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Physical review. B, Condensed matter

Spin-Peierls transitions inS>1/2 Heisenberg chains

Dandan Guo, Tom Kennedy, S. Mazumdar

Published May 1, 1990• 48 citations
Publisher

Abstract

Exact diagonalizations are done to calculate the energy gain from bond alternation in finite spin chains as a function of chain length. Calculations are done for S=1/2, 1, 3/2, and 2. Qualitative differences are found between half-integer and interger spins. Our results (i) indicate the absence of the spin-Peierls transition in systems with integer spins and (ii) suggest a transition in systems with half-integer spins.

DOI
10.1103/physrevb.41.9592

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Exact diagonalizations are done to calculate the energy gain from bond alternation in finite spin chains as a function of chain length. Calculations are done for S=1/2, 1, 3/2, and 2. Qualitative differences are found between half-integer and interger spins. Our results (i) indicate the absence of the spin-Peierls transition in systems with integer spins and (ii) suggest a transition in systems with half-integer spins.
Finite-size scaling calculations for deformable Heisenberg chains with spin value S=(1/2, 1, (3/2, and 2 indicate that as far as the spin-Peierls transition is concerned, the model with any spin S>(1/2 behaves differently from the S=(1/2 situation. For S=(1/2 the ground state of the isotropic antiferromagnetic or planar models is dimerized, irrespective of the strength of the elastic constant. On the other hand, from the extrapolation of finite-chain results it seems that for any higher-spin value the dimerized and uniform phases are separated by a finite critical value of the elastic constant, and the transition is discontinuous.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.
We study the spin-Peierls transition in a Heisenberg spin chain coupled to optical bond phonons. Quantum Monte Carlo results for systems with up to N=256N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}256N=256 spins show unambiguously that the transition occurs only when the spin-phonon coupling $\ensuremath{\alpha}$ exceeds a critical value ${\ensuremath{\alpha}}_{c}$. Using sum rules, we show that the phonon spectral function has divergent (for $N\ensuremath{\rightarrow}\ensuremath{\infty}$) weight extending to zero frequency for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. The phonon correlations decay with distance rrr as 1/r1/r1/r. This behavior is characteristic for all $0<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$ and the $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}$ phonon does not soften (to zero frequency) at $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{c}$.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.
According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.