Ground-State Phase Diagram of an Anisotropic S=1 Ferromagnetic-Antiferromagnetic Bond-Alternating Chain

By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an $S=1$ ferromagnetic-antiferromagnetic bond-alternating chain with the $XXZ$ and the on-site anisotropies. This system can be mapped onto an anisotropic spin-2 chain when the ferromagnetic interaction is much stronger than the antiferromagnetic interaction. Since there are many quantum parameters in this system, we numerically obtained the GSPD on the plane of the magnitude of the antiferromagnetic coupling versus its $XXZ$ anisotropy, by use of the exact diagonalization, the level spectroscopy as well as the phenomenological renormalization group. The obtained GSPD consists of six phases. They are the $XY$1, the large-$D$ (LD), the intermediate-$D$ (ID), the Haldane (H), the spin-1 singlet dimer (SD), and the N\'eel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the symmetry-protected topological phase. The former three are smoothly connected without any quantum phase transitions. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the anisotropic spin-2 chain with the $XXZ$ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.


I. INTRODUCTION
In recent years, low dimensional quantum spin systems have been attracting increasing attention because they provide rich physics even when models are rather simple. Several years ago, we investigated 1-4 the T = 2 quantum spin chain with the XXZ and on-site anisotropies described by where T µ j (µ = x, y, z) represents the µ-component of the spin-2 operator T j at the jth site, and ∆ and D 2 are, respectively, the XXZ anisotropy parameter of the nearest-neighbor interactions and the on-site anisotropy parameter. We obtained the ground-state phase diagram 1,2 (GSPD) mainly by the use of the exact diagonalization and the level spectroscopy (LS) analysis [5][6][7][8] . The remarkable features of the GSPD are: (a) there exists the intermediate-D (ID) phase which was first predicted by Oshikawa 9 in 1992 and has been believed to be absent for about two decades until our finding 1-3 in 2011; (b) the Haldane (H) state and the large-D (LD) state belong to the same phase. These features are consistent with the discussion by Pollmann et al. 10,11 . Namely, the ID state is a symmetry-protected topological (SPT) state protected by (i) the time-reversal symmetry S j → −S j , as well as by (ii) the space inversion symmetry with respect to a bond, while the H state and the LD state are trivial states. Slightly after our works, the ID phase was also discussed by Tzeng 12 and Kjäll et al. 13 .
Considering these situations, we investigate the GSPD of the S = 1 ferromagneticantiferromagnetic bond-alternating chain, since it is thought that this chain can be mapped onto the spin-2 model in the strong ferromagnetic coupling limit. We describe our model in §2, and the numerically determined GSPD are shown in §3. In §4 the perturbation theory from the strong ferromagnetic coupling limit is developed. Section 5 is devoted to concluding remarks.

II. MODEL
We investigate the model described by Here, S µ j (µ = x, y, z) is the µ-component of the spin-1 operator S j acting on the jth site; J F (> 0.0) and J AF (≥ 0.0) denote, respectively, the magnitudes of exchange interaction constants for the ferromagnetic and antiferromagnetic bonds; ∆ F and ∆ AF are, respectively, the parameters representing the XXZ anisotropies of the former and latter interactions.  Among them, the LD, H, and SD phases are the trivial phases, while the ID phase is the SPT phase. Interestingly, the former three are smoothly connected without any quantum phase transitions between the LD and H phases and between the H and SD phases, and therefore they belong to the same phase. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the Hamiltonian (1) 1-3 .  Firstly, the phase transition between the LD and ID phases and that between the ID and H phases are of the Gaussian type. Therefore, as is well known, the phase boundary lines can be accurately estimated by Kitazawa's level spectroscopy (LS) method 7 . Namely, we numerically solve the equation, to calculate the finite-size critical values. It is noted that, at the N → ∞ limit, E Then, we solve the following equation to calculate the finite-size critical values: where P = −1 or P = +1 depending upon whether the transitions are associated with the ID phase or with the LD, H, and SD phases.

IV. PERTURBATION THEORY FROM THE STRONG FERROMAGNETIC COUPLING LIMIT
In the strong ferromagnetic coupling limit, it is thought that the present system can be mapped onto the spin-2 model. Here we take the unperturbed Hamiltonian as The ground states of h 2j,2j+1 are five-fold degenerate, which are interpreted as isolated spin-2 states, expressed by the spin-2 operator T . We note that, if we include the XXZ anisotropy in H (0) , we cannot treat lowest five states as isolated spin-2 states, In the lowest order perturbation theory, we obtain where It is interesting thatD 4 (T z j ) 4 term appears. Since we have set ∆ F = 0.8, and Unfortunately, the GSPD of the Hamiltonian (10) with the parameter set given by eq. (14) has not been reported in the literature. However, that with ∆ F = 0.8, and D 2 = −1/15 (namely, −D 2 =D 4 = 1/60J AF ) is shown in Fig.3(a) of our previous paper 4 , Thus, as the second-best plan, we are going to compare our present GSPD with that in ref. 4 . The GSPD of Fig.3(a) of 4 can be recasted into Fig.4.
When the XXZ anisotropy of the ferromagnetic interaction is introduced (namely, when β = 0), the five-fold degenerate states of ground states of h 2j,2j+1 are split into three levels with T z = 0, T z = ±1 and T z = ±2. We note that this effect is expressed as theD 2 (T z j ) 2 andD 4 (T z j ) 4 terms in the effective Hamiltonian (12). For our parameter set (β = −0.2 and D 2 = −1/30), these energies are When J AF is much smaller than the splitting energy (for instance, 0.1), the states with T z = ±2 will be strongly suppressed. In this case, it is appropriate to neglect the T z = ±2, which leads to the mapping onto the σ = 1 spin system. A straightforward calculation leads The GSPD for the effective Hamiltonian (16)  the Haldane (H), the spin-1 singlet dimer (SD), and the Néel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the SPT phase. We have also developed the perturbation theory from the strong ferromagnetic coupling limit to map onto the T = 2 effective model, which quantitatively explains the numerically obtained GSPD.
We can see a considerably wider region of the ID phase in Fig.2 than that for the model described by the Hamiltonian (1), in which the ID phase was numerically observed for the first time. The reason for the wider ID region in Fig.2 is the existence of theD 4 (T z j ) 4 term in eq.(10). We have already shown that the addition of the D 4 (T z j ) 4 term with D 4 > 0 to the Hamiltonian (1) drastically widen the ID region 4 . We believe that the finding of the wider ID region in our GSPD provides a guiding principle to find or synthesize real materials in which the ID phase could be experimentally observed.