Building upon Dyson's fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V , a space of endomorphisms in End(V ⊕ V * ), a bilinear form on V ⊕ V * which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V * . Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.

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...

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Abstract:We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using the Kostant-Sternberg theorem, we show that separable states form a unique symplectic orbit, whereas orbits of entangled states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement. The above statements remain true for systems with an arbitrary number of components, moreover the presented method is general and can be applied also under different additional symmetry conditions stemming, e.g., from the indistinguishability of particles. We show how to calculate the degeneracy for various multiparticle systems providing also simple criteria of separability.

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