Symmetry Classes Research Articles

Multi-dimensional optimal system and conservation laws for Chaplygin gas Cargo-LeRoux model

The well-known Cargo-LeRoux model for the isentropic Chaplygin gas is investigated using classical Lie symmetry method. Optimal systems up to six dimensions are constructed using the adjoint transformation and the invariants of the admitted Lie algebras. We develop exact solutions to the Cargo-LeRoux model by using the one-dimensional optimal system and discussed the physical behavior of such solutions graphically. Conservation laws to the governing equations are also obtained by using the nonlinear self-adjointness of the model. The performance of the developed solutions is explored through the evolutionary behavior of a discontinuity wave.

Anyonic topological order in twisted equivariant differential (TED) K-theory

While the classification of noninteracting crystalline topological insulator phases by equivariant K-theory has become widely accepted, its generalization to anyonic interacting phases — hence to phases with topologically ordered ground states supporting topological braid quantum gates — has remained wide open. On the contrary, the success of K-theory with classifying noninteracting phases seems to have tacitly been perceived as precluding a K-theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-theory. Here, we provide a detailed argument for the classification of symmetry protected/enhanced [Formula: see text]-anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-theory of configuration spaces of points in the complement of nodal points inside the crystal’s Brillouin torus orbi-orientifold. We argue, in particular, that : (1) topological 2d semi-metal phases modulo global mass terms are classified by the flat differential twisted equivariant K-theory of the complement of the nodal points; (2) [Formula: see text]-electron interacting phases are classified by the K-theory of configuration spaces of [Formula: see text] points in the Brillouin torus; (3) the somewhat neglected twisting of equivariant K-theory by “inner local systems” reflects the effective “fictitious” gauge interaction of Chen, Wilczeck, Witten and Halperin (1989), which turns fermions into anyonic quanta; (4) the induced [Formula: see text]-anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations. A tight dictionary relates these arguments to those for classifying defect brane charges in string theory [H. Sati and U. Schreiber, Anyonic defect branes in TED-K-theory, arXiv:2203.11838], which we expect to be the images of momentum-space [Formula: see text]-anyons under a nonperturbative version of the AdS/CMT correspondence.

Entanglement spectrum and entropy in Floquet topological matter

Entanglement is one of the most fundamental features of quantum systems. In this work, we obtain the entanglement spectrum and entropy of Floquet noninteracting fermionic lattice models and build their connections with Floquet topological phases. Topological winding and Chern numbers are introduced to characterize the entanglement spectrum and eigenmodes. Correspondences between the spectrum and topology of entanglement Hamiltonians under periodic boundary conditions and topological edge states under open boundary conditions are further established. The theory is applied to Floquet topological insulators in different symmetry classes and spatial dimensions. Our work thus provides a useful framework for the study of rich entanglement patterns in Floquet topological matter.

Topological classification of higher-order topological phases with nested band inversion surfaces

Higher-order topological phases (HOTPs) hold gapped bulk bands and topological boundary states localized in boundaries with codimension higher than one. In this paper, we provide a unified construction and topological characterization of HOTPs for the full Altland-Zirnbauer tenfold symmetry classes, based on a method known as nested band inversion surfaces (BISs). Specifically, HOTPs built on this method are decomposed into a series of subsystems, and higher-order topological boundary states emerges from the interplay of their first-order topology. Our analysis begins with a general discussion of HOTPs in continuous Hamiltonians for each symmetry class, then moves on to several lattice examples illustrating the topological characterization based on the nested-BIS method. Despite the example minimal models possessing several spatial symmetries, our method does not rely on any spatial symmetry, and can be easily extended into arbitrary orders of topology in dimensions. Furthermore, we extend our discussion to systems with asymmetric boundary states induced by two different mechanisms, namely, crossed BISs that break a $\mathcal{C}_4$ rotation symmetry, and non-Clifford operators that break certain chiral-mirror symmetries of the minimal models.

