Symmetry Classes Research Articles

Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise

It is known that knowledge of a symmetry of a scalar Ito stochastic differential equations leads, thanks to the Kozlov substitution, to its integration. In the present paper we provide a classification of scalar autonomous Ito stochastic differential equations with simple noise possessing symmetries; here "simple noise" means the noise coefficient is of the form $\s (x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to a standard form via a well known transformation; for such standard forms we also provide the integration of the symmetric equations. Our work extends previous classifications in that it also consider recently introduced types of symmetries, in particular standard random symmetries, not considered in those.

Emergence of Fermi's Golden Rule.

Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.

Generalized multifractality in the spin quantum Hall symmetry class with interaction

Scaling of various local observables with a system size at Anderson transition criticality is characterized by a generalized multifractality. We study the generalized multifractality in the spin quantum Hall symmetry class (class C) in the presence of interaction. We employ Finkel'stein nonlinear sigma model and construct the pure scaling derivativeless operators for class C in the presence of interaction. Within the two-loop renormalization group analysis we compute the anomalous dimensions of the pure scaling operators and demonstrate that they are affected by the interaction. We find that the interaction breaks exact symmetry relations between generalized multifractal exponents known for a noninteracting problem.

Universality classes of the Anderson transitions driven by quasiperiodic potential in the three-dimensional Wigner-Dyson symmetry classes

Quasiperiodic system is an intermediate state between periodic and disordered systems with unique delocalization-localization transition driven by the quasiperiodic potential (QP). One of the intriguing questions is whether the universality class of the Anderson transition (AT) driven by QP is similar to that of the AT driven by the random potential in the same symmetry class. Here, we study the critical behavior of the ATs driven by QP in the three-dimensional (3D) Anderson model, Peierls phase model, and Ando model, which belong to the Wigner-Dyson symmetry classes. The localization length and two-terminal conductance have been calculated by the transfer matrix method, and we argue that their error estimations in statistics suffer from the correlation of QP. With the correlation under control, the critical exponents $\nu$ of the ATs driven by QP are estimated by the finite size scaling analysis of conductance, which are consistent with $\nu$'s of the ATs driven by the random potential. Moreover, the critical conductance distribution and the level spacing ratio distribution have been studied. We also find that a convolutional neural network trained by the localized/delocalized wavefunctions in a disordered system predicts the localized/delocalized wavefunctions in quasiperiodic systems. Our numerical results strongly support that the universality classes of the ATs driven by QP and random potential are similar in the 3D Wigner-Dyson symmetry classes.

Threshold conditions for transversal modes of tunable plasmonic nanolasers shaped as single and twin graphene-covered circular quantum wires

We implement the lasing eigenvalue problem (LEP) approach to study the electromagnetic field in the presence of a circular quantum wire (QW) made of a gain material and wrapped in graphene cover and a dimer of two identical graphene-covered QWs, at the threshold of stationary emission. LEP delivers the mode-specific eigenvalue pairs, namely the frequencies and the threshold values of the QW gain index for the plasmon and the wire modes of such nanolasers. In our analysis, we use quantum Kubo formalism for the graphene conductivity and classical Maxwell boundary-value problem for the field functions. The technique involves the resistive boundary conditions, the separation of variables in the local coordinates, and, for the dimer, the addition theorem for the cylindrical functions. For single-wire plasmonic laser, we derive approximate engineering expressions for the lasing frequencies and threshold values of the gain index that complement the full-wave computations. For the dimer, we derive separate determinantal equations for four different classes of symmetry of the lasing supermodes and solve them numerically. Our investigation of the mode frequencies and thresholds versus the graphene and QW parameters shows that plasmon modes or, for the dimer, plasmon supermodes have lower frequencies and thresholds than the wire modes provided that the QW radius is smaller than 10 μm, however in thicker wires they are comparable. Only the plasmon-mode characteristics are well-tunable using the graphene chemical potential. In the dimer, all lasing supermodes form closely located quartets, however, they quickly approach the single-wire case if the inter-wire separation becomes comparable to the radius. These results open a way for building essentially single-mode plasmonic nanolasers and their arrays and suggest certain engineering rules for their design.

