Continuous Symmetry Research Articles

5d to 3d compactifications and discrete anomalies

Much insight into the dynamics of quantum field theories can be gained by studying the relationship between field theories in different dimensions. An interesting observation is that when two theories are related by dimensional reduction on a compact surface, their ’t Hooft anomalies corresponding to continuous symmetries are also related: the anomaly polynomial of the lower-dimensional theory can be obtained by integrating that of the higher-dimensional one on the compact surface. Naturally, this relation only holds if both theories are even dimensional. This raises the question of whether similar relations can also hold for the case of anomalies in discrete symmetries, which might be true even in odd dimensions. The natural generalization to discrete symmetries is that the anomaly theories, associated with the lower and higher dimensional theories, would be related by reduction on the compact surface. We explore this idea for compactifications of 5d superconformal field theories (SCFTs) to 3d on Riemann surfaces with global-symmetry fluxes. In this context, it can be used both as a check for these compactification constructions and for discovering new anomalies in the 5d SCFTs. This opens the way to applying the same idea of dimensional reduction of the anomaly theory to more general types of compactifications.

Accelerated discovery of machine-learned symmetries: Deriving the exceptional Lie groups G2, F4 and E6

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups G2, F4, and E6. A third post-processing algorithm renders the found generators in sparse form. We benchmark the performance improvement of the new algorithms relative to the standard approach. Given the significant complexity of the exceptional Lie groups, our results demonstrate that this machine-learning method for discovering symmetries is completely general and can be applied to a wide variety of labeled datasets.

Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model

The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/rα), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2D. However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of SU(2) symmetry in the ferromagnetic Heisenberg model with 1/rα-form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.

Realization of an inherent time crystal in a dissipative many-body system

Time crystals are many-body states that spontaneously break translation symmetry in time the way that ordinary crystals do in space. While experimental observations have confirmed the existence of discrete or continuous time crystals, these realizations have relied on the utilization of periodic forces or effective modulation through cavity feedback. The original proposal for time crystals is that they would represent self-sustained motions without any external periodicity, but realizing such purely self-generated behavior has not yet been achieved. Here, we provide theoretical and experimental evidence that many-body interactions can give rise to an inherent time crystalline phase. Following a calculation that shows an ensemble of pumped four-level atoms can spontaneously break continuous time translation symmetry, we observe periodic motions in an erbium-doped solid. The inherent time crystal produced by our experiment is self-protected by many-body interactions and has a measured coherence time beyond that of individual erbium ions.

Efficient post-processing of electromagnetic plane wave simulations to model arbitrary structured beams incident on axisymmetric structures

The study of an optical beam interacting with material structures is a fundamental of nanophotonics. Computational electromagnetic solvers facilitate the rapid calculation of the scattering from material structures with arbitrary geometry and complexity, but have limited efficiency when employing structured excitation fields. We have developed a post-processing method and package that can efficiently calculate the full three-dimensional electric and magnetic fields for any optical beam incident on a particle or structure with at least one axis of continuous rotational symmetry, called an axisymmetric body (such as a sphere, cylinder, cone, torus or surface). Provided an initial batch of plane wave simulations is computed, this open-source package combines data from computational electromagnetic solvers in a post-processing fashion using the angular spectrum representation to create arbitrarily structured beams, including vector vortex beams. Any and all possible incident beams can be generated from the initial batch of PWSs, without the need for further simulations. This allows for efficiently performing parameter sweeps such as changing the angle of illumination or translating the particle position relative to the beam, all in post-processing, with no need for additional time-consuming simulations. We demonstrate some applications by numerically calculating optical force and torque maps for a spherical plasmonic nanoparticle in a tightly focused Gaussian beam, a plasmonic nanocone in an azimuthally polarised beam and compute the fields of a non-paraxial Laguerre–Gaussian vortex beam reflecting on a multilayered surface. We believe this package, called BEAMS, is a valuable tool for rapidly quantifying electromagnetic systems that are beyond traditional analytical methods.