Leptogenesis and dark matter through relativistic bubble walls with observable gravitational waves

We study a scenario where both dark matter and heavy right handed neutrino (RHN) responsible for leptogenesis acquire masses by crossing the relativistic bubble walls formed as a result of a TeV scale supercooled first order phase transition (FOPT). While this leads to a large out-of-equilibrium abundance of right handed neutrino inside the bubble sufficient to produce the required lepton asymmetry, the dark matter being lighter can still remain in equilibrium with its relic being set by subsequent thermal freeze-out. A classical conformal symmetry ensures the origin of mass via FOPT induced by a singlet scalar while also ensuring supercooling leading to enhanced gravitational wave amplitude within the sensitivity of the LISA experiment. A minimal scenario with three RHN, one inert scalar doublet and one singlet scalar as additional fields beyond the standard model is sufficient to realize this possibility which also favours inert RHN dark matter over inert scalar doublet.

Topological states of generalized dissipative Majorana wires

We study the generalized one-dimensional (1D) quantum dissipative models corresponding to a Majorana wire which can possess more than one Majorana bound state at each end. The system consists of a 1D fermionic open quantum system whose dynamics is governed by a quadratic Lindblad equation. Using the adjoint Lindblad equation for the fermionic two-point correlations, we find the gaps in the damping and purity spectra of a generic 1D model. Then, using the symmetry-based classification, we show that a winding number as the topological invariant can be defined which distinguishes different steady states of the system in the presence of damping and purity gaps. Then we focus on certain models with different Lindblad quantum jump terms and explore their phase diagrams by calculating the damping and the purity gaps as well as the winding number. In particular, we show that by inclusion of quantum jumps between next-nearest-neighbor sites, higher winding numbers and equivalently more Majorana bound states can be achieved. Also, by introducing imbalanced couplings we can switch between states with negative and positive winding numbers. Finally, we should mention that since our formulation is based on the fermionic correlations rather than the Majorana operators, it can be easily extended to the dissipative topological phases belonging to other symmetry classes.

Topological properties of subsystem-symmetry-protected edge states in an extended quasi-one-dimensional dimerized lattice

We investigate theoretically the topological properties of dimerized quasi-one-dimensional (1D) lattice comprising of multi legs $(L)$ as well as multi sublattices $(R)$. The system has main and subsidiary exchange symmetries. In the basis of latter one, the system can be divided into $L$ 1D subsystems each of which corresponds to a generalized $SSH_R$ model having $R$ sublattices and on-site potentials. Chiral symmetry is absent in all subsystems except when the axis of main exchange symmetry coincides on the central chain. We find that the system may host zero- and finite-energy topological edge states. The existence of zero-energy edge state requires a certain relation between the number of legs and sublattices. As such, different topological phases, protected by subsystem symmetry, including zero-energy edge states in the main gap, no zero-energy edge states, and zero-energy edge states in the bulk states are characterized. Despite the classification symmetry of the system belongs to $BDI$ but each subsystem falls in either $AI$ or $BDI$ symmetry class.

Nonclassical potential symmetry analysis and exact solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct exact solutions to a quasilinear system of hyperbolic partial differential equations which governs the dynamics of a thin film of a perfectly soluble anti-surfactant solution. The symmetry analysis for this system is performed first time in the literature to the best of our knowledge. In fact, we present a detailed and comprehensive study of symmetry analysis for the governing system and compute classical symmetries, nonclassical symmetries, nonlocal symmetries and nonclassical potential symmetries. Nonclassical potential symmetries appear to be very much useful in terms of obtaining several new hidden solutions of the system that can not be established by using classical symmetry reductions, nonclassical symmetry method, or potential symmetry analysis. By using the direct multipliers we demonstrate several conserved quantities of the governing system those yield associated nonlocally related potential systems to the given system. Further, we prove that the given system admits a nonlocal symmetry that arises from the symmetry-based method and consequently we obtain a family of exact solutions. We analyze the physical interpretation of some of the obtained solutions which include various soliton type solutions such as kink type solitons, breather type solitons, multiple solitons, singular kink type solitons and etc. Lastly, as an application, we analyze the evolutionary property of characteristic shock, weak discontinuity and collision between them by using one of the obtained solutions.

FRACTIONAL COMPLEX TRANSFORMS, REDUCED EQUATIONS AND EXACT SOLUTIONS OF THE FRACTIONAL KRAENKEL–MANNA–MERLE SYSTEM

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions.

