Continuous Symmetry Breaking Research Articles

Fractional Optical Angular Momentum and Multi-Defect-Mediated Mode Renormalization and Orientation Control in Photonic Crystal Microring Resonators.

Whispering gallery modes (WGMs) in circularly symmetric optical microresonators exhibit integer quantized angular momentum numbers due to the boundary condition imposed by the geometry. Here, we show that incorporating a photonic crystal pattern in an integrated microring can result in WGMs with fractional optical angular momentum. By choosing the photonic crystal periodicity to open a photonic band gap with a band-edge momentum lying between that of two WGMs of the unperturbed ring, we observe hybridized WGMs with half-integer quantized angular momentum numbers (m∈Z+1/2). Moreover, we show that these modes with fractional angular momenta exhibit high optical quality factors with good cavity-waveguide coupling and an order of magnitude reduced group velocity. Additionally, by introducing multiple artificial defects, multiple modes can be localized to small volumes within the ring, while the relative orientation of the delocalized band-edge states can be well controlled. Our Letter unveils the renormalization of WGMs by the photonic crystal, demonstrating novel fractional angular momentum states and nontrivial multimode orientation control arising from continuous rotational symmetry breaking. The findings are expected to be useful for sensing and metrology, nonlinear optics, and cavity quantum electrodynamics.

Continuous symmetry breaking along the Nishimori line

We prove continuous symmetry breaking in three dimensions for a special class of disordered models described by the Nishimori line. The spins take values in a group, such as S1, SU(n) or SO(n). Our proof is based on a theorem about group synchronization proved by Abbe et al. [Math. Stat. Learn. 1(3), 227–256 (2018)]. It also relies on a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Prog. Theor. Phys. 66(4), 1169–1181 (1981)]. The proof does not use reflection positivity. The correlation inequalities of Messager et al. [Commun. Math. Phys. 58(1), 19–29 (1978)] imply symmetry breaking for the classical XY model without disorder.

A model of persistent breaking of continuous symmetry

We consider a UV-complete field-theoretic model in general dimensions, including d=2+1, which consists of two copies of the long-range vector models, with O(m) and O(N-m) global symmetry groups, perturbed by double-trace operators. Using conformal perturbation theory we find weakly-coupled IR fixed points for N\geq 6N≥6 that reveal a spontaneous breaking of global symmetry. Namely, at finite temperature the lower rank group is broken, with the pattern persisting at all temperatures due to scale-invariance. We provide evidence that the models in question are unitary and invariant under full conformal symmetry. Furthermore, we show that this model exhibits a continuous family of weakly interacting field theories at finite N.

Disorder- driven gradual transition of the continuous symmetry- breaking phase transition

We study the influence of random anisotropy type quenched disorder on the phase behavior of the system, which exhibits in undistorted case the 2nd order continuous symmetry breaking phase transition. Invoking the central limit theorem we express the free energy of the system in terms of the order parameter η and the characteristic length ξ of the gauge field ϕ. The latter exhibits the Goldstone fluctuation mode and is consequently extremely susceptible to the imposed disorder. In case of negligible distribution width ΔT of the local transition temperatures the disorder converts the 2nd order transition into a discontinuous one for W<WC, where Wrepresents the disorder strength. Above the critical disorder strength WC the transition becomes gradual. However for the finite width ΔT the transition becomes gradual for any W>0. We demonstrate that for large enough values of ΔT the system behavior is dominated by the distribution of temperatures, while the details of the random field interaction term play a secondary role. The influence of distribution of local (quasi) phase transitions is in most theoretical approaches dealing with randomly perturbed systems neglected from the outset.

Long-range order and quantum criticality in antiferromagnetic chains with long-range staggered interactions.

We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.

The effect of elastic anisotropy on the symmetry selection of irradiation-induced void superlattices in cubic metals

Self-organized microstructures and patterns have been widely observed in non-equilibrium physical systems. In particular, irradiation in metals creates far-from-equilibrium environments, in which the competing dynamics of defect production and annihilation can lead to unique self-organized superlattice structures, e.g., void and gas bubble superlattices. From a physical point of view, the superlattice structures are dictated by the intrinsic symmetry breaking in the metals, i.e., anisotropy caused by the breaking of continuous rotational symmetry. In the literature, two distinctive anisotropies, elastic anisotropy and diffusion anisotropy of interstitials, have been proposed to be the origins of superlattice formation. However, it is still unclear which anisotropy dominates the symmetry selection of superlattice structures. In this paper, we study elastic anisotropy and its effect on the symmetry of void superlattices. By using theoretical analyses and phase field simulations, we show that elastic anisotropy in cubic metals can lead to either face-centered cubic or simple cubic superlattices depending on the Zener anisotropy ratio. The superlattices formed under this elastic anisotropy mechanism must form under the influence of spinodal decomposition, as the mechanism requires perturbations in the vacancy concentration field to develop into spatially-static concentration waves. We compare to existing work on symmetry selection in superlattices via diffusion anisotropy and to experimental observations, and we suggest that concentration wave development under the influence of elastic anisotropy is not the mechanism for symmetry selection during the formation of irradiation-induced void superlattices, but that diffusion anisotropy could be the dominant mechanism.

