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arXiv:cond-mat/0612165 · Physical Review B

Coexistence of vector chiral order and Tomonaga-Luttinger liquid in the frustrated three-leg spin tube in a magnetic field

Masahiro Sato

Published May 4, 2007• 29 citations
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Abstract

The frustrated three-leg antiferromagnetic spin-$\frac{1}{2}$ tube with a weak interchain coupling in a magnetic field is investigated by means of Abelian bosonization techniques. It is clearly shown that a vector chiral long-range order and a one-component Tomonaga-Luttinger liquid coexist in a wide magnetic-field region from a state with a small magnetization to a nearly saturated one. The chiral order is predicted to still survive in the intermediate plateau state. We further predict that (even) when the strength of one bond in the three rung couplings is decreased (increased), an Ising-type quantum phase transition takes place and the chirality vanishes (no singular phenomena occur and the chiral order is maintained). Even without magnetic fields, the chiral order would also be present if the spin tube possesses easy-plane anisotropy.

DOI
10.1103/physrevb.75.174407

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We study the ground state phase diagram of the twisted three-leg spin tube in magnetic fields by the density matrix renormalization group (DMRG) method. The twisted spin tube is composed of triangular unit cells and possesses strong quantum fluctuations under geometrical frustration. We apply the sine square deformation method to remove the strong boundary effects and obtain smooth magnetization curves without steps of finite systems. With the analysis of the magnetization curves and correlation functions we determine the ground state phase diagram consisting of (a) a Tomonaga-Luttinger (TL) liquid characterized by spin-32\frac{3}{2}23​ Heisenberg model, (b) 3-sublattice state named UUD with 1/3 magnetization and (c) TL-liquid of massless chirality with 1/3 magnetization plateau, (d) TL-liquid of massless spin mode with or without chirality quasi long-range order.
We study the magnetization plateau state of the three-leg spin-12\frac{1}{2}21​ tube in the strong rung coupling region, where S3{S}_{3}S3​ symmetry breaking and the low-energy chirality degree of freedom play crucial roles. On the basis of the effective chirality model and density matrix renormalization group, we clarify that, as the leg coupling increases, the chirality liquid with gapless nonmagnetic excitations, the spin-imbalance phase, and the vector-spin-chirality ordered phase emerge without closing the plateau spin gap. The relevance of these results to experiments is also discussed.
We study the ground state phase diagram of a frustrated spin tube in a strong external magnetic field. This model can be viewed as two coupled zigzag spin chains, or as a two-leg spin ladder with frustrating next-nearest-neighbor couplings along the legs, and its study is motivated by the physics of such materials as sulfolane-Cu2Cl4{\mathrm{Cu}}_{2}{\mathrm{Cl}}_{4}Cu2​Cl4​ and BiCu2PO6{\mathrm{BiCu}}_{2}{\mathrm{PO}}_{6}BiCu2​PO6​. In magnetic fields right below the saturation, the system can be effectively represented as a dilute gas of two species of bosonic quasiparticles that correspond to magnons with inequivalent incommensurate momenta at two degenerate minima of the magnon dispersion. Using the method previously proposed and tested for frustrated spin chains, we calculate effective interactions in this two-component Bose gas. On this basis, we establish the phase diagram of nearly saturated frustrated spin tube, which is shown to include the two-component Luttinger liquid, two types of vector chiral phases, and phases whose physics is determined by the presence of bound magnons. We study the phase diagram of the model numerically by means of the density matrix renormalization group technique, and find a good agreement with our analytical predictions.
We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings ${J}_{\ensuremath{\perp}}^{\ensuremath{'}}$ while the other two are kept constant ${J}_{\ensuremath{\perp}}$. We focus on the strong rung-coupling regime ${J}_{\ensuremath{\perp}}\ensuremath{\gg}{J}_{\ensuremath{\parallel}}$, where ${J}_{\ensuremath{\parallel}}$ is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point $|{J}_{\ensuremath{\perp}}^{\ensuremath{'}}\