Continuous Symmetry Research Articles

Lorentz Violation in Astroparticles and Gravitational Waves

Lorentz invariance is one of the fundamental continuous symmetries of the laws of nature [...]

Bipolar fuzzy metric spaces with application

The concept of a bipolar metric space is presented in this article. In bipolar fuzzy metric space, the notions of continuous t-norms and bipolar continuous symmetry t_s-conorms employs important role. Requirements in metric space, triangular norms are used to extrapolate with the probability density function of triangle inequality. Dual processes of triangular norms are known as triangular conorms. Continuous t-norms and continuous symmetry t_s-conorms are used to define bipolar metric space in this paper. Besides, a number of topological and functional properties of the bipolar metric space have been studied. Afterwards, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. After that, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed. Finally, the validity of the proposed MADM method for selecting the best proper form of vaccine is also established.

RG flows with global symmetry breaking and bounds from chaos

We discuss general aspects of renormalization group (RG) flows between two conformal fixed points in 4D with a broken continuous global symmetry in the UV. Every such RG flow can be described in terms of the dynamics of Nambu-Goldstone bosons of broken conformal and global symmetries. We derive the low-energy effective action that describes this class of RG flows from basic symmetry principles. We view the theory of Nambu-Goldstone bosons as a theory in anti--de Sitter space with the flat space limit. This enables an equivalent 3D CFT (${\mathrm{CFT}}_{3}$) formulation of these 4D RG flows in terms of spectral deformations of a generalized free ${\mathrm{CFT}}_{3}$. We utilize this dual description to impose further constraints on the low-energy effective action associated with unitary RG flows in 4D by invoking the chaos bound in 3D. This approach naturally provides a set of independent monotonically decreasing $C$-functions for 4D RG flows with global symmetry breaking by explicitly relating 4D $C$-functions with certain out-of-time-order correlators that diagnose chaos in 3D. We also comment on a more general connection between RG and chaos in QFT.

Resonance-enhanced spectral funneling in Fabry–Perot resonators with a temporal boundary mirror

Abstract A temporal boundary refers to a specific time at which the properties of an optical medium are abruptly changed. When light interacts with the temporal boundary, its spectral content can be redistributed due to the breaking of continuous time-translational symmetry of the medium where light resides. In this work, we use this principle to demonstrate, at terahertz (THz) frequencies, the resonance-enhanced spectral funneling of light coupled to a Fabry–Perot resonator with a temporal boundary mirror. To produce a temporal boundary effect, we abruptly increase the reflectance of a mirror constituting the Fabry–Perot resonator and, correspondingly, its quality factor in a step-like manner. The abrupt increase in the mirror reflectance leads to a trimming of the coupled THz pulse that causes the pulse to broaden in the spectral domain. Through this dynamic resonant process, the spectral contents of the input THz pulse are redistributed into the modal frequencies of the high-Q Fabry–Perot resonator formed after the temporal boundary. An energy conversion efficiency of up to 33% was recorded for funneling into the fundamental mode with a Fabry–Perot resonator exhibiting a sudden Q-factor change from 4.8 to 48. We anticipate that the proposed resonance-enhanced spectral funneling technique could be further utilized in the development of efficient mechanically tunable narrowband terahertz sources for diverse applications.

Restrictions on realizable unitary operations imposed by symmetry and locality

According to an elementary result in quantum computing, any unitary transformation on a composite system can be generated using 2-local unitaries, i.e., those that act only on two subsystems. Beside its fundamental importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. We ask if such universality remains valid in the presence of conservation laws and global symmetries. In particular, can k-local symmetric unitaries on a composite system generate all symmetric unitaries on that system? Interestingly, it turns out that the answer is negative in the case of continuous symmetries, such as U(1) and SU(2): generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries. In fact, the difference between the dimensions of the manifold of all symmetric unitaries and the submanifold of unitaries generated by k-local symmetric unitaries, constantly increases with the system size. On the other hand, we find that this no-go theorem can be circumvented using ancillary qubits. For instance, any unitary invariant under rotations around z can be implemented using Hamiltonian XX+YY together with local Z Hamiltonian on the ancillary qubit. Moreover, any globally energy-conserving unitary on a composite system can be implemented using a sequence of 2-local energy-conserving unitaries, provided that one can use a single ancillary qubit (catalyst).