Resonance-enhanced spectral funneling in Fabry–Perot resonators with a temporal boundary mirror

Abstract A temporal boundary refers to a specific time at which the properties of an optical medium are abruptly changed. When light interacts with the temporal boundary, its spectral content can be redistributed due to the breaking of continuous time-translational symmetry of the medium where light resides. In this work, we use this principle to demonstrate, at terahertz (THz) frequencies, the resonance-enhanced spectral funneling of light coupled to a Fabry–Perot resonator with a temporal boundary mirror. To produce a temporal boundary effect, we abruptly increase the reflectance of a mirror constituting the Fabry–Perot resonator and, correspondingly, its quality factor in a step-like manner. The abrupt increase in the mirror reflectance leads to a trimming of the coupled THz pulse that causes the pulse to broaden in the spectral domain. Through this dynamic resonant process, the spectral contents of the input THz pulse are redistributed into the modal frequencies of the high-Q Fabry–Perot resonator formed after the temporal boundary. An energy conversion efficiency of up to 33% was recorded for funneling into the fundamental mode with a Fabry–Perot resonator exhibiting a sudden Q-factor change from 4.8 to 48. We anticipate that the proposed resonance-enhanced spectral funneling technique could be further utilized in the development of efficient mechanically tunable narrowband terahertz sources for diverse applications.

Amplitude Mode in Quantum Magnets via Dimensional Crossover.

We investigate the amplitude (Higgs) mode associated with longitudinal fluctuations of the order parameter at the continuous spontaneous symmetry breaking phase transition. In quantum magnets, due to the fast decay of the amplitude mode into low-energy Goldstone excitations, direct observation of this mode represents a challenging task. By focusing on a quasi-one-dimensional geometry, we circumvent the difficulty and investigate the amplitude mode in a system of weakly coupled spin chains with the help of quantum MonteCarlo simulations, stochastic analytic continuation, and a chain-mean field approach combined with a mapping to the field-theoretic sine-Gordon model. The amplitude mode is observed to emerge in the longitudinal spin susceptibility in the presence of a weak symmetry-breaking staggered field. A conventional measure of the amplitude mode in higher dimensions, the singlet bond mode, is found to appear at a lower than the amplitude mode frequency. We identify these two excitations with the second (first) breather of the sine-Gordon theory, correspondingly. In contrast to higher-dimensional systems, the amplitude and bond order fluctuations are found to carry significant spectral weight in the quasi-1D limit.

Conflict-Based Increasing Cost Search

Two popular optimal search-based solvers for the multi-agent pathfinding (MAPF) problem, Conflict-Based Search (CBS) and Increasing Cost Tree Search (ICTS), have been extended separately for continuous time domains and symmetry breaking. However, an approach to symmetry breaking in continuous time domains remained elusive. In this work, we introduce a new algorithm, Conflict-Based Increasing Cost Search (CBICS), which is capable of symmetry breaking in continuous time domains by combining the strengths of CBS and ICTS. Our experiments show that CBICS often finds solutions faster than CBS and ICTS in both unit time and continuous time domains.

Discreteness of space from anisotropic spin\u2013orbit interaction

Various approaches to Quantum Gravity suggest an existence of a minimal measurable length. The cost to have such minimal length could be modified uncertainty principle, modified dispersion relation, non-commutative geometry or breaking of continuous Lorentz symmetry. In this paper, we propose that minimal length can be obtained naturally through spin–orbit interaction. We consider Dresselhaus anisotropic spin–orbit interaction as the perturbative Hamiltonian. When applied to a particle, it implies that the space, which seizes this particle, should be quantized in terms of units that depend on particle’s mass. This suggests that all measurable lengths in the space are quantized in units depending on existent mass and the Dresselhaus coupling constant. On one side, this indicates a breakdown of the space continuum picture near the scale of tabletop experiments, and on the other side, it proposes that spin–orbit interaction is a possible quantum gravity effect at low energy scale that leads naturally to space quantization.

