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arXiv:cond-mat/9809292 · Physical review. B, Condensed matter

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

Tota Nakamura

Published Mar 1, 1999• 10 citations
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Abstract

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.

DOI
10.1103/physrevb.59.r6589

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Localization length of a free-spin soliton from a nonmagnetic impurity is deduced in a general double-spin-chain model (J0\ensuremath−J1\ensuremath−J2\ensuremath−J3{(J}_{0}{\ensuremath{-}J}_{1}{\ensuremath{-}J}_{2}{\ensuremath{-}J}_{3}(J0​\ensuremath−J1​\ensuremath−J2​\ensuremath−J3​ 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 (J2\ensuremath−J3){(J}_{2}\ensuremath{-}{J}_{3})(J2​\ensuremath−J3​) with an exponent \ensuremath−1/3;\ensuremath{-}1/3;\ensuremath−1/3; \ensuremathξ\ensuremath∼(J2\ensuremath−J3)\ensuremath−1/3.\ensuremath{\xi}\ensuremath{\sim}{(J}_{2}\ensuremath{-}{J}_{3}{)}^{\ensuremath{-}1/3}.\ensuremathξ\ensuremath∼(J2​\ensuremath−J3​)\ensuremath−1/3. This explains why NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​ does not show the antiferromagnetic order, while CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ 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.
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 J1{\mathrm{J}}_{1}J1​-J2{\mathrm{J}}_{2}J2​ model, or the frustrated ladder model.
The effects of spin-1 impurities introduced in dimerized antiferromagnetic spin chains are studied using Lanczos diagonalization and quantum Monte Carlo simulations. First, the magnitude of the interaction between a Ni ion replacing a Cu ion is derived for the case study system Cu-O-Cu. Then, it is shown that the introduction of S=1S=1S=1 impurities leads to the suppression of the original spin gap of the dimerized system and to the enhancement of the antiferromagnetic correlations close to the impurity. These features are interpreted in terms of a valence bond picture of the ground state of both the pure and the doped system. The similitudes and differences between the underlying mechanisms that produce these effects in the cases of nonmagnetic and S=1S=1S=1 impurities are analyzed. It is stressed that a singlet formed between the spin-1 impurity site and the two spin-1/2 neighbors spins behaves as an effective nonmagnetic vacancy which may be partially responsible for the enhancement of the antiferromagnetic correlations on the whole system. Implications of these results for experiment are also discussed.
The linear spin-12\frac{1}{2}21​ Heisenberg antiferromagnet with exchanges J1{J}_{1}J1​ and J2{J}_{2}J2​ between first and second neighbors has a bond-order wave (BOW) phase that starts at the fluid-dimer transition at J2/J1=0.2411{J}_{2}/{J}_{1}=0.2411J2​/J1​=0.2411 and is particularly simple at J2/J1=1/2{J}_{2}/{J}_{1}=1/2J2​/J1​=1/2. The BOW phase has a doubly degenerate singlet ground state, broken inversion symmetry, and a finite-energy gap Em{E}_{m}Em​ to the lowest-triplet state. The interval 0.4<J2/J1<1.00.4<{J}_{2}/{J}_{1}<1.00.4<J2​/J1​<1.0 has large Em{E}_{m}Em​ and small finite-size corrections. Exact solutions are presented up to N=28N=28N=28 spins with either periodic or open boundary conditions and for thermodynamics up to N=18N=18N=18. The elementary excitations of the BOW phase with large Em{E}_{m}Em​ are topological spin-12\frac{1}{2}21​ solitons that separate BOWs with opposite phase in a regular array of spins. The molar spin susceptibility ${\ensuremath{\chi}}_{M}(T)$ is exponentially small for T⪡EmT⪡{E}_{m}T⪡Em​ and increases nearly linearly with TTT to a broad maximum. J1{J}_{1}J1​ and J2{J}_{2}J2​ spin chains approximate the magnetic properties of the BOW phase of Hubbard-type models and provide a starting point for modeling alkali-tetracyanoquinodimethane salts.
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.
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Spin models that have been proposed to describe dimerized chains, ladders, two-dimensional antiferromagnets, and other compounds are studied here when some spins are replaced by spinless vacancies, such as it occurs by Zn doping. A small percentage of vacancies rapidly destroys the spin gap, and their presence induces enhanced antiferromagnetic correlations near those vacancies. The study is performed with computational techniques which includes Lanczos, world-line Monte Carlo, and the density-matrix renormalization-group methods. Since the phenomenon of enhanced antiferromagnetism is found to occur in several models and cluster geometries, a common simple explanation for its presence may exist. It is argued that the resonating-valence-bond character of the spin correlations at short distances of a large variety of models is responsible for the presence of robust staggered spin correlations near vacancies and lattice edges. The phenomenon takes place regardless of the long distance properties of the ground state, and it is caused by a ``pruning'' of the available spin singlets in the vicinity of the vacancies. The effect produces a broadening of the low-temperature NMR signal for the compounds analyzed here. This broadening should be experimentally observable in the structurally dimerized chain systems $\mathrm{Cu}({\mathrm{NO}}_{3}{)}_{2}\ensuremath{\cdot}2.5{\mathrm{H}}_{2}\mathrm{O}$, CuWO4{\mathrm{CuWO}}_{4}CuWO4​, (VO)2P2O7(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}(VO)2​P2​O7​, and Sr14Cu24O41,{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},Sr14​Cu24​O41​, in ladder materials such as SrCu2O3{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}SrCu2​O3​, in the spin-Peierls systems CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ and NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​, and in several others since it is a universal effect common to a wide variety of models and compounds. In addition, it is argued that the N\'eel order observed in SrCu2O3{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}SrCu2​O3​ upon Zn doping is induced by the local antiferromagnetic order discussed in this paper, enhanced by a favorable ratio between the actual Heisenberg couplings along chains and rungs, as reported in recent experimental literature. Based on this reasoning it is predicted here that other ladder materials such as Zn-doped Cu2(C5H12N2)2Cl4{\mathrm{Cu}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}Cu2​(C5​H12​N2​)2​Cl4​ will not present N\'eel order at small Zn concentrations.