(Color online). Phase diagram for the fully anisotropic XYZ model. We have assumed x = cos(θ) and y = sin(θ). The phase boundaries between dimer phase and FM phases are infinite-fold degenerate, while the boundaries between the FM phases are critical and gapless with central charge c = 1. The black dots are boundaries determined by order parameters (see Eq. 7), with accuracy better than 3.0 × 10 −4. In dimer phase, the deep red regimes can not be explained by mixing of two anisotrpic dimer models. The classical limits are denoted as H(1, 0, 0), H(0, 1, 0) and H(0, 0, 1) and the dashed lines are conditions for exact FM states. 

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... boundary for dimer phase. Our determined phase diagram for the dimer phase is presented in Fig. 1. This regime has the advantage to be determined exactly with even a small lattice sites with periodic boundary condition, provided that the wave functions are in form of Eq. 2. We consider the simplest case with L = ...

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... is the major phase boundary determined for the dimer phase (see boundaries in Fig. 1). From the first equation, we may always assume that x + y > 0, then these two equations give rise to z > −xy/(x + y). The same analysis can be performed for L = 6, which can also be solved exactly and give the same phase boundary. By this result, the GS energy for the dimer phase in a length L system (L is an even ...

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... which can always be fulfilled for the given condition. Nevertheless, not all dimer state defined by Eq. 4 can be explained in this way. In Eq. 6, one may replace the XXZ model by anisotropic XYZ model and we prove that this decoupling only allows solution when (2)], and z ≥ −2 sin(θ) for θ ∈ [3π/2 + arctan(2), 2π] (see the light red regime in Fig. 1), beyond which it can not be understood by mechanism of Eq. 6, indicating of nontriviality for this ...

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... degeneracy at the boundary by Eq. 4. The boundary in Eq. 4 automatically satisfies the permutation symmetry of H. This boundary is numerically verified with extraordinary high accuracy (see Fig. 1). A typical transition from the dimer phase to the z-FM phase is presented in Fig. 2a, which is characterized by dimer order ∆ d [51,53] and magnetization M η ...

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... the exact dimer phase, ∆ d = 3/4, and M z = 0, while in the z-FM phase, M z − L/2 ∝ 1/z 2 (from second-order perturbation theory), and ∆ d = 0. The boundary determined in these orders is precisely that from Eq. 4, with difference less than 3 × 10 −4 . The similar accuracy has been found for all black dots in Fig. 1. In Fig. 2b, we show that at the phase boundary, all excitation gaps, δE n1 = E n − E 1 , collapse to zero, indicating of infinite-fold degeneracy extended to infinite volume. In the phase boundary, we have three classical points: H(1, 0, 0), H(0, 1, 0) and H(0, 0, 1). Here, H(0, 0, 1) is relevant to the boundary defined in Eq. 4 in ...

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... FM state spontaneously polarized along z direction. This state can be mapped to the exact FM state along other two directions by dual rotation, R η = i exp i π 2 s η for η = x, y, which induces permutation among the three directions. We find that the other two exact FM states at z = max(x, y) and z < x = y = −1/ √ 2. These lines are presented in Fig. 1 with dashed lines. The arrows mark the evolution of these dual mapping start from z → −∞. One should be noticed that when z → −∞, it equals to −H(0, 0, 1), and can be mapped to −H(1, 0, 0) and −H(0, 1, 0) by dual rotation. The GSs of these points should be two-fold degenerate with exact FM states in Eq. 10. This exact two-fold ...