Excited-state quantum phase transition in light-matter systems with discrete and continuous symmetries

Excited-state quantum phase transition in light-matter systems with discrete and continuous symmetries

A Goldstone theorem for continuous non-invertible symmetries

We study systems with an Adler-Bell-Jackiw anomaly in terms of non-invertible symmetry. We present a new kind of non-invertible charge defect where a key role is played by a local current operator localized on the defect. The charge defects are now labeled by elements of a continuous (1). We use this construction to prove an analogue of Goldstone’s theorem for such non-invertible symmetries. We comment on possible applications to string theory.

EVOLUTION AND DISSEMINATION OF SPECIALIZED STRATEGIES, METHODS, AND TECHNIQUES OF SYNCHRONOUS PULSEWUDTH MODULATION FOR CONTROL OF VOLT-AGE SOURCE INVERTERS AND INVERTER-BASED SYSTEMS

This publication provides a brief historical overview of the development of methods and techniques of pulsewidth modulation (PWM) for voltage source inverters, published mainly in Ukrainian publishing houses and in Ukrainian periodicals. Accent has been done to review of results of investigation of alternative methods and techniques of synchronous space-vector-based multi-zone PWM for inverters with low switching frequency. In particular, in the mentioned publications, the basic strategies, schemes, and algorithms of synchronous multi-zone modulation have been further developed, modernized, modified, and disseminated in relation to new promising topologies of power conversion systems, including: two-inverter-based electric drives with open-end winding of electrical motor; dual three-phase electric drives of symmetrical and asymmetric types; 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 PWM, applied for control of inverter-based systems, provide continuous synchronization and symmetry of basic voltage waveforms of systems during the whole control range including zone of overmodulation of inverters. It provides minimization of even harmonics and undesirable subharmonics (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 characteristics of the phase and line voltages of 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. References 30, tables 2, figures 25.

Landau-Forbidden Quantum Criticality in Rydberg Quantum Simulators.

The Landau-Ginzburg-Wilson theory of phase transitions precludes a continuous transition between two phases that spontaneously break distinct symmetries. However, quantum mechanical effects can intertwine the symmetries, giving rise to an exotic phenomenon called deconfined quantum criticality (DQC). In this Letter, we study the ground state phase diagram of a one-dimensional array of individually trapped neutral atoms interacting strongly via Rydberg states, and demonstrate through extensive numerical simulations that it hosts a variety of symmetry-breaking phases and their transitions including DQC. We show how an enlarged, emergent continuous symmetry arises at the DQCs, which can be experimentally observed in the joint distribution of two distinct order parameters, obtained within measurement snapshots in the standard computational basis. Our findings highlight quantum simulators of Rydberg atoms not only as promising platforms to experimentally realize such exotic phenomena, but also as unique ones allowing access to physical properties not obtainable in traditional experiments.

Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations

As a typical application of deep learning, physics-informed neural network (PINN) has been successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. We test the effectiveness of SPINN for the forward problem via two groups of ten independent numerical experiments using different numbers of collocation points and neurons per layer for the Korteweg-de Vries (KdV) equation, breaking soliton equation, heat equation, and potential Burgers equations respectively, and for the inverse problem by considering different layers and neurons as well as different numbers of training points with different levels of noise for the Burgers equation in potential form. The numerical results show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network, and in particular, exhibits superiorities than the PINN method and the two-stage PINN method of Lin and Chen by considering the Sawada-Kotera equation. Furthermore, we discuss the computational overhead of SPINN in terms of the relative computational cost to PINN and show that the training time of SPINN has no obvious increases, even less than PINN for certain cases.

Symmetry-reduced low-dimensional representation of large-scale dynamics in the asymptotic suction boundary layer

An important feature of turbulent boundary layers are persistent large-scale coherent structures in the flow. Here, we use Dynamic Mode Decomposition (DMD), a data-driven technique designed to detect spatio-temporal coherence, to construct optimal low-dimensional representations of such large-scale dynamics in the asymptotic suction boundary layer (ASBL). In the ASBL, fluid is removed by suction through the bottom wall, resulting in a constant boundary layer thickness in streamwise direction. That is, the streamwise advection of coherent structures by the mean flow ceases to be of dynamical importance and can be interpreted as a continuous shift symmetry in streamwise direction. However, this results in technical difficulties, as DMD is known to perform poorly in presence of continuous symmetries. We address this issue using symmetry-reduced DMD (Marensi et al., 2023), and find the large-scale dynamics of the ASBL to be low-dimensional indeed and potentially self-sustained, featuring ejection and sweeping events at large scale. Interactions with near-wall structures are captured when including only a few more modes.

