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Publication# Exact diagonalization of SU( N ) Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains using the full SU( N ) symmetry

Abstract

We present a method for the exact diagonalization of the SU(N) Heisenberg interaction Hamiltonian using Young tableaus to work directly in each irreducible representation of the global SU(N) group. This generalized scheme is applicable to chains consisting of several particles per site, with any SU(N) symmetry at each site. Extending some of the key results of substitutional analysis, we demonstrate how basis states can be efficiently constructed for the relevant SU(N) subsector, which, especially with increasing values of N or numbers of sites, has a much smaller dimension than the full Hilbert space. This allows us to analyze systems of larger sizes than can be handled by existing techniques. We apply this method to investigate the presence of edge states in SU(N) Heisenberg and Affleck-Kennedy-Lieb-Tasaki Hamiltonians.

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Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be

Werner Heisenberg

Werner Karl Heisenberg (ˈvɛʁnɐ kaʁl ˈhaɪzn̩bɛʁk; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechan

Frédéric Mila, Pierre Marcel Nataf

Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labeled by the set of 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 us 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).

2014Frédéric Mila, Pierre Marcel Nataf

Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg SU(N) Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of SU(N). The method, based on the use of standard Young tableaux, takes advantage of the full SU(N) symmetry, allowing one to work directly in each irreducible representation of the global SU(N) group. Since the SU(N) singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to N=10 and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [SU(2) with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the SU(N) level k Wess-Zumino-Witten universality class, where k is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the SU(N) level 1 WZW universality class.

2016