Exact Symmetry Research Articles

Enforcing exact permutation and rotational symmetries in the application of quantum neural networks on point cloud datasets

Recent developments in the field of quantum machine learning have promoted the idea of incorporating physical symmetries in the structure of quantum circuits. A crucial milestone in this area is the realization of Sn-permutation equivariant quantum neural networks (QNNs) that are equivariant under permutations of input objects. In this paper, we focus on encoding the rotational symmetry of point cloud datasets into the QNN. The key insight of the approach is that all rotationally invariant functions with vector inputs are equivalent to a function with inputs of vector inner products. We provide a structure of the QNN that is exactly invariant to both rotations and permutations, with its efficacy demonstrated numerically in the problems of two-dimensional image classifications and identifying high-energy particle decays, produced by proton-proton collisions, with the SO(1,3) Lorentz symmetry. Published by the American Physical Society 2024

Exact lattice chiral symmetry in 2D gauge theory

We construct symmetry-preserving lattice regularizations of 2D QED with one and two flavors of Dirac fermions, as well as the “3450” chiral gauge theory, by leveraging bosonization and recently proposed modifications of Villain-type lattice actions. The internal global symmetries act just as locally on the lattice as they do in the continuum, the anomalies are reproduced at finite lattice spacing, and in each case we find a sign-problem-free dual formulation. Published by the American Physical Society 2024

Symmetry group detection of point clouds in 3D via a decomposition method

Analyzing the symmetries present in point clouds, which represent sets of 3D coordinates, is important for understanding their underlying structure and facilitating various applications. In this paper, we propose a novel decomposition-based method for detecting the entire symmetry group of 3D point clouds. Our approach decomposes the point cloud into simpler shapes whose symmetry groups are easier to find. The exact symmetry group of the original point cloud is then derived from the symmetries of these individual components. The method presented in this paper is a direct extension of the approach recently formulated in Bizzarri et al. (2022a) for discrete curves in plane. The method can be easily modified also for perturbed data. This work contributes to the advancement of symmetry analysis in point clouds, providing a foundation for further research and enhancing applications in computer vision, robotics, and augmented reality.

Conjectures about the Chiral Phase Transition in QCD from Anomalous Multi-Instanton Interactions.

To date numerical simulations of lattice QCD have not found a chiral phase transition of first order that is expected to occur for sufficiently light pions. We show how the restoration of an exact global chiral symmetry can strongly decrease the breaking of the approximate, anomalous U_{A}(1) symmetry. This is testable on the lattice through simulations for one through four flavors. In QCD a small breaking of the U_{A}(1) symmetry in the chirally symmetric phase generates novel experimental signals.

Vector dark matter with Higgs portal in type II seesaw framework

We study the phenomenology of a vector dark matter (VDM) in a U(1)X gauged extension of the Standard Model (SM), which is set in a type II seesaw framework. When this U(1)X symmetry is spontaneously broken by the vacuum expectation value (VEV) of a newly introduced complex scalar singlet, the gauge boson Z′ becomes massive. The stability of the dark matter (DM) is ensured by the presence of an exact charge conjugation symmetry resulting from the structure of the Lagrangian. On the other hand, the SU(2)L triplet scalar facilitates light neutrino masses via the type II seesaw mechanism. We have studied the phenomenology of the usual WIMP DM, considering all possible theoretical and experimental constraints that are applicable. Due to the presence of a triplet scalar, the present framework can also accommodate the observed 2σ deviation in h→Zγ decay rate, recently measured at the LHC. The possibility of nonthermal production of DM from the decay of the singlet scalar has also been briefly discussed. Published by the American Physical Society 2024

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Perfectly spherical Bloch hyper-spheres from quantum matrix geometry

Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics.

Emergent Noninvertible Symmetries in N=4 Supersymmetric Yang-Mills Theory.

