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In this paper, we study a generalized Biot–Savart law for suitable velocity that possibly diverges at infinity, and then show its application to the 2D general incompressible inviscid fluids. We first prove the generalized Biot–Savart law for the active scalar equations in a discrete rotational symmetry framework, which allows the velocity grow almost linearly at infinity. Based on this, we further obtain a unique global symmetric solution to Euler equation under the Yudovich type regularity. Additionally, we investigate the local well-posedness for the Boussnesq equation, SQG equation, and especially for the IPM equation which enjoys particular symmetric property in our setting. The proof mainly relies on the Fourier model analysis and some refined estimates to the singular integral operator. [ABSTRACT FROM AUTHOR]

In this work we discuss the deformed relativistic wave equations, namely the Klein–Gordon and Dirac equations in a doubly special relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert space for the Klein–Gordon case. [ABSTRACT FROM AUTHOR]

Although highly successful, the truncated Wigner approximation (TWA) does not account for genuine many-body (MB) quantum interference between different solutions of the mean-field equations of a bosonic MB system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck’s constant â„Ź. The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being compared with ‘exact’ quantum calculations that do not involve any a priori approximation. A rather prominent cause for such quantum effects that are not accounted for by the TWA is the constructive interference between the contributions of symmetry-related trajectories, which would occur in the presence of discrete symmetries provided the phase-space distribution of the initial state and the observable to be evaluated feature a strong localization about the corresponding symmetry subspaces. Here we show how one can conceive an augmented version of the TWA which can account for this particular effect. This augmented TWA effectively amounts to complementing conventional TWA calculations by separate truncated Wigner simulations that are restricted to symmetric subspaces and involve weight factors that account for the dynamical stability of sampling trajectories with respect to perpendicular deviations from those subspaces. We illustrate the validity of this method at pre- as well as post-Ehrenfest time scales in prototypical Boseâ€"Hubbard systems displaying chaotic classical dynamics, where it also reveals the existence of additional MB interference effects. [ABSTRACT FROM AUTHOR]

We construct a B âˆ' L model with A 4 Ă— Z 3 Ă— Z 4 flavor symmetry that accounts for the recent 3 + 1 active-sterile neutrino data. The tiny neutrino mass and the mass hierarchy are obtained by the type-I seesaw mechanism. The hierarchy of the lepton masses is satisfied by a factor of v H v l Λ 2 ∼ 1 0 âˆ' 4 G e V of the electron mass compared to the muon and tau masses of the order of v H v l Λ ∼ 1 0 âˆ' 1 G e V. The 3 + 1 active-sterile neutrino mixings are predicted to be 0.015 â©˝ | U 14|2 â©˝ 0.045, 0.004 â©˝ | U 24|2 â©˝ 0.012 and 0.004 â©˝ | U 34|2 â©˝ 0.014 for normal hierarchy while 0.020 â©˝ | U 14|2 â©˝ 0.045, 0.008 â©˝ | U 24|2 â©˝ 0.018 and 0.008 â©˝ | U 34|2 â©˝ 0.022 for inverted hierarchy. Sterile neutrino masses are predicted to be 0.7 ≲ m s  (eV) ≲ 3.16 for normal hierarchy and 2.6 ≲ m s  (eV) ≲ 7.1 for inverted hierarchy. For three neutrino scheme the model predicts 0.3401 â©˝ sin2  θ 12 â©˝ 0.3415, 0.460 â©˝ sin2  θ 23 â©˝ 0.540, âˆ'0.60 â©˝ sin  δ CP â©˝ âˆ'0.20 for normal hierarchy and 0.3402 â©˝ sin2  θ 12 â©˝ 0.3416, 0.434 â©˝ sin2  θ 23 â©˝ 0.610, âˆ'0.95 â©˝ sin  δ CP â©˝ âˆ' 0.60 for inverted hierarchy. [ABSTRACT FROM AUTHOR]

