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arXiv:1201.2030 · Physical Review B

Mass ratio of elementary excitations in frustrated antiferromagnetic chains with dimerization

Shintaro Takayoshi, Masaki Oshikawa

Published Oct 11, 2012• 2 citations
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Abstract

Excitation spectra of $S=1/2$ 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 $S=1/2$ chain. The $\text{SU}(2)$-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 $\text{SU}(2)$-triplet and a $\text{SU}(2)$-singlet of elementary excitation, and the mass ratio $r$ of the singlet to the triplet is $\sqrt{3}$. However, our numerical calculation with the infinite time-evolving block decimation method shows that $r$ depends on the frustration (next-nearest-neighbor coupling) and is generally different from $\sqrt{3}$. 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 $r=\sqrt{3}$ holds. We derive the mass ratio $r$ 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 $\sqrt{3}$ 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.

DOI
10.1103/physrevb.86.144408

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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 \ensuremathδ=\ensuremathδc\ensuremath{\delta}={\ensuremath{\delta}}_{\mathrm{c}}\ensuremathδ=\ensuremathδc​, where \ensuremathδc=0{\ensuremath{\delta}}_{\mathrm{c}}=0\ensuremathδc​=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 \ensuremathδ\ensuremath∼\ensuremathδc\ensuremath{\delta}\ensuremath{\sim}{\ensuremath{\delta}}_{c}\ensuremathδ\ensuremath∼\ensuremathδc​. However, such a region is extremely small and would be difficult to observe numerically.
We study the finite-size behavior of the low-lying excitations of spin-12\frac{1}{2}21​ Heisenberg chains with dimerization and next-to-nearest neighbors interaction J2{J}_{2}J2​. 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 LLL is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At J2=J2c{J}_{2}={J}_{2c}J2​=J2c​, 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.
Dynamics of S=1S=1S=1 antiferromagnetic bond-alternating chains in the dimer phase, in the vicinity of the critical point with the Haldane phase, is studied by a field theoretical method. This model is considered to represent the compound Ni(C9{}_{9}9​H24{}_{24}24​N4{}_{4}4​)(NO2{}_{2}2​)ClO4{}_{4}4​ (abbreviated as NTENP). We derive a sine-Gordon (SG) field theory as a low-energy effective model of this system, starting from a Tomonaga-Luttinger liquid at the critical point. Using the exact solution of the SG theory, we give a field theoretical picture of the low-energy excitation spectrum of NTENP. Results derived from our picture are in a good agreement with results of inelastic neutron scattering experiments on NTENP and numerical calculation of the dynamical structure factor. Furthermore, on the basis of the obtained theoretical picture, we predict that the sharp peaks corresponding to a single elementary excitation are absent in the Raman scattering spectrum of NTENP in contrast to the inelastic neutron scattering spectrum.
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