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Phase Diagram of S = 1 Bond-Alternating XXZ Chains Reflecting the Hidden Z2 × Z2 Symmetry

Atsuhiro Kitazawa, K. Nomura, Received February

Published Jan 1, 2008• 0 citations
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Abstract

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.

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Phase Diagram of S=1 Bond-Alternating XXZ Chains Reflecting the Hidden Z_2\times Z_2 Symmetry

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Phase diagram of theS=1/2 quantum spin chain with bond alternation

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We study the ground-state properties of the bond-alternating S=1/2 quantum spin chain whose Hamiltonian is H= ${\mathit{tsum}}_{\mathit{j}}$ (${\mathit{S}}_{2\mathit{j}}^{\mathit{x}}$${\mathit{S}}_{2\mathit{j}+1}^{\mathit{x}}$ +${\mathit{S}}_{2\mathit{j}}^{\mathit{y}}$${\mathit{S}}_{2\mathit{j}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathit{S}}_{2\mathit{j}}^{\mathit{z}}$${\mathit{S}}_{2\mathit{j}+1}^{\mathit{z}}$) +\ensuremath{\beta}t${sum}_{\mathit{j}}$${\mathbf{S}}_{2\mathit{j}\mathrm{\ensuremath{-}}1}$\ensuremath{\cdot}${\mathbf{S}}_{2\mathit{j}}$. 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 ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry. The ground-state phase diagram is obtained by series expansions.

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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.
Critical properties of the spin-1 bond-alternating XXZ chain are studied numerically. This model is appropriate to study the VBS picture and hidden Z 2 × Z 2 symmetry concerning to the Haldane gap problem. The possible phases are the ferromagnetic, the X Y , 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 × 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.
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 ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
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.