A Sagnac interferometer’s ring structure causes electromagnetic waves traversing it to periodically encounter the same optical elements. Due to this discrete translational symmetry, the frequency spectrum of the clockwise and counter-clockwise modes acquire a band structure with a characteristic band gap. When the interferometer is rotated, an additional non-reciprocal phase shift between the counter propagating modes arises and it results in the loss of time reversal symmetry. While prior understanding of the impact of Sagnac rotation on the band structure exists, the prevalence of topological geometric phase in Sagnac interferometer under rotation has not been prominently discussed. We propose a coupled mode theory with the required time reversal symmetry properties which influences the Berry curvature and we show that it leads to the accumulation of Berry phase with tunable topological charge. [ABSTRACT FROM AUTHOR]

A manifestly Lorentz-covariant formulation of loop quantum gravity (LQG) is given in terms of finite-dimensional representations of the Lorentz group. The formulation accounts for discrete symmetries, such as parity and time-reversal, and it establishes a link with Wigner classification of particles. The resulting quantum model can be seen as LQG-like with the internal SU (2) ⊗ SU (2) group and it is free of the Immirzi parameter, while the scalar constraint is just the Euclidean part. [ABSTRACT FROM AUTHOR]

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups. [ABSTRACT FROM AUTHOR]

We discuss the phenomenology of type-II seesaw by extending the standard model with additional Higgs doublets, scalar triplets as well as a scalar singlet invoked with discrete flavor symmetry Δ(27), for the explanation of non-zero neutrino masses and mixing, matter–antimatter asymmetry, and lepton flavor violation (LFV). The light neutrino masses can be obtained via type-II seesaw mechanism by transforming scalar triplets as 3 and scalar doublets as various singlet representations like 1 1 , 1 2 , 1 3 along with scalar singlet as 1 1 under Δ(27) symmetry. With the implementation of this discrete flavor symmetry Δ(27), we get simpler structure for light neutrino mass matrix, which helps in analyzing neutrino phenomenology analytically as well as numerically. We further demonstrate the detailed numerical analysis by scanning the input model parameters in agreement with neutrino oscillation data like non-zero reactor mixing angle (θ 13), CP-violating phase (δ CP), the sum of the light neutrino masses, two mass squared differences, and its implication to neutrino-less double beta decay. The model also explains the matter–antimatter asymmetry of the Universe through resonant leptogenesis with the decay of TeV scale scalar triplets and variation of CP-asymmetry with input model parameters. Finally, we comment on the implication to LFV decays like μ → eγ, μ → 3 e processes via the mediation of these TeV scale scalar triplets. We have studied collider physics briefly. [ABSTRACT FROM AUTHOR]

Given its briefness and predictability, the minimal seesaw—a simplified version of the canonical seesaw mechanism with only two right-handed neutrino fields—has been studied in depth and from many perspectives, and now it is being pushed close to a position of directly facing experimental tests. This article is intended to provide an up-to-date review of various phenomenological aspects of the minimal seesaw and its associated leptogenesis mechanism in neutrino physics and cosmology. Our focus is on possible flavor structures of such benchmark seesaw and leptogenesis scenarios and confronting their predictions with current neutrino oscillation data and cosmological observations. In this connection particular attention will be paid to the topics of lepton number violation, lepton flavor violation, discrete flavor symmetries, CP violation and antimatter of the Universe. [ABSTRACT FROM AUTHOR]

In this paper, we develop the quantum theory that has discrete Poincaré symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry which coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincaré group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green's function of the theory. The spacetime points where the Green's function is nonzero display a lattice structure. During the propagation, the particle stays localized on a single or a few sites to preserve the Lorentz symmetry. [ABSTRACT FROM AUTHOR]

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres–Douglas theories on S1 × M3 with a non-trivial holonomy of a discrete global symmetry along the S1. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern–Simons and Rozansky–Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the and Chern–Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d theory. We test our proposal by applying localization techniques to the conjectural UV Lagrangian descriptions of the (A1, A2), (A1, A3) and (A1, D3) theories. [ABSTRACT FROM AUTHOR]

