Reflection Symmetry Research Articles

When the moduli space is an orbifold: spontaneous breaking of continuous non-invertible symmetries

We investigate theories of Nambu-Goldstone bosons where the spontaneously broken continuous symmetry is non-invertible. In such theories, the vacua generically parameterize an orbifold. We study in detail the simplest example of a single free scalar with shift symmetry, modded by reflection symmetry. At singular points of the vacuum manifold, we show that the spectrum of NG excitations is reduced, in particular there are no single-particle states. At the smooth points, on the other hand, single NG modes are present. We show that this is a consequence of the fact that at those points one can construct invertible operators implementing the continuous symmetry on the Hilbert space.

Learning analytics in mathematics education: the case of feedback use in a digital classification task on reflective symmetry

Learning Analytics is concerned with the use of data collected in educational settings to support learning processes. We take a Learning Analytics approach to study the use of immediate feedback in digital classification tasks in mathematics. Feedback serves as an opportunity for learning, however its mere existence does not guarantee its use and effectiveness, as what matters is how learners interact with it. Therefore, our research questions are focused on that interaction. The data consisted of 266 object movements for classifying polygons, and 524 shape movements for classifying traffic signs, under the topic of symmetry. Participants included 29 elementary school students (9–12 years old) from Israel and Germany. Analyzing students’ success, feedback use, and the associations between them, we demonstrate how not acting upon feedback is negatively associated with success, and how this undesired behavior slightly reduces along the learning process.

Hydrogenation-induced superconductivity in monolayer

Here, we construct a new two-dimensional hydrogenated transition metal dichalcogenide material, the Janus WSH monolayer, which is created by replacing the top-layer S atoms in the 2H-WS2 monolayer with H atoms. Then we use first-principles calculations to investigate its electronic structure, phonon dispersion, and superconductivity. The results show that hydrogenation breaks the reflection symmetry, which helps orbital hybridization and to flatten the electronic bands. Thus, it leads to a high electronic density of states near the Fermi level. Additionally, the electron-phonon coupling is enhanced by the softening of phonon modes from the in-plane vibrations of W. The strong interactions between electrons and phonons result in phonon-mediated superconductivity in Janus WSH monolayer. The calculated critical temperature (T c ) is approximately 23.1 K at atmospheric pressure. This T c is about twice higher than that of existing WS2-based materials.

The brain does not process horizontal reflection when attending to vertical reflection, and vice versa.

Previous work has found that feature attention can modulate electrophysiological responses to visual symmetry. In the current study, participants observed spatially overlapping clouds of black and white dots. They discriminated vertical symmetry from asymmetry in the target dots (e.g., black or white) and ignored the regularity of the distractor dots (e.g., white or black). We measured an electroencephalography component called the sustained posterior negativity (SPN), which is known to be generated by visual symmetry. There were five conditions with different combinations of target and distractor regularity. As well as replicating previous results, we found that an orthogonal axes of reflection in the distractor dots had no effect on SPN amplitude. We conclude that the visual system can processes reflectional symmetry in independent axis-orientation specific channels.

Tunable chiral bound states in a dimer chain of coupled resonators

We study an excitation hopping on a one-dimensional (1D) dimer chain of coupled resonators with the alternate on-site photon energies, which interacts with a two-level emitter (TLE) by a coupling point or two adjacent coupling points. In the single-excitation subspace, this system not only possesses two energy bands with propagating states, but also possesses photonic bound states. The number of bound states depends on the coupling forms between the TLE and the dimer chain. It is found that when the TLE is locally coupled to one resonator of the dimer chain, the bound-state that has mirror reflection symmetry. When the TLE is nonlocally coupled to two adjacent resonators, three bound states with preferred direction arise due to the mirror symmetry breaking. By using chirality to measure the asymmetry, it is found that the chirality of these bound states can be tuned by changing the energy differences of single photon in the adjacent resonators, the coupling strengths and the transition energy of the TLE.

Light-Driven Spontaneous Phonon Chirality and Magnetization in Paramagnets.