General Landau theory of non-symmetry-breaking and symmetry-breaking spin transition materials

Spin-transition materials, including the families of spin-crossover and charge-transfer systems, and more generally molecular-based materials exhibiting electronic and/or structural bistability, may undergo various types of phase transitions. The change of electronic state is stabilized by molecular reorganizations and both phenomena, which are usually non-symmetry breaking, can be described through the evolution of an order parameter q. Due to symmetry, q linearly couples to volume change. It is known that such elastic interactions are responsible for cooperative phenomena in non-symmetry-breaking spin-transitions. However, spin-transition materials may also exhibit symmetry-breaking phenomena related to various types of orders such as structural order as well as spin-state concentration waves. The universal framework of the Landau theory of phase transition is relevant for describing such ordering processes through the evolution of a symmetry-breaking order parameter η. The simultaneous or sequential occurrence of spin-transition and symmetry-breaking phenomena are reported for numerous spin-transition materials, and the coupling between these two types of instabilities is responsible for the emergence of various types of functions. In this work, we use the Landau approach to describe both symmetry-breaking phenomena and non-symmetry-breaking spin transition. We discuss how their coupling can generate sequences of phase transitions, from simple spin-crossover to spin-transition, continuous or discontinuous symmetry breaking, including ferroelasticity or stepwise spin transitions.

Low- and high-β lasers in the class-A limit: photon statistics, linewidth, and the laser-phase transition analogy

Nanocavity lasers are commonly characterized by the spontaneous coupling coefficient β that represents the fraction of photons emitted into the lasing mode. While β is conventionally discussed in semiconductor lasers where the photon lifetime is much shorter than the carrier lifetime (class-B lasers), little is known about β in atomic lasers where the photon lifetime is much longer than the other lifetimes and only the photon degree of freedom exists (class-A lasers). We investigate the impact of the spontaneous coupling coefficient β on lasing properties in the class-A limit by extending the well-known Scully–Lamb master equation. We demonstrate that in the class-A limit all the photon statistics are uniquely characterized by β and that the laser phase transition-like analogy becomes transparent. In fact, β perfectly represents the “system size” in phase transition. Finally, we investigate the laser-phase transition analogy from the standpoint of a quantum dissipative system. Calculating a Liouvillian gap, we clarify the relation between β and the continuous phase symmetry breaking.

Cavity QED with quantum gases: new paradigms in many-body physics

We review the recent developments and the current status in the field of quantum-gas cavity QED. Since the first experimental demonstration of atomic self-ordering in a system composed of a Bose-Einstein condensate coupled to a quantized electromagnetic mode of a high-$Q$ optical cavity, the field has rapidly evolved over the past decade. The composite quantum-gas--cavity systems offer the opportunity to implement, simulate, and experimentally test fundamental solid-state Hamiltonians, as well as to realize non-equilibrium many-body phenomena beyond conventional condensed-matter scenarios. This hinges on the unique possibility to design and control in open quantum environments photon-induced tunable-range interaction potentials for the atoms using tailored pump lasers and dynamic cavity fields. Notable examples range from Hubbard-like models with long-range interactions exhibiting a lattice-supersolid phase, over emergent magnetic orderings and quasicrystalline symmetries, to the appearance of dynamic gauge potentials and non-equilibrium topological phases. Experiments have managed to load spin-polarized as well as spinful quantum gases into various cavity geometries and engineer versatile tunable-range atomic interactions. This led to the experimental observation of spontaneous discrete and continuous symmetry breaking with the appearance of soft-modes as well as supersolidity, density and spin self-ordering, dynamic spin-orbit coupling, and non-equilibrium dynamical self-ordered phases among others. In addition, quantum-gas--cavity setups offer new platforms for quantum-enhanced measurements. In this review, starting from an introduction to basic models, we pedagogically summarize a broad range of theoretical developments and put them in perspective with the current and near future state-of-art experiments.

Fifth forces and discrete symmetry breaking

Modifications of general relativity often involve coupling additional scalar fields to the Ricci scalar, leading to scalar-tensor theories of Brans-Dicke type. If the additional scalar fields are light, they can give rise to long-range fifth forces, which are subject to stringent constraints from local tests of gravity. In this talk, we show that Yukawa-like fifth forces only arise for the Standard Model (SM) due to a mass mixing of the additional scalar with the Higgs field, and we emphasise the pivotal role played by discrete and continuous symmetry breaking. Quite remarkably, if one assumes that sufficiently light, non-minimally coupled scalar fields exist in nature, the non-observation of fifth forces has the potential to tell us about the structure of the SM Higgs sector and the origin of its symmetry breaking. Moreover, with these observations, we argue that certain classes of scalar-tensor theories are, up to and including their dimension-four operators, equivalent to Higgs-portal theories. In this way, ultra-light dark matter models may also exhibit fifth-force phenomenology, and we consider the impact on the dynamics of disk galaxies as an example.

