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arXiv:cond-mat/0608475 · Physical Review B

Gapless points of dimerized quantum spin chains: Analytical and numerical studies

V. Ravi Chandra, Diptiman Sen, Naveen Surendran

Published Nov 21, 2006• 2 citations
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Abstract

We study the locations of the gapless points which occur for quantum spin chains of finite length (with a twisted boundary condition) at particular values of the nearest-neighbor dimerization, as a function of the spin $S$ and the number of sites. For strong dimerization and large values of $S$, a tunneling calculation reproduces the same results as those obtained from more involved field theoretic methods using the nonlinear $\ensuremath{\sigma}$-model approach. A different analytical calculation of the matrix element between the two N\'eel states gives a set of gapless points; for strong dimerization, these differ significantly from the tunneling values. Finally, the exact diagonalization method for a finite number of sites yields a set of gapless points which are in good agreement with the N\'eel state calculations for all values of the dimerization, but the agreement with the tunneling values is not very good even for large $S$. This raises questions about possible corrections to the tunneling results.

DOI
10.1103/physrevb.74.184424

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We study the locations of the gapless points which occur for quantum spin chains of finite length (with a twisted boundary condition) at particular values of the nearest-neighbor dimerization, as a function of the spin SSS and the number of sites. For strong dimerization and large values of SSS, a tunneling calculation reproduces the same results as those obtained from more involved field theoretic methods using the nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model approach. A different analytical calculation of the matrix element between the two N'eel states gives a set of gapless points; for strong dimerization, these differ significantly from the tunneling values. Finally, the exact diagonalization method for a finite number of sites yields a set of gapless points which are in good agreement with the N'eel state calculations for all values of the dimerization, but the agreement with the tunneling values is not very good even for large SSS. This raises questions about possible corrections to the tunneling results.
We establish exact results for coupled spin-1/2 chains for special values of the four-spin interaction VVV and dimerization parameter $\ensuremath{\delta}$. The first exact result is at $\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2$. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with $V\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}4$ and gives evidence that the system has a spontaneously dimerized ground state. Numerical diagonalization and bosonization techniques indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\ensuremath{\le}V<\ensuremath{-}2$.
We establish exact results for the one-dimensional spin-orbital model for special values of the four-spin interaction VVV and dimerization parameter δδδ. The first exact result is at δ=1/2δ=1/2δ=1/2 and V=−2V=-2V=−2. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with V=−4V=-4V=−4 and gives evidence that the system has a spontaneously dimerized ground state. Numerical results indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\leq V
Phases and phase transitions in a dimerized spin-12\mathbf{\frac{1}{2}}21​ XXZ chain
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We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-1/21/21/2 Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.
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.