Magnetic shape fabric analysis from syntectonic granites: a study based on the eigenvalue method

AbstractWe investigate the shape and strength of the magnetic fabrics (anisotropy of magnetic susceptibility (AMS) data) of various massive granitic plutons from different parts of India, using the eigenvalue method. The study aims to analyse eigenvalues and establish their relationship with various deformational attributes. It involves: (1) calculating eigenvectors and their corresponding eigenvalues from magnetic fabric datasets; (2) finding a link between the geometrical appearance of eigenvectors and the mechanistic issues involved with a specific deformation scenario; and (3) determining shape and strength parameters from the magnetic foliation data distribution.The statistical analysis for the unimodal magnetic fabric dataset of orthorhombic symmetry class implies that the plane, consisting of intermediate (V2) and minimum (V3) eigenvectors with pole V1, accurately traces the instantaneous stretching axis (ISAmax) of a particular material flow system under a pure shear regime. Moreover, for the distributions of similar symmetry and modality, we infer that the rotational characteristics of eigenvectors with respect to a fixed coordinate cause a distinct shift of such planes (V2–V3) from the ISAmax of a steady-state flow system under simple shear, where a substantial amount of rotational strain is involved. However, our findings also suggest that variation in symmetry and modality of magnetic fabric data distribution of different studied granitoids can directly influence the relative disposition of V2–V3 with respect to the direction of ISAmax. We conclude that eigenvalue analysis of magnetic fabrics is a powerful approach, which can be utilized while studying the salient deformational aspects of any syntectonic massive granitic body.

Generalized multifractality at metal-insulator transitions and in metallic phases of two-dimensional disordered systems

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Anderson-localization transitions are considered. By using the nonlinear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signaling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related to a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.

Bound states in the continuum in waveguide arrays within a symmetry classification scheme.

We study a photonic implementation of a modified Fano-Anderson model - a waveguide array with two additional waveguides and by using the coupled mode theory we calculate its spectral and scattering properties. We classify eigenmodes according to vertical symmetry of the structure given by self-coupling coefficients of the additional waveguides and establish the conditions for bound states in the continuum (BIC) existence. The main predictions drawn from the theoretical model are verified by rigorous full-wave simulations of realistic structures. We use the Weierstrass factorization theorem and interpret the scattering spectra of the systems with broken symmetry in terms of the eigenmodes. The Fano resonance related with excitation of quasi-BIC is explained as arising from the interference between this mode and another leaky mode.

Lie Symmetry Classification and Qualitative Analysis for the Fourth-Order Schrödinger Equation

The Lie symmetry analysis for the study of a 1+n fourth-order Schrödinger equation inspired by the modification of the deformation algebra in the presence of a minimum length is applied. Specifically, we perform a detailed classification for the scalar field potential function where non-trivial Lie symmetries exist and simplify the Schrödinger equation. Then, a qualitative analysis allows for the reduced ordinary differential equation to be analysed to understand the asymptotic dynamics.

Periodic Clifford symmetry algebras on flux lattices

Real Clifford algebras play a fundamental role in the eight real Altland-Zirnbauer symmetry classes and the classification tables of topological phases. Here, we present another elegant realization of real Clifford algebras in the $d$-dimensional spinless rectangular lattices with $\pi$ flux per plaquette. Due to the $T$-invariant flux configuration, real Clifford algebras are realized as projective symmetry algebras of lattice symmetries. Remarkably, $d$ mod $8$ exactly corresponds to the eight Morita equivalence classes of real Clifford algebras with eightfold Bott periodicity, resembling the eight real Altland-Zirnbauer classes. The representation theory of Clifford algebras determines the degree of degeneracy of band structures, both at generic $k$ points and at high-symmetry points of the Brillouin zone. Particularly, we demonstrate that the large degeneracy at high-symmetry points offers a rich resource for forming novel topological states by various dimerization patterns, including a $3$D higher-order semimetal state with double-charged bulk nodal loops and hinge modes, a $4$D nodal surface semimetal with $3$D surface solid-ball zero modes, and $4$D M\"{o}bius topological insulators with a eightfold surface nodal point or a fourfold surface nodal ring. Our theory can be experimentally realized in artificial crystals by their engineerable $\mathbb{Z}_2$ gauge fields and capability to simulate higher dimensional systems.

Symmetry classification of plane deformations of hyperelastic compressible materials

The governing equations for plane deformations of isotropic compressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. For materials known as harmonic materials, only one solution of any generality is known. It is natural to ask whether the governing equations possess some special property for this particular material. In this paper, we show that the governing equations admit an extensive class of symmetries for these so-called harmonic materials and further show that there exists a second class of strain-energy functions, where the governing equations admit a general class of symmetries. We further show that for this strain-energy function, the governing equations are linearizable.

Local invariants identify topology in metals and gapless systems

Although topological band theory has been used to discover and classify a wide array of novel topological phases in insulating and semimetal systems, it is not well suited to identifying topological phenomena in metallic or gapless systems. Here, we develop a theory of topological metals based on the system's spectral localizer and associated Clifford pseudospectrum, which can both determine whether a system exhibits boundary-localized states despite the presence of degenerate bulk bands and provide a measure of these states' topological protection even in the absence of a bulk band gap. We demonstrate the generality of this method across symmetry classes in two lattice systems, a Chern metal and a higher-order topological metal, and prove the topology of these systems is robust to relatively strong perturbations. The ability to define invariants for metallic and gapless systems allows for the possibility of finding topological phenomena in a broad range of natural, photonic, and other artificial materials that could not be previously explored.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously.

