Obtaining eigenvalues of permutations acting on the product space of N representations of SU( n ) usually involves either diagonalizing their representation matrices on total-weight subspaces or decomposing their characters, which can be obtained from Frobenius' formula or via graphical methods using Young tableaux (YT). For products of fundamental representations of SU( n ), Schuricht and one of us proposed the method of extended YT (eYT), which allows reading the eigenvalues of the cyclic permutation C N directly off the, slightly modified, standard YT labelling an irreducible SU( n ) representation. Here we generalize the method to all symmetric representations of SU( n ), and show that C N eigenvalue computation based on eYT is at least linearly faster than the standard methods mentioned.
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).
We develop an efficient method to perform density matrix renormalization group simulations of the SU(
N) Heisenberg chain with open boundary conditions taking full advantage of the SU(
N) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to
N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with
N, but it is still satisfactory for
N=8 (six digits). Central charges
c are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the
SU(N)1 Wess-Zumino-Witten conformal field theories.
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.
We show how to perform exact diagonalizations of SU(N) Fermi-Hubbard models on L-site clusters separately in each irreducible representation (irrep) of SU(N). Using the representation theory of the unitary group U(L), we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of semistandard Young tableaux (or, equivalently, the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some SU(N) phases predicted in the Heisenberg limit upon decreasing the on-site interaction U on various lattices of size L≤12 and for 2≤N≤6. In particular, we show that a long-range color ordered phase emerges for intermediate U for N=4 at filling 1/4 on the triangular lattice.
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.