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Journal of Physics A Mathematical and General

Critical properties of the spin- Heisenberg chain with frustration and bond alternation

Kiyomi Okamoto, Tota Nakamura

Published Sep 21, 1997• 6 citations
Publisher

Abstract

We study the Heisenberg chain with frustration and alternation which is expressed by , both analytically and numerically. We focus on the case, where there is no marginal operator which brings about the logarithmic corrections in various quantities. By using the bosonization method, we calculate the energy gap, the change in the ground-state energy due to the alternation (so-called energy gain), the spin correlation and the string correlation (not only their exponents but also their amplitudes), and compare them with the results of the numerical diagonalization for finite systems. We point out the existence of the logarithmic correction in the energy gain despite the absence of the marginal operator. Taking into account this logarithmic correction, we can obtain a reasonable hyperscaling relation between the critical exponents of the energy gap and an energy gain from the numerical data.

DOI
10.1088/0305-4470/30/18/012

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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 65,65,65, 172408 (2002)], we find that for small frustration (J2/J1){(J}_{2}{/J}_{1})(J2​/J1​) 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 effect of the marginal operator on the critical exponents of the one-dimensional S=1/2 Heisenberg model, using the relation between the critical exponents and the energy gaps of the finite system. The energy gaps behave as \ensuremath{\Delta}E\ensuremath{\propto}1/L with the logarithmic corrections from the marginally irrelevant operator. The numerical results obtained with the Bethe ansatz are well explained by the two-loop renormalization of the marginal coupling. Thus the discrepancies between the one-loop renormalization prediction and numerical results are resolved, though it is found that the nonuniversal constant of the logarithmic correction in the ground state does not agree with the asymptotic expansion of the Bethe ansatz.
In conformal field theory, key properties of spin-12\frac{1}{2}21​ 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)(J=1)(J=1) and next-nearest-neighbor interactions (J2)({J}_{2})(J2​), for J2c=0.2411{J}_{2\mathrm{c}}=0.2411J2c​=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 (J2<0.2411J)({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})(J2​<0.2411J) are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately 3∕43∕43∕4 to 2∕32∕32∕3 for the spin gap and from approximately 3∕23∕23∕2 to 4∕34∕34∕3 for the energy per site, and error bars in the exponent also decrease as J2{J}_{2}J2​ approaches to J2c{J}_{2\mathrm{c}}J2c​.
One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D≥0. The ground states of this model contain the Haldane state, large-D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.
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.
One dimensional S =1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥0. The ground states of this model contain the Haldane state, large- D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.