We explore an extended quantum Rabi model describing the interaction between a two-mode bosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom model is found when the ratio η of the atom energy scale to the bosonic field frequency approaches infinity. An analytical solution is provided when the two lowest-energy levels are degenerate. According to it, we recognize that the phase diagram of the model consists of three regions: one normal phase and two superradiant phases. The quantum phase transitions between the normal phase and the two superradiant phases are of second order relating to the spontaneous breaking of the discrete Z2 symmetry. On the other hand, the quantum phase transition between the two different superradiant phases is discontinuous with a phase boundary line relating to the continuous U(1) symmetry. For a large enough but finite η, the scaling function and critical exponents are derived analytically and verified numerically, from which the universality class of the model is identified. [ABSTRACT FROM AUTHOR]

Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, an exotic new phase of matter. We present observations characteristic of discrete time crystalline order in a driven system of paramagnetic P-donor impurities in isotopically enriched 28Si cooled below 10 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of discrete time crystalline order. This signal has a finite lifetime of ∼100 Floquet periods, but this effect is long-lived relative to coherent spin–spin interaction timescales, lasting ∼104 times longer. We present simulations of the system based on the paradigmatic central spin model and show good agreement with experiment. We investigate the role of dissipation and interactions within this model, and show that both are capable of giving rise to discrete time crystal-like behaviour. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study an emergent semiclassical time crystal composed of two interacting driven-dissipative bosonic modes. The system has a discrete spatial symmetry which, depending on the strength of the drive, can be broken in the time-crystalline phase or it cannot. An exact semiclassical mean-field analysis, numerical simulations in the quantum regime, and the spectral analysis of the Liouvillian are combined to show the emergence of the time crystal and to prove the robustness of the oscillation period against quantum fluctuations. [ABSTRACT FROM AUTHOR]

We recently introduced a class of graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential–difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated with the 2D Toda lattice. [ABSTRACT FROM AUTHOR]

This article provides a coarse-grained review on the experimental tests of fundamental discrete symmetries, focusing on those which appear to be deeply ingrained in nature, or at least rooted into the description of it which we are currently capable of providing, namely those related to space-time inversions and particle/anti-particle exchange. A broad selection of tests is discussed, not in the (vain) pursuit of completeness, but rather with the aim of providing an overview of the variety of approaches, and of the underlying common principles. [ABSTRACT FROM AUTHOR]

Since its inception, no decisive departure from the predictions of the standard model (SM) has been reported. However, recently various experiments have observed a few hints of possible departure from SM predictions in lepton flavor universality observables such as , , muon (g–2), , etc. Many of these are observable where deviation from the SM in the range of (2–4)σ is observed related to the muon (μ) lepton. So these deviations may hint at a possible new physics (NP) in the muon sector. In this work we extend the SM by introducing two SM singlet heavy charged leptons (Fe, Fμ), whose left-handed components are charged under a new U(1)F gauge symmetry, one color triplet lepto-quark (ϕQ) doublet under SU(2)L, one inert Higgs doublet (ϕl), and three very heavy Majorana neutrinos (NiR), all of which are odd under a Z2 discrete symmetry. One more scalar (ϕ) charged only under the U(1)F, whose VEV gives masses to the U(1)F gauge boson as well as the heavy leptons. With these new particles, we show that the observed anomalies in the muon sector as well as small neutrino masses can be explained taking into account all the other experimental and theoretical constraints to date. [ABSTRACT FROM AUTHOR]

We investigate the possibility that the first column of the lepton mixing matrix U is given by u1 = (2, - 1, - 1)T / √6. In a purely group-theoretical approach, based on residual symmetries in the charged-lepton and neutrino sectors and on a theorem on vanishing sums of roots of unity, we discuss the finite groups which can enforce this. Assuming that there is only one residual symmetry in the Majorana neutrino mass matrix, we find the almost unique solution ℤq x S4 where the cyclic factor ℤq with q = 1, 2, 3, ... is irrelevant for obtaining u1 in U. Our discussion also provides a natural mechanism for achieving this goal. Finally, barring vacuum alignment, we realize this mechanism in a class of renormalizable models. [ABSTRACT FROM AUTHOR]

The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second-order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary 'quantum of space', and a transparent asymptotic picture of the semiclassical and classical regimes emerges. The definition of coordinates adapted to the Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsionlike modes. [ABSTRACT FROM AUTHOR]

