Bosonic realization of SU(3) chiral Haldane phases

TL;DR

The paper develops a Holstein-Primakoff bosonic realization of the SU(3) antiferromagnetic chain in an alternating fundamental/anti-fundamental representation and analyzes its phase diagram under staggered couplings and $T^3$,$T^8$ anisotropy. It identifies two chiral Haldane SPT phases of opposite chirality and a $\mathbb{Z}_3$-breaking trivial phase, with a first-order chiral-reversed transition and a novel excited-state SPT near the Heisenberg point. A variational ansatz explains the $\mathbb{Z}_3$ symmetry-breaking transition, and the work proposes a feasible optical-lattice realization using two-species spin-1/2 bosons, enabling experimental observation of string orders and edge states. These results illuminate how higher SU($N$) symmetries enrich Haldane-like physics and offer practical routes to explore SPT order in cold-atom platforms.

Abstract

We give a bosonic realization of the SU(3) antiferromagnetic Heisenberg (AFH) chain in the alternating conjugate representation, and study its phase diagram as a function of staggered interactions and anisotropy along the $T^3$ and $T^8$ directions. Unlike the SU(2) case, we observe a chiral-reversed quantum phase transition, where each competing phase is adiabatically connected to one of the chiral Haldane phases predicted in the SU(3) AFH chain with local adjoint representation. In the vicinity of the Heisenberg point, we identify a symmetry-protected topological state that appears at the first excited energy level. We also study the spontaneous $\mathbb{Z}_3$ symmetry breaking of the system, and provide a variational wavefunction that captures the transition from the topological phase to the trivial phase. Finally, we propose an experimental realization of our bosonic model by two spin-1/2 bosons in an optical lattice.

Figures (25)

• Figure 1: The weight diagram of SU(3) triplet state in the Holstein-Primakoff bosons representation. The Hilbert space is spanned by two bosonic states $\vert a\rangle$ and $\vert b\rangle$, together with a vacuum state $\vert 0\rangle$, corresponding to the $|u\rangle, |d\rangle, |s\rangle$ states in the fundamental representation $\bm{3}$, or $|\overline{u}\rangle, |\overline{d}\rangle, |\overline{s}\rangle$ states in the anti-fundamental representation $\overline{\bm{3}}$. The rising and lowering operators can be constructed by the creation or annihilation of bosonic particles.
• Figure 2: The phase diagram as a function of staggered interaction $J_R/J_L$ and anisotropy $g$. The color scale indicates the magnitude of the string order parameter with system size $L = 180$ and bond dimension $m=600$ under open boundary conditions. From the string order, we observe two SPT states with different chiralities and a trivial phase. There is a chiral-reversed transition at $J_R/J_L=1$, and the solid curve denotes the phase boundary between two SPT phases to the trivial phase that is determined by the change of average particle number. This transition can be qualitatively explained by our variational ansatz (the red dashed lines) as detailed in Sec. \ref{['ansatz']}.
• Figure 3: (a) The string order parameter for different system sizes at $J_R/J_L = 1$ and $g = 0$ (OBC, $m = 800$). The inset shows a second-order polynomial fitting as a function of $1/L$, which gives a finite string order of $\mathcal{O}_{0}^{\text{str}} = 0.090 \pm 0.001$ at the thermodynamic limit. (b) The correlation function as a function of distance ($L=180$, OBC, $m=800$). The correlation function at odd data points (dashed red line) shows a well-behaved exponential decay, and the exponential fitting in the inset gives a correlation length of approximately 15.8. (c) The subsystem entanglement entropy for $J_R / J_L = 1$ and $g = 0$ (PBC, $L = 78$, $m = 1400$). The blue solid line corresponds to our fitting using Eq. (\ref{['eq:entropy']}), performed around the chain center (see the vertical dashed lines), which gives a central charge ($c \approx 1.17$) that is below the value predicted by SU(3)$_1$ WZNW critical theory.
• Figure 4: The energy gap as a function of $g$ at $J_R/J_L = 1$ (OBC, $m = 800$). The data points show the results by polynomial fitting the gap to the thermodynamic limit, as illustrated in the inset. At the point $g = 0$, we find an energy gap of approximately $0.066 \pm 0.001$.
• Figure 5: The first and second-order derivatives of the ground-state energy $E_0$ to $J_R$ as a function of $J_R/J_L$ at $g=1$ (PBC, $L=64$, $m=800$). The discontinuities of both derivatives at $J_R/J_L=1$ signal a first-order quantum phase transition.