We map out the ground-state phase diagram of a one-dimensional
SU(N) Kondo-Heisenberg lattice model at half-filling and in the fully antisymmetric self-adjoint representation as a function of
N1 and Kondo coupling
Jk/t. On the basis of auxiliary field quantum Monte Carlo (QMC) simulations with even
N up to 8, we show that the enlarged $\mathrm{SU}(N\ensuremath{\ge}4$) symmetry realizes a quantum phase transition separating a valence bond solid (VBS) phase occupying a weak-coupling part of the phase diagram and the Kondo insulator (KI) state dominating in the strong
Jk/t limit. Along the phase boundary, we always observe critical exponents that belong to a two-dimensional classical Ising universality class. We next trace the evolution of the composite fermion and spin spectral functions across the phase boundary and conclude that VBS order triggers a bond order wave state of conduction electrons, both coexisting with Kondo screening. Upon further reducing
Jk/t, we observe that composite quasiparticles lose their spectral weight indicating that VBS order gradually liberates localized
f spins from forming Kondo singlets with conduction electron spins. We contrast the QMC results with a static large-
N approximation, and we show that in the limit of infinite degeneracy $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the order-disorder transition becomes of first order and is accompanied by a full decoupling of conduction electrons and localized
f spins. We complete our analysis by considering the limit of a vanishing Heisenberg coupling
Jh=0, and we provide evidence that the enlarged
SU(N) symmetry and low dimensionality of the Ruderman-Kittel-Kasuya-Yosida exchange interaction are not sufficient to induce VBS order in the conventional
SU(N) Kondo chain, which hosts solely a KI phase. Finally, we quantify the role of increasing
N in reducing the separation of charge and spin timescales specific to the KI state, and we show that the latter is adiabatically connected to a trivial band insulator captured by the large-
N approximation.
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.
The
SU(N) symmetric antiferromagnetic Heisenberg model with multicolumn representations on the two-dimensional square lattice is investigated by quantum Monte Carlo simulations. For the representation of a Young diagram with two columns, we confirm that a valence-bond solid (VBS) order appears as soon as the N\'eel order disappears at
N=10, indicating no intermediate phase. In the case of the representation with three columns, there is no evidence for either the N\'eel or the VBS ordering for $N\ensuremath{\ge}15$. This is actually consistent with the large-
N theory, which predicts that the VBS state immediately follows the N\'eel state, because the expected spontaneous order is too weak to be detected.
In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.
We investigate the phase diagram of the half-filled
SU(N) Hubbard-Heisenberg model with hopping
t, exchange
J, and Hubbard
U, on a two-dimensional square lattice. In the large-
N limit, and as a function of decreasing values of
t∕J, the model shows a transition from a
d-density wave state to a spin dimerized insulator. A similar behavior is observed at
N=6 whereas at
N=2 a spin density wave insulating ground state is stabilized. The
N=4 model, has a
d-density wave ground state at large values of
t∕J which as a function of decreasing values of
t∕J becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective
(T=0) quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.
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.