A scheme is proposed for detection of the topology in the one-dimensional Afeck-Kennedy-Lieb-Tasaki model, based on ultracold spinor atomic gas in an optical lattice. For this purpose, a global operation O(θ) is introduced with respect to the breaking of spinrotational symmetry. Consequently, the topology can be manifested unambiguously by identifying the special values of θ where the expectation value of the global operator with degenerate ground states is vanishing. Furthermore, experimentally θ can be detected readily by the interference of ultracold atomic gases. This scheme can be implemented readily in experiment since it does not need the addressing of individual atoms or the probing of a boundary.
We exploit a natural projected entangled-pair state (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On the kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a
Z2 spin liquid, by identifying the four topological sectors and computing the topological entropy. Furthermore, we construct a one-parameter family of PEPS states interpolating between the RAL state and a short-range resonating valence bond state and find a critical point, consistent with the fact that the two states belong to two different phases. We also perform a variational study of the spin-1 kagome Heisenberg model using this one-parameter PEPS.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of
SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating
SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its
SU(2) analog. We discuss the SPT feature of the state.