Continuous Symmetry Research Articles

Observation of a phase transition from a continuous to a discrete time crystal

Discrete (DTCs) and continuous time crystals (CTCs) are novel dynamical many-body states, that are characterized by robust self-sustained oscillations, emerging via spontaneous breaking of discrete or continuous time translation symmetry. DTCs are periodically driven systems that oscillate with a subharmonic of the external drive, while CTCs are continuously driven and oscillate with a frequency intrinsic to the system. Here, we explore a phase transition from a continuous time crystal to a discrete time crystal. A CTC with a characteristic oscillation frequency ωCTC is prepared in a continuously pumped atom-cavity system. Modulating the pump intensity of the CTC with a frequency ωdr close to 2ωCTC leads to robust locking of ωCTC to ωdr/2 , and hence a DTC arises. This phase transition in a quantum many-body system is related to subharmonic injection locking of non-linear mechanical and electronic oscillators or lasers.

Floquet systems with continuous dynamical symmetries: characterization, time-dependent noether charge, and solvability

We study quantum Floquet (periodically-driven) systems having continuous dynamical symmetry (CDS) consisting of a time translation and a unitary transformation on the Hilbert space. Unlike the discrete ones, the CDS strongly constrains the possible Hamiltonians H(t) and allows us to obtain all the Floquet states by solving a finite-dimensional eigenvalue problem. Besides, Noether’s theorem leads to a time-dependent conservation charge, whose expectation value is time-independent throughout evolution. We exemplify these consequences of CDS in the seminal Rabi model, an effective model of a nitrogen-vacancy center in diamonds without strain terms, and Heisenberg spin models in rotating fields. Our results provide a systematic way of solving for Floquet states and explain how they avoid hybridization in quasienergy diagrams.

Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond

We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.

CSM Software: Continuous Symmetry and Chirality Measures for Quantitative Structural Analysis.

We present a comprehensive and updated Python-based open software to calculate continuous symmetry measures (CSMs) and their related continuous chirality measure (CCM) of molecules across chemistry. These descriptors are used to quantify distortion levels of molecular structures on a continuous scale and were proven insightful in numerous studies. The input information includes the coordinates of the molecular geometry and a desired cyclic symmetry point group (i.e., Cs, Ci, Cn, or Sn). The results include the coordinates of the nearest symmetric structure that belong to the desired symmetry point group, the permutation that defines the symmetry operation, the direction of the symmetry element in space, and a number, between zero and 100, representing the level of symmetry or chirality. Rather than treating symmetry as a binary property by which a structure is either symmetric or asymmetric, the CSM approach quantifies the level of gray between black and white and allows one to follow the course of change. The software can be downloaded from https://github.com/continuous-symmetry-measure/csm or used online at https://csm.ouproj.org.il.

The numerical solution of the eigenvalue problem in dielectric thin films

In this study, the eigenvalue problem for one-dimensional thin films made of alternating layers of two dielectric materials with dielectric constants ε1 and ε2 was solved for both transverse electric (TE) and transverse magnetic (TM) modes to determine the allowed and forbidden frequencies in the bandgap. The dispersion relation for the one-dimensional thin films was obtained, and the eigenvalue problems were solved using a MATLAB program. The results show that the photonic band structure for the film in a homogeneous medium (ε1 = ε2) has no forbidden frequencies between the bands due to the continuous translational symmetry. However, when the dielectric constant is changed (ε1 ≠ ε2), gaps and bands appear alternately on the frequency axis. The first nine gaps and ten bands were obtained, and it is possible to determine the forbidden and allowed frequencies through the photonic band structure.

