Haldane and Dimer Gap in General Double Spin-Chain Models

We have obtained an analytic expression for the k-dependence of the excitation energy gap for an arbitrary double S = 1/2 spin chain by using the non-local unitary transformation and the variational method. We check that it explains the gap behaviour of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar - Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar - Ghosh point, where a local triplet is decomposed into two S = 1/2 free spins moving among the singlet dimers.We have obtained an analytic expression for the k-dependence of the excitation energy gap for an arbitrary double S = 1/2 spin chain by using the non-local unitary transformation and the variational method. We check that it explains the gap behaviour of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar - Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar - Ghosh point, where a local triplet is decomposed into two S = 1/2 free spins moving among the singlet dimers.

arXiv:cond-mat/9702182Journal of Physics Condensed MatterJul 28, 19976 citations

Closing of the Haldane gap in a spin-1 XXZ chain

Chan Yu, Ji-Woo Lee

Journal of the Korean Physical SocietyAug 30, 20212 citations

Ground-state phase diagram of a spin- 12 frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases

Shunsuke Furukawa, Masahiro Sato, Shigeki Onoda +1 more

The ground-state phase diagram of a spin- $\frac{1}{2}$ XXZ chain with competing ferromagnetic nearest-neighbor ( ${J}_{1}<0$ ) and antiferromagnetic second-neighbor ( ${J}_{2}>0$ ) 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.The ground-state phase diagram of a spin-$\frac{1}{2}$ XXZ chain with competing ferromagnetic nearest-neighbor (${J}_{1}<0$) and antiferromagnetic second-neighbor (${J}_{2}>0$) 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.

arXiv:1207.1059Physical Review BSep 12, 2012120 citations

Localization length of a soliton from a nonmagnetic impurity in a general double-spin-chain model

Tota Nakamura

Localization length of a free-spin soliton from a nonmagnetic impurity is deduced in a general double-spin-chain model ${(J}_{0}{\ensuremath{-}J}_{1}{\ensuremath{-}J}_{2}{\ensuremath{-}J}_{3}$ model). We have solved a variational problem which employs the nearest-neighbor singlet-dimer basis. The wave function of a soliton is expressed by the Airy function, and the localization length (\ensuremath{\xi}) is found to obey a power law of the dimerization ${(J}_{2}\ensuremath{-}{J}_{3})$ with an exponent $\ensuremath{-}1/3;$ $\ensuremath{\xi}\ensuremath{\sim}{(J}_{2}\ensuremath{-}{J}_{3}{)}^{\ensuremath{-}1/3}.$ This explains why ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ does not show the antiferromagnetic order, while ${\mathrm{CuGeO}}_{3}$ does by impurity doping. When the gap exists by the bond dimerization, a soliton is localized and no order is expected. On the contrary, there is a possibility of the order when the gap is mainly due to frustration.Localization length of a free-spin soliton from a nonmagnetic impurity is deduced in a general double-spin-chain model ${(J}_{0}{\ensuremath{-}J}_{1}{\ensuremath{-}J}_{2}{\ensuremath{-}J}_{3}$ model). We have solved a variational problem which employs the nearest-neighbor singlet-dimer basis. The wave function of a soliton is expressed by the Airy function, and the localization length (\ensuremath{\xi}) is found to obey a power law of the dimerization ${(J}_{2}\ensuremath{-}{J}_{3})$ with an exponent $\ensuremath{-}1/3;$ $\ensuremath{\xi}\ensuremath{\sim}{(J}_{2}\ensuremath{-}{J}_{3}{)}^{\ensuremath{-}1/3}.$ This explains why ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ does not show the antiferromagnetic order, while ${\mathrm{CuGeO}}_{3}$ does by impurity doping. When the gap exists by the bond dimerization, a soliton is localized and no order is expected. On the contrary, there is a possibility of the order when the gap is mainly due to frustration.

arXiv:cond-mat/9809292Physical review. B, Condensed matterMar 1, 199910 citations

The ground-state phase diagram of a spin-

\frac{1}{2}

XXZ chain with competing ferromagnetic nearest-neighbor (

{J}_{1}<0

) and antiferromagnetic second-neighbor (

{J}_{2}>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.

{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}

does not show the antiferromagnetic order, while

{\mathrm{CuGeO}}_{3}

does by impurity doping. When the gap exists by the bond dimerization, a soliton is localized and no order is expected. On the contrary, there is a possibility of the order when the gap is mainly due to frustration.