Discrete Symmetry Group Research Articles

Finding discrete symmetry groups via machine learning

Finding discrete symmetry groups via machine learning

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials.

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${M}_{{W}_{R}}$$\\end{document} dependence of leptogenesis in minimal Left-Right Symmetric Model with different strengths of Type-II seesaw mass

Left Right Symmetric Model (LRSM) being an extension of the Standard model of particle physics incorporates within itself Type-I and Type-II seesaw mass terms naturally. Both the mass terms can have significant amount of contribution to the resulting light neutrino mass within the model and hence on the different phenomenology associated within. In this paper, we have thoroughly analyzed and discussed the implications of specifying different weightages to both the mass terms and also the study has been carried out for different values of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${M}_{{W}_{R}}$$\\end{document} which is mass of the right-handed gauge boson. This paper also gives a deeper insight into the new physics contributions of Neutrinoless Double Beta Decay (0νββ) and their variations with the net baryon asymmetry arising out of the model. Therefore, the main objective of the present paper rests on investigating the implications of imposing different weightage to the type-I and type-II seesaw terms and different values of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${M}_{{W}_{R}}$$\\end{document} on the new physics contributions of 0νββ and net baryon asymmetry arising out as a result of resonant leptogenesis. LRSM in this work has been realized using modular group of level 3, Γ(3) which is isomorphic to non-abelian discrete symmetry group A4, the advantage being the non-requirement of flavons within the model and hence maintaining the minimality of the model.

Minimal left–right symmetric model with A4 modular symmetry

In this paper, we have realized the left–right symmetric model (LRSM) with modular symmetry. Most of the previous works on LRSM have been done considering discrete flavor symmetry, but this work has been carried out using [Formula: see text](3) modular group which is isomorphic to nonabelian discrete symmetry group [Formula: see text]. The advantage of using modular symmetry is the nonrequirement for the use of extra particles called “flavons”. In this model, the Yukawa couplings are expressed in terms of modular forms [Formula: see text]. In this work, we have studied minimal LRSM for both type-I and type-II dominances. Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses. The results obtained are well within the experimental limits for the desired values of sum of neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions and the correlation of lepton flavor violation and lightest neutrino mass within the framework of modular symmetric LRSM.

A unifying framework for BRST and BRST-related symmetries

We propose a general framework to study BRST-related transformations. We investigate different forms of BRST and BRST-related symmetries, realized within a prototypical first-class system, including ordinary BRST, anti-BRST, dual-BRST, anti-dual-BRST and additional sets of new BRST-related symmetries. We identify a precise discrete group of symmetries of the ghost sector, responsible for connecting the various forms of BRST-related transformations. Their distinct roles in different Hamiltonian and Lagrangian approaches are clarified. As a unifying framework, we use a gauge invariant prototypical first-class system encompassing an extensive class of physical models.

Exploring models with modular symmetry in neutrino oscillation experiments

Our study aims to investigate the viability of neutrino mass models that arise from discrete non-Abelian modular symmetry groups, i.e., ΓN with (N = 1, 2, 3, . ) in the future neutrino experiments T2HK, DUNE, and JUNO. Modular symmetry reduces the usage of flavon fields compared to the conventional discrete flavor symmetry models. Theories based on modular symmetries predict the values of leptonic mixing parameters, and therefore, these models can be tested in future neutrino experiments. In this study, we consider three models based on the A4 modular symmetry, i.e., Model-A, B, and C such a way that they predict different values of the oscillation parameters but still allowed with respect to the current data. In the future, it is expected that T2HK, DUNE, and JUNO will measure the neutrino oscillation parameters very precisely, and therefore, some of these models can be excluded in the future by these experiments. We have estimated the prediction of these models numerically and then used them as input to scrutinize these models in the neutrino experiments. Assuming the future best-fit values of θ23 and δCP remain the same as the current one, our results show that at 5σ C.L, Model-A can be excluded by T2HK whereas Model-B can be excluded by both T2HK and DUNE. Model-C cannot be excluded by T2HK and DUNE at 5σ C.L. Further; our results show that JUNO alone can exclude Model-B at an extremely high confidence level if the future best-fit of θ12 remains at the current-one. We have also identified the region in the θ23 - δCP parameter space, for which Model-A cannot be separated from Model-B in T2HK and DUNE.

The other variants of mixed μ-τ symmetry

Two new neutrino mass matrix textures exhibiting the mixed μ-τ symmetry are proposed. The mass matrices hint for a promising neutrino mixing schemes and find their connections with Δ(27) and A4 discrete symmetry groups respectively.

