We present an algorithm for the explicit numerical calculation of SU(N ) and SL(N, C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2). I. INTRODUCTIONClebsch-Gordan coefficients (CGCs) arise when decomposing the tensor product V S ⊗ V S of the representation spaces of two irreducible representations (irreps) S and S of some group into a direct sum V S 1 ⊕ · · · ⊕ V S r of irreducible representation spaces. They describe the corresponding basis transformation from a tensor product basis {|M ⊗ M } to a basis {|M } which explicitly accomplishes this decomposition.CGCs are familiar to physicists in the context of angular momentum coupling, in which the direct product of two irreps of the SU(2) group is decomposed into a direct sum of irreps. SU(3) Clebsch-Gordan coefficients arise, for example, in the context of quantum chromodynamics, while SU(N ) CGCs, for general N , appear in the construction of unifying theories whose symmetries contain the SU(3) × SU(2) × U (1) standard model as a subgroup 1 . SU(N ) CGCs are also useful for the numerical treatment of models with SU(N ) symmetry, where they arise when exploiting the Wigner-Eckart theorem to simplify the calculation of matrix elements of the Hamiltonian. Such a situation arises, for example, in the numerical treatment of SU(N )-symmetric quantum impurity models using the numerical renormalization group 2 . Such models can be mapped onto SU(N )-symmetric, half-infinite quantum chains, with hopping strengths that decrease exponentially along the chain. The Hamiltonian is diagonalized numerically in an iterative fashion, requiring the explicit calculation of matrix elements of the Hamiltonian of subchains of increasing length. The efficiency of this process can be increased dramatically by exploiting the Wigner-Eckart theorem, which requires knowledge of the relevant Clebsch-Gordan coefficients. (Details of how to implement SU(N ) symmetries within the context of the numerical renormalization group will be published elsewhere.) Similarly, tremendous gains in efficiency would result from developing SU(N )-symmetric implementations of the density matrix renormalization group for treating generic quantum chain models 3,4 , or generalizations of this approach for treating two-dimensional tensor network models 5 .For explicit calculations with models having SU(N ) symmetry, explicit tables of SU(N ) Clebsch-Gordan coefficients are needed. Their calculation is a problem of applied representation theory of Lie groups that has been solved, in principle, long ago 6-10 . For example, for SU(2) Racah 11 has found an explicit formula that gives the CGCs for the direct product decomposition of two arbitrary irreps S and S . For SU(N ), explicit CGC formulas exist for certain special cases, e.g. where S is the defining representation 12-14 . Moreover, symbolic packages su...

We report on the direct observation of the transmission phase shift through a Kondo correlated quantum dot by employing a new type of two-path interferometer. We observed a clear π/2-phase shift, which persists up to the Kondo temperature TK. Above this temperature, the phase shifts by more than π/2 at each Coulomb peak, approaching the behavior observed for the standard Coulomb blockade regime. These observations are in remarkable agreement with two-level numerical renormalization group calculations. The unique combination of experimental and theoretical results presented here fully elucidates the phase evolution in the Kondo regime.

We show how the density-matrix numerical renormalization group method can be used in combination with nonAbelian symmetries such as SU(N ). The decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.