Optimal system, invariant solutions and conservation laws of the hyperbolic geometry flow with time-dependent dissipation

Based on the classical Lie symmetry method, the hyperbolic geometry flow with time-dependent dissipation is discussed. In addition, we construct the optimal system of one-dimensional subalgebras, and invariant solutions of two different dissipation equations are obtained, which include traveling wave solutions. Finally, the conservation laws are obtained for geometry flow with different two dissipation term.

Symmetry methods for a hyperbolic model for a class of populations

We investigate a hyperbolic system that describe the dispersal dynamics of a population, introduced by Méndez and Camacho (1997)[1], in Lie symmetry classification perspective. A Lie group classification is provided for different forms of two constitutive functions: the propagation coefficient D(u) and the reaction term r(u). We establish the Lie symmetry determining equations by utilizing an equivalence generator and the projection theorem. Several extensions of principal Lie algebra are found for different forms of D(u) and r(u). By performing classical reductions we obtain several exact solutions.

Stress concentration around a circular rigid inclusion in incompressible nonlinearly elastic transversely isotropic and orthotropic thin sheets under uniaxial extension

In this paper, we study the stress concentration around a circular rigid inclusion in an incompressible anisotropic homogeneous nonlinearly elastic thin sheet subjected to in-plane finite deformation uniaxial extension. Two classes of material symmetry are considered: transverse isotropy and orthotropy. For the case of transverse isotropy, a three-parameter constitutive relation that under infinitesimal deformations reduces appropriately to the classical incompressible linearized elastic transversely isotropic constitutive relation is studied. Similarly, a six-parameter constitutive relation is considered for the case of orthotropic material symmetry. These constitutive relations may be suitable to describe the response of transversely isotropic and orthotropic bodies subjected to moderate deformations.

Radiatively generated fermion mass hierarchy from flavor nonuniversal gauge symmetries

A framework based on a class of abelian gauge symmetries is proposed in which the masses of only the third generation quarks and leptons arise at the tree level. The fermions of the first and second families receive their masses through radiative corrections induced by the new gauge bosons in the loops. It is shown that the class of abelian symmetries which can viably implement this mechanism are flavour non-universal in nature. Taking the all-fermion generalization of the well-known leptonic $L_\mu-L_\tau$ and $L_e - L_\mu$ symmetries, we construct an explicit renormalizable model based on two $U(1)$ which is shown to reproduce the observed fermion mass spectrum of the Standard Model. The first and second generation fermion masses are loop suppressed while the hierarchy between these two generations results from a gap between the masses of two vector bosons of the extended gauge symmetries. Several phenomenological aspects of the flavourful new physics are discussed and lower limits on the masses of the vector bosons are derived.

Correspondence of topological classification between quantum graph extra dimension and topological matter

In this paper, we study five-dimensional Dirac fermions of which extra-dimension is compactified on quantum graphs. We find that there is a non-trivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of non-interacting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in non-interacting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.

Quartic asymmetric exchange for two-dimensional ferromagnets with trigonal prismatic symmetry

We suggest a possible origin of noncollinear magnetic textures in ferromagnets (FMs) with the $D_{3h}$ point group symmetry. The suggested mechanism is different from the Dzyaloshinskii-Moriya interaction (DMI) and its straightforward generalizations. The considered symmetry class is important because a large fraction of all single-layer intrinsic FMs should belong to it. In particular, so does a monolayer Fe$_3$GeTe$_2$. At the same time, DMI vanishes identically in materials described by this point group, in the continuous limit. We use symmetry analysis to identify the only possible contribution to the free energy density in two dimensions that is of the fourth order with respect to the local magnetization direction and linear with respect to its spatial derivatives. This contribution predicts long-range conical magnetic spirals with both the average magnetization and the average chirality dependent on the spiral propagation direction. We relate the predicted spirals to a recent experiment on Fe$_3$GeTe$_2$. Finally, we demonstrate that, for easy-plane materials, the same mechanism may stabilize bimerons.

