Generalization of the Haldane conjecture to SU(n) chains

Abstract

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-Shultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions. arXiv:1910.08196v2 [cond-mat.str-el]

Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach

“…We now switch to a Lagrangian description to study the infrared properties of the perturbed CFT (13) and to make connection to the semiclassical field theory derived recently for SU(N ) Heisenberg chain (1) in totally symmetric representations. 30,32,59 In this respect, we consider the following action first introduced in Ref. 60:…”

“…which are integers. However, the topological charges are not all independent due to the orthormalization constraint: Φ * i • Φ j = δ ij and satisfy: 30,61 It implies that model ( 18) is left invariant by shifting all topological angles by a same amount: θ a → θ a + θ for all a. There are thus N − 1 independent topological angles θ a = 2πa (a = 1, .…”

“…The generalization of the Haldane conjecture for SU(N ) is described by three different cases depending on the value of n Y with respect to N . [29][30][31] When n Y and N are coprime, both a semiclassical approach of model (1) in Refs. 30-32 and a CFT analysis 33,34 have shown that a quantum critical behavior in the SU(N ) 1 universality class with central charge c = N − 1 emerges.…”

Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice

“…In this paper, we consider the SU (3) antiferromagnetic spins on the triangular lattice, and generalize the Néel-VBS transition to SU (3) spin systems. Such a generalization from SU (2) spins to SU (N ) spins has been recently considered in the context of spin chains [5][6][7][8][9][10][11][12], which leads to the SU (N ) generalization of the Haldane conjecture. On the triangular lattice, the classical SU (3) spins can be set to the Néel order without any frustrations thanks to the tripartite lattice structure.…”

Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain

“…Indeed, ever since this prediction was confirmed experimentally and numerically, the community has thought differently about antiferromagnets, and the actual value of the spin (and not simply its length as a measure of quantum fluctuations) has become a defining parameter along with the dimensionality of space, the topology of the lattice and the isotropy of the couplings [3][4][5][6][7][8][9][10][11]. With the progress made in ultracold fermionic experiments [12][13][14][15][16][17][18][19][20][21][22][23], the SU(N ) cousins of the SU(2) Heisenberg model have become the focus of a lot of attention, and quite naturally the question of whether and how Haldane's conjecture can be generalized has been addressed by many authors [24][25][26][27][28][29][30][31]. The generalization of the AKLT construction has proven to be a very useful guide in predicting which model may or may not be gapped, as well as the concept of spinon confinement [29,[32][33][34][35][36].…”