Continuous and discrete symmetries of renormalization group equations for neutrino oscillations in matter

Three-flavor neutrino oscillations in matter can be described by three effective neutrino masses (for i = 1, 2, 3) and the effective mixing matrix V αi (for α = e,μ, τ and i = 1, 2, 3). When the matter parameter is taken as an independent variable, a complete set of first-order ordinary differential equations for and |V αi |2 have been derived in the previous works. In the present paper, we point out that such a system of differential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the differential equations and looking for differential invariants are discussed.

Construction of Nahm data and BPS monopoles with continuous symmetries

We study solutions to Nahm’s equations with continuous symmetries, and under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.

Effects of the Symmetries and Related Orders to the Thermalization of Many-Body Localized System

In this paper, we discuss the effects of the symmetries and related topological orders to the thermalization of many-body localized system. We consider the one-dimensional fermion chain system with open (or periodic) boundary condition, whose boundaries are characterized by Sachdev-Ye-Kitaev (SYK) intercation. Just like in the SYK model and the tendor models, there are many-body quantum chaos in out-of-time-ordered correlation when the system is being thermalized by the interactions (usually nonuniform and being randomly distributed), and satisfies the eigenstate thermalization hypothesis (ETH). While the continuous or discrete symmetries usually protect the related topological orders against the thermalization, which may lead to the localization of quantum states and generate large degeneracy. We discuss these effects in terms of the fermionic or spin languages. In many-body localized phase, a large number of degenerate ETH-violated states can be found in both the frustration-free AKLT model or the integrable biquadratic (Wishart) SYK model. The quantum scars appear when such degenerate states are embeded into an ETH-satisfying spectrum of the Hilbert space enlarged by a much larger number of bosonic flavors (e.g., SU(M) multiplets degeneracy). Usually, the emergence of quantum chaos require large limits of boson flavor M and the number of coupled (undegenerate) states (N; which related to Z N symmetry). Further, the boson (or excitations) flavor number M can often to related to the fermion number N through the duality transformations, in which case the Z M symmetry is possible to generted by SU(M) model and leads to asymptotic degeneracy in large-M limit.

Continuous Symmetry Analysis of the Effects of City Infrastructures on Invariant Metrics for House Market Volatilities

The invariant metrics of the effects of park size and distance to public transportation on housing value volatilities in Boston, Milwaukee, Taipei and Tokyo are investigated. They reveal a Cobb-Douglas-like behavior. The scale-invariant exponents corresponding to the percentage of a green area (a) are 7.4, 8.41, 14.1 and 15.5 for Boston, Milwaukee, Taipei and Tokyo, respectively, while the corresponding direct distances to the nearest metro station (d) are −5, −5.88, −10 and −10, for Boston, Milwaukee, Taipei and Tokyo, respectively. The multiphysics-based analysis provides a powerful approach for the symmetry characterization of market engineering. The scaling exponent ratio between park area percentages and distances to metro stations is approximately 3/2. The scaling exponent ratio expressed in the perceptual stimuli will remain invariant under group transformation. According to Stevens' power law, the perception-dependent feature spaces for parks and public transportation can be described as two- and three-dimensional conceptual spaces. Based on the prolongation structure of the Schrödinger equation, the SL(2, R) models are used to analyze the house-price volatilities. Consistent with Shepard’s law, the rotational group leads to a Gaussian pattern, exhibiting an extension of the special linear group structure by embedding SO(3) ⊗ R(3) in SL(2, R). The influencing factors related to cognitive functioning exhibit substantially different scale-invariant characteristics corresponding to the complexity of the socio-economic features. Accordingly, the contour shapes of the price volatilities obtained from the group-theoretical analysis not only corroborate the impact of the housing pricing estimation in these cities but also reveal the invariant features of their housing markets are faced with the forthcoming sustainable development of big data technologies and computational urban science research.

Invisible strings. The first single crystal of the cTSAP form of [Eu(DOTA)(H2O)]- has an electronic structure similar to one of the reported cSAP forms.