Total ankle arthroplasty improves discrete and continuous stance phase gait symmetry

BackgroundTotal ankle arthroplasty (TAA) is used to treat symptomatic end-stage ankle arthritis (AA). However, little is known about TAA's effects on gait symmetry. Research questionDetermine if symmetry changes from before surgery through two years following TAA utilizing the normalized symmetry index (NSI) and statistical parametric mapping (SPM). Methods141 patients with end-stage unilateral AA were evaluated from a previously collected prospective database, where each participant was tested within two weeks of surgery (Pre-Op), one year and two years following TAA. Walking speed, hip extension angle and moment, hip flexion angle, ankle plantarflexion angle and moment, ankle dorsiflexion angle, weight acceptance (GRF1), and propulsive (GRF2) vertical ground reaction forces were calculated for each limb. Gait symmetry was assessed using the NSI. A linear mixed effects model with a single response for each gait symmetry variable was used to examine the fixed effect of follow-up time (Pre-Op, Post-1 yr, Post-2 yr) and the random effect of participant with gait speed as a covariate in the model. A one-dimensional repeated measures analysis of variance (ANOVA) statistical parameter mapping (SPM) was completed to examine differences in the time-series NSI to determine regions of significant differences between follow-up times. ResultsRelative to Pre-Op values, GRF1, and GRF2 showed increased symmetry for discrete metrics and the time-series NSI across sessions. Hip extension moment had the largest symmetry improvement. Ankle plantarflexion angle was different between Pre-Op and Post-2 yr (p=0.010); and plantarflexion moment was different between Pre- Op and each post-operative session (p<0.001). The time-series Ankle Angle NSI was greater during the early stance phase in the Pre-Op session compared to Post-2 yr. SignificanceSymmetry across most of the stance phase improved following TAA indicating that TAA successfully improves gait symmetry and future work should determine if these improvements restore symmetry to levels equivalent with health age-match controls.

Evaluation of Point Group Symmetry in Lanthanide(III) Complexes: A New Implementation of a Continuous Symmetry Operation Measure with Autonomous Assignment of the Principal Axis.

The structure of molecular systems dictates the physical properties, and symmetry is the determining factor for all electronic properties. This makes group theory a powerful tool in quantum mechanics to compute molecular properties. For inorganic compounds, the coordination geometry has been estimated as idealized polyhedra with high symmetry, which, through ligand field theory, provides predictive capabilities. However, real samples rarely have ideal symmetry, and although continuous shape measures (CShM) can be used to evaluate deviation from an ideal reference structure σideal, this often fails for lanthanide(III) complexes with high coordination numbers, no obvious choice of principal axes, and no obvious reference structure. In lanthanide complexes, the unique electronic structures and associated properties are intricately tied to the symmetry around the lanthanide center. Therefore, robust methodologies to evaluate and estimate point group symmetry are instrumental for building structure-property relationships. Here, we have demonstrated an algorithmic approach that orients a molecular structure Q in the best possible way to the symmetry axis of any given point group G and computes a deviation from the ideal symmetry σsym(G,Q). This approach does not compute the deviation from an ideal reference system, but the intrinsic deviation in the structure induced by symmetry operations. If the structure contains the symmetry operation, there is no deviation and σsym(G,Q) = 0. The σsym deviation is generated from all of the symmetry operation ÔS in a point group G using the most correct orientation of the sample structure in each group G. The best orientation is found by an algorithm that minimizes the orientation of the structure with respect to G. To demonstrate the methodology, we have investigated the structure and symmetry of 8- and 9-coordinated lanthanide(III) aqua complexes and correlated the luminescence from 3 europium(III) crystals to their actual symmetry. To document the methodology, the approach has been tested on 26 molecules with different symmetries. It was concluded that the method is robust and fully autonomous.

Hyperbolic monopoles with continuous symmetries

We provide a framework to classify hyperbolic monopoles with continuous symmetries and find a Structure Theorem, greatly simplifying the construction of all those with spherically symmetry. In doing so, we reduce the problem of finding spherically symmetric hyperbolic monopoles to a problem in representation theory. Additionally, we determine constraints on the structure groups of such monopoles. Using these results, we construct novel spherically symmetric Sp(n) hyperbolic monopoles.

