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 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 analyze the antiferromagnetic
SU(3) Heisenberg chain by means of the density-matrix renormalization group. The results confirm that the model is critical and the computation of its central charge and the scaling dimensions of the first-excited states show that the underlying low-energy conformal field theory is the
SU(3)1 Wess-Zumino-Novikov-Witten model.
We compute the dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced by Dalla Piazza et al. [Nat. Phys. 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the $S(k,\ensuremath{\omega})$ to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)
1 behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the
SU(N) Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at $k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3$.
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.