We study the integrable discretization of the generalized coupled integrable dispersionless equations. Two semi-discrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. [ABSTRACT FROM AUTHOR]

The concept of electric and magnetic field lines is intrinsically non-relativistic. Nonetheless, for certain types of fields satisfying certain geometric properties, field lines can be defined covariantly. More precisely, two Lorentz-invariant 2D surfaces in spacetime can be defined such that magnetic and electric field lines are determined, for any observer, by the intersection of those surfaces with spacelike hyperplanes. An instance of this type of field is constituted by the so-called Hopf-Rañada solutions of the source-free Maxwell equations, which have been studied because of their interesting topological properties, namely, linkage of their field lines. In order to describe both geometric and topological properties in a succinct manner, we employ the tools of geometric algebra (aka Clifford algebra) and use the Clebsch representation for the vector potential as well as the Euler representation for both magnetic and electric fields. This description is easily made covariant, thus allowing us to define electric and magnetic field lines covariantly in a compact geometric language. The definitions of field lines can be phrased in terms of 2D surfaces in space. We display those surfaces in different reference frames, showing how those surfaces change under Lorentz transformations while keeping their topological properties. As a byproduct we also obtain relations between optical helicity, optical chirality and generalizations thereof, and their conservation laws. [ABSTRACT FROM AUTHOR]

Periodically driven quantum systems manifest various non-equilibrium features which are absent at equilibrium. For example, discrete time-translation symmetry can be broken in periodically driven quantum systems leading to an exotic phase of matter, called discrete time crystal (DTC). For open quantum systems, previous studies showed that DTC can be found only when there exists a meta-stable state in the undriven system. However, by investigating the simplest Bose–Hubbard model with dissipation and time periodically tunneling, we find in this paper that a 2T DTC can appear even when the meta-stable state is absent in the undriven system. This observation extends the understanding of DTC and shed more light on the physics behind the DTC. Besides, by the detailed analysis of simplest two-sites model, we show further that the two-sites model can be used as basic building blocks to construct large rings in which a nT DTC might appear. These results might find applications into engineering exotic phases in driven open quantum systems. [ABSTRACT FROM AUTHOR]

Measuring the plasma equilibrium is essential for real time plasma control and for the investigation of an abundance of physics questions. However, many existing kinetic equilibria reconstruction techniques rely heavily on integrated magnetic measurements and neutron-sensitive diagnostics, which will be difficult to design and operate in ITER and DEMO. Here we present and test a conceptual design for a non-magnetic equilibrium reconstruction method using data from a radial electron cyclotron emission diagnostic and an image of the plasma boundary—diagnostics which can be robustly designed in high-neutron environments. This technique is based on the measurement of a discrete q-profile and the symmetry mapping of the electron temperature profile, both of which can be acquired with ECE. [ABSTRACT FROM AUTHOR]

We explore the density and spin self-ordering of driven spin-1/2 collisionless fermionic atoms coupled to the electromagnetic fields of a ring resonator. The two spin states are two-photon Raman-coupled via a pair of degenerate counterpropagating cavity modes and two transverse pump fields. In this one-dimensional configuration the coupled atom-field system possesses a continuous U(1) translational symmetry and a discrete Z2 spin inversion symmetry. At half filling for sufficiently strong pump strengths, the combined U(1) × Z2 symmetry is spontaneously broken at the onset of a superradiant phase transition to a state with self-ordered density and spin structures. We predominately find an antiferromagnetic lattice order at the cavity wavelength. The self-ordered states exhibit unexpected positive momentum pair correlations between fermions with opposite spin. These strong cavity-mediated correlations vanish at higher pump strength. [ABSTRACT FROM AUTHOR]

This work is a brief presentation of the theory based on the gauge group in the presence of heavy leptons. Recent studies [1] have considered a set of four possible variants for the 3-3-1HL, whose content arises according to the so-denoted variable β. Since it has been argued about the presence of stable charged particles in this sort of model, we divide the different sectors of the Lagrangian between universal and specific vertices, and conclude that the omission of β-dependent terms in the potential may induce discrete symmetry for the versions defined by . In the context of , where the new degrees of freedom have the same standard electric charges, additional Yukawa interactions may create decay channels into the SM sector. Furthermore, motivated by a general consequence of the Goldstone theorem, a method of diagonalization by parts is introduced in the Scalar sector and provides a clarification on the definition of mass eigenstates. In summary, we develop the most complete set of terms allowed by the symmetry group and resolve their definitive pieces in order to justify the model description present in the literature. [ABSTRACT FROM AUTHOR]

