Methods Of Group Theory Research Articles

Symmetry considerations in exact diagonalization: spin-1/2 pyrochlore magnets

Abstract We describe how the methods of group theory (symmetry) are used to optimize the problem of exact diagonalization of a quantum system on a 16-site pyrochlore lattice. By analytically constructing a complete set of symmetrized states, the Hamiltonian can be completely block-diagonalized. As an example, we consider a spin-1/2 system with nearest-neighbour exchange interactions. We present the energy spectrum of a small system for a range of model parameters in the vicinity of the spin ice phase.

Anderson Localization Transition in Disordered Hyperbolic Lattices.

We study Anderson localization in disordered tight-binding models on hyperbolic lattices. Such lattices are geometries intermediate between ordinary two-dimensional crystalline lattices, which localize at infinitesimal disorder, and Bethe lattices, which localize at strong disorder. Using state-of-the-art computational group theory methods to create large systems, we approximate the thermodynamic limit through appropriate periodic boundary conditions and numerically demonstrate the existence of an Anderson localization transition on the {8,3} and {8,8} lattices. We find unusually large critical disorder strengths, determine critical exponents, and observe a strong finite-size effect in the level statistics.

Symmetry Determining Equations and Lie Groups of the Neutron Transport Equation

The Grigoriev-Meleshko Method, an indirect Lie group theory method, is used to derive the symmetry determining equations (SDEs) of the neutron transport equation (NTE) and the coupled delayed neutron precursor equations (DNPEs). A solution to the SDEs is a Lie group of transformations that can be used to reduce the order of the NTE and DNPEs or outright solve the equations. We found several solutions of the SDEs and worked through the mathematical algorithm to demonstrate relationships of instantiations of the NTE and its known solutions with the Lie groups. Examples of solutions include the Lie group that allows for the transformation of the differential form of the NTE to the integral form of the NTE; the Lie groups that permit Case’s solution; and the Lie group used to transform from the NTE to the α -eigenvalue form of the NTE.

Surface Magnetization in Antiferromagnets: Classification, Example Materials, and Relation to Magnetoelectric Responses

We use symmetry analysis and density-functional theory to determine and characterize surface terminations that have a finite equilibrium magnetization density in antiferromagnetic materials. A nonzero magnetic dipole moment per unit area or “surface magnetization” can arise on particular surfaces of many antiferromagnets due to the bulk magnetic symmetries. Such surface magnetization underlies intriguing physical phenomena like interfacial magnetic coupling and can be used as a readout method of antiferromagnetic domains. However, a universal description of antiferromagnetic surface magnetization is lacking. We first introduce a classification system based on whether the surface magnetization is either sensitive or robust to roughness and on whether the magnetic dipoles at surface of interest are compensated or uncompensated when the bulk magnetic order is retained at the surface. We show that roughness-sensitive categories can be identified by a simple extension of a previously established group-theory formalism for identifying roughness-robust surface magnetization. We then map the group-theory method of identifying surface magnetization to a novel description in terms of bulk magnetic multipoles, which are already established as symmetry indicators for bulk magnetoelectric responses at both linear and higher orders. We use density-functional calculations to illustrate that nominally compensated surfaces in magnetoelectric Cr2O3 and centrosymmetric altermagnetic FeF2 develop a finite magnetization density at the surface, in agreement with our predictions based on both group theory and the ordering of the bulk multipoles. Our analysis provides a comprehensive basis for understanding the surface magnetic properties and their intimate correspondence to bulk magnetoelectric effects in antiferromagnets and has important implications for technologically relevant phenomena such as exchange-bias coupling. Published by the American Physical Society 2024

Invariant subalgebras of von Neumann algebras arising from negatively curved groups

Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group Γ its von Neumann algebra L(Γ) satisfies the so-called ISR property: any von Neumann subalgebraN⊆L(Γ)that is normalized by all group elements in Γ is of the formN=L(Σ)for a normal subgroupΣ◁Γ. In particular, this applies to all groups Γ in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at least three generators, most graph product groups arising from simple graphs without visual splitting, etc. This result answers positively an open question of Amrutam and Jiang from [2].In the second part of the paper we obtain similar results for factors associated with groups that admit nontrivial (quasi)cohomology valued into various natural representations. In particular, we establish the ISR property for all icc, nonamenable groups that have positive first L2-Betti number and contain an infinite amenable subgroup.