Neutrino masses and mixing in minimal inverse seesaw using A4 modular symmetry

In this paper, we construct a model with the help of modular symmetry in the framework of minimal inverse seesaw [ISS(2,3)]. We have used [Formula: see text] modular group which is isomorphic to non-Abelian discrete symmetry group [Formula: see text]. In this group, there are three Yukawa modular forms of weight 2. In this model, we study neutrino masses and mixings for both normal and inverted hierarchies. Use of modular symmetry reduces the need for more number of extra flavons and their specific VEV alignments, as such, minimality of the model is maintained to a great extent. Along with [Formula: see text] symmetry group, we have used [Formula: see text] to restrict certain interaction terms in the Lagrangian. Further, we calculate the effective mass to address the phenomena of neutrinoless double-beta decay ([Formula: see text]). The values of effective mass are found to lie within the bound ([Formula: see text][Formula: see text]eV) as predicted by different [Formula: see text] experiments.

Exploring Dynamics of A4 Flavour Symmetry Using Low Scale Seesaw Mechanisms

Low scale seesaw models, like low scale type II, inverse (ISS), and linear seesaw (LSS) models provide an interesting mechanism to obtain light neutrino masses and mixings, as they can be tested in future TeV scale experiments. Discrete flavour symmetry groups like $A_4$ can be incorporated to explain the flavour structure of particles. But, so far, nothing is known about dynamics of flavour symmetry - scale of its breaking, or VEV alignment of the flavon fields. In a recent study [1], we have investigated and shed light on how to pinpoint the favoured VEV alignment of the flavon field using light neutrino oscillation data. In this work, for the first time, we present an analysis on dependence of light neutrino masses on scale of flavon VEV in these three seesaw models, and comment on how this information can be used to discriminate among them. We also discuss about the estimated value of the constant $F$ which can constrain various coupling constants of the model, cut-off scale of the theory and scale of flavour symmetry breaking. This study can provide useful insight into the hitherto unknown dynamics of flavour symmetry and hence can contribute as an important ingredient in the model building for future studies.

Characteristic matrix functions for delay differential equations with symmetry

A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.

5d Higgs branches from M-theory on quasi-homogeneous cDV threefold singularities

We classify rank zero 5d SCFTs geometrically engineered from M-theory on quasi-homogeneous compound Du Val isolated threefold singularities. For all such theories, we characterize the Higgs Branch, by computing the dimension, the continuous and discrete symmetry groups, as well as more refined details such as the charges of the hypermultiplets under these groups. We derive these data by means of a gauge-theoretic method, that we have recently introduced, based on establishing a correspondence between an adjoint Higgs field and the M-theory geometry. As a byproduct, this further allows us to construct several T-brane backgrounds, that yield inequivalent 5d spectra but are associated with the same geometry.

Searching for exotic Higgs bosons at the LHC

We analyse in a model independent way the possibilities of digging out neutral exotic Higgs states, should they exist endowed with unconventional couplings with ordinary matter and gauge fields, at the 14 TeV run of the Large Hadron Collider (LHC), adding some comparative studies for 13.6 and 13 TeV runs. Flavor models, based on some discrete symmetry groups, with extended scalar sectors are known to yield exotic spin-0 states, both CP-even and CP-odd, with purely flavor off-diagonal Yukawa couplings. The gauge interaction of one such CP-even state is also unusual that, unlike the Standard Model Higgs boson, it does not couple to gauge boson pairs. Such unconventional properties immune these exotic states from receiving traditional collider and electroweak constraints, and hence those states could be light. Without committing to any specific model, exploiting their peculiar Yukawa and gauge properties, we explore the discovery potential of those exotic Higgs states through some interesting topologies by figuring out some specific kinematic variables that suppress the backgrounds.

Group-Invariant Quantum Machine Learning

Quantum machine learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries. These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group G associated with the dataset. We present theoretical results underpinning the design of G-invariant models, and exemplify their application through several paradigmatic QML classification tasks, including cases when G is a continuous Lie group and also when it is a discrete symmetry group. Notably, our framework allows us to recover, in an elegant way, several well-known algorithms for the literature, as well as to discover new ones. Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.3 MoreReceived 13 May 2022Accepted 24 August 2022DOI:https://doi.org/10.1103/PRXQuantum.3.030341Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMachine learningQuantum algorithmsQuantum computationQuantum entanglementQuantum Information

Constructing viable quark mass matrices in the flavor basis using S3 × Z3 symmetry

Non-Abelian discrete symmetry group [Formula: see text] has been used to construct quark mass matrices in the flavor basis. Interestingly, the case when [Formula: see text] is nondiagonal and [Formula: see text] is diagonal, is very much compatible with the presently refined data regarding the quark masses and mixing. However, the other case, when [Formula: see text] is nondiagonal and [Formula: see text] is diagonal, looks to be incompatible with the data.

