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arXiv:0808.0278 · Journal of Physics Conference Series

Basic properties of three-leg Heisenberg tube

Satoshi Nishimoto, Mitsuhiro Arikawa

Published Jan 1, 2009• 2 citations
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Abstract

We study three-leg antiferromagnetic Heisenberg model with the periodic boundary conditions in the rung direction. Since the rungs form regular triangles, spin frustration is induced. We use the density-matrix renormalization group method to investigate the ground state. We find that the spin excitations are always gapped to remove the spin frustration as long as the rung coupling is nonzero. We also visibly confirm spin-Peierls dimerization order in the leg direction. Both the spin gap and the dimerization order are basically enhanced as the rung coupling increases.

DOI
10.1088/1742-6596/145/1/012041

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We study three-leg antiferromagnetic Heisenberg model with the periodic boundary conditions in the rung direction. Since the rungs form regular triangles, spin frustration is induced. We use the density-matrix renormalization group method to investigate the ground state. We find that the spin excitations are always gapped to remove the spin frustration as long as the rung coupling is nonzero. We also visibly confirm spin-Peierls dimerization order in the leg direction. Both the spin gap and the dimerization order are basically enhanced as the rung coupling increases.
Abstract. We study three-leg antiferromagnetic Heisenberg model with the periodic boundary conditions in the rung direction. Since the rungs form regular triangles, spin frustration is induced. We use the density-matrix renormalization group method to investigate the ground state. We find that the spin excitations are always gapped to remove the spin frustration as long as the rung coupling is nonzero. We also visibly confirm spin-Peierls dimerization order in the leg direction. Both the spin gap and the dimerization order are basically enhanced as the rung coupling increases. 1.
Using the (dynamical) density-matrix renormalization group method, we study the low-energy physics of three-leg antiferromagnetic Heisenberg model where the periodic boundary conditions are applied in the rung direction. We confirm that the spin excitations are always gapped as long as the intraring couplings form a regular triangle. From precise finite-size-scaling analyses of the spin gap and dimerization order parameter, we also find that the spin gap is collapsed by very small asymmetric modulation of the intraring couplings. Moreover, the dynamical spin structure factors on the intraleg and interleg correlations are calculated. It is demonstrated that the low-lying structure of the interleg spectra is particularly affected by the asymmetric modulation.
The antiferromagnetic Heisenberg spin systems on the three-leg ladder are investigated. Periodic boundary condition is imposed in the rung direction. The system has an excitation gap for all antiferromagnetic inter-chain coupling (J⊥>0J_{\perp}>0J⊥​>0). The estimated gap for the strong coupling limit (J⊥/J1→∞J_{\perp}/J_1 \to \inftyJ⊥​/J1​→∞) is 0.28J1J_1J1​. Although the interaction is homogeneous and only nearest-neighbor, the ground states of the system are dimerized and break the translational symmetry in the thermodynamic limit. Introducing the next-nearest neighbor coupling (J2J_2J2​), we can see that the system is solved exactly. The ground state wave function is completely dimer-ordered. Using density matrix renomalization group algorithm, we show numerically that the original model (J2=0J_2=0J2​=0) has the same nature with the exactly solvable model. The ground state properties of the ladder with a higher odd number of legs are also discussed.
The ground-state properties of the three-leg S=32S=\frac{3}{2}S=23​ straight Heisenberg tube are studied using the density-matrix renormalization group method. We find that the spin-excitation gap associated with a spontaneous dimerization opens for the whole coupling regime, as seen in the three-leg S=12S=\frac{1}{2}S=21​ straight Heisenberg tube. However, in contrast to the case of the S=12S=\frac{1}{2}S=21​ straight tube, the gap increases very slowly with increasing the rung coupling, and its size is only a few percent\text{percent}percent or less of the leg exchange interaction in the weak- and intermediate-coupling regimes. We thus argue that, unless the rung coupling is substantially larger than the leg coupling, the gap may be quite hard to be observed experimentally. We also calculate the quantized Berry phase to show that there exist three kinds of valence-bond-solid states depending on the ratio of leg and rung couplings.
Using the density-matrix renormalization group and exact diagonalization methods the ground-state properties of the S = 3/2 three-leg Heisenberg tube with leg ( J ||) and rung ( J ⊥) exchange couplings are studied. We find that the spin-excitation gap associated with a spontaneous dimerization opens in the entire region of the coupling strength, as seen in the S = 1/2 three-leg tube. However, in contrast to the case of the S = 1/2 tube, the gap develops very slowly with increasing the rung coupling strength in the weak-coupling regime and its size remains only about 5 % of the leg coupling strength even at J ⊥/ J || = 5. We also calculate the quantized Berry phase and suggest that there exist three types of the valence-bond-solid states depending on the ratio of leg and rung coupling strengths. Moreover, an effective model is derived by the perturbative expansion to check the numerical results in the strong rung-coupling limit.
We study the ground-state properties of three-leg S = 1/2 Heisenberg tube with antiferromagnetic interleg couplings J ⊥ (> 0). The spin gap, dimerization-order parameter, and Berry phase are calculated with the density-matrix renormalization group method. We find that the system is spontaneously dimerized and all the spin excitations are gapped whether intraleg couplings J || are antiferromagnetic or ferromagnetic. Also, the phase diagram is divided into three regimes according to topological configuration of the spin-singlet pairs: (i) Haldane-gap-like regime for J ⊥ / J || < 0, (ii) intraleg-singlet regime 0 < J ⊥ J || ≤ 5, and (iii) intrarung-singlet regime for J ⊥ J || ≥ 5.