One of the simplest examples of noninvertible symmetries in higher dimensions appears in 4D Maxwell theory, where its SL(2,Z) duality group can be combined with gauging subgroups of its electric and magnetic 1-form symmetries to yield such defects at many different values of the coupling. Even though N=4 supersymmetric Yang-Mills (SYM) theory also has an SL(2,Z) duality group, it only seems to share two types of such noninvertible defects with Maxwell theory (known as duality and triality defects). Motivated by this apparent difference, we begin our investigation of the fate of these symmetries by studying the case of 4D N=4 U(1) gauge theory, which contains Maxwell theory in its content. Surprisingly, we find that the noninvertible defects of Maxwell theory give rise, when combined with the standard U(1) symmetry acting on the free fermions, to defects that act on local operators as elements of the U(1) outer automorphism of the N=4 superconformal algebra, an operation that was referred to in the past as the "bonus symmetry." Turning to the non-Abelian case of N=4 SYM theory, the bonus symmetry is not an exact symmetry of the theory, but is known to emerge at the supergravity limit. Based on this observation, we study this limit and show that, if it is taken in a certain way, noninvertible defects that realize different elements of the bonus symmetry emerge as approximate symmetries, in analogy to the Abelian case.

Color symmetry and confinement as an underlying superconformal structure in holographic QCD

Dedicated to the memory of our colleague, Harald Fritzsch, who, together with Murray Gell-Mann, introduced the color quantum number as the exact symmetry responsible for the strong interaction, thus establishing quantum chromodynamics (QCD) as a fundamental non-Abelian gauge theory. A basic understanding of hadron properties, however, such as confinement and the emergence of a mass scale, from first principles QCD has remained elusive: Hadronic characteristics are not explicit properties of the QCD Lagrangian and perturbative QCD, so successful in the large transverse momentum domain, is not applicable at large distances. In this article, we shall examine how this daunting obstacle is overcome in holographic QCD with the introduction of a superconformal symmetry in anti de Sitter (AdS) space which is responsible for confinement and the introduction of a mass scale within the superconformal group. When mapped to light-front coordinates in physical spacetime, this approach incorporates supersymmetric relations between the Regge trajectories of meson, baryon and tetraquark states which can be visualized in terms of specific [Formula: see text] color representations of quarks. We will also briefly discuss here the implications of holographic models for QCD color transparency in view of the present experimental interest.

Chiral Gauge Theory at the Boundary between Topological Phases.

I demonstrate how chiral fermions with an exact gauge symmetry can appear on the d-dimensional boundary of a finite volume (d+1)-dimensional manifold, without any light mirror partners. The condition for the d-dimensional boundary theory to be local is that gauge anomalies cancel and that the volume be large. This can likely be achieved on a lattice and provides a new paradigm for the lattice regularization of chiral gauge theories.

Black holes, conformal symmetry, and fundamental fields

Cosmic censorship protects the outside world from black hole singularities and paves the way for assigning entropy to gravity at the event horizons. We point out a tension between cosmic censorship and the quantum backreacted geometry of Schwarzschild black holes, induced by vacuum polarization and driven by the conformal anomaly. A similar tension appears for the Weyl curvature hypothesis at the Big Bang singularity. We argue that the requirement of exact conformal symmetry resolves both conflicts and has major implications for constraining the set of fundamental constituents of the Standard Model.

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom.

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

UV and IR effects in axion quality control

Motivated by recent discussions and the absence of exact global symmetries in UV completions of gravity we re-examine the axion quality problem (and naturalness issues more generally) using antisymmetric Kalb-Ramond (KR) fields rather than their pseudoscalar duals, as suggested by string and higher dimensional theories. Two types of axions can be identified: a model independent S-type axion dual to a two form Bμν in 4D and a T-type axion coming directly as 4D scalar Kaluza-Klein (KK) components of higher-dimensional tensor fields. For T-type axions our conclusions largely agree with earlier workers for the axion quality problem, but we also reconcile why T-type axions can couple to matter localized on 3-branes with Planck suppressed strength even when the axion decay constants are of order the KK scale. For S-type axions, we review the duality between form fields and massive scalars and show how duality impacts naturalness arguments about the UV sensitivity of the scalar potential. In particular UV contributions on the KR side suppress contributions on the scalar side by powers of m/M with m the axion mass and M the UV scale. We re-examine how the axion quality problem is formulated on the dual side and compare to recent treatments. We study how axion quality is affected by the ubiquity of p-form gauge potentials (for both p = 2 and p = 3) in string vacua and identify two criteria that can potentially lead to a problem. We also show why most fields do not satisfy these criteria, but when they do the existence of multiple fields also provides mechanisms for resolving it. We conclude that the quality problem is easily evaded.

New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.