Spin-phonon coupling enables the mutual manipulation of phonon and spin degrees of freedom in solids. In this study, we reveal the inherent nonlinearity within this coupling. Using a paramagnet as an illustration, we demonstrate the nonlinearity by unveiling spontaneous symmetry breaking under a periodic drive. The drive originates from linearly polarized light, respecting a mirror reflection symmetry of the system. However, this symmetry is spontaneously broken in the steady state, manifested in the emergence of coherent chiral phonons accompanied by a nonzero magnetization. We establish an analytical self-consistency equation to find the parameter regime where spontaneous symmetry breaking occurs. Furthermore, we estimate realistic parameters and discuss potential materials that could exhibit this behavior. Our findings shed light on the exploration of nonlinear phenomena in magnetic materials and present possibilities for on-demand control of magnetization.

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In recent decades, the Altland-Zirnabuer (AZ) table has proven incredibly powerful in delineating constraints for topological classification of a given band-insulator based on dimension and (nonspatial) symmetry class, and has also been expanded by considering additional crystalline symmetries. Nevertheless, realizing a three-dimensional (3D), time-reversal symmetric (class AII) topological insulator (TI) in the absence of reflection symmetries, with a classification beyond the Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2}$$\\end{document} paradigm remains an open problem. In this work we present a general procedure for constructing such systems within the framework of projected topological branes (PTBs). In particular, a 3D projected brane from a “parent” four-dimensional topological insulator exhibits a Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}$$\\end{document} topological classification, corroborated through its response to the inserted bulk monopole loop. More generally, PTBs have been demonstrated to be an effective route to performing dimensional reduction and embedding the topology of a (d+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(d+1)$$\\end{document}-dimensional “parent” Hamiltonian in d dimensions, yielding lower-dimensional topological phases beyond the AZ classification without additional symmetries. Our findings should be relevant for the metamaterial platforms, such as photonic and phononic crystals, topolectric circuits, and designer systems.

Stacking-induced phonon transport engineering of siligene

Tunable phonon transport properties of two-dimensional materials are desirable for effective heat management in various application scenarios. Here, we demonstrate by first-principles calculations and Boltzmann transport theory that the lattice thermal conductivity of siligene could be efficiently engineered by forming various stacking configurations. Unlike few-layer graphene, the stacked siligenes are found to be covalently bonded along the out-of-plane direction, which leads to unique dependence of the thermal conductivity on both the stacking order and layer number. Due to the restricted flexural phonon scattering induced by the horizontal reflection symmetry, the AA stacking configuration of bilayer siligene exhibits obviously higher thermal conductivity compared with the AB stacking. In addition, we observe increasing thermal conductivity with the layer number, as evidenced by the reduced phonon scattering phase space and Grüneisen parameter. Interestingly, the Fuchs-Sondheimer model works well for the thickness-dependent thermal conductivity of stacked siligenes.

Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems

In interacting dynamical systems, specific local interaction rules for system components give rise to diverse and complex global dynamics. Long dynamical cycles are a key feature of many natural interacting systems, especially in biology. Examples of dynamical cycles range from circadian rhythms regulating sleep to cell cycles regulating reproductive behavior. Despite the crucial role of cycles in nature, the properties of network structure that give rise to cycles still need to be better understood. Here, we use a Boolean interaction network model to study the relationships between network structure and cyclic dynamics. We identify particular structural motifs that support cycles, and other motifs that suppress them. More generally, we show that the presence of dynamical reflection symmetry in the interaction network enhances cyclic behavior. In simulating an artificial evolutionary process, we find that motifs that break reflection symmetry are discarded. We further show that dynamical reflection symmetries are over-represented in Boolean models of natural biological systems. Altogether, our results demonstrate a link between symmetry and functionality for interacting dynamical systems, and they provide evidence for symmetry’s causal role in evolving dynamical functionality.

Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2}ℤ2×ℤ2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2}ℤ2⊂ℤ2×ℤ2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZXYZ chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n}ℤn clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n}ℤn×ℤn global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n}ℤn charge and dipole symmetries.

