Chaos in celestial CFT

The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} currents. This naturally gives rise to a overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} current algebra living on the celestial sphere. The generators of the overline{mathrm{SL}left(2,mathrm{mathbb{C}}right)} current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this algebra. To be more precise, 1) The subleading terms in the OPE are determined in terms of the leading OPE coefficient if we demand that both sides of the OPE transform in the same way under this local symmetry algebra. 2) Positive helicity gravitons have null states under this local algebra whose decoupling leads to differential equations for MHV amplitudes. An n point MHV amplitude satisfies two systems of (n − 2) linear first order PDEs corresponding to (n − 2) positive helicity gravitons. We have checked, using Hodges’ formula, that one system of differential equations is satisfied by any MHV amplitude, whereas the other system has been checked up to six graviton MHV amplitude. 3) One can determine the leading OPE coefficients from these differential equations.This points to the existence of an autonomous sector of the Celestial CFT which holographically computes the MHV graviton scattering amplitudes and is completely defined by this local symmetry algebra. The MHV-sector of the Celestial CFT is like a minimal model of 2-D CFT.

We investigate the phenomenology of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of a system of coupled nonlinear Schrödinger equations(NLSEs) with a combination of linear and nonlinear double-well potentials (DWPs), including self- and cross-phase-modulation (SPM and XPM) nonlinear terms. The DWP structure is represented by a symmetric pair of delta functions. This setup can be realized in optics and for matter waves in Bose-Einstein condensates. It is commonly known that SSB states are completely stable in the single one-dimensional NLSE with a combination of DWP and attractive SPM nonlinearity. The objective of the present work is to elaborate on the SSB and SASB phenomenology in the two-component system. One factor that affects the phenomenology (actually, the stability of stationary states) in this context is the relative sign of the two components, the states with identical/opposite signs defined as ones with in-phase/out-of-phase components. We demonstrate that the SSB/SASB, with both components originating from spatially symmetric or antisymmetric modes, of the out-of-phase type, in the coupled NLSEs including the SPM and XPM terms with opposite signs, are similar to their counterparts in the single NLSE. On the other hand, the SSB/SASB of in-phase states in the coupled NLSEs, including the SPM and XPM terms with opposite signs, are quite different from the ones in the single NLSE. In particular, we find that the in-phase SSB/SASB states are unstable in a part of the intervals of their propagation constantif the system includes SPM and XPM terms with opposite signs. As another feature that is possible only in the two-component system, we address mixed-parity states, originally built of spatially symmetric and antisymmetric components, and find that such states, featuring SSB or SASB, are stable in a part of their interval of the propagation constant. We clarify that the spatially antisymmetric component is a dominant one in determining the asymmetry of the established state if it is coupled to the symmetric one, in the case of the repulsive SPM. On the other hand, the symmetric component dominates if it is coupled to the antisymmetric one in the case of the attractive SPM.

Spontaneous chiral symmetry breaking in QCD with massless quarks at infinite volume can be seen in a finite box by studying, for instance, the dependence of the chiral condensate from the volume and the quark mass. We perform a feasibility study of this program by computing the quark condensate on the lattice in the quenched approximation of QCD at small quark masses. We carry out simulations in various topological sectors of the theory at several volumes, quark masses and lattice spacings by employing fermions with an exact chiral symmetry, and we focus on observables which are infrared stable and free from mass-dependent ultraviolet divergences. The numerical calculation is carried out with an exact variance-reduction technique, which is designed to be particularly efficient when spontaneous symmetry breaking is at work in generating a few very small low-lying eigenvalues of the Dirac operator. The finite-size scaling behaviour of the condensate in the topological sectors considered agrees, within our statistical accuracy, with the expectations of the chiral effective theory. Close to the chiral limit we observe a detailed agreement with the first Leutwyler-Smilga sum rule. By comparing the mass, the volume and the topology dependence of our results with the predictions of the chiral effective theory, we extract the corresponding low-energy constant.

Spontaneous chiral symmetry breaking is a ubiquitous property in nature and diverse fields of modern physics. However, such symmetry breaking has been elusive experimentally in the optical systems, which usually demands multiple identical subsystems [1]. As a prominent photonic device, the ultrahigh-Q whispering-gallery mode (WGM) microresonator supports clockwise (CW) and counterclockwise (CCW) propagating waves coupled to each other, leading to symmetric and antisymmetric standing-wave modes with equal CW and CCW amplitudes. The demonstrations of such overall chirality have to rely on external perturbations to a resonator, either by breaking the parity or time-reversal symmetry [2,3]. The chirality with unbalanced CW and CCW components not only attracts general interest in physics, but also is of importance in novel devices [2,3]. Here, we experimentally demonstrate the spontaneous chirality in a single WGM microresonator (Fig. 1) without any explicit breaking of parity or time-reversal symmetry.

