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.
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 first show that, for the SU(N) Heisenberg model with isotropic two-site interactions on an arbitrary lattice of Ns sites, the full spectrum in its singlet sector (when N divides Ns) is the negative of the full spectrum in its singlet sector of the SU(Ns/N) Heisenberg model defined on the same lattice. This theorem, valid for models with the fundamental irreducible representation (irrep) at each lattice site, follows naturally from a transposition operation in the basis of Standard Young Tableaux (SYTs). Second, this basis -- previously recognized as particularly convenient for exact diagonalization (ED) of SU(N) Heisenberg models -- admits an efficient, memory-saving implementation, thereby enabling large-scale diagonalizations for unprecedented system sizes Ns and numbers of colors N>3. We apply these ideas to an ED study of the SU(4) and SU(7) Heisenberg models on the triangular lattice (TL), extending the calculations up to Ns=28 sites. In particular, starting from the SU(4) case, whose nearest-neighbor ground state remains debated, we confirm that next-nearest-neighbor couplings stabilize four-sublattice long-range order. We further find that increasing the number of colors from N=4 to N=7 provides another route toward a Néel-like ground state, even without such additional couplings. N = 4 to N = 7 provides another route toward a Néel-like ground state, even without such additional couplings.
We show how to perform exact diagonalization of SU(
N)-symmetric Bose-Hubbard models separately in each irreducible representation of SU(
N), by exploiting the Howe duality between U(
L) and U(
N). Three structural consequences follow for any lattice and any Hamiltonian built from the U(
L) generators: (i) the spectrum in a given orbital sector is independent of
N and of the particle statistics, (ii) the canonical partition functions for bosons and fermions obey the exact identity
Zb(−N)=(−1)NbZf(N), (iii) with unfrustrated hopping the groundstate is always a color ferromagnet, so that the color-resolved method we introduce is useful to study excited sectors, frustrated lattices and finite temperatures. As applications, we first derive the finite-temperature phase diagram of attractive SU(
N) bosons on few-site rings at large
N for arbitrary filling. We then show the emergence of singlet-cluster states from frustrated hopping, the counterpart of fermionic ferromagnetism, with the example of the sawtooth and kagome lattice.
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.