Generalization of the Haldane conjecture to SU(3) chains

TL;DR

The paper generalizes Haldane's SU(2) conjecture to SU(3) chains with symmetric p-box representations by mapping to a flag-manifold SU(3)/[U(1)×U(1)] nonlinear sigma-model with two topological angles, θ = 2πp/3. It predicts a gapped phase for p divisible by 3 and a massless SU(3)_1 critical line for p not divisible by 3, supported by Monte Carlo simulations that show a finite gap at θ = 0 and a vanishing gap near θ = ±2π/3 for small coupling. RG analysis reveals asymptotic freedom and a relevant λ-term that can influence the low-energy behavior, while a lattice gauge formulation clarifies the strong-coupling limit and phase structure. The work connects rigorous results (LSMA, Bethe Ansatz, AKLT states) with field-theory predictions, offering a unified framework for understanding SU(3) spin chains and their topological sectors, and outlining open questions on critical exponents and cross-over scales.

Abstract

We apply field theory methods to $\mbox{SU}(3)$ chains in the symmetric representation, with $p$ boxes in the Young tableau, mapping them into a flag manifold non-linear $σ$-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 $p=3m\pm 1$ (for integer $m$), corresponding to a massless phase of the $σ$-model at $θ=\pm 2π/3$. We confirm this with Monte Carlo calculations on the $σ$-model.

Figures (12)

• Figure 1: (a) The renormalization group flow diagram of the $\hbox{O}(3)$ nonlinear $\sigma$-model, as proposed in Ref. AffleckLesHouches1988. At $\theta=\pi$ the system undergoes a phase transition from a gapless phase at $g<g_c$ into a gapped phase with a spontaneously broken $\mathbb{Z}_2$ symmetry at $g>g_c$. For $\theta\neq \pi$ the system is gapped with a unique ground state for all values of $g$. (b) Proposed renormalization group flow diagram for the $\hbox{SU}(3)/[U(1)\times U(1)]$ nonlinear $\sigma$-model in the special case where the two topological angles are equal and opposite. At $\theta=2\pi/3$ and $4\pi/3$ the system undergoes a phase transition from a gapless phase at $g<g_c$ into a gapped phase with a spontaneously broken $\mathbb{Z}_3$ symmetry at $g>g_c$. For $2\pi/3<\theta <4\pi/3$ the system is gapped with a spontaneously broken $\mathbb{Z}_2$ symmetry, while for $\theta<2\pi/3$ and $\theta> 4\pi/3$ the system is gapped with a unique ground state for all values of $g$.
• Figure 2: Illustration of the exact ground states discussed by GreiterRachelSchuricht2007. (a) Threefold degenerate trimerized ground states in the $p=1$ case, and (b) the uniqe ground state of an AKLT construction for the $p=3$ case. See sections III.A and VIII.B of Ref. GreiterRachelSchuricht2007 for the construction of the corresponding Hamiltonians.
• Figure 3: Sketch of the low energy fluctuations around the classical three sublattice ground state. The spin states are given by Eq. \ref{['eq:spinstates0']}. The lattice constant $a$ is the distance between neighbouring sites.
• Figure 4: Breaking translational symmetry of the system by weakening the nearest neighbour bonds between sublattices 1 and 3. The $R_{13}$ mirror symmetry and a three site translational symmetry still remains.
• Figure 5: (a)The different $\mathcal{R}_{m,n}$ sector on the $\theta_1 -\theta_3$ plane derived from the $g\to\infty$ calculations, and (b) a zoomed in version with the expected RG flow. The phase diagram is similar for all g values, the flow in the $\theta_1 -\theta_3$ plane is everywhere complemented with a flow towards $g\to\infty$. The transition lines correspond to a conserved parity symmetry, which is spontaneously broken on the solid, but not dotted, lines. At the intersection of the lines, at $(2\pi/3 , -2\pi/3)$ a $\mathbb{Z}_3$ symmetry is also present.