Journal of Physics Condensed Matter

Analytical approach to dimerized and frustrated Heisenberg chains

Xue‐Fan Jiang, Haoxin Chen, D. Y. Xing, Jiang Dong

Published Jul 12, 2001• 6 citations

Abstract

We present a self-consistent quantum-theoretical analysis of the ground-state energy and low-energy excitation spectrum of a dimerized and frustrated Heisenberg chain by the bosonization representation in the continuum-limit approach. The frustration effects on the ground-state energy and energy gap are investigated for α>αc where α is the frustration parameter and αc is its critical value, above which a frustration-induced energy gap is opened. It is shown that as α increases, the ground-state energy decreases and the energy gap increases. We find that the dimerization dependence of the ground-state energy and energy gap for small dimerization obeys the Cross-Fisher power law only at α = αc, but departs significantly from it when α>αc. The present results are in good agreement with numerical analyses from the density-matrix renormalization-group and exact diagonalization methods. The relevance of our calculated results to recent experiments on the spin-Peierls compounds CuGeO3 is also discussed.

DOI: 10.1088/0953-8984/13/30/307

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Journal of Physics Condensed Matter

Analytical approach to dimerized and frustrated Heisenberg chains

Xue‐Fan Jiang, Haoxin Chen, D. Y. Xing, Jiang Dong

Published Jul 12, 2001• 6 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/13/30/307

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We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet $(S=0)$ and triplet $(S=1)$ modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet $(S=0)$ and triplet $(S=1)$ modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.

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Ground state of a Heisenberg chain with next-nearest-neighbor bond alternation

Luca Capriotti, Federico Becca, Sandro Sorella +1 more

We investigate the ground-state properties of the spin-half ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg chain with a next-nearest-neighbor spin-Peierls dimerization using conformal field theory and Lanczos exact diagonalizations. In agreement with the results of a recent bosonization analysis by Sarkar and Sen [Phys. Rev. B $65,$ 172408 (2002)], we find that for small frustration ${(J}_{2}{/J}_{1})$ the system is in a Luttinger spin-fluid phase, with gapless excitations, and a finite spin-wave velocity. In the regime of strong frustration the ground state is spontaneously dimerized and the bond alternation reduces the triplet gap, leading to a slight enhancement of the critical point separating the Luttinger phase from the gapped one. An accurate determination of the phase boundary is obtained numerically from the study of the excitation spectrum.We investigate the ground-state properties of the spin-half ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg chain with a next-nearest-neighbor spin-Peierls dimerization using conformal field theory and Lanczos exact diagonalizations. In agreement with the results of a recent bosonization analysis by Sarkar and Sen [Phys. Rev. B $65,$ 172408 (2002)], we find that for small frustration ${(J}_{2}{/J}_{1})$ the system is in a Luttinger spin-fluid phase, with gapless excitations, and a finite spin-wave velocity. In the regime of strong frustration the ground state is spontaneously dimerized and the bond alternation reduces the triplet gap, leading to a slight enhancement of the critical point separating the Luttinger phase from the gapped one. An accurate determination of the phase boundary is obtained numerically from the study of the excitation spectrum.

arXiv:cond-mat/0304302Physical review. B, Condensed matterMay 12, 200310 citations

We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin

S=1/2

Heisenberg chain. In particular, we show that in the commensurate spin-Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory. We discuss the relevance of our data in connection with recent experiments on

{\mathrm{CuGeO}}_{3},

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

and

(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}

In conformal field theory, key properties of spin-

\frac{1}{2}

chains, such as the ground state energy per site and the excitation gap, scale with dimerization $\ensuremath{\delta}$ as ${\ensuremath{\delta}}^{\ensuremath{\alpha}}$ with known exponents $\ensuremath{\alpha}$ and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest

(J=1)

and next-nearest-neighbor interactions

({J}_{2})

, for

{J}_{2\mathrm{c}}=0.2411

. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of $\ensuremath{\alpha}$, and the scaling relation is valid up to the physically realized range, $\ensuremath{\delta}\ensuremath{\sim}0.1$. However, chains with logarithmic corrections

({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})

are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately

3∕4

2∕3

for the spin gap and from approximately

3∕2

4∕3

for the energy per site, and error bars in the exponent also decrease as

{J}_{2}

approaches to

{J}_{2\mathrm{c}}

(S=0)

and triplet

(S=1)

modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.

65,

172408 (2002)], we find that for small frustration

{(J}_{2}{/J}_{1})

the system is in a Luttinger spin-fluid phase, with gapless excitations, and a finite spin-wave velocity. In the regime of strong frustration the ground state is spontaneously dimerized and the bond alternation reduces the triplet gap, leading to a slight enhancement of the critical point separating the Luttinger phase from the gapped one. An accurate determination of the phase boundary is obtained numerically from the study of the excitation spectrum.