Quantum phase diagram of the spin-1 Kitaev chain with a tunable SIA in which −1.5 ≤ D ≤ 0 and 0 ≤ ϑ ≤ π/2. The spin-nematic (SN) phase is gapped and can be classified into two types termed SN-I and SN-II, based on the degeneracy of their first excited states. They undergo a crossover rather than a phase transition as the lowest excitation gap never closes. The transitions from the dimerized phase and AFM phase to the spin-nematic phase belongs to the Ising universality class. In particular, the quantum critical point is −0.6551(2) for the [001]-type SIA (i.e., ϑ = 0), while it is −0.6035(2) for the [111]-type SIA (i.e., ϑ = tan −1 ( √ 2)).

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... the calculation, we set the TrotterSuzuki step τ = 0.01 and the block states m = 1024. Figure 1 illustrates the quantum phase diagram in the region of D ∈ [−1.5, 0.0] and ϑ ∈ [0, π/2] in the spin-1 Kitaev chain with tunable SIA. Firstly, by calculating the four-spin correlation function pertaining to the spinnematic order, we find that the small-D region, including the Kitaev limit whose ground state is previously termed Kitaev phase [29], exhibits a nonzero spin-nematic correlation over the vanishing magnetic moment. ...

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... for the first excited ground state. The first excited ground state is unique in the wide region, as compared to the twofold case observed in a specific area where |D| and ϑ are small. Therefore, we distinguish the spin-nematic phase as type-I and type-II, respectively, based on its degeneracy of the first excited state (for illustration, see Fig. 1). However, since the lowest excitation gap of the spin-nematic phase does not close throughout its whole region, there is not a QPT but a likely crossover between the two. To illustrate it, we have calculated the phase factor ϕ at fixed SIA, saying D = −0.3. The derivative of ϕ with respect to ϑ shows a broad hump and suffers from an ...

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... hallmark of the difference between the type-I and type-II spin-nematic phases. It is in this sense that we can identify the crossover boundary of the two by the standard deviation of ⟨W [1] i ⟩, i.e., σ W . In our calculation on three closed chains of length L = 24, 48, and 72, the quantity σ W undergoes a sharp jump at ϑ/π ≈ 0.13, as depicted in Fig. 1. We note that the periodicity of ⟨W [l] i ⟩ in the excited states should be different as we change the chain length, and such a periodicity can be discerned by the discrete Fourier transform of ⟨W [l] i ⟩. Nevertheless, the most remarkable feature that the curves of ⟨W [l] i ⟩ (l > 0) are smooth and discrete, respectively, in the ...