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arXiv:cond-mat/9604107 · Journal of the Physical Society of Japan

Hidden Order and Dimerization Transition inS=2Chains

Masanori Yamanaka, Masaki Oshikawa, Seiji Miyashita

Published Jun 15, 1996• 18 citations
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Abstract

We study ground state properties of the $S=2$ quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden $Z_2 \times Z_2$ symmetry is broken for $0.3 < |δ| < 0.5$ while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden $Z_2 \times Z_2$ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.

DOI
10.1143/jpsj.65.1562

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We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a bond alternation [ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath SSS}{j} \cdot \mbox{\boldmath SSS}{j+1} ] by a Quantum Monte Carlo calculation. We find that the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry is broken for 0.3<∣δ∣<0.50.3 < |δ| < 0.50.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
The successive phase transitions of the Heisenberg spin chain with bond alternation, H = ∑ i =1 L -1 [1-(-1) i δ] S i · S i +1 , for the spin S =3/2 case under the open boundary condition are investigated by the density-matrix renormalization-group method proposed by White. Besides the Heisenberg point, we find that another gapless point exists at δ c = 0.42 ±0.02. This result qualitatively supports the field-theoretical prediction by Affleck and Haldane. However, our estimation of δ c deviates from the field-theoretical prediction, δ c = 2/3. We also find a fourfold degeneracy of the ground state in the region 0 <δ< δ c in the thermodynamic limit. Our results show that the degeneracy is described by the edge excitations, which is essentially similar to the one in the S =1 case. We calculate the string correlation function and find that its behavior is consistent with the valence-bond-solid (VBS) picture.
We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J${\ensuremath{\sum}}_{\mathrm{i}}$[1-(-1)i{)}^{\mathrm{i}})i\ensuremath{\delta}]Si{\mathbf{S}}_{\mathrm{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-\ensuremath{\sigma}-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter, Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta})=${\mathrm{lim}}_{\mathrm{L}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L) with Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L)=〈SL/4z{\mathrm{S}}_{\mathrm{L}\mathrm{/}4}^{\mathrm{z}}SL/4z​${\ensuremath{\prod}}_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{\ensuremath{-}}1}$exp[i\ensuremath{\theta}Sjz{\mathrm{S}}_{\mathrm{j}}^{\mathrm{z}}Sjz​]S3L/4z{\mathrm{S}}_{3\mathrm{L}\mathrm{/}4}^{\mathrm{z}}S3L/4z​〉. It turns out that the transition points are successfully detected by observing the overall behavior of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta}) at various values of \ensuremath{\delta}. Investigating the dependences of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L) on \ensuremath{\theta}, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.
The ground-state properties of a spin S=1/2S=1/2S=1/2 tetrameric Heisenberg antiferromagnetic chain with alternating couplings AF1-AF2-AF1-F{\text{AF}}_{1}{\text{-AF}}_{2}{\text{-AF}}_{1}\text{-F}AF1​-AF2​-AF1​-F (AF and F denote antiferromagnetic and ferromagnetic couplings, respectively) are studied by means of the density-matrix renormalization-group method. Two plateaus of magnetization mmm are found at m=0m=0m=0 and 1/4. It is shown that in such a spin-1/2 AF system, there is a gap from the singlet ground state to the triplet excited states in the absence of a magnetic field. The spin-spin correlation function decays exponentially, and the gapped states have a nonvanishing string order, which measures a hidden symmetry in the system. By a dual transformation, the string order is transformed into a ferromagnetic order and the hidden symmetry is unveiled to be a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ discrete symmetry, which is fully broken in the gapped states. This half-integer spin Heisenberg AF chain is in a Haldane-like phase suggesting that the present findings extend the substance of Haldane's scenario. A valence-bond-solid state picture is also proposed for the gapped states.
We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry. When the chain is in the Haldane phase, this Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S=1 chain.
We study the S=1/2 spin chain with the Hamiltonian H=${\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}\mathrm{\ensuremath{-}}1}$ {${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{x}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{x}}$+${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{y}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}^{\mathit{z}}$${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}+1}$} +\ensuremath{\beta}Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ (1-${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathrm{\ensuremath{\sigma}}}_{2\mathit{j}}$), where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$=(${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{x}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{y}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}^{\mathit{z}}$) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry of the system. It has been argued that a similar hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry of the S=1 chain is fully broken when and only when the system exhibits the Haldane gap. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry in a range of the parameter space including the line \ensuremath{\beta}=0, \ensuremath{\lambda}>-1. We argue that the range with such properties indeed extends to the limit \ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty} in which the present model reduces to the S=1 spin chain. This observation provides support of Hida's conclusion that the Haldane gap in the S=1 chain is continuously connected to the gap in the decoupled S=1/2 system with \ensuremath{\beta}=0.
The author studies integer S>1 spin chains. He extends the Kennedy-Tasaki nonlocal unitary transformation for S=1 to arbitrary integer S. He shows the main results of Kennedy and Tasaki (1992) are maintained for S>1: Heisenberg-type Hamiltonians are transformed to Hamiltonians of nearest-neighbour interactions with Z 2 *Z 2 symmetry, and the den Nijs-Rommelse string observables are transformed to the ferromagnetic correlation observables. He asserts that in general values of integer S there exist several phases with the hidden Z 2 *Z 2 symmetry breaking. The den Nijs-Rommelse string order parameters, which measure the hidden Z 2 *Z 2 symmetry breaking, are calculated explicitly for several variants of the VBS-type states. In the standard VBS state, the hidden Z 2 *Z 2 symmetry breaks down when S is odd but remains unbroken when S is even. His results for partially dimerized VBS states suggest that the hidden Z 2 *Z 2 symmetry breaking can be used to detect the successive dimerization transitions predicted by Affleck and Haldane (1987). Some new anisotropic VBS-type states are investigated. The result suggests that there are successive phase transitions when he increases the uniaxial anisotropy in a Heisenberg-type model. Other new VBS-type states with long-range order are considered, and their relevance to the phase diagram of the Heisenberg XXZ model and the magnetization process of antiferromagnets is investigated. He introduces an extended string order parameter which possesses a characteristic behavior in the partially dimerized VBS states.