The coordination geometry of lanthanide(III) ions is extremely sensitive to perturbation from the surrounding environment. Changes in the crystal field can be observed as spectral variations in the emission spectra of luminescent lanthanide(III) ions. Europium(III) ions are commonly used to correlate luminescence properties to the crystal field. In solution, kinetically inert complexes as [Ln(DOTA)(H2O)]- can fluctuate and give rise to different diastereomers (H4DOTA = 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid). The [Ln(DOTA)(H2O)]- complex adopts a capped square antiprism (cSAP) geometry but can rotate into a capped twisted square antiprism (cTSAP). The time scale of the solution dynamics in [Ln(DOTA)(H2O)]- is shorter than that of luminescence emission, thus, structural averages are observed in the emission spectra. For the first time, we were able to crystallise both forms of the [Eu(DOTA)(H2O)]- diastereomers. The single crystal structure was combined with single crystal luminescence spectroscopy to reveal the electronic structure of Eu(III) in each form of the complex. The coordination geometry of the crystallized cSAP and cTSAP forms of the complex was compared to ideal coordination polyhedra using a continuous symmetry measure in the AlignIt code. The diastereomers of [Eu(DOTA)(H2O)]- all demonstrate very little deviation from ideal geometries yet nearly identical electronic properties were observed from the two different forms.

Democratic Lagrangians for Nonlinear Electrodynamics.

We construct a Lagrangian for general nonlinear electrodynamics that features electric and magnetic potentials on equal footing. In the language of this Lagrangian, discrete and continuous electric-magnetic duality symmetries can be straightforwardly imposed, leading to a simple formulation for theories with the SO(2) duality invariance. When specialized to the conformally invariant case, our construction provides a manifestly duality-symmetric formulation of the recently discovered ModMax theory. We briefly comment on a natural generalization of this approach to p-forms in 2p+2 dimensions.

Two Nambu-Goldstone zero modes for rotating Bose-Einstein condensates

We consider rotating finite-size vortex arrays in Bose-Einstein condensates that are confined by cylindrically symmetric external potentials. We show that such systems possess two exact Nambu-Goldstone zero modes associated with two spontaneously broken continuous symmetries of the system. We verify our analytical result via direct numerical diagonalizations of the Bogoliubov--de Gennes equations. We conclude by comparing rotating vortex lattices in superfluids to supersolids and discrete time crystals.

Spontaneous Symmetry Breaking via Inhomogeneities and the Differential Surface Tension.

We discuss spontaneously broken quantum field theories with a continuous global symmetry group via the constraint effective potential. Employing lattice simulations with constrained values of the order parameter, we demonstrate explicitly that the path integral is dominated by inhomogeneous field configurations and that these are unambiguously related to the flatness of the effective potential in the broken phase. We determine characteristic features of these inhomogeneities, including their topology and the scaling of the associated excess energy with their size. Concerning the latter we introduce the differential surface tension-the generalization of the concept of a surface tension pertaining to discrete symmetries. Within our approach, spontaneous symmetry breaking is captured merely via the existence of inhomogeneities, i.e., without the inclusion of an explicit breaking parameter and a careful double limiting procedure to define the order parameter. While here we consider the three-dimensional O(2) model, we also elaborate on possible implications of our findings for the chiral limit of QCD.

1-form symmetry, isolated mathcal{N} = 2 SCFTs, and Calabi-Yau threefolds

We systematically study 4D mathcal{N} = 2 superconformal field theories (SCFTs) that can be constructed via type IIB string theory on isolated hypersurface singularities (IHSs) embedded in ℂ4. We show that if a theory in this class has no mathcal{N} = 2-preserving exactly marginal deformation (i.e., the theory is isolated as an mathcal{N} = 2 SCFT), then it has no 1-form symmetry. This situation is somewhat reminiscent of 1-form symmetry and decomposition in 2D quantum field theory. Moreover, our result suggests that, for theories arising from IHSs, 1-form symmetries originate from gauge groups (with vanishing beta functions). One corollary of our discussion is that there is no 1-form symmetry in IHS theories that have all Coulomb branch chiral ring generators of scaling dimension less than two. In terms of the a and c central charges, this condition implies that IHS theories satisfying a<frac{1}{24}left(15r+2fright) and c<frac{1}{6}left(3r+fright) (where r is the complex dimension of the Coulomb branch, and f is the rank of the continuous 0-form flavor symmetry) have no 1-form symmetry. After reviewing the 1-form symmetries of other classes of theories, we are motivated to conjecture that general interacting 4D mathcal{N} = 2 SCFTs with all Coulomb branch chiral ring generators of dimension less than two have no 1-form symmetry.