Group Theory in Number Theory

The theory of groups exists in many fields of mathematics and has made a great impact on many fields of mathematics. In this article, this paper first introduces the history of group theory and elementary number theory, and then lists the definitions of group, ring, field and the most basic prime and integer and divisor in number theory that need to be used in this article. Then from the definitions, step by step, Euler's theorem, Bzout's lemma, Wilson's theorem and Fermat's Little theorem in elementary number theory are proved by means of definitions of group theory, cyclic groups, and even polynomials over domains. Finally, some concluding remarks are made. Many number theory theorems can be proved directly by the method of group theory without the action of tricks in number theory. Number theory is the thinking of certain special groups (e.g., (Z,+),(Z,)), so the methods of group theory work well inside number theory.

Some applications of group-theoretic Rips constructions to the classification of von Neumann algebras

In this paper we study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin in geometric group theory \cite{BO06}. Specifically, developing a new interplay between Popa's deformation/rigidity theory \cite{Po07} and geometric group theory methods we show that several algebraic features of these groups are completely recognizable from the von Neumann algebraic structure. In particular, we obtain new infinite families of pairwise non-isomorphic property (T) group factors thereby providing positive evidence towards Connes' Rigidity Conjecture. In addition, we use the Rips construction to build examples of property (T) II$_1$ factors which posses maximal von Neumann subalgebras without property (T) which answers a question raised in an earlier version of \cite{JS19} by Y. Jiang and A. Skalski.

Exact diagonalization for a 16-site spin-1/2 pyrochlore cluster

We find exact solutions to the Hamiltonian of a 16-site spin-1/2 pyrochlore cluster with nearest neighbour exchange interactions. The methods of group theory (symmetry) are used to completely block-diagonalize the Hamiltonian, yielding precise details about symmetry of the eigenstates, in particular those components which are spin ice states, in order to evaluate the spin ice density at finite temperature. At low enough temperatures, a ‘perturbed’ spin ice phase, where the ‘2-in-2-out’ ice rule is largely obeyed, is clearly outlined within the four parameter space of the general model of exchange interactions. The quantum spin ice phase is expected to exist within these boundaries.

Энергетический спектр и спектр оптического поглощения экзоэдрального фуллерена С-=SUB=-70-=/SUB=-Br-=SUB=-10-=/SUB=- в модели Хаббарда

Energy spectrum exohedral fullerene С70Br10 with the Cs symmetry groups have been obtained within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecule С70Br10 have been determined. On the basis of the energy spectrum of exohedral fullerene С70Br10 is given an interpretation of experimentally observed optical absorption bands molecule С70Br10.

Energy Spectrum and Optical Absorption Spectra of exohedral Fullerene C-=SUB=-70-=/SUB=-Br-=SUB=-10-=/SUB=- within the Hubbard Model

Energy spectrum exohedral fullerene C70Br10 with the Cs symmetry groups have been obtained within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecule C70Br10 have been determined. On the basis of the energy spectrum of exohedral fullerene C70Br10 is given an interpretation of experimentally observed optical absorption bands molecule C70Br10. Keywords: Hubbard model, Green's functions, energy spectrum, fullerenes, nanosystems, fullerene C70, exohedral fullerene C70Br10.

Энергетический спектр и оптические свойства бромида фуллерена С-=SUB=-70-=/SUB=-Br-=SUB=-10-=/SUB=- в модели Хаббарда

The energy spectrum of molecule C70Br10 of symmetry group Cs were obtained within the Hubbard model in the mean-field approximation. Using group theory methods, the classification of energy states was carried out, and the allowed transitions in the energy spectrum of bromide fullerene C70Br10 of symmetry group Cs were determined. On the basis of this spectrum, an interpretation of experimentally observed optical absorption bands of molecule C70Br10 is proposed.

