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We study a three-leg spin-1/2 ladder with geometrically frustrated interleginteractions. We call this model an anisotropic triangular-strip (ATS) model.We numerically and field-theoretically show that its ground state belongs to agapless symmetry-protected topological (SPT) phase. The numerical approach isbased on density-matrix renormalization group analyses of the entanglemententropy and the entanglement spectrum. Whereas the entanglement entropyexhibits a critical behavior, the entanglement spectrum is nontriviallydegenerate. These entanglement properties imply that the ground state is agapless topological phase. We investigate the ATS model using a quantum fieldtheory to support the numerical findings. When the frustrated interchaininteraction is deemed a perturbation acting on the three spin chains, thefrustrated interchain interaction almost isolates the second chain from theother two chains. However, at the same time, the second chain mediates aferromagnetic interaction between the first and third chains. Therefore, theground state of the ATS model is a gapless Tomonaga-Luttinger liquid weaklycoupled to a spin-1 Haldane chain with irrelevant interactions. Last but notleast, we show that the gapless SPT phase of the ATS model is asymmetry-protected critical phase. We point out that the symmetry protection ofcriticality is essential in characterization of the gapless SPT phase.

Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip