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arXiv:cond-mat/9705263 · Journal of the Physical Society of Japan

Phase Transitions ofS=3/2andS=2XXZ Spin Chains with Bond Alternation

Atsuhiro Kitazawa, K. Nomura

Published Nov 15, 1997• 25 citations
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Abstract

We study the phase diagram of S=3/2 and S=2 bond-alternating spin chains numerically. In previous papers, the phase diagram of S=1 XXZ spin chain with bond-alternation was shown to reflect the hidden $Z_{2}\times Z_{2}$ symmetry. But for the higher S Heisenberg spin chain, the successive dimerization transition occurs, and for anisotropic spin chains the phase structure will be more colorful than the S=1 case. Using recently developed methods, we show directly that the phase structure of the anisotropic spin chains relates to the $Z_{2}\times Z_{2}$ symmetry.

DOI
10.1143/jpsj.66.3379

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We study the phase diagram of S=3/2 and S=2 bond-alternating spin chains numerically. In previous papers, the phase diagram of S=1 XXZ spin chain with bond-alternation was shown to reflect the hidden Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry. But for the higher S Heisenberg spin chain, the successive dimerization transition occurs, and for anisotropic spin chains the phase structure will be more colorful than the S=1 case. Using recently developed methods, we show directly that the phase structure of the anisotropic spin chains relates to the Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry.
The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Neel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Neel and dimer-Neel). We determine the phase boundary and universality class using the recent developed techniques by the authors.
The phase diagram of the spin-1 bond-alternating XXZ chain is studied numerically. This model is appropriate to study the VBS picture and the hidden Z_2\times Z_2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the XY, the Haldane, the dimer, and the Néel ones. The critical properties of this model can be interpreted as the Ashkin-Teller type reflecting the hidden Z_2\times Z_2 symmetry. The phase transitions are of the Berezinskii-Kosterlitz-Thouless type (XY-Haldane and XY-dimer), of the 2D Gaussian type (Haldane-dimer), and of the 2D Ising type (Haldane-Néel and dimer-Néel). We determine the phase boundary and universality class using the recent developed techniques by the authors.
Phase transitions between the XY, the dimer, and the Haldane phases of the spin-1 bond-alternating XXZ chain are studied numerically. By use of the ``level spectroscopy'' we overcome the difficulties in treating the Berezinskii-Kosterlitz-Thouless transition which occurs at the XY-dimer and XY-Haldane boundaries. We find that the XY-Haldane boundary lies just on $\ensuremath{\Delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ (pure XY line). The critical properties can be interpreted as the Ashkin-Teller type reflecting the hidden ${\mathrm{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{Z}}_{2}$ symmetry. We make a conjecture on the phase diagram of the present model for arbitrary SSS.
By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an S=1S=1S=1 ferromagnetic-antiferromagnetic bond-alternating chain with the XXZXXZXXZ 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 XXZXXZXXZ 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 $XY1,thelarge−1, the large-1,thelarge−D$ (LD), the intermediate-DDD (ID), 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 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 XXZXXZXXZ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.
We study the ground-state properties of the bond-alternating S=1/2 quantum spin chain whose Hamiltonian is H= tsumj{\mathit{tsum}}_{\mathit{j}}tsumj​ (S2jx{\mathit{S}}_{2\mathit{j}}^{\mathit{x}}S2jx​S2j+1x{\mathit{S}}_{2\mathit{j}+1}^{\mathit{x}}S2j+1x​ +S2jy{\mathit{S}}_{2\mathit{j}}^{\mathit{y}}S2jy​S2j+1y{\mathit{S}}_{2\mathit{j}+1}^{\mathit{y}}S2j+1y​ +\ensuremath{\lambda}S2jz{\mathit{S}}_{2\mathit{j}}^{\mathit{z}}S2jz​S2j+1z{\mathit{S}}_{2\mathit{j}+1}^{\mathit{z}}S2j+1z​) +\ensuremath{\beta}tsumj{sum}_{\mathit{j}}sumj​${\mathbf{S}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}S2j{\mathbf{S}}_{2\mathit{j}}S2j​. When \ensuremath{\beta}=0, the ground state is a collection of local singlets with a finite excitation gap. In the limit of strong ferromagnetic coupling \ensuremath{\beta}\ensuremath{\rightarrow}-\ensuremath{\infty}, this is equivalent to the S=1 XXZ Hamiltonian. It has several ground-state phases in the \ensuremath{\lambda}-\ensuremath{\beta} plane including the Haldane phase which has a finite excitation gap above the ground. They are characterized by a full breakdown, partial breakdowns, and a nonbreakdown of the hidden discrete Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry. The ground-state phase diagram is obtained by series expansions.
We study the S=1S=1S=1 quantum spin chain with bond alternation ${\cal H}=\sum _i (1-(-1)^iδ)\vect{S}_i\cdot \vect{S}_{i+1}$ by the density matrix renormalization group method recently proposed by Steven R. White (\PRL{69}{3844}{1993}). We find a massless point at δc=0.25±0.01δ_c =0.25 \pm 0.01δc​=0.25±0.01. We also find the edge states in the region δδcδδ_{c}δδc​. At the massless point, the spin wave velocity vsv_svs​ is 3.66±0.103.66 \pm 0.103.66±0.10 and the central charge ccc is 1.0±0.151.0\pm 0.151.0±0.15. Our results indicate that a continuous phase transition occurs at the massless point $δ=δ_c $ accompanying breaking of the hidden Z2×Z2Z_2\times Z_2Z2​×Z2​ symmetry.