Классификация геосистем: аксиоматический подход

Using the experience of solving classification problems in quantum mechanics and biology, the basic principles (axioms) that the natural classification of geosystems should satisfy are formulated. The science resolutes research problems from three independent positions: describing phenomena, highlighting fundamental laws and classifying objects. Classification is the relation structure of various types of objects (taxa), usually in the form of a graph. Geography classifies not territorial objects, but varieties of the environment (geomers) in which these objects manifest themselves to varying degrees. The classification theory is an intertheory that describes processes and phenomena in nature and society in a through way. Mathematical methods of feature space bundle, group theory and functional spaces are used to model the structure of typology and classification. The intertheory is created on the general theory of systems at various levels of formalization: from systems of quantum numbers, eigenvalues and variables to systems of geoinformation functions, probability estimation functionals, function transformation operators for describing states (taxonomic positions) and their quantum transitions. At the lower level of quantization, the axiomatization scheme assumes the existence of a universal facet space of series of integers of the classification code, restrictions on the length of the series and rules for changing the numbers of the code during quantum transitions. Triangular classification structures satisfying these requirements were identified at the systematics of the facies of the southern taiga of the Lower Angara region.

Applications of Group Theory

Groups play a fundamental role in Algebra: many algebraic structures, including rings, fields, and modules, can be seen as formed by adding new operations and axioms based on groups. Researchers often use group theory to explain many kinds of phenomena. In recent years, group theory has been introduced into crystallography to further explore the macroscopic symmetry of crystals from a mathematical point of view. In this paper, the applications of group theory in crystallography and magic cubic will be discussed. Basic definitions and models of these fields are demonstrated. A finite group is a group with a finite number of elements, which are the important contents of group theory. Besides, this paper proves that n−1 elements in a n order group can completely decide the nth element and gives a method of the nth element in a commutative group of order n. The analysis suggests that the research method of group theory has an important influence on other subjects.

Applications of a Group Theoretical Method on Biomagnetic Fluid Flow and Heat Transfer for Different Shapes of Fe3O4 Magnetic Particles under the Influence of Thermal Radiation and a Magnetic Dipole over a Cylinder

The flow and heat characteristics of an unsteady, laminar biomagnetic fluid, namely blood containing Fe3O4 magnetic particles, under the influence of thermal radiation and a magnetic dipole over a cylinder with controlled boundary conditions using a group theory method are investigated in the present study. The mathematical formulation of the problem is constructed with the aid of biomagnetic fluid dynamics (BFD) which combines principles of ferrohydrodynamics (FHD) and magnetohydrodynamics (MHD). It is assumed that blood exhibits polarization as well as electrical conductivity. Additionally, the shape of the magnetic particles, namely cylindrical and spherical, is also considered. Moreover, in this model, a group theoretical transformation, namely a two-parameter group technique, is applied. By applying this group transformation, the governing system of partial differential equations (PDEs) along with applicable boundary conditions are reduced to one independent variable and, consequently, converted into a system of ordinary differential equations (ODEs) with suitable boundary conditions. An efficient numerical technique is applied to solve the resultant ODEs and this technique is based on three essential features, namely (i) a common finite differences method with central differencing, (ii) tridiagonal matrix manipulation and (iii) an iterative procedure. The flow and heat characteristics of blood-Fe3O4 are found to be dependent on some physical parameters such as the particle volume fraction, the ferromagnetic interaction parameter, the magnetic field parameter, and the thermal radiation parameter. An ample parametric study is accomplished to narrate the influences of such physical parameters on velocity, temperature distributions as well as the coefficient of skin friction and rate of heat transfer. From the numerical results, it is deduced that the fluid velocity is enhanced for the ferromagnetic number and the temperature profile is decreased as the ferromagnetic number is gradually increased. It is also obtained that for the cylindrical shape of magnetic particles, the fluid temperature is more enhanced than that of the spherical shape. Both the skin friction coefficient and the local Nusselt number are increased for increasing values of the ferromagnetic interaction parameter, where the heat transfer rate of blood-Fe3O4 is significantly increased by approximately 33.2% compared to that of pure blood, whereas the coefficient of skin friction is reduced by approximately 6.82%.