Spectral properties of disordered interacting non-Hermitian systems

Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two of the models considered are short-ranged and have different symmetries. The third model is long-ranged, whose hermitian counterpart has itself become a subject of growing interest. All these models exhibit a deep connection with the non-hermitian Random Matrix Theory of corresponding symmetry classes at relatively weak disorder. At relatively strong disorder, the models show the absence of complex eigenvalue correlation, thereby, corresponding to Poisson statistics. Our thorough analysis is expected to play a crucial role in understanding disordered open quantum systems in general.

Lie symmetry analysis for two-phase flow with mass transfer

Abstract We perform a complete symmetry classification for the hyperbolic system of partial differential equations, which describes a drift-flux two-phase flow in a one-dimensional pipe, with a mass-transfer term between the two different phases of the fluid. In addition, we consider the polytropic equation of states parameter and gravitational forces. For general values of the polytropic indices, we find that the fluid equations are invariant under the elements of a three-dimensional Lie algebra. However, additional Lie point symmetries follow for specific values of the polytropic indices. The one-dimensional systems are investigated in each case of the classification scheme, and the similarity transformations are calculated in order to reduce the fluid equations into a system of ordinary differential equations. Exact solutions are derived, while the reduced systems are studied numerically.

Mixture of the nearest- and next-nearest-neighbor d + id-wave pairings on the honeycomb lattice

Symmetry of superconducting pairing is one of the cardinal issues in superconductivity. In general, the nearest-neighbor and next-nearest-neighbor pairings cannot spontaneously mix with each other constrained by the distinct symmetry classification, for example the - and d xy -wave pairings in the square lattice. The honeycomb lattice provides a special platform to study the mixture of the nearest-neighbor and the next-nearest-neighbor pairing components due to its unique crystal structure. Here, using the variational quantum Monte Carlo simulations, we report a robust superconducting ground state with the intrinsic mixture of the nearest-neighbor and next-nearest-neighbor chiral d + id-wave pairings on the honeycomb lattice. We numerically identify that the two pairing channels promote each other. The proportion of the next-nearest-neighbor d + id component can be readily tuned by doping density and the short-range interactions. Our results provide the efficient ways to manipulate the real-space pairings.

Boosting to BMS

Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, are intrinsically associated to null manifolds and in two dimensions can be obtained as an Inönü-Wigner contraction of the two-dimensional (2d) relativistic conformal algebra. Instead of performing contractions, we demonstrate in this paper how this transmutation of symmetries can be achieved by infinite boosts or degenerate linear transformations on coordinates. Taking explicit cues from the worldsheet theory of null strings, we show boosting the system is equivalent to adding a current-current deformation term to the Hamiltonian. As the strength of this deformation term reaches a critical value, the classical symmetry algebra “flows” from two copies of Virasoro to the BMS algebra. We further explore the situation where the CFT coordinates are asymmetrically transformed, and degenerate limits lead to chiral theories.

Karyotype characterization of three important aromatic Cinnamomum L. species with special emphasis on reversible chromosome banding

Conventional stain (orcein) and DNA base-specific fluorochromes (CMA and DAPI) were used to characterize three medicinally and economically important aromatic Cinnamomum L. species namely, C. camphora, C. tamala and C. zeylanicum. Despite having identical somatic chromosome numbers 2n = 24, the studied Cinnamomum L. species had distinct centromeric formulae, such as 24m in C. camphora, 18m + 6sm in C. tamala, and 8m + 16sm in C. zeylanicum. When karyotype assessment criteria were taken into account, these three studied species also displayed distinctiveness, and C. zeylanicum represented a more heterogeneous karyotype than the other two species. The karyotypic diversification was reflected in the symmetry classes of Stebbins’ classification, with C. camphora and C. tamala having 1B karyotype while C. zeylanicum was found to have a 2A karyotype. CMA- and DAPI-banding patterns of these species were different and species-specific regarding the number, location and percentages of GC- and AT-rich repetitive sequences in the genome. Reversible fluorescent banding patterns revealed the stain-specific nature of satellite in C. camphora L. By comparing karyotypic traits and reversible fluorescent banding patterns, characterization was thereby encountered for three Cinnamomum L. species from Bangladesh.