Presence Of Symmetry Research Articles

Spin-space groups and magnon band topology

Band topology is both constrained and enriched by the presence of symmetry. The importance of anti-unitary symmetries such as time reversal was recognized early on leading to the classification of topological band structures based on the ten-fold way. Since then, lattice point group and non-symmorphic symmetries have been seen to lead to a vast range of possible topologically nontrivial band structures many of which are realized in materials. In this paper we show that band topology is further enriched in many physically realizable instances where magnetic and lattice degrees of freedom are wholly or partially decoupled. The appropriate symmetry groups to describe general magnetic systems are the spin-space groups. Here we describe cases where spin-space groups are essential to understand the band topology in magnetic materials. We then focus on magnon band topology where the theory of spin-space groups has its simplest realization. We consider magnetic Hamiltonians with various types of coupling including Heisenberg and Kitaev couplings revealing a hierarchy of enhanced magnetic symmetry groups depending on the nature of the lattice and the couplings. We describe, in detail, the associated representation theory and compatibility relations thus characterizing symmetry-enforced constraints on the magnon bands revealing a proliferation of nodal points, lines, planes and volumes.

Symmetry and Symmetry Breaking in Physics: From Geometry to Topology

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle physics, and quantum physics, respectively. The latter groups now play an important role in many branches of mathematics. In more recent years, we have been faced with the impact of topological quantum field theory (TQFT). Topology and symmetry have deep connections, but topology is inherently broader and more complex. While the presence of symmetry in physical phenomena imposes strong constraints, topology seems to be related to low-energy states and is very likely to provide information about the different dynamical trajectories and patterns that particles can follow. For example, regarding the relationship of topology to low-energy states, Hodge’s theory of harmonic forms shows that the zero-energy states (for differential forms) correspond to the cohomology. Regarding the relationship of topology to particle trajectories, a topological knot can be seen as an orbit with complex properties in spacetime. The various deformations or embeddings of the knot, performed in low or high dimensions, allow defining different equivalence classes or topological types, and interestingly, it is possible from these types to study the symmetries associated with the deformations and their changes. More specifically, in the present work, we address two issues: first, that quantum geometry deforms classical geometry, and that this topological deformation may produce physical effects that are specific to the quantum physics scale; and second, that mirror symmetry and the phenomenon of topological change are closely related. This paper was aimed at understanding the conceptual and physical significance of this connection.

Spectral Geometry in the Presence of Symmetry

Spectral Geometry in the Presence of Symmetry

Invariance or equivalence: a tale of two principles

The presence of symmetries in physical theories implies a pernicious form of underdetermination. In order to avoid this theoretical vice, philosophers often espouse a principle called Leibniz Equivalence, which states that symmetry-related models represent the same state of affairs. Moreover, philosophers have claimed that the existence of non-trivial symmetries motivates us to accept the Invariance Principle, which states that quantities that vary under a theory’s symmetries aren’t physically real. Leibniz Equivalence and the Invariance Principle are often seen as part of the same package. I argue that this is a mistake: Leibniz Equivalence and the Invariance Principle are orthogonal to each other. This means that it is possible to hold that symmetry-related models represent the same state of affairs whilst having a realist attitude towards variant quantities. Various arguments have been presented in favour of the Invariance Principle: a rejection of the Invariance Principle is inter alia supposed to cause indeterminism, undetectability or failure of reference. I respond that these arguments at best support Leibniz Equivalence.

Quantum Orbit Method in the Presence of Symmetries

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering

The paper aims to study a concrete structure, currently used in civil engineering, which has certain symmetries. This type of problem is common in engineering practice, especially in civil engineering. There are many reasons why structures with identical elements or certain symmetries are used in industry, related to economic considerations, shortening the design time, for constructive, simplicity, cost or logistical reasons. There are many reasons why the presence of symmetries has benefits for designers, builders, and beneficiaries. In the end, the result of these benefits materializes through short execution times and reduced costs. The paper studies the eigenvalue and eigenmode properties of vibration for components of the constructions’ structure, often encountered in current practice. The identification of such properties allows the simplification and easing of the effort necessary for the dynamic analysis of such a structure.