Continuous symmetry breaking in adaptive quantum dynamics

Continuous symmetry breaking in adaptive quantum dynamics

Higgs Oscillations in a Unitary Fermi Superfluid.

Symmetry-breaking phase transitions are central to our understanding of states of matter. When a continuous symmetry is spontaneously broken, new excitations appear that are tied to fluctuations of the order parameter. In superconductors and fermionic superfluids, the phase and amplitude can fluctuate independently, giving rise to two distinct collective branches. However, amplitude fluctuations are difficult to both generate and measure, as they do not couple directly to the density of fermions and have only been observed indirectly to date. Here, we excite amplitude oscillations in an atomic Fermi gas with resonant interactions by an interaction quench. Exploiting the sensitivity of Bragg spectroscopy to the amplitude of the order parameter, we measure the time-resolved response of the atom cloud, directly revealing amplitude oscillations at twice the frequency of the gap. The magnitude of the oscillatory response shows a strong temperature dependence, and the oscillations appear to decay faster than predicted by time-dependent Bardeen-Cooper-Schrieffer theory applied to our experimental setup.

Spontaneous breaking of the SO(2N) symmetry in the Gross-Neveu model

The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N, at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1, at which SO(2N)→SO(N)×SO(N). To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single (L=1) 4N-component real (Majorana) fermion, and then extend the theory to L&gt;1. This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1, where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N)-singlet mass-order-parameter, (2) SO(2N)-symmetric-tensor mass-order-parameters, and (3) SO(2N)-adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding (SO(2N)-singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1. We interpret this metamorphosis of the SO(2N)-symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024

Maximising CP Violation in naturally aligned Two-Higgs Doublet Models

The Two-Higgs Doublet Model (2HDM) is a well-motivated theoretical framework that provides additional sources of CP Violation (CPV) beyond the Standard Model (SM). After studying the vacuum topology of a general (convex) 2HDM potential, we unambiguously identify three origins of CPV: (i) Spontaneous CPV (SCPV), where the vacuum manifold has at least two degenerate CPV minima disconnected by domain walls, (ii) Explicit CPV (ECPV) with one single CPV ground state, and (iii) Mixed Spontaneous and Explicit CPV (MCPV), where the theory possesses more than one non-degenerate CPV local minimum. Most importantly, we define a novel complex parameter rCP whose norm and phase control the three different realisations of CPV, at least at the tree level. In all these scenarios, only two CPV phases can be made independent, as any third CPV parameter will always be constrained via the CP-odd tadpole condition. Since ECPV vanishes in 2HDMs where SM Higgs alignment is achieved naturally through accidental continuous symmetries, we analyse the possibility of maximising CPV through soft and explicit breaking of these symmetries. We derive upper limits on key CPV parameters that quantify the degree of SM misalignment from constraints due to the non-observation of an electron Electric Dipole Moment (EDM). Finally, we delineate the CP-violating parameter space of the so-constrained naturally aligned 2HDMs that can further be probed at the CERN Large Hadron Collider (LHC).

Percolation transition for random forests in d⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d\\geqslant 3$\\end{document}

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Critical Casimir effect in a disordered O(2)-symmetric model.

The critical Casimir effect appears when critical fluctuations of an order parameter interact with classical boundaries. We investigate this effect in the setting of a Landau-Ginzburg model with continuous symmetry in the presence of quenched disorder. The quenched free energy is written as an asymptotic series of moments of the model's partition function. Our main result is that, in the presence of a strong disorder, Goldstone modes of the system contribute either with either an attractive or a repulsive force. This result was obtained using the distributional zeta-function method without relying on any particular ansatz in the functional space of the moments of the partition function.