Time reversal and spin flip are discrete symmetry operations of substantial importance to quantum information and quantum computation. Spin flip arises in the context of separability, quantification of entanglement and the construction of universal NOT gates. The present work investigates the relationship between the quantum state of a multiqubit system represented by the discrete Wigner function (DWFs) and its spin-flipped counterpart. The two are shown to be related through a Hadamard matrix that is independent of the choice of the quantum net used for the tomographic reconstruction of the DWF. These results are of interest to cases involving the direct tomographic reconstruction of the DWF from experimental data, and in the analysis of entanglement related properties purely in terms of the DWF. [ABSTRACT FROM AUTHOR]

We study the dynamical properties of the quantum Rabi model using a systematic expansion method. Based on the observation that the parity symmetry of the Rabi model is kept during evolution of the states, we decompose the initial state and the time-dependent one into positive and negative parity parts expanded by superposition of the coherent states. The evolutions of the corresponding positive and the negative parities are obtained, in which the expansion coefficients in the dynamical equations are known from the derived recurrence relation. [ABSTRACT FROM AUTHOR]

In this work, we derive an analytical procedure that allows us to write the multidimensional washboard ratchet potential (MDWBP) Uf for a two-dimensional Josephson junction array. The array has an applied perpendicular magnetic field. The magnetic field is given in units of the quantum flux per plaquette or frustration of the form , where Φ0 is the flux quantum. The derivation is done under the assumption that the checkerboard pattern ground state or unit cell of a two-dimensional Josephson junction array is preserved under current biasing. The resistively and capacitively shunted Josephson junction model with a white noise term describes the dynamics for each junction in the array. The multidimensional potential is the unique expression of the collective effects that emerge from the array in contrast to the single junction. The first step in the procedure is to write the equation for the phases for the unit cell. In doing this, one takes into account the constraints imposed for the gauge invariant phases due to frustration. Second, and the key idea of the procedure, is to perform a variable transformation from the original systems of stochastic equations to a system of variables where the condition for the equality of mixed second partial happens. This is achieved via Poincaré's theorem for differential forms. In this way, we find to a nonlinear matrix equation (equation (9) in the text), that permits us to find the new coordinate variables xf where the potential exists. The transformation matrix also permits the correct transformation of the original white noise terms of each junction to the intensities in the xf variables. The commensurate symmetries of the ground state pinned vortex lattice leads to discrete symmetries to the part of the washboard potential that does not contain a tilt due to the external bias current (equation (11) in the text). In this work we apply the procedure for the important cases . For , we show that previously efforts for finding the potential are restricted, leading to a reduced dimension of the potential. The correct potential is given in equation (21). We examine this issue in detail. New physics emerge when currents are applied in the x and y directions, in particular, we confirm analytically previous numerical work for , concerning the border of stationary states, a landmark of the potential. For , we give a generalization of previous work, in which we include both the currents in the x and y directions as well the noise terms. We find the MDWBP realizes tilted ratchets analogous to a combustion motor. [ABSTRACT FROM AUTHOR]

We find new classes of exact solutions of the initial momentum constraint for vacuum Einstein’s equations. Considered data are either invariant under a continuous symmetry or they are assumed to have the exterior curvature tensor of a simple form. In general the mean curvature H is non-constant and g is not conformally flat. In the generic case with the symmetry we obtain general solution in an explicit form. In other cases solutions are given up to quadrature. We also find a class of explicit solutions without symmetries which generalizes data induced by the Kerr metric or other metrics related to the Ernst equation. The conformal method of Lichnerowicz, Choquet-Bruhat and York is used to prove solvability of the Hamiltonian constraint if H vanishes. Existence of marginally outer trapped surfaces in initial manifold is discussed. [ABSTRACT FROM AUTHOR]