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.

AbsractSymmetry degree is utilized to characterize the asymmetry of a physical system with respect to a symmetry group. The scalar form of symmetry degree (SSD) based on Frobenius-norm has been introduced recently to present a quantitative description of symmetry. Here we present the vector form of the symmetry degree (VSD) which possesses more advantages than the SSD. Mathematically, the dimension of VSD is defined as the conjugacy class number of the symmetry group, the square length of the VSD gives rise to the SSD and the direction of VSD is determined by the orders of the conjugacy classes. The merits of applying VSD both for finite and infinite symmetry groups include the additional information of broken symmetry operators with single symmetry breaking perturbation, and the capability of distinguishing distinct symmetry breaking perturbations which exactly give rise to degenerate SSD. Additionally, the VSD for physical systems under symmetry breaking perturbations can be regarded as a projection of the initial VSD without any symmetry breaking perturbations, which can be described by an evolution equation. There are the same advantages by applying VSD for the accidental degeneracy and spontaneous symmetry breaking.

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

Spontaneous discrete symmetry breaking during inflation and the NMSSM domain wall problem

We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Molmer [1]).

We show how trinification models based on the gauge group SU(3)C× SU(3)L× SU(3)R realized near the TeV scale can provide naturally a variety of dark matter (DM) candidates. These models contain a discrete T parity which may remain unbroken even after spontaneous symmetry breaking. The lightest T-odd particle, which could be a fermion, a scalar, or a gauge boson, can constitute the dark matter of the universe. This framework naturally admits a doublet-singlet fermionic DM, a singlet scalar DM, or a vector boson DM. Here we develop the vector boson DM scenario wherein the DM couples off-diagonally with the usual fermions and vector-like fermions present in the theory. We show consistency of this framework with dark matter relic abundance and direct detection limits as well as LHC constraints. We derive upper limits of 900 GeV on the vector gauge boson DM mass and 4.5 TeV on the vector-like quark masses. We also show the consistency of spontaneous gauge symmetry breaking down to Standard Model times an extra U(1) while preserving the T-parity.

The phenomena of spontaneous PT symmetry breaking, associated with non-Hermitian Hamiltonians, are investigated. It is shown that spontaneous breakdown of PT symmetry is accompanied by the explicit breakdown of pseudo-supersymmetry. We also discuss in detail the resulting structure.

The Higgs mechanics is very powerful, it furnishes a description of the electroweak theory in the standard Model which has a convincing experimental verification. But although the Higgs mechanism had been applied successfully, the conceptual background is not clear. The Higgs mechanism is often presented as spontaneous breaking of local gauge symmetry. But local gauge symmetry is rooted in redundancy of description, gauge transformation connect states that cannot be physically distinguished. Gauge symmetry is therefore not symmetry of nature, but of our description of nature. The spontaneous breaking of such symmetry cannot be expected to have physical effects since asymmetries are not reflected in the physics. If spontaneous gauge symmetry breaking cannot have physical effects, this causes conceptual problems for the Higgs mechanism, if taken to be described as spontaneous gauge symmetry breaking. In a gauge invariant theory, gauge fixing is necessary to retrieve the physics from the theory. This means that also in a theory with spontaneous gauge symmetry breaking, a gauge should be fixed. But gauge fixing itself breaks the gauge symmetry, and thereby obscures the spontaneous breaking of the symmetry. It suggests that spontaneous gauge symmetry breaking is not part of the physics, but an unphysical artefact of the redundancy in description.

Quantum field theories with exact but spontaneously broken conformal invariance have an intriguing feature: their vacuum energy (cosmological constant) is equal to zero. Up to now, the only known ultraviolet complete theories where conformal symmetry can be spontaneously broken were associated with supersymmetry (SUSY), with the most prominent example being the N=4 SUSY Yang-Mills. In this Letter we show that the recently proposed conformal “fishnet” theory supports at the classical level a rich set of flat directions (moduli) along which conformal symmetry is spontaneously broken. We demonstrate that, at least perturbatively, some of these vacua survive in the full quantum theory (in the planar limit, at the leading order of 1/Nc expansion) without any fine tuning. The vacuum energy is equal to zero along these flat directions, providing the first non-SUSY example of a four-dimensional quantum field theory with “natural” breaking of conformal symmetry.

We will discuss a method of finding essential conservation laws for equations possessing an infinite symmetry group whose generators contain arbitrary functions of time. We will show how to generate a finite number of conserved quantities corresponding to infinite symmetries for the equation of non-stationary transonic gas flows in two dimensions, and discuss a critical role of boundary conditions.