Spinor bosons realization of the SU(3) Haldane phase with adjoint representation

TL;DR

The work provides a concrete route to realize the nontrivial $SU(3)$ Haldane phase with adjoint representation using a two-species spinor Bose gas, mapped via Schwinger bosons to a spin-2 bosonic model. It maps out the strong-coupling phase diagram, identifying a chiral Haldane (VBS) phase and a transition to a dimer phase, with edge modes, string order, and entanglement signatures thoroughly analyzed by DMRG and analytical constructions. An explicit ground-state ansatz for the dimer point clarifies the phase's physics, while an experimental proposal outlines a feasible optical-lattice setup and Raman schemes to realize and probe the phase, including edge-state detection and string-order measurements. Overall, the work extends $SU(N)$ symmetry-protected topological physics beyond $SU(2)$ to $SU(3)$ in a bosonic platform, with concrete observables and a viable experimental path.

Abstract

The Haldane phase with local SU(3) adjoint representation constitutes the simplest non-trivial symmetry-protected topological phases in the SU($N>2$) Heisenberg spin chains. In this paper, we propose to realize this phase by a two-species spinor Bose gas, with each species labeling the quark or antiquark states of SU(3) symmetry. In the strong-coupling limit, we determine the ground-state phase diagram, and identify a quantum phase transition from the Haldane phase to a dimer phase. We show how to characterize the Haldane phase through its edge excitations. We also explain the physics at the dimer phase, by constructing an explicit ground-state ansatz at the dimer point.

Figures (14)

• Figure 1: An illustration of the AKLT valence-bond states with local SU(3) adjoint representation [2 1 0] shown as their Young diagrams, where the solid lines map two neighbour virtual states to a SU(3) singlet, and the dashed circles map the local virtual states to a physical octet state. The appearance of two inequivalent virtual states, as shown in dashed circles, gives rise to two distinct chiral topological states with different edge modes.
• Figure 2: Two sets of Schwinger bosons that label the fundamental representation of SU(3) symmetry (shown as their corresponding Young diagrams). Three different internal hyperfine states ($\sigma=2, -2, 0$ as illustrated in Fig. \ref{['fig:8']}b of each boson $a_\sigma$ and $b_\sigma$ are included to construct the triplet (quark) and antitriplet (antiquark) representations.
• Figure 3: Characteristic phase diagram in our parameter regime. The Heisenberg limit and VBS limit in the Haldane phase are shown as the green and orange points. As explained in the Appendix, the experimentally accessible regime is for $t$ positive. The regime at $\delta/g_0>1$ is not shown, as it corresponds to its inverse chiral counterpart, and the phase diagram is symmetric along the line $\delta=g_0$.
• Figure 4: (a) The log-linear plot of the ground state string order at the dimer point, which shows an exponential behavior at long distances as indicated by the red line, with the fitting giving $C^{Str}(r)=0.023\times e^{-0.08r}$. (b) The energy gap at the dimer point as a function of inverse system size $1/L$. The dashed line corresponds to a second-order polynomial fitting, which gives an energy gap of around 0.16.
• Figure 5: The order parameters obtained from DMRG calculations with system size $L=128$. The squares and triangles label the string and dimer orders respectively, with their sizes corresponding to their magnitudes. A translational symmetry-breaking dimer phase is observed at a small regime near the dimer point $t/g_0=0, \delta/g_0=1.0$.