The SUSY partners of the QES sextic potential revisited

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential Vqes(x)=νx6+2νμx4+μ2−(4N+3)νx2 , N∈Z+ , are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable z = x 2, the underlying sl2(R) hidden algebra of V qes(x) is inherited by its SUSY partner potential V 1(x) only for N = 0. At fixed N > 0, the algebraic polynomial operator h(x, ∂ x ; N) that governs the N exact eigenpolynomial solutions of V 1 is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential V 1 can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by V qes with a different non-integer (cohomology) parameter N1=N−32 . A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking N as a continuous real constant and using the Lagrange-mesh method, highly accurate values (∼20 s. d.) of the energy E n = E n (N) in the interval N ∈ [ − 1, 3] are calculated for the three lowest states n = 0, 1, 2 of the system. The critical value N c above which tunneling effects (instanton-like terms) can occur is obtained as well. At N = 0, the non-algebraic sector of the spectrum of V qes is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.

Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states

We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are arranged in a spatially symmetric pattern and the states have maximal entanglement for all bipartitions that follow from the reflection symmetries of the given geometry. We consider those cases where the state itself is invariant with respect to the geometrical symmetry group. The simplest examples are those dual unitary operators which are also self dual and reflection invariant, but we also consider the generalizations in the hexagonal, cubic, and octahedral geometries. We provide a number of constructions and concrete examples for these objects for various local dimensions. All of our examples can be used to build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple equivalent choices for the “direction of time”.

Nonequilibrium Schwinger-Keldysh formalism for density matrix states: Analytic properties and implications in cosmology

Motivated by cosmological Hartle-Hawking and microcanonical density matrix prescriptions for the quantum state of the Universe we develop a Schwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systems with the initial density matrix. We build the generating functional of in-in Green’s functions and expectation values for a generic density matrix of the Gaussian type and show that the requirement of particle interpretation selects a distinguished set of positive/negative frequency basis functions of the wave operator of the theory, which is determined by the density matrix parameters. Then we consider a special case of the density matrix determined by the Euclidean path integral of the theory, which in the cosmological context can be considered as a generalization of the no-boundary pure state to the case of the microcanonical ensemble, and show that in view of a special reflection symmetry its Wightman Green’s functions satisfy Kubo-Martin-Schwinger periodicity conditions which hold despite the nonequilibrium and nonstationary nature of the physical setup. Rich analyticity structure in the complex plane of the time variable reveals the combined Euclidean-Lorentzian evolution of the theory, which depending on the properties of the initial density matrix can be interpreted as a decay of a classically forbidden quantum state. Published by the American Physical Society 2024

Soft colloidal monolayers with reflection symmetry through confined drying.

Colloidal monolayers serve as fundamental building blocks in fabricating diverse functional materials, pivotal for surface modifications, chemical reactivity, and controlled assembly of nanoparticles. In this article, we report the formation of colloidal monolayers generated by drying an aqueous droplet containing soft colloids confined between two hydrophilic parallel plates. The analysis of the kinetics of evaporation in this confined mode showed that: (i) for a significant portion of the drying time, the drops adopt a catenoid configuration; (ii) in the penultimate stage of drying, the catenoid structure undergoes division into two daughter droplets; (iii) the three-phase contact line remains pinned at a specific location while it continuously slips at all other locations. The interplay between interface-assisted particle deposition onto the solid substrate and the time evolution of particle concentration within the droplet during evaporation results in unique microstructural features in the deposited patterns. Notably, these deposit patterns exhibit reflection symmetry. The microstructural features of the dried deposits are further quantified by calculating the particle number density, inter-particle separation, areal disorder parameter, and bond orientational order parameter. The variation of these parameters for deposits formed under different conditions, such as by altering the spacing between parallel plates and the concentration of microgel particles in the droplet, is discussed.

К теории спиральной турбулентности немагнитного астрофизического диска. Образование крупномасштабных вихревых структур

A closed system of three-dimensional hydrodynamic equations of the mean motion scale is presented for modeling spiral turbulence in a rotating astrophysical disk. The diffusion equations for the averaged vortex and the integral vortex spirality transport equation are derived. A general concept of the emergence of energy-consuming mezoscale coherent vortex structures in a thermodynamically open subsystem of turbulent chaos, associated with the realization of the inverse cascade of kine-tic energy in mirror-nonsymmetric disk turbulence, is formulated. It is shown that negative viscosity in the rotating disk three-dimensional system is apparently a manifestation of cascade processes in spiral turbulence, when an inverse energy transfer from small vortices to larger ones is realized. It is also shown that a relatively long decay of turbulence in the disk is associated with the absence of reflective symmetry of the anisotropic field of turbulent velocities relative to its equatorial plane. The work is of a review character, made with the aim of improving new models of astrophysical nonmagnetic disks for which the effects of spiral turbulence play a determining role.