Critical drag as a mechanism for resistivity

A quantum many-body system with a conserved electric charge can have a DC resistivity that is either exactly zero (implying it supports dissipationless current) or nonzero. Exactly zero resistivity is related to conservation laws that prevent the current from degrading. In this paper, we carefully examine the situations in which such a circumstance can occur. We find that exactly zero resistivity requires either continuous translation symmetry, or an internal symmetry that has a certain kind of "mixed anomaly" with the electric charge. (The symmetry could be a generalized global symmetry associated with the emergence of unbreakable loop or higher dimensional excitations.) However, even if one of these is satisfied, we show that there is still a mechanism to get nonzero resistivity, through critical fluctuations that drive the susceptibility of the conserved quantity to infinity; we call this mechanism "critical drag". Critical drag is thus a mechanism for resistivity that, unlike conventional mechanisms, is unrelated to broken symmetries. We furthermore argue that an emergent symmetry that has the appropriate mixed anomaly with electric charge is in fact an inevitable consequence of compressibility in systems with lattice translation symmetry. Critical drag therefore seems to be the only way (other than through irrelevant perturbations breaking the emergent symmetry, that disappear at the renormalization group fixed point) to get nonzero resistivity in such systems. Finally, we present a very simple and concrete model -- the "Quantum Lifshitz Model" -- that illustrates the critical drag mechanism as well as the other considerations of the paper.

Long range order in atomistic models for solids

The emergence of long range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read– Shockley law for grain boundaries, which we rigorously derive in a simple, explicit geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza–Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques.

Steady plasma flows in a periodic non-symmetric domain

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Phys. Plasmas, vol. 21, 2014, p. 022515) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by Hameiri (J. Math. Phys., vol. 22, 1981, pp. 2080–2088, § III), to obtain a generalized Grad–Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we can obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

Holographic Duals of Argyres-Douglas Theories.

We propose the first explicit holographic duals for a class of superconformal field theories of Argyres-Douglas type, which are inherently strongly coupled and provide a window onto remarkable nonperturbative phenomena (such as mutually nonlocal massless dyons and relevant operators of fractional dimension). The theories under examination are realized by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. In the dual 11d supergravity solutions, the irregular puncture is realized as an internal M5-brane source.

Reparameterization Invariant Model of a Supersymmetric System: BRST and Supervariable Approaches

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one 0 + 1 -dimensional (1D) model of a free massive spinning relativistic particle (i.e., a supersymmetric system) by exploiting its classical infinitesimal and continuous reparameterization symmetry transformations. We use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the target space variables and the (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction for the 1D model of our supersymmetric (SUSY) system. The nilpotent (anti-)BRST symmetry transformations for other variables of our model are derived by using the (anti-)chiral supervariable approach (ACSA) to BRST formalism. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e., a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e., 1D diffeomorphism) invariant models of the (non-)relativistic particles.

Symmetry and Symmetry Breaking in Physics: From Geometry to Topology

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle physics, and quantum physics, respectively. The latter groups now play an important role in many branches of mathematics. In more recent years, we have been faced with the impact of topological quantum field theory (TQFT). Topology and symmetry have deep connections, but topology is inherently broader and more complex. While the presence of symmetry in physical phenomena imposes strong constraints, topology seems to be related to low-energy states and is very likely to provide information about the different dynamical trajectories and patterns that particles can follow. For example, regarding the relationship of topology to low-energy states, Hodge’s theory of harmonic forms shows that the zero-energy states (for differential forms) correspond to the cohomology. Regarding the relationship of topology to particle trajectories, a topological knot can be seen as an orbit with complex properties in spacetime. The various deformations or embeddings of the knot, performed in low or high dimensions, allow defining different equivalence classes or topological types, and interestingly, it is possible from these types to study the symmetries associated with the deformations and their changes. More specifically, in the present work, we address two issues: first, that quantum geometry deforms classical geometry, and that this topological deformation may produce physical effects that are specific to the quantum physics scale; and second, that mirror symmetry and the phenomenon of topological change are closely related. This paper was aimed at understanding the conceptual and physical significance of this connection.