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arXiv:cond-mat/0503539 · Physical Review B

Scaling of excitations in dimerized and frustrated spin- 12 chains

Davide Controzzi, C. Degli Esposti Boschi, F. Ortolani, Stefano Pasini

Published Nov 17, 2005• 4 citations
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Abstract

We study the finite-size behavior of the low-lying excitations of spin-$\frac{1}{2}$ Heisenberg chains with dimerization and next-to-nearest neighbors interaction ${J}_{2}$. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where $L$ is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At ${J}_{2}={J}_{2c}$, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.

DOI
10.1103/physrevb.72.172409

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Excitation spectra of S=1/2S=1/2S=1/2 and 1 frustrated Heisenberg antiferromagnetic chains with bond alternation (explicit dimerization) are studied using a combination of analytical and numerical methods. The system undergoes a dimerization transition at a critical bond alternation parameter $\ensuremath{\delta}={\ensuremath{\delta}}_{\mathrm{c}}$, where ${\ensuremath{\delta}}_{\mathrm{c}}=0$ for the S=1/2S=1/2S=1/2 chain. The SU(2)\text{SU}(2)SU(2)-symmetric sine-Gordon theory is known to be an effective field theory of the system except at the transition point. The sine-Gordon theory has a SU(2)\text{SU}(2)SU(2)-triplet and a SU(2)\text{SU}(2)SU(2)-singlet of elementary excitation, and the mass ratio rrr of the singlet to the triplet is 3\sqrt{3}3​. However, our numerical calculation with the infinite time-evolving block decimation method shows that rrr depends on the frustration (next-nearest-neighbor coupling) and is generally different from 3\sqrt{3}3​. This can be understood as an effect of marginal perturbation to the sine-Gordon theory. In fact, at the critical frustration separating the second-order and first-order dimerization transitions, the marginal operator vanishes and r=3r=\sqrt{3}r=3​ holds. We derive the mass ratio rrr analytically using form-factor perturbation theory combined with a renormalization-group analysis. Our formula agrees well with the numerical results, confirming the theoretical picture. The present theory also implies that, even in the presence of a marginally irrelevant operator, the mass ratio approaches 3\sqrt{3}3​ in the very vicinity of the second-order dimerization critical point $\ensuremath{\delta}\ensuremath{\sim}{\ensuremath{\delta}}_{c}$. However, such a region is extremely small and would be difficult to observe numerically.
We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin S=1/2S=1/2S=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 CuGeO3,{\mathrm{CuGeO}}_{3},CuGeO3​, NaV2O5,{\mathrm{NaV}}_{2}{\mathrm{O}}_{5},NaV2​O5​, and (VO)2P2O7(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}(VO)2​P2​O7​.
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)(S=0)(S=0) and triplet (S=1)(S=1)(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 study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.
Using the Bond-Operator Technique (BOT), we have studied the low energy excitation spectrum of a frustrated dimerized antiferromagnetic Heisenberg chain. In particular, we have compared our analytical results with previous Exact Diagonalization (ED) data. Qualitatively, the BOT results are in good agreement with the ED data. And even a very good quantitative agreement is obtained in some parameter region. It is clearly shown that there is only one elementary excitation branch (lowest triplet branch) and that the two other well defined excitations which appear below the continuum, one singlet and one triplet, are bound states of two elementary triplets.