Symmetry and instability of marginally outer trapped surfaces

We consider an initial data set having a continuous symmetry and a marginally outer trapped surface (MOTS) that is not preserved by this symmetry. We show that such a MOTS is unstable except in an exceptional case. In non-rotating cases we provide a Courant-type lower bound on the number of unstable eigenvalues. These results are then used to prove the instability of a large class of exotic MOTSs that were recently observed in the Schwarzschild spacetime. We also discuss the implications for the apparent horizon in data sets with translational symmetry.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H=(xp+px)/2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

Steven Weinberg and Higgs physics

As a tribute to Steven Weinberg, we summarize the immense impact that he had on the understanding of the mechanism of spontaneous symmetry breaking and on the physics of the Higgs boson. In particular, four landmark contributions to this field are highlighted. A first one is his early work with Goldstone and Salam on spontaneously broken continuous symmetries that paved the way to the Higgs mechanism. A second towering breakthrough is his model of leptons which later became the Standard Model of particle physics and for which he was awarded the Nobel prize with Glashow and Salam. A third seminal work is the so-called Weinberg-Linde lower bound on the Higgs boson mass that was derived from the requirement of the stability of the electroweak vacuum. Finally, we summarize his important contributions in model-building of new physics with extended Higgs sectors and their possible impact in flavor physics and CP-violation. The historical aspects as well as the contemporary way of viewing these four major topics are summarized and their impact on today Higgs physics, and more generally particle physics, is highlighted.

Symmetry and symmetric transformations in mathematical imaging

The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetriesreflection, rotational, and translationalexemplified through concrete mathematical expressions. Evariste Galoiss Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrots fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetrys practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetrys foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.

Mode Shape-Based Damage Detection of Twin Skewed Bridges Based on OMA and FE Model Updating

A comparative study of the dynamic behavior of twin single-span bridges was carried out based on operational modal analysis and a linear finite element model updating with the aim of finding a potential damage indicator. The bridges are skewed, 30 m long and with prestressed concrete girders as the main support structure, but one of them were collided and suffered damage in two of its I-shape girders. Ambient vibration tests were conducted by deploying 14 uniaxial accelerometers along the sidewalks, experimental modal parameters were identified using the COVariance-driven Stochastic Subspace Identification method (SSI-COV). Six vertical modes were clearly identified for each structure. Additionally, some extra modes with non-proportional damping were identified. It was found that the modal frequencies are sensitive to the support conditions and the overall stiffness of the bridge but not to the localized girder damage, which, on the contrary, is reflected in the discrepancy in the fundamental bending mode shape amplitude between the lateral sides of the bridge, and a so-called Continuous Symmetry Measure adopted as a symmetry index seems to be a good damage indicator for this type of straight-alignment simply supported bridges, given the symmetric nature of the bending mode shapes and the asymmetric nature of the damage.

Methods, Techniques, and Algorithms of Synchronous Multi-Zone Modulation of Signals of Voltage Source Inverters (literature review)

This publication provides a brief overview of the results of research work in the field of developing alternative methods, schemes, and algorithms of synchronous multi-zone modulation of signals of inverters of power conversion systems with reduced switching frequency of power switches. In particular, in the mentioned researches, the basic strategies, schemes, and algorithms of synchronous multi-zone modulation have been fur-ther developed, modernized, modified, and disseminated in relation to new promising topologies of power conversion systems, including: two-inverter-based electric drives with open windings of electrical motor; drive systems based on electric motor with two stator windings; dual three-phase electric drives of symmetrical and asymmetric type; five-phase power conversion systems, powerful six-phase systems based on four inverters, and two-inverter-based and three-inverter-based photovoltaic installations with multi-winding transformer. It is shown that the developed schemes and algorithms of synchronous space-vector modulation applied for control of inverter-based systems provide continuous synchronization and symmetry of the basic voltage waveforms of systems during the whole control range. It provides minimization of even harmonics and undesirable subhar-monics (of the fundamental frequency) in spectra of the basic voltages of systems, leading to reducing of losses in systems and to increasing of its efficiency. Based on a comparative analysis of the integral spectral character-istics of the phase and line voltages of the systems, recommendations are formulated for the rational choice of schemes and algorithms of synchronous modulation for the relevant installations, depending on the modes of their operation.