arXiv:cond-mat/9612066 · Physical review. B, Condensed matter

Behavior of a frustrated quantum spin chain with bond dimerization

Tota Nakamura, Satoshi Takada

Published Jun 1, 1997• 5 citations

Abstract

We clarified behavior of the excitation gap in a frustrated S=1/2 quantum spin chain with bond dimerization by using the numerical diagonalization of finite systems and a variational approach. The model interpolates between the independent dimer model and the S=1 spin chain by changing a strength of the dimerization. The energy gap is minimum at the fully frustrated point, where a localized kink and a freely mobile antikink govern the low-lying excitations. Away from the point, a kink and an antikink form a bound state by an effective triangular potential between them. They finally collapse to a local triplet at a sufficient value of the dimerization. The wave function of the bound state, the consequential gap enhancement, and the localization length of the bound state were obtained exactly in the continuous limit. The gap enhancement obeys a power law with exponent 2/3. We also obtained the dispersion relation of the local triplet excitation for the entire phase space. The method and the obtained results are common to other frustrated double spin-chain systems, such as the one-dimensional ${\mathrm{J}}_{1}$-${\mathrm{J}}_{2}$ model, or the frustrated ladder model.

DOI: 10.1103/physrevb.55.14413

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arXiv:cond-mat/9612066 · Physical review. B, Condensed matter

Behavior of a frustrated quantum spin chain with bond dimerization

Tota Nakamura, Satoshi Takada

Published Jun 1, 1997• 5 citations

arXiv HTML PDF

Abstract

{\mathrm{J}}_{1}

{\mathrm{J}}_{2}

model, or the frustrated ladder model.

DOI: 10.1103/physrevb.55.14413

Related papers

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Tetsuya Nakamura, Satoshi Takada, Kazuki Okamoto +1 more

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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

{\mathrm{J}}_{1}

{\mathrm{J}}_{2}

model, or the frustrated ladder model.

We study the finite-size behavior of the low-lying excitations of spin-

\frac{1}{2}

Heisenberg chains with dimerization and next-to-nearest neighbors interaction

{J}_{2}

. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where

L

is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At

{J}_{2}={J}_{2c}

, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

{t}_{1,2,3}

{J}_{1,2,3}

model a diagrammatic approach is presented that employs a mapping of the spin Hamiltonian onto a pseudofermion bond-boson model. Results for the single-hole spectrum are detailed. A finite quasiparticle weight is observed and studied for a variety of system parameters. A comparison with existing exact-diagonalization data is performed and good agreement is found.

We perform a systematic investigation on an asymmetric zigzag spin ladder with interleg exchange

{J}_{1}

and different exchange integrals ${J}_{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}$ on both legs. In the weak frustration limit, the spin model can be mapped to a revised double frequency sine-Gordon model by using bosonization. Renormalization-group analysis shows that the Heisenberg critical point flows to an intermediate-coupling fixed point with gapless excitations and a vanishing spin velocity. When the frustration is large, a spin gap opens and a dimer ground state is realized. Fixing

{J}_{2}{=J}_{1}/2,

we find, as a function of $\ensuremath{\delta},$ a continuous manifold of Hamiltonians with dimer product ground states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model. While the ground state is independent of the alternating next-nearest-neighbor exchange $\ensuremath{\delta},$ the gap size of excitations is found to decrease with increasing $\ensuremath{\delta}.$ We also extend our study to a two-dimensional double layer model with an exactly known ground state.

{\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.