Detecting Emergent Continuous Symmetries at Quantum Criticality.

New or enlarged symmetries can emerge at the low-energy spectrum of a Hamiltonian that does not possess the symmetries, if the symmetry breaking terms in the Hamiltonian are irrelevant under the renormalization group flow. We propose a tensor network based algorithm to numerically extract lattice operator approximation of the emergent conserved currents from the ground state of any quantum spin chains, without the necessity to have prior knowledge about its low-energy effective field theory. Our results for the spin-1/2 J-Q Heisenberg chain and a one-dimensional version of the deconfined quantum critical points demonstrate the power of our method to obtain the emergent lattice Kac-Moody generators. It can also be viewed as a way to find the local integrals of motion of an integrable model and the local parent Hamiltonian of a critical gapless ground state.

High-order field theory and a weak Euler–Lagrange–Barut equation for classical relativistic particle-field systems

In both quantum and classical field systems, conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties. A broadly accepted approach to deriving conservation laws is built using Noether’s method. However, this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines. In the present study, we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems. In contrast to quantum systems, where particles are viewed as quantum fields, classical relativistic particle-field systems present specific challenges. These challenges arise from two sides. The first comes from the mass-shell constraint. To deal with the mass-shell constraint, the Euler–Lagrange–Barut (ELB) equation is used to determine the particle’s world lines in the four-dimensional (4D) Minkowski space. Besides, the infinitesimal criterion, which is a differential equation in formal field theory, is reconstructed by an integro-differential form. The other difficulty is that fields and particles depend on heterogeneous manifolds. To overcome this challenge, we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems. By applying a weak ELB equation to classical relativistic particle-field systems, we can systematically derive local conservation laws by examining the underlying symmetries of the system. Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.

Disconnected 0-form and 2-group symmetries

Quantum field theories can have both continuous and finite 0-form symmetries. We study global symmetry structures that arise when both kinds of 0-form symmetries are present. The global structure associated to continuous 0-form symmetries is described by a connected Lie group, which captures the possible backgrounds of the continuous 0-form symmetries the theory can be coupled to. Finite 0-form symmetries can act as outer-automorphisms of this connected Lie group. Consequently, possible background couplings to both continuous and finite 0-form symmetries are described by a disconnected Lie group, and we call the resulting symmetry structure a disconnected 0-form symmetry. Additionally, finite 0-form symmetries may act on the 1-form symmetry group. The 1-form symmetries and continuous 0-form symmetries may combine to form a 2-group, which when combined with finite 0-form symmetries leads to another type of 2-group, that we call a disconnected 2-group and the resulting symmetry structure a disconnected 2-group symmetry. Examples of arbitrarily complex disconnected 0-form and 2-group symmetries in any spacetime dimension are furnished by gauge theories: with 1-form symmetries arising from the center of the gauge group, continuous 0-form symmetries arising as flavor symmetries acting on matter content, and finite 0-form symmetries arising from outer-automorphisms of gauge and flavor Lie algebras.

Oracle-Preserving Latent Flows

A fundamental task in data science is the discovery, description, and identification of any symmetries present in the data. We developed a deep learning methodology for the simultaneous discovery of multiple non-trivial continuous symmetries across an entire labeled dataset. The symmetry transformations and the corresponding generators are modeled with fully connected neural networks trained with a specially constructed loss function, ensuring the desired symmetry properties. The two new elements in this work are the use of a reduced-dimensionality latent space and the generalization to invariant transformations with respect to high-dimensional oracles. The method is demonstrated with several examples on the MNIST digit dataset, where the oracle is provided by the 10-dimensional vector of logits of a trained classifier. We find classes of symmetries that transform each image from the dataset into new synthetic images while conserving the values of the logits. We illustrate these transformations as lines of equal probability (“flows”) in the reduced latent space. These results show that symmetries in the data can be successfully searched for and identified as interpretable non-trivial transformations in the equivalent latent space.