Corrections to “Test Statistics for Reflection Symmetry: Applications to Quad-Polarimetric SAR Data for Detection of Man-Made Structures”

Corrections to “Test Statistics for Reflection Symmetry: Applications to Quad-Polarimetric SAR Data for Detection of Man-Made Structures”

Role of four-phonon processes in thermal conductivity of two-dimensional materials and thermal-transport enhancement arising from interconnected nanofiller networks in polymer/nanofiller composites.

Recent research has shed light on the importance of four-phonon scattering processes in the thermal conductivity (k) of 2D materials. The inclusion of 4 phonon scattering processes from first-principles has been shown to lead to a thermal conductivity of ∼1290 W m-1 K-1 in graphene at 300 K, significantly lower than the values predicted to be in excess of 4000 W m-1 K-1 based only on 3 phonon scattering processes. Four phonon processes are shown to be most significant for flexural ZA phonon modes, where the reflection symmetry selection rule (RSSR) is less restrictive for 4-phonon than 3-phonon scattering processes. This combined with the low frequencies of ZA phonon modes, leading to high populations, leads to higher 4-phonon than 3-phonon scattering of low frequency ZA phonon modes in graphene at 300 K. In this review, the role of parameters such as atomic structure, phonon dispersion and temperature on 4-phonon scattering processes in a wide range of 2D materials is reviewed. Materials such as graphene nanoplatelets (GnPs) have been extensively investigated for enhancement of the thermal conductivity of polymer composites. However, such enhancement is limited by the poor interfacial thermal conductance between the polymer and filler material. Interconnected filler networks overcome this issue through highly efficient continuous percolative heat transfer paths throughout the composite. Such 3D networks have been shown to enable ultra-high polymer thermal conductivities, approaching ∼100 W m-1 K-1, and even exceeding those of several metals. In this review, different techniques used to achieve such interconnected 3D filler networks, namely, aerogels, foams, ice-templating, expanded graphite, hot pressing of filler coated polymer particles, the synergistic effect between multiple fillers, and the stitching of filler sheets, are discussed and their impact on thermal conductivity enhancement are presented.

Four-phonon scattering and thermal transport in 2H–MoTe2

Among two-dimensional (2D) transition metal dichalcogenides, molybdenum ditelluride (MoTe2) is of great interest to achieving unique functionalities in diverse electronic, photonic, and phononic devices. Despite extensive studies over the past decade, the thermal transport properties and mechanisms of MoTe2 remain largely unclear. Here, we grow few-layer (7L, 10L, and 17L) single-crystalline 2H–MoTe2 and measure their in-plane thermal conductivities using an optothermal Raman technique. The room-temperature values range from 4 to 9 W m−1 K−1, substantially lower than theoretical predictions considering only three-phonon scattering. By employing first-principles calculations and machine learning-assisted molecular dynamics simulations, we reveal that four-phonon scattering could suppress the thermal conductivity of 2H–MoTe2 by 80%, which agrees with experimental measurement and is mainly attributed to the flexural acoustic modes. Thickness-dependent suppression is observed owing to the varying reflection symmetry. Our work highlights higher-order phonon-phonon interactions in 2D materials and offers guidance for the thermal design of emerging devices.

Exact Mobility Edges for Almost-Periodic CMV Matrices via Gauge Symmetries

Abstract We investigate the symmetries of the so-called generalized extended Cantero–Moral–Velázquez (CMV) matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero–Grünbaum–Moral–Velázquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them.

Geometrical structures of nested polyhedra

The polyhedra with A 3, B 3/C 3, H 3 reflection symmetry group G in the real 3D space are considered. The recursive rules for finding orbits with smaller radii, which provide the structures of nested polytopes, are demonstrated.