One dimensional SU(n) chains with the same irreducible representation R at each site are considered. We determine which R admit low-energy mappings to a SU(n)/[U(1)]n−1 flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each R, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreps that also possess a Zn symmetry that acts transitively on the SU(n)/[U(1)]n−1 fields. Such SU(n) chains have an 't Hooft anomaly in certain cases, allowing for a generalisation of Haldane's conjecture to these novel representations. In particular, for even n and for representations whose Young tableaux have two rows, of lengths p1 and p2 satisfying p1≠p2, we predict a gapless ground state when p1+p2 is coprime with n. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if p1+p2 is not a multiple of n.
One dimensional SU(n) chains with the same irreducible representation R at each site are considered. We determine which R admit low-energy mappings to a SU(n)/[U(1)]n−1 flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each R, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreps that also possess a Zn symmetry that acts transitively on the SU(n)/[U(1)]n−1 fields. Such SU(n) chains have an 't Hooft anomaly in certain cases, allowing for a generalisation of Haldane's conjecture to these novel representations. In particular, for even n and for representations whose Young tableaux have two rows, of lengths p1 and p2 satisfying p1≠p2, we predict a gapless ground state when p1+p2 is coprime with n. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if p1+p2 is not a multiple of n.
Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-p symmetric ones with p not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU(n) chains in the symmetric representation. For a rank-p symmetric representation with n and p coprime, we predict gapless excitations above the ground state. If p is a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if p and n have a greatest common divisor 1<q<n, we predict a ground state degeneracy of n/q, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU(n) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Schultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions.
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.
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.
We consider two sets of related models: initially, these are SU(2) antiferromagnetic spin chains with N sites of spin S, and the O(3) nonlinear sigma model in two dimensions with topological coefficient Θ a multiple of π (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with Θ=2πS. We study these models with N odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the Z2 inversion symmetry n→−n, and amounts to a flux of a Z2 gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the Z2 inversion symmetry. We show directly that, if and only if (−1)Θ/π=−1, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space RP2≅S2/Z2, and hence this model does not exist for Θ=π (mod 2π). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on N modulo 4 and 2S modulo 2. For a large class of spin chains and associated sigma models we find similar results, but now (−1)Θ/π is replaced by the value ±1 of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.
It was recently argued that SU(3) chains in the p-box symmetric irreducible representation (irrep) exhibit a ``Haldane gap'' when p is a multiple of 3 and are otherwise gapless [Nucl. Phys. B 924, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with p columns of length 2 and p columns of length 1 in the Young tableau (p=1 corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for p odd but not even.