The extension of the linear flavor-wave theory to fully antisymmetric irreducible representations (irreps) of
SU(N) is presented in order to investigate the color order of
SU(N) antiferromagnetic Heisenberg models in several two-dimensional geometries. The square, triangular, and honeycomb lattices are considered with
m fermionic particles per site. We present two different methods: the first method is the generalization of the multiboson spin-wave approach to
SU(N) which consists of associating a Schwinger boson to each state on a site. The second method adopts the Read and Sachdev bosons which are an extension of the Schwinger bosons that introduces one boson for each color and each line of the Young tableau. The two methods yield the same dispersing modes, a good indication that they properly capture the semiclassical fluctuations, but the first one leads to spurious flat modes of finite frequency not present in the second one. Both methods lead to the same physical conclusions otherwise: long-range N\'eel-type order is likely for the square lattice for SU(4) with two particles per site, but quantum fluctuations probably destroy order for more than two particles per site, with
N=2m. By contrast, quantum fluctuations always lead to corrections larger than the classical order parameter for the tripartite triangular lattice (with
N=3m) or the bipartite honeycomb lattice (with
N=2m) for more than one particle per site,
m>1, making the presence of color very unlikely except maybe for
m=2 on the honeycomb lattice, for which the correction is only marginally larger than the classical order parameter.
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.
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.
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 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 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 {\it 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.