Condensed matter physics in time crystals

Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.

Theory of generalized Josephson effects

Abstract The DC Josephson effect is the flow of supercurrent across a weak link between two superconductors with different values of their order parameters, the phase. We formulate this notion for any kind of spontaneous continuous symmetry breaking. The quantity that flows between the two systems is the zero-wavenumber Noether current associated with the broken symmetry. The AC Josephson effect is the oscillation of current due to the energy difference between the two systems caused by an imposed asymmetric chemical potential of Noether charge. As an example of novel physics, a Josephson effect is predicted between two crystalline solids, potentially measurable as a force periodic in the separation distance.

S3 -inspired three-Higgs-doublet models: A class with a complex vacuum

In this paper we analyse in detail an $S_3$-symmetric three-Higgs-doublet model with a specific vacuum configuration. This analysis allows us to illustrate important features of models with several Higgs doublets, such as the possibility of having spontaneous CP violation. We start with a real potential and pick a particularly interesting complex vacuum configuration, which does not violate CP before adding soft breaking terms to the potential. We study the r\^ ole played by different soft symmetry breaking terms. These are essential for our choice of vacuum in order to remove unwanted massless scalars which arise from the spontaneous breaking of an accidental continuous symmetry. We list scalar sector and scalar-gauge sector-couplings for the particular case we consider in detail in this work. Results presented in this paper will be useful for model building, in particular for implementations of models with $S_3$ symmetry and spontaneous CP violation, extensions of the fermionic sector with realistic Yukawa couplings and for Dark Matter studies.

Landau ordering phase transitions beyond the Landau paradigm

Continuous phase transitions associated with the onset of a spontaneously broken symmetry are thought to be successfully described by the Landau-Ginzburg-Wilson-Fisher theory of fluctuating order parameters. In this work we show that such transitions can admit new universality classes which cannot be understood in terms of a theory of order parameter fluctuations. We explicitly demonstrate continuous time reversal symmetry breaking quantum phase transitions of $3+1$-D bosonic systems described by critical theories expressed in terms of a deconfined gauge theory with massless Dirac fermions instead of the fluctuating Ising order parameter. We dub such phase transitions "Landau transitions beyond Landau description" (LBL). A key feature of our examples is that the stability of the LBL fixed points requires a crucial global symmetry, which is non-anomalous, unbroken, and renders no symmetry protected topological phase throughout the phase diagram. Despite this, there are elementary critical fluctuations of the phase transition that transform projectively under this symmetry group. We also construct examples of other novel quantum critical phenomena, notably a continuous Landau-forbidden deconfined critical point between two Landau-allowed phases in $3+1$-D.

‘Place-cell’ emergence and learning of invariant data with restricted Boltzmann machines: breaking and dynamical restoration of continuous symmetries in the weight space

Distributions of data or sensory stimuli often enjoy underlying invariances. How and to what extent those symmetries are captured by unsupervised learning methods is a relevant question in machine learning and in computational neuroscience. We study here, through a combination of numerical and analytical tools, the learning dynamics of restricted Boltzmann machines (RBM), a neural network paradigm for representation learning. As learning proceeds from a random configuration of the network weights, we show the existence of, and characterize a symmetry-breaking phenomenon, in which the latent variables acquire receptive fields focusing on limited parts of the invariant manifold supporting the data. The symmetry is restored at large learning times through the diffusion of the receptive field over the invariant manifold; hence, the RBM effectively spans a continuous attractor in the space of network weights. This symmetry-breaking phenomenon takes place only if the amount of data available for training exceeds some critical value, depending on the network size and the intensity of symmetry-induced correlations in the data; below this ‘retarded-learning’ threshold, the network weights are essentially noisy and overfit the data.

Entanglement and correlations in a one-dimensional quantum spin- 12 chain with anisotropic power-law long-range interactions

The correlations, entanglement entropy, and fidelity susceptibility are calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law long range interactions by employing the density matrix renormalization group method. In particular, this long-range interaction is assigned to ferromagnetic for transversal components, while it can be either ferro- or antiferromagnetic for the longitudinal spin component. Two ground-state phase diagrams are established versus the anisotropy of the interactions which not only changes the phase boundaries of the counterparts with short-range interactions, but also leads to the emergence of exotic phases. We found that the long-range interactions of the z-component results in a Wigner crystal phase, whereas the transversal one may break a continuous symmetry, resulting in a continuous symmetry breaking phase.