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.
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 investigated the properties of the
SU(N) Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with
m<N particles per site. These models have been studied for arbitrary
N and
m with non-Abelian bosonization [I. Affleck, Nucl. Phys. B 265, 409 (1986); I. Affleck, Nucl. Phys. B 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with $N\ensuremath{\le}10$ and $m\ensuremath{\le}N/2$, and we have shown that the opening of a gap is associated to a spontaneous dimerization or trimerization depending on the value of
m and
N. We have also investigated the marginal cases where Abelian bosonization did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents consistent with conformal field theory.
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 have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU(
N) Fermi-Hubbard chain, fully leveraging the SU(
N) symmetry of the problem. This method extends a previously developed approach for the SU(
N) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with
N compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of
U, i.e., the atom-atom interaction amplitude, up to
N=6 for filling
1/N (one particle per site in average), and for one atom (respectively, hole) away from filling
1/N. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value
Uc (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e.,
Uc>0) and to increase with
N. Central charges
c are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the
SU(N)1 Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and
c=N in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.
Motivated by recent experimental progress in the context of ultra-cold multi-colour fermionic atoms in optical lattices, this thesis investigates the properties of the antiferromagnetic SU(N) Heisenberg models with fully antisymmetric irreducible representations labelled by a Young tableau with
m boxes on each site of various lattices. Thanks to an intensive use of variational Monte Carlo method based on Gutzwiller projected fermionic wave functions, we aim to determine the ground state and its related properties of systems which are beyond the capabilities of other numerical methods. This typically means that we will study large lattices with many particles per site for
N>2. The first part of this thesis will introduce the basic theoretical tools connected to the SU(N) Heisenberg model. A very short review of group theory introducing Young tableaux, will clarify the concept of fully antisymmetric irreducible representation of the SU(N) Lie algebra. Some properties of the Heisenberg model will be discussed and a mean-field solution will be given. The numerical methods used during this thesis and their limitations wil also be introduced. We will investigate the properties of 1-dimensional systems in the second part of this work. In particular, the chains have been studied for arbitrary
N and
m with non-abelian bosonisation leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. We have been able to verify these predictions for a representative number of cases with
N≤10 and
m≤N/2, and we have shown that the opening of a gap is associated with a spontaneous dimerisation or trimerisation depending on the value of
m and
N. We have also investigated the marginal cases, where non-abelian bosonisation did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents that are consistent with conformal field theory. The other 1-dimensional system that has been studied is the ladder for which field-theory gives some predictions. For instance, the ground state of the SU(4) case in the fundamental irreducible representation is believed to consist of a 4-site plaquette state for any non-zero inter-leg coupling. We have confirmed the presence of this state, even if in the weak coupling regime, the optimal variational state is gapless. The study has been extended to various
N and
m values providing interesting results. The last part will present the results for 2-dimensional lattices. The square lattice will first be investigated. A mean-field analysis predicts a plaquette ground state for
3≤N/m<5 and of a chiral ground state for
N/m≥5. By systematically studying the SU(3m) and SU(6m) Heisenberg model, we determined that the mean-field solution is valid even at small
m (
m>1 for SU(3m) and
m>2 for SU(6m)). Then, for the SU(6m) Heisenberg model with
m particles per site on a honeycomb lattice, we make the connection between the
m=1 case, for which the ground state has recently been shown to be in a plaquette singlet state, and the
m→∞ limit, where a mean-field approach has established that the ground state has chiral order. We were able to show that this system has a clear tendency toward the chiral order as soon as
m≥2. This demonstrates that the chiral phase can be stabilised for not too large values of
m, opening the way to its experimental realisations in other lattices. We will finally present results for the SU(3m) on the triangular lattice for which we find a plaquette state, when
m>1.