A Role of Symmetries in Evaluation of Fundamental Bounds

A problem of the erroneous duality gap caused by the presence of symmetries is solved in this paper utilizing point group theory. The optimization problems are first divided into two classes based on their predisposition to suffer from this deficiency. Then, the classical problem of Q-factor minimization is shown in an example where the erroneous duality gap is eliminated by combining solutions from orthogonal sub-spaces. Validity of this treatment is demonstrated in a series of subsequent examples of increasing complexity spanning the wide variety of optimization problems, namely minimum Q-factor, maximum antenna gain, minimum total active reflection coefficient, or maximum radiation efficiency with self-resonant constraint. They involve problems with algebraic and geometric multiplicities of the eigenmodes, and are completed by an example introducing the selective modification of modal currents falling into one of the symmetry-conformal sub-spaces. The entire treatment is accompanied with a discussion of finite numerical precision, and mesh grid imperfections and their influence on the results. Finally, the robust and unified algorithm is proposed and discussed, including advanced topics such as the uniqueness of the optimal solutions, dependence on the number of constraints, or an interpretation of the qualitative difference between the two classes of the optimization problems.

Adding potentials to superintegrable systems with symmetry

In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

Revealing New Candidate Genes for Teat Number Relevant Traits in Duroc Pigs Using Genome-Wide Association Studies.

Simple SummaryNumber of teats is very important for lactating sows. We conducted genome-wide association studies (GWAS) and estimated the genetic parameters for traits related to teat number. Results showed that there were nine and 22 SNPs exceeding genome-wide significance and suggestive significance levels, respectively. Eighteen genes annotated near them were concentrated on chromosomes 7 and 10. Among them, three new candidate genes were located on the genomic regions around the significant SNPs. Our findings provide new insight into investigating the complex genetic mechanism of traits related to teat number in pigs.The number of teats is related to the nursing ability of sows. In the present study, we conducted genome-wide association studies (GWAS) for traits related to teat number in Duroc pig population. Two mixed models, one for counted and another for binary phenotypic traits, were employed to analyze seven traits: the right (RTN), left (LTN), and total (TTN) teat numbers; maximum teat number on a side (MAX); left minus right side teat number (LR); the absolute value of LR (ALR); and the presence of symmetry between left and right teat numbers (SLR). We identified 11, 1, 4, 13, and 9 significant SNPs associated with traits RTN, LTN, MAX, TTN, and SLR, respectively. One significant SNP (MARC0038565) was found to be simultaneous associated with RTN, LTN, MAX and TTN. Two annotated genes (VRTN and SYNDIG1L) were located in genomic region around this SNP. Three significant SNPs were shown to be associated with TTN, RTN and MAX traits. Seven significant SNPs were simultaneously detected in two traits of TTN and RTN. Other two SNPs were only identified in TTN. These 13 SNPs were clustered in the genomic region between 96.10—98.09 Mb on chromosome 7. Moreover, nine significant SNPs were shown to be significantly associated with SLR. In total, four and 22 SNPs surpassed genome-wide significance and suggestive significance levels, respectively. Among candidate genes annotated, eight genes have documented association with the teat number relevant traits. Out of them, DPF3 genes on Sus scrofa chromosome (SSC) 7 and the NRP1 gene on SSC 10 were new candidate genes identified in this study. Our findings demonstrate the genetic mechanism of teat number relevant traits and provide a reference to further improve reproductive performances in practical pig breeding programs.

Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge-dependent, their sum is gauge-independent. From this we conclude that the edge polarization is not by itself a physical observable, and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.

Berry phase in quantum field theory: Diabolical points and boundary phenomena

We study aspects of Berry phase in gapped many-body quantum systems by means of effective field theory. Once the parameters are promoted to spacetime-dependent background fields, such adiabatic phases are described by Wess-Zumino-Witten (WZW) and similar terms. In the presence of symmetries, there are also quantized invariants capturing generalized Thouless pumps. Consideration of these terms provides constraints on the phase diagram of many-body systems, implying the existence of gapless points in the phase diagram which are stable for topological reasons. We describe such diabolical points, realized by free fermions and gauge theories in various dimensions, which act as sources of "higher Berry curvature" and are protected by the quantization of the corresponding WZW terms or Thouless pump terms. These are analogous to Weyl nodes in a semimetal band structure. We argue that in the presence of a boundary, there are boundary diabolical points---parameter values where the boundary gap closes---which occupy arcs ending at the bulk diabolical points. Thus the boundary has an "anomaly in the space of couplings" in the sense of C\'ordova et al. Consideration of the topological effective action for the parameters also provides some new checks on conjectured infrared dualities and deconfined quantum criticality in 2+1d.

The Noether–Bessel-Hagen symmetry approach for dynamical systems

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

A relational perspective on the Wigner-Araki-Yanase theorem

We present a novel interpretation of the Wigner-Araki-Yanase (WAY) theorem based on a relational view of quantum mechanics. Several models are analysed in detail, backed up by general considerations, which serve to illustrate that the moral of the WAY theorem may be that in the presence of symmetry, a measuring apparatus must fulfil the dual purpose of both reflecting the statistical behaviour of the system under investigation, and acting as a physical reference system serving to define those quantities which must be understood as relative.