Symmetry effects on the static and dynamic properties of coupled magnetic oscillators

The effect of symmetry on the resonance spectra of antiferromagnetically coupled oscillators has attracted new interest with the discovery of symmetry-breaking induced anti-crossings. Here, we experimentally characterise the resonance spectrum of a synthetic antiferromagnet Pt/CoFeB/Ru/CoFeB/Pt, where we are able to independently tune the effective magnetisation of the two coupled magnets. To model our results we apply the mathematical methods of group theory to the solutions of the Landau Lifshitz Gilbert equation. This general approach, usually applied to quantum mechanical systems, allows us to identify the main features of the resonance spectrum in terms of symmetry breaking and to make a direct comparison with crystal antiferromagnets.

Энергетический спектр и спектр оптического поглощения эндоэдральных фуллеренов Lu-=SUB=-3-=/SUB=-N@C-=SUB=-80-=/SUB=- и Y-=SUB=-3-=/SUB=-N@C-=SUB=-80-=/SUB=- в модели Хаббарда

Anticommutator Green’s functions and energy spectra of fullerene C80, endohedral fullerenes Lu3N@С80 and Y3N@С80 with the Ih symmetry groups have been obtained in an analytical form within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C80, Lu3N@С80 and Y3N@С80have been determined. On the basis of these spectra, an interpretation of experimentally observed optical absorption bands endohedral fullerenes Lu3N@С80 and Y3N@С80.

Energy Spectrum and Optical Absorption Spectraof endohedral Fullerenes Lu-=SUB=-3-=/SUB=-N\@C-=SUB=-80-=/SUB=- and Y-=SUB=-3-=/SUB=-N\@C-=SUB=-80-=/SUB=-within the Hubbard Model

Anticommutator Green's functions and energy spectra of fullerene C80,endohedral fullerenes Lu3N@C80 and Y3N@C80 with the Ih symmetry groups have been obtained in an analytical form within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C80, Lu3N@C80 andY3N@C80 have been determined. On the basis ofthese spectra, an interpretation of experimentally observed opticalabsorption bands endohedral fullerenes Lu3N@C80 andY3N@C80. Keywords: Hubbard model, Green's functions, energy spectrum,nanosystems, fullerene C80, endohedral fullerene Lu3N@C80, endohedral fullerene Y3N@C80.\

Energy Spectrum and Optical Absorption Spectrum of Fullerene C-=SUB=-26-=/SUB=- within the Hubbard Model

Anticommutator Green's functions and energy spectra of fullerene C26 and endohedral fullerene U@C26 with the D3h symmetry groups have been obtained in an analytical form within the Hubbard model in the approximation of static fluctuations. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C26 and U@C26 with the D3h symmetry groups have been determined. Keywords: Hubbard model, Green's functions, energy spectrum, nanosystems, fullerene C26.

Energy Spectrum and Optical Absorption Spectrum of exohedral Fullerene C-=SUB=-50-=/SUB=-Cl-=SUB=-10-=/SUB=- within the Hubbard Model

Energy spectra of fullerene C50 and exohedral fullerene C50Cl10 with the D5h symmetry groups have been obtained within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C50 and C50Cl10 have been determined. On the basis of the energy spectrum of exohedral fullerene C50Cl10 is given an interpretation of experimentally observed optical absorption bands molecule C50Cl10. Keywords: Hubbard model, Green's functions, energy spectrum, fullerenes, nanosystems, fullerene C50, exohedral fullerene C50Cl10.

Energy Spectrum and Optical Absorption Spectra of endohedral Fullerenes Lu-=SUB=-3-=/SUB=-N\@C-=SUB=-80-=/SUB=- and Y-=SUB=-3-=/SUB=-N\@C-=SUB=-80-=/SUB=- within the Hubbard Model

Anticommutator Green's functions and energy spectra of fullerene C80, endohedral fullerenes Lu3N@C80 and Y3N@C80 with the Ih symmetry groups have been obtained in an analytical form within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C80, Lu3N@C80 and Y3N@C80 have been determined. On the basis of these spectra, an interpretation of experimentally observed optical absorption bands endohedral fullerenes Lu3N@C80 and Y3N@C80. Keywords: Hubbard model, Green's functions, energy spectrum, nanosystems, fullerene C80, endohedral fullerene Lu3N@C80, endohedral fullerene Y3N@C80.