-conjugation is to formulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.

Structure of the Order Parameter of Topological Superconductors Based on 3d and 4d Transition Metal Oxides

We consider experimental data on the order parameter structure in superconductors with a layered perovskite structure: La2CuO4 and Sr2RuO4. Experimental data are analyzed using the space group approach to two-electron states in crystals. From the crystalline symmetry of Sr2RuO4, we derive two-electron functions with Eu symmetry, having nodal planes consistent with experimental data. In the case of La2CuO4 and high-Tc materials, the order parameter corresponds to the irreducible representation B1g with hidden symmetry D4h. It is also shown that the charge density waves found in pseudogap states of high-Tc materials can be thought of as two-electron states of equivalent electrons with a nonzero momentum.

Scalar-metric-affine theories: Can we get ghost-free theories from symmetry?

We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian. For this purpose, we study scalar-tensor theories in the metric-affine (Palatini) formalism of gravity, which we call scalar-metric-affine theories for short, where the metric and the connection are independent. We show that the projective symmetry, a local symmetry under a shift of the connection, can provide a ghost-free structure of scalar-metric-affine theories. The ghostly sector of the second-order derivative of the scalar is absorbed into the projective gauge mode when the unitary gauge can be imposed. Incidentally, the connection does not have the kinetic term in these theories and then it is just an auxiliary field. We can thus (at least in principle) integrate the connection out and obtain a form of scalar-tensor theories in the Riemannian geometry. The projective symmetry then hides in the ghost-free scalar-tensor theories. As an explicit example, we show the relationship between the quadratic order scalar-metric-affine theory and the quadratic U-degenerate theory. The explicit correspondence between the metric-affine (Palatini) formalism and the metric one could be also useful for analyzing phenomenology such as inflation.

Mini-Review: Hydrogen Atoms in a High-Frequency Laser Field

Because of the continuing advances in developing lasers in the far-ultraviolet and x-ray ranges, studies of the behavior of atoms under a high-frequency laser field are of theoretical and practical interest. In the present paper, we review various analytical results obtained by the method of separating rapid and slow subsystems for various polarizations of the laser field. Specifically, we review the corresponding analytical results both in terms of the quantum description of the phenomena involved and in terms of the classical description of the phenomena involved. We point out that, for the classical description of hydrogen atoms in a high-frequency laser field, there are interesting celestial analogies. We discuss hidden symmetries of these physical systems, the advantages of this analytical method, and the connection between these results and the transition to chaos.

The Three-body Problem in Riemannian Geometry. Hidden Irreversibility of the Classical Dynamical System

The classical three-body problem is formulated as a problem of geodesic flows on a Riemannian manifold. It is proved that a curved space allows to detect new hidden symmetries of the internal motion of a dynamical system and reduces the three-body problem to the system of 6th order. It is shown that the equivalence of the original Newtonian three-body problem and the developed representation provides coordinate transformations together with an underdetermined system of algebraic equations. The latter makes the system of geodesic equations relative to the evolution parameter (internal time), i.e. to the arc length of the geodesic curve, irreversible.

Hidden Symmetries in Bowtie Nanocavities and Diabolo Nanoantennas

Symmetries play an important role in many branches of physics and enable simplification of the mathematical description of problems. In some cases, symmetries are hidden and are only evident under ...

Eisenhart lift of 2-dimensional mechanics

The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy–momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux–Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.