Bases For Irreducible Representations Research Articles

Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries.

Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parametrization of a quantum wave function with non-abelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the non-abelian symmetries of the system. We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-1/2 and spin-1degrees of freedom, and obtain a substantially higher accuracy than when using the s_{z} basis as an input to the neural network. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model. We also generalize the framework to the case of anyonic spin chains.

Essential signatures and canonical bases of irreducible representations of the group G 2

We consider representations of simple Lie algebras and the problem of constructing a “canonical” weight basis in an arbitrary irreducible finite-dimensional highest-weight module. Vinberg suggested a method for constructing such bases by applying the lowering operators corresponding to all negative roots to the highest-weight vector and put forward a number of conjectures on the parametrization and structure of such bases. It follows from papers by Feigin, Fourier, and Littelmann that these conjectures are true for the cases of A n and C n . In the present paper, we prove these conjectures for the case of G 2 by using a different approach suggested by Vinberg.

THE TETRAHEDRAL–OCTAHEDRAL BASES FOR THE GENERALIZED ROTOR

The rotational basis for irreducible representations of the point groups T d, O and O h for nuclear angular momenta J = 0, 1…, 5 is constructed.

An atom-photon cluster localized in a microcavity

It is shown that our description of the dynamics of interaction between atoms with microcavity mode quanta and an external electromagnetic field under conditions of two-photon resonance follows from a simple theory of one-photon resonance interaction between two-level atoms and the same external electromagnetic field, if we substitute su(2) algebra elements of the one-photon resonance theory for third-order elements of polynomial algebra. Commutation relations of polynomial algebra determine the features of processes with the participation of a single object, i.e. an atom-photon cluster localized in a microcavity. Its states are the basis for irreducible representations of this polynomial algebra.

Effect of time reversal symmetry on states of quasiparticles in crystals

The problem of additional time reversal (TR) degeneracy and the transformation properties of ‘‘time reversed' wave functions are investigated. In crystals the one electron Hamiltonian is invariant with respect to a space group of a crystal. The Bloch functions are the basis for irreducible representations (irrp's) of the wave vector groups k. When an irrp is complex the TR symmetry must be considered. Using Herring's criterion adopted to space groups we have investigated several hexagonal materials for optoelectronic devices such as ZnO, GaN, 6H-SiC. We have found many complex irrps. We discuss the effect of the TR on vibrational and electronic states together with optical selection rules in ZnO and GaN. In addition we list all complex irrps of the 32 crystallographic double point groups. Spin is taken into account throughout the whole discussion.

Conversion equations and domain states in ferroic phase transitions

Tensorial covariants whose components are linear combinations of the cartesian tensor components which transform like bases for irreducible representations of parent groups facilitate determination of tensorial parameters that onset at ferroic phase transitions. Conversion equations express cartesian tensor components in terms of these parameters. In conjunction with existing tables of fine domain structures these equations enable a complete analysis of tensor properties of domain states in each ferroic phase transition. This includes the tensor distinction of pairs of domain states from which to continue to considerations of tensorial properties when crossing a domain wall is quite straightforward.

The embeddingU'q(so3)⊂Uq(sl3)

We study the embedding U'q(so3)?Uq(sl3), where Uq(sl3) is a well known Drinfeld-Jimbo quantum algebra and the algebra U'q(so3) is the cyclically symmetric q-deformation of the universal enveloping algebra U(so3) of the Lie algebra so3 which is not a Drinfeld-Jimbo quantum algebra. Finite-dimensional irreducible representations of Uq(sl3) are decomposed into irreducible representations of U'q(so3). An explicit expression for the matrix of the transition from the Gel'fand-Tsetlin basis for Uq(sl3) to the bases of irreducible representations of U'q(so3) is calculated for representations of Uq(sl3) with highest weights (l,0,0). Entries of this matrix are expressed in terms of products of dual q-Krawtchouk polynomials and dual q-Hahn polynomials. Expressions for representation operators of Uq(sl3) in the U'q(so3) basis are given.

Canonical Bases for Irreducible Representations of Quantum GL n , II

Canonical Bases for Irreducible Representations of Quantum GL_<i>n</i> , II

Basis of irreducible representations of orthogonal groups

A method for constructing a basis of irreducible representations of unitary groupsUn is generalized to orthogonal groupsOn. The corresponding algorithm is given for the chain for groupsUn ⊃On and also for the chain which is of great relevance in physics.

Matrix representation of the generators of symplectic algebras. I. The case of sp(4,R)

Because of their many physical applications, there has been considerable interest lately in the basis for irreducible representations (irreps) of symplectic groups and in the matrix elements of the generators of these groups with respect to this basis. In the present paper the authors carry out this programme for the irreps in the positive discrete series of sp(4,R) by building up this basis from powers of the raising generators applied to the lowest weight state. By using the Dyson boson realisation of the generators, their matrix representation can be obtained by direct differentiation, while the overlap of the states of the basis can be determined with the help of coherent states. The extension of the analysis to symplectic groups of arbitrary dimension is straightforward and will be implemented explicitly for sp(6,R) in a later paper.

Basis states for equivalent electrons. II. States for the Lie group Sp(4l + 2)

The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of the Lie group U(4l + 2). States which are eigenfunctions of the total spin and total orbital angular momentum form the basis for irreducible representations of SO(3) × SU(2). In this paper the intermediate group Sp(4l + 2) is studied. The basis states for irreducible representations of Sp(4l + 2) are expressed in terms of the Slater basis states.

Gradient invariants and incommensurate phase transitions

Gradient invariants which couple the bases for irreducible representations are tabulated for all magnetic crystal point groups. These invariants are necessary for the construction of thermodynamic potentials used in the phenomenological description of incommensurate phase transitions. As an example, the description is given of such a transition in a ferroelectric crystal.

Three applications of the Clebsch-Gordan reduction in phenomenological crystal physics

The use of a method based on the Clebsch-Gordan reduction in terms of typical variables is described for the solution of three important problems of crystal physics and of the structural phase transition theory: 1. decomposition of any tensor into bases of irreducible representations of symmetry group, 2. construction of thermodynamic potential of given symmetry in various variables, 3. construction of gradient invariants which are used in the theory of incommensurate structures. The method is elucidated on an example of crystallographic point groupD4 and on the family of its isomorphs. On the basis of a table of Clebsch-Gordan products for the groupD4 there are given: 1. tables of tensorial bases up to fourth rank tensors for the groupsD4,C4v, andD2d, 2. the so-called “extended integrity basis” for the groupD4 in components of polarization and of strain tensor, 3. the so-called “typical extended integrity basis” for the isomorphic familyD4 On the basis of which any extended integrity basis for groups of the familyD4 and in any set of variables can be deduced, 4. the so-called Lifshits-type invariants and quadratic gradient invariants for all magnetic groups of the Laue classD4.

Electron-muon puzzle and the electromagnetic coupling constant

On the basis of a heuristic model we argued in an earlier paper (paper C of this series) electric field (and of course the magnetic field, too) of a lepton or of a quark may be formulated in terms of a closed loop of quantized magnetic flux whose alternative forms (''loopforms'') are superposed with probability amplitudes so as to represent the electromagnetic field of that lepton or quark. The Zitterbewegung of a single stationary (''elementary'') particle suggests a kind of quasiextension, which is assumed, in the present theory, to permit concepts of structuralization of the electromagnetic field even for leptons. Mesons and baryons may be represented by linked quantized flux loops, i.e., quark loops (as in paper B). The central problem now (in this paper D) is to formulate those probability-amplitude distributions in terms of wave functions to characterize the internal structure of the lepton or quark in question. As probability-amplitude functions one may choose bases of irreducible representations of the group with respect to which the model is to be invariant. It is seen that this implies the SO(4) group. As both the electron-muon mass ratio and the electromagnetic coupling constant depend, in this flux-quantization model, on the correctmore » formulation of the structuralization of probability-amplitude distributions, we should expect to get an insight into both these puzzles from finding the right probability-amplitude wave functions. Furthermore, it is seen that this same structuralization of probability-amplitude distributions also permits one to estimate the rate of weak interactions, thus relating them to electromagnetic interactions.« less

Invariant functions and homogeneous bases of irreducible representations of the crystal point groups, with applications to thermodynamic properties of crystals under strain

A general method is described for deriving the integrity bases of finite transformation groups. A generalization of an integrity basis which determines the homogeneous bases of irreducible representations of such groups is also given. It is shown that such bases can be expressed as linear combinations of a finite number of these bases with coefficients which are homogeneous polynomials that are invariant to the Coxeter group containing the group in question. Tables are given for the integrity bases for polynomial functions of a symmetric second-order tensor which are invariant to the crystal point groups. It is shown how these results may be applied to the finite strain tensor and hence to the thermodynamics of anisotropic crystals.

The Dimensions of Irreducible Representations of Linear Groups

The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensorsforms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformationis associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].

Orbital fractional parentage coefficients for the harmonic oscillator shell model

The Hamiltonian ofn particles moving in a common harmonic oscillator potential has as its symmetry group the unitary groupU(3n) in 3n dimensions,n particle states of the harmonic oscillator shell model can be characterized as bases of irreducible representations (BIR) of the groupU(3n) and of certain subgroups of this group. Use is made of these subgroups for the factorization and calculation of 2, 3, and 4 particle fractional parentage coefficients (fpc) of the harmonic oscillator shell model. Recoupling coefficients for subgroup chains of the symmetric groupS (n) appear as factors in the fpc. These coefficients are analyzed and calculated explicitly. The 2, 3 and 4 particle fpc of the 1s 1p shell configuration are obtained as products of these recoupling coefficients with known reduced Wigner coefficients of the unitary groupU(3) in 3 dimensions. Possible applications are indicated.

POLYNOMIAL BASES OF GROUP SU&lt;sub&gt;3&lt;/sub&gt; AND AN EXPLICIT EXPRESSION FOR ITS CLEBSCH-GORDAN COEFFICIENTS

Using the properties of the Young diagrams, we have constructed a complete set of polynomial bases for representing the SUn group. The basis of irreducible representations in the product space of two irreducible representations is then discussed. Finally by methods of construction of an invariant, an explicit expression for the Clebsch-Gordan coefficients of group SU3 is obtained.

Lattice Theory of Spin Configuration

In the frame of a given crystallographic symmetry a matrix ζ(k) is constructed, the eigenvectors and eigenvalues of which are directly related to the spin configuration and magnetic exchange energy, respectively. For Bravais lattices the matrix equation reduces to Villain's equation. When chemical and magnetic unit cells coincide, the eigenvectors of ζ(0) are shown to be identical to the ``basis of irreducible representations'' used by Turov and Dzialoshinski in the construction of an invariant Hamiltonian. The usefulness of both methods is discussed. The matrix theory is more general in the sense that it remains valid for magnetic cells different from the chemical one. Although the theory presented here starts from conventional crystallographic symmetry, it contains all the configurations possible in the so-called magnetic groups and even those not contained there (helical configurations). The theory includes isotropic as well as anisotropic (crystalline field, dipolar, pseudodipolar, antisymmetric) coupling in the order 2 approximation. Various examples of magnetic structures in the fields of perovskites, ilmenites, spinels, garnets, etc., are presented.