Generalization of the Haldane conjecture to SU(n) chains

In this thesis, we study the low energy properties of SU(n) chains in various representations. We are motivated by Haldane's conjecture about antiferromagnets, namely that integer spin chains exhibit a finite energy gap, while half-odd integer spin chains have gapless excitations. Haldane was led to this conclusion by deriving a sigma model description of the antiferromagnet, and this is what we generalize here to SU(n). We find that most representations of SU(n) admit a mapping to a sigma model with target space equal to the complete flag manifold of SU(n). These theories are not automatically relativistic, but we show that at low energies, their renormalization group flow leads to Lorentz invariance. We also show explicitly in SU(3) that the theory is asymptotically free, and contains a novel two-form operator that is relevant at low energies. For all n, these sigma models are equipped with n-1 topological angles which depend on the SU(n) representation at each site of the chain. For the rank-p symmetric representations, which generalize the spin representations of the antiferromagnet, these angles are all nontrivial only when gcd(n,p)=1. This observation, together with recent 't Hooft anomaly matching conditions, and various exact results known about SU(n) chains, allow us to formulate the following generalization of Haldane's conjecture to SU(n) chains in the rank-p symmetric representation: When p is coprime with n, a gapless phase occurs at weak coupling; for all other values of p, there is a finite energy gap with ground state degeneracy equal to n/gcd(n,p). We offer an intuitive explanation of this behaviour in terms of fractional topological excitations. We also predict a similar gapless phase for two-row representations with even n. The topological content of these chains is the same as the symmetric ones, with p now equal to the sum of row lengths of the representation. Finally, we show that the most generic representation of SU(n) will admit a sigma model with both linear and quadratic dispersion; such theories requires further understanding before their low energy spectra can be characterized.Science, Faculty ofPhysics and Astronomy, Department ofGraduat

Renormalization group analysis of phase transitions in the two dimensional Majorana-Hubbard model

A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified epsilon-expansion, confirming the previous conclusions.Science, Faculty ofPhysics and Astronomy, Department ofGraduat

Generalization of the Haldane conjecture to SU(3) chains

We apply field theory methods to SU(3) chains in the symmetric representation, with p boxes in the Young tableau, mapping them into a flag manifold nonlinear σ-model with a topological angle θ=2πp/3. Generalizing the Haldane conjecture, we argue that the models are gapped for p=3m but gapless for p=3m±1 (for integer m), corresponding to a massless phase of the σ-model at θ=±2π/3. We confirm this with Monte Carlo calculations on the σ-model

Generalization of the Haldane conjecture to SU(n) chains

Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-psymmetric ones with pnot 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-psymmetric representation with nand pcoprime, we predict gapless excitations above the ground state. If pis a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if pand nhave 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. (C) 2020 The Author(s). Published by Elsevier B.V

Self-conjugate representation SU(3) chains

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