Codebase release 3.4 for NetKet

We introduce version 3 of NetKet, the machine learning toolbox for many-body quantum physics. NetKet is built around neural quantum states and provides efficient algorithms for their evaluation and optimization. This new version is built on top of JAX, a differentiable programming and accelerated linear algebra framework for the Python programming language. The most significant new feature is the possibility to define arbitrary neural network ansätze in pure Python code using the concise notation of machine-learning frameworks, which allows for just-in-time compilation as well as the implicit generation of gradients thanks to automatic differentiation. NetKet 3 also comes with support for GPU and TPU accelerators, advanced support for discrete symmetry groups, chunking to scale up to thousands of degrees of freedom, drivers for quantum dynamics applications, and improved modularity, allowing users to use only parts of the toolbox as a foundation for their own code.

NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems

We introduce version 3 of NetKet, the machine learning toolbox for many-body quantum physics. NetKet is built around neural quantum states and provides efficient algorithms for their evaluation and optimization. This new version is built on top of JAX, a differentiable programming and accelerated linear algebra framework for the Python programming language. The most significant new feature is the possibility to define arbitrary neural network ansätze in pure Python code using the concise notation of machine-learning frameworks, which allows for just-in-time compilation as well as the implicit generation of gradients thanks to automatic differentiation. NetKet 3 also comes with support for GPU and TPU accelerators, advanced support for discrete symmetry groups, chunking to scale up to thousands of degrees of freedom, drivers for quantum dynamics applications, and improved modularity, allowing users to use only parts of the toolbox as a foundation for their own code.

Origin of nonsymmorphic bosonization formulas in generalized antiferromagnetic Kitaev spin- 12 chains from a renormalization-group perspective

Recently, in the Luttinger liquid phase of the one-dimensional generalized antiferromagnetic Kitaev spin-1/2 model, it has been found that the Abelian bosonization formulas of the local spin operators only respect the exact discrete nonsymmorphic symmetry group of the model, not the emergent U(1) symmetry. In this work, we perform a renormalization group (RG) study to provide explanations for the origin of the U(1) breaking terms in the bosonization formulas. We find that the lack of U(1) symmetry originates from the wave-function renormalization effects in the spin operators along the RG flow induced by the U(1) breaking interactions in the microscopic Hamiltonian. In addition, the RG analysis can give predictions to the signs and order of magnitudes of the coefficients in the bosonization formulas. Our work is helping to understand the rich nonsymmorphic physics in one-dimensional Kitaev spin models.

Deviation from Tribimaximal mixing using A4 flavour model with five extra scalars

A neutrino mass model with five extra scalars is constructed using A4×Z3×Z2 discrete symmetry group. The resultant mass matrix is able to give necessary deviation from Tribimaximal mixing which reproduces the current neutrino masses and mixing data with good accuracy. The model gives testable prediction for the future measurements of the neutrinoless double-beta decay parameter |mββ|. The analysis is consistent with latest cosmological bound Σmi≤0.12 eV.

Scalar gauge-Higgs models with discrete Abelian symmetry groups.

We investigate the phase diagram and the nature of the phase transitions of three-dimensional lattice gauge-Higgs models obtained by gauging the Z_{N} subgroup of the global Z_{q} invariance group of the Z_{q} clock model (N is a submultiple of q). The phase diagram is generally characterized by the presence of three different phases, separated by three distinct transition lines. We investigate the critical behavior along the two transition lines characterized by the ordering of the scalar field. Along the transition line separating the disordered-confined phase from the ordered-deconfined phase, standard arguments within the Landau-Ginzburg-Wilson framework predict that the behavior is the same as in a generic ferromagnetic model with Z_{p} global symmetry, p being the ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2) universality class for p=2 and p≥4, respectively, while for p=3 only first-order transitions are possible. The results of Monte Carlo simulations confirm these predictions. There is also a second transition line, which separates two phases in which gauge fields are essentially ordered. Along this line we observe the same critical behavior as in the Z_{q} clock model, as it occurs in the absence of gauge fields.

2-Group symmetries and their classification in 6d

We uncover 2-group symmetries in 6d superconformal field theories. These symmetries arise when the discrete 1-form symmetry and continuous flavor symmetry group of a theory mix with each other. We classify all 6d superconformal field theories with such 2-group symmetries. The approach taken in 6d is applicable more generally, with minor modifications to include dimension specific operators (such as instantons in 5d and monopoles in 3d), and we provide a discussion of the dimension-independent aspects of the analysis. We include an ancillary mathematica code for computing 2-group symmetries, once the dimension specific input is provided. We also discuss a mixed 't Hooft anomaly between discrete 0-form and 1-form symmetries in 6d