Fermion Disorder Operator at Gross-Neveu and Deconfined Quantum Criticalities.

The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang etal., arXiv:2209.07103]. Here we study, by means of large-scale quantum MonteCarlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.

Photonic Dirac cone and topological transition in a moving dielectric slab

The moving media theory is applied to a photonic confined structure which is a continuous dielectric slab waveguide with the uniaxial anisotropy and without the discrete translational symmetry. The moving effect not only brings about non-reciprocity to the whole photonic band structure in the co-moving and counter-moving directions, but also leads to the topological transition of local degenerate points within the band diagram. We demonstrate through calculation that the type-II Dirac point can be turned into type-I Dirac point when the uniaxial slab is moving over certain speed. Our results provide a new approach to regulate the topology of degeneracy for two-dimensional photonic bands in the continuous translational symmetry condition.

Chiral Nelson-Barr models: Quality and cosmology

It was recently shown that domain walls from the spontaneous breaking of CP symmetry are exactly stable, and must be inflated away to recover a viable cosmology. We investigate the phenomenological implications of this result in Nelson--Barr solutions of the strong CP problem. Combined with the upper bound on the scale of spontaneous CP breaking necessary to suppress contributions from dangerous, nonrenormalizable operators to $\bar{\theta}$, this puts an upper bound on the scale of inflation and the reheating temperature after inflation. Minimal Nelson--Barr models are therefore in tension with thermal leptogenesis, models of large-field inflation, or potential future observations of signals from topological remnants of an unrelated, subsequent phase transition. We study how extending Nelson--Barr models with a new, continuous chiral gauge symmetry can ameliorate this tension by forbidding the dangerous dimension-five operators. In particular, we show that gauging a linear combination of baryon number and hypercharge allows for an economic, anomaly-free extension of the minimal Nelson--Barr model, and discuss the phenomenological implications.

Cellular automaton for spinor gravity in four dimensions

Certain fermionic quantum field theories are equivalent to probabilistic cellular automata, with fermionic occupation numbers associated to bits. We construct an automaton that represents a discrete model of spinor gravity in four dimensions. Local Lorentz symmetry is exact on the discrete level and diffeomorphism symmetry emerges in the naive continuum limit. Our setting could serve as a model for quantum gravity if diffeomorphism symmetry is realized in the true continuum limit and suitable collective fields for vierbein and metric acquire non-vanishing expectation values. The discussion of this interesting specific model reveals may key qualitative features of the continuum limit for probabilistic cellular automata. This limit obtains for a very large number of cells if the probabilistic information is sufficiently smooth. It is associated to coarse graining. The automaton property that every bit configuration is updated at every discrete time step to precisely one new bit configuration does no longer hold on the coarse grained level. A coarse grained configuration of occupation numbers can evolve into many different configurations with certain probabilities. This characteristic feature of quantum field theories can come along with the emergence of continuous space-time symmetries.

Disorder-induced spin-charge separation in the one-dimensional Hubbard model

Many-body localization is believed to be generically unstable in quantum systems with continuous non-Abelian symmetries, even in the presence of strong disorder. Breaking these symmetries can stabilize the localized phase, leading to the emergence of an extensive number of quasilocally conserved quantities known as local integrals of motion, or $l$ bits. Using a sophisticated nonperturbative technique based on continuous unitary transforms, we investigate the one-dimensional Hubbard model subject to both spin and charge disorder, compute the associated $l$ bits and demonstrate that the disorder gives rise to a novel form of spin-charge separation. We examine the role of symmetries in delocalizing the spin and charge degrees of freedom, and show that while symmetries generally lead to delocalization through multiparticle resonant processes, certain subsets of states appear stable.