Quantum current and holographic categorical symmetry

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C}𝒞:=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C}𝒞 and symmetric operators are morphisms in \mathcal{C}𝒞. The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C})Z1(𝒞). The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C})Z1(𝒞) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

Symmetries of differential delay systems with applications to observer design

AbstractIn this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers).

Correlated order at the tipping point in the kagome metal CsV3Sb5

Spontaneously broken symmetries are at the heart of many phenomena of quantum matter and physics more generally. However, determining the exact symmetries that are broken can be challenging due to imperfections such as strain, in particular when multiple electronic orders are competing. This is exemplified by charge order in some kagome systems, where evidence of nematicity and flux order from orbital currents remains inconclusive due to contradictory measurements. Here we clarify this controversy by fabricating highly symmetric samples of a member of this family, CsV3Sb5, and measuring their transport properties. We find that a measurable anisotropy is absent at any temperature in the unperturbed material. However, a pronounced in-plane transport anisotropy appears when either weak magnetic fields or strains are present. A symmetry analysis indicates that a perpendicular magnetic field can indeed lead to in-plane anisotropy by inducing a flux order coexisting with more conventional bond order. Our results provide a unifying picture for the controversial charge order in kagome metals and highlight the need for materials control at the microscopic scale in the identification of broken symmetries.

Diraxiogenesis

The family of Dirac Seesaw models offers an intriguing alternative explanation for the smallness of neutrino masses without necessarily requiring microscopic lepton number violation, when compared to the more familiar class of Majorana Seesaws. A global U(1)D symmetry, that is explicitly broken by a higher dimensional scalar operator, ensures that the right handed neutrino does not couple directly to the Standard Model like Higgs and an exact gauged or residual lepton number symmetry prohibits all Majorana masses. We demonstrate that all three Dirac Seesaws possess a Pseudo-Nambu-Goldstone boson associated with the U(1)D symmetry, that we call the Diraxion, whose cosmological dynamics have so far been left unexplored. Furthermore we illustrate that a Dirac-Leptogenesis version of the recently proposed Lepto-Axiogenesis scenario can be realized in this class of models, leading to a unified origin of the observed baryon asymmetry and dark matter relic abundance. Explaining only the baryon asymmetry can lead to potentially observable amounts of right handed neutrino dark radiation with ∆Neff. ≲ 0.028. On the other hand, if we only fix the dark matter abundance via the kinetic misalignment mechanism, this set-up could lead to detectable signatures in proposed cosmic neutrino background experiments via decays of eV-scale Diraxions to neutrinos. Here there is no domain wall problem, since topological defects decay to a subleading fraction of relic Diraxions. A key ingredient of all Axiogenesis scenarios is the dynamics of relatively light scalar called the Saxion, that in our case has a mass at the GeV-scale and which might reveal itself in heavy meson decays or collider searches. Our setup predicts isocurvature perturbations in baryons, dark matter and dark radiation sourced by fluctuations of the Saxion.

Generalized Ginsparg-Wilson relations

We give a general derivation of Ginsparg-Wilson relations for both Dirac and Majorana fermions in any dimension. These relations encode continuous and discrete chiral, parity and time-reversal anomalies and will apply to the various classes of free-fermion topological insulators and superconductors (in the framework of a relativistic quantum field theory in Euclidean spacetime). We show how to formulate the exact symmetries of the lattice action and the relevant index theorems for the anomalies. Published by the American Physical Society 2024

Automorphism Faithfulness Metrics for Symmetric Graph Drawings.

In this article, we present new quality metrics for symmetric graph drawing based on group theory. Roughly speaking, the new metrics are faithfulness metrics, i.e., they measure how faithfully a drawing of a graph displays the ground truth (i.e., geometric automorphisms) of the graph as symmetries. More specifically, we introduce two types of automorphism faithfulness metrics for displaying: (1) a single geometric automorphism as a symmetry (axial or rotational), and (2) a group of geometric automorphisms (cyclic or dihedral). We present algorithms to compute the automorphism faithfulness metrics in O(n logn) time. Moreover, we also present efficient algorithms to detect exact symmetries in a graph drawing. We then validate our automorphism faithfulness metrics using deformation experiments. Finally, we use the metrics to evaluate existing graph drawing algorithms to compare how faithfully they display geometric automorphisms of a graph as symmetries.