Finite electro-elasticity with physics-augmented neural networks

In the present work, a machine learning based constitutive model for electro-mechanically coupled material behavior at finite deformations is proposed. Using different sets of invariants as inputs, an internal energy density is formulated as a convex neural network. In this way, the model fulfills the polyconvexity condition which ensures material stability, as well as thermodynamic consistency, objectivity, material symmetry, and growth conditions. Depending on the considered invariants, this physics-augmented machine learning model can either be applied for compressible or nearly incompressible material behavior, as well as for arbitrary material symmetry classes. The applicability and versatility of the approach is demonstrated by calibrating it on transversely isotropic data generated with an analytical potential, as well as for the effective constitutive modeling of an analytically homogenized, transversely isotropic rank-one laminate composite and a numerically homogenized cubic metamaterial. These examinations show the excellent generalization properties that physics-augmented neural networks offer also for multi-physical material modeling such as nonlinear electro-elasticity.

Effective theory of quantum black holes

We explore the quantum nature of black holes by introducing an effective framework that takes into account deviations from the classical results. The approach is based on introducing quantum corrections to the classical Schwarzschild geometry in a way that is consistent with the physical scales of the black hole and its classical symmetries. This is achieved by organizing the quantum corrections in inverse powers of a physical distance. By solving the system in a self-consistent way we show that the derived physical quantities, such as event horizons, temperature and entropy can be expressed in a well-defined expansion in the inverse powers of the black hole mass. The approach captures the general form of the quantum corrections to black hole physics without requiring us to commit to a specific model of quantum gravity.

Triple nodal points characterized by their nodal-line structure in all magnetic space groups

Triple nodal points are threefold degeneracies of energy bands in momentum space and therefore constitute intermediates between Dirac and Weyl points. Here, the authors study these band nodes in spinless band structures arising in any space group, possibly magnetic and nonsymmorphic, and derive a complete symmetry classification according to the nodal-line structure in the vicinity of the triple nodal point. This provides criteria for the stability of such band nodes and predicts the intricate nodal structures observed in data obtained by first-principles calculations for real materials.

Lie symmetry analysis, exact solutions and power series solutions of the logarithmic Monge–Ampère flow evolution equation

In this paper, the logarithmic Monge–Ampère flow evolution equation is studied by the classical Lie symmetry analysis method. First, the optimal system was obtained by the adjoint representation table. Second, by performing symmetric transformation on the optimal system, the corresponding ordinary differential equations are obtained, and the Jacobian elliptic function solution, the periodic solution and the power series solution are constructed. Finally, the dynamical behaviors of the solutions are described by choosing arbitrary parameters.

Distance to plane elasticity orthotropy by Euler–Lagrange method

Constitutive tensors are of common use in mechanics of materials. To determine the relevant symmetry class of an experimental tensor is still a tedious problem. For instance, it requires numerical methods in three-dimensional elasticity. We address here the more affordable case of plane (bi-dimensional) elasticity, which has not been fully solved yet. We recall first Vianello's orthogonal projection method, valid for both the isotropic and the square symmetric (tetragonal) symmetry classes. We then solve in a closed-form the problem of the distance to plane elasticity orthotropy, thanks to the Euler-Lagrange method.

Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

Axion anomalies

We study fermions derivatively coupled to axion-like or pseudoscalar fields, and show that the axial vector current of the fermions is not conserved in the limit where the fermion is massless. This apparent violation of the classical chiral symmetry is due to the background axion field. We compute the contributions to this anomalous Ward identity due to the pseudoscalar field alone, which arise in Minkowski space, as well as the effects due to an interaction with an external gravitational field. For the case of massless fermions, these interactions induce terms in the axion effective action that can be removed by the addition of local counterterms. We demonstrate that these counterterms are generated by the transformation of the path integral measure when transforming the theory from a form where the chiral symmetry is manifest to one where the symmetry is only apparent after using the classical equations of motion. We work perturbatively in Minkowski space and include the effects of interactions with a linearized gravitational field. Using the heat kernel method, we study the transformation properties of the path integral measure, and include the effects of non-linear gravity as well as interactions with gauge fields. Finally, we verify our relation by considering derivatively coupled fermions during pseudoscalar-driven inflation and computing the divergence of the axial current in de Sitter spacetime.