Modified Harris Hawks Optimizer for Solving Machine Scheduling Problems

Scheduling can be described as a decision-making process. It is applied in various applications, such as manufacturing, airports, and information processing systems. More so, the presence of symmetry is common in certain types of scheduling problems. There are three types of parallel machine scheduling problems (PMSP): uniform, identical, and unrelated parallel machine scheduling problems (UPMSPs). Recently, UPMSPs with setup time had attracted more attention due to its applications in different industries and services. In this study, we present an efficient method to address the UPMSPs while using a modified harris hawks optimizer (HHO). The new method, called MHHO, uses the salp swarm algorithm (SSA) as a local search for HHO in order to enhance its performance and to decrease its computation time. To test the performance of MHHO, several experiments are implemented using small and large problem instances. Moreover, the proposed method is compared to several state-of-art approaches used for UPMSPs. The MHHO shows better performance in both small and large problem cases.

Necessary and Sufficient Conditions for Time Reversal Symmetry in Presence of Magnetic Fields

Time reversal invariance (TRI) of particles systems has many consequences, among which the celebrated Onsager reciprocal relations, a milestone in Statistical Mechanics dating back to 1931. Because for a long time it was believed that (TRI) dos not hold in presence of a magnetic field, a modification of such relations was proposed by Casimir in 1945. Only in the last decade, the strict traditional notion of reversibility that led to Casimir’s work has been questioned. It was then found that other symmetries can be used, which allow the Onsager reciprocal relations to hold without modification. In this paper we advance this investigation for classical Hamiltonian systems, substantially increasing the number of symmetries that yield TRI in presence of a magnetic field. We first deduce the most general form of a generalized time reversal operation on the phase space of such a system; secondly, we express sufficient conditions on the magnetic field which ensure TRI. Finally, we examine common examples from statistical mechanics and molecular dynamics. Our main result is that TRI holds in a much wider generality than previously believed, partially explaining why no experimental violation of Onsager relations has so far been reported.

Inaccessible entanglement in symmetry protected topological phases

We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G, which we call G-LOCC, and we define the inaccessible entanglement Einacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on Einacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely , where Dω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of Einacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.

Entanglement equipartition in critical random spin chains

Quantum many-body systems naturally present intricate entanglement patterns between the extensive number of degrees of freedom. In the presence of symmetry, the entanglement contribution of each symmetry sector gives a powerful lens through which to observe these patterns, both in theory and in experiment. It has been proposed that, in clean systems, an entanglement equipartition generically emerges, meaning that the different sectors contribute equally. This work shows that such equipartition is, not obviously, robust against disorder, at least in the random singlet phase of quantum spin chains. Furthermore, the authors identify the subleading term breaking the equipartition.

Variational integrators for stochastic dissipative Hamiltonian systems

Abstract Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange–d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange–d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange–d’Alembert Runge–Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behaviour compared to nongeometric methods. The Vlasov–Fokker–Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

Second Homotopy Group and Invariant Geometry of Flag Manifolds

We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these 2-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates how “rubber-band” topology can, in the presence of symmetry, single out very rigid objects. We characterize when these 2-spheres in the same homotopy class have the same geometry for all invariant metrics. This is done by exploring the action of Weyl group of the flag manifold, generalizing results of Patrao and San Martin (Indag. Math. 26:547–579, 2015) and de Siebenthal (Math. Helvetici 44(1):1–3, 1969). This illustrates how some aspects of “continuum” invariant geometry can, in the presence of symmetry, be reduced to the study of discrete objects. We remark that the topology singling out very rigid objects and the study of a continuum object being reduced to discrete ones is a characteristic of situations with a lot of symmetry and, thus, are recurring themes in Lie theory.

Mesoscopic Limit Cycles in Coupled Nanolasers.

Two coupled nanolasers exhibit a mode switching transition, theoretically described by mode beating limit cycle oscillations. Their decay rate is vanishingly small in the thermodynamic limit, i.e., when the spontaneous emission noise tends to zero. We provide experimental statistical evidence of mesoscopic limit cycles (∼10^{3} intracavity photons). Specifically, we show that the order parameter quantifying the limit cycle amplitude can be reconstructed from the mode intensity statistics. We observe a maximum of the averaged amplitude at the mode switching, accounting for limit cycle oscillations. We finally relate this maximum to a dip of mode cross-correlations, reaching a minimum of g_{ij}^{(2)}=2/3, which we show to be a mesoscopic limit. Coupled nanolasers are thus an appealing test bed for the investigation of spontaneous breaking of time translation symmetry in the presence of strong quantum fluctuations.