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Register a new userA complementary group technique for a resolution of the outer multiplicity problem of SU(n): (I) Littlewood rule and a complementary group of SU(n)
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A complementary group to SU(n) is found that realizes all features of the Littlewood rule for Kronecker products of SU(n) representations. This is accomplished by considering a state of SU(n) to be a special Gelfand state of the complementary group {cal U}(2n-2). The labels of {cal U}(2n-2) can be used as the outer multiplicity labels needed to distinguish multiple occurrences of irreducible representations (irreps) in the SU(n)times SU(n)downarrow SU(n) decomposition that is obtained from the Littlewood rule. Furthermore, this realization can be used to determine SU(n)supset SU(n-1)times U(1) Reduced Wigner Coefficients (RWCs) and Clebsch-Gordan Coefficients (CGCs) of SU(n), using algebraic or numeric methods, in either the canonical or a noncanonical basis. The method is recursive in that it uses simpler RWCs or CGCs with one symmetric irrep in conjunction with standard recoupling procedures. New explicit formulae for the multiplicity for SU(3) and SU(4) are used to illustrate the theory.
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Feng Pan
, J. P. Draayer (Department of Physics & Astronomy
, n Louisiana State University
1997
A general procedure for the derivation of SU(3)supset U(2) reduced Wigner coefficients for the coupling (lambda_{1}mu_{1})times (lambda_{2}mu_{2})downarrow (lambdamu)^{eta}, where eta is the outer multiplicity label needed in the decomposition, is proposed based on a recoupling approach according to the complementary group technique given in (I). It is proved that the non-multiplicity-free reduced Wigner coefficients of SU(n) are not unique with respect to canonical outer multiplicity labels, and can be transformed from one set of outer multiplicity labels to another. The transformation matrices are elements of SO(m), where m is the number of occurrence of the corresponding irrep (lambdamu) in the decomposition (lambda_{1}mu_{1})times (lambda_{2}mu_{2})downarrow (lambdamu). Thus, a kind of the reduced Wigner coefficients with multiplicity is obtained after a special SO(m) transformation. New features of this kind of reduced Wigner coefficients and the differences from the reduced Wigner coefficients with other choice of the multiplicity label given previously are discussed. The method can also be applied to the derivation of general SU(n) Wigner or reduced Wigner coefficients with multiplicity. Algebraic expression of another kind of reduced Wigner coefficients, the so-called reduced auxiliary Wigner coefficients for SU(3)supset U(2), are also obtained.
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.
We propose a lattice model for two-dimensional SU(N) N=(2,2) super Yang-Mills model. We start from the CKKU model for this system, which is valid only for U(N) gauge group. We give a reduction of U(1) part keeping a part of supersymmetry. In order to suppress artifact vacua, we use an admissibility condition.
The resonant eigenmodes of a nitrogen-implanted iron {alpha}-FeN characterized by weak stripe domains are investigated by Brillouin light scattering and broadband ferromagnetic resonance experiments, assisted by micromagnetic simulations. The spectrum of the dynamic eigenmodes in the presence of the weak stripes is very rich and two different families of modes can be selectively detected using different techniques or different experimental configurations. Attention is paid to the evolution of the mode frequencies and spatial profiles under the application of an external magnetic field, of variable intensity, in the direction parallel or transverse to the stripes. The different evolution of the modes with the external magnetic field is accompanied by a distinctive spatial localization in specific regions, such as the closure domains at the surface of the stripes and the bulk domains localized in the inner part of the stripes. The complementarity of BLS and FMR techniques, based on different selection rules, is found to be a fruitful tool for the study of the wealth of localized mag-netic excitations generally found in nanostructures.
Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labelled by the set of {it standard Young tableaux} in which the matrix of the Heisenberg SU(N) model (the quantum permutation of N-color objects) takes an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of the singlet space on $n$ sites increases very fast with N, this formulation allows to extend exact diagonalizations of finite clusters to much larger values of N than accessible so far. Using this method, we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid for SU(10).
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