Kronecker product in terms of Hubbard operators and the Clebsch-Gordan decomposition of SU(2)xSU(2)

Abstract

We review the properties of the Kronecker (direct, or tensor) product of square matrices $A\otimes B\otimes C\cdots$ in terms of Hubbard operators. In its simplest form, a Hubbard operator $X_n^{i,j}$ can be expressed as the $n$-square matrix which has entry 1 in position $(i,j)$ and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch-Gordan decomposition of the product group $SU(2)\times SU(2)$. Finally, the $n$-dimensional irreducible representations so obtained are used to derive closed forms of the Clebsch-Gordan coefficients that rule the addition of angular momenta. Our results can be further developed in many different directions.

64 pages, no figures. Some references and two new sections have been included. The applications have been extended to the Hubbard and t-J models (Section 2.4) as well as to the diagonalization of the Hamiltonian in the Heisenberg model and to the solving of the Jaynes-Cummings problem in Hubbard notation (Section 3.4)

References in corpus (2)

• One-electron self energies and spectral functions for the t-J model in the large-N limit
• Decoherence and Dephasing in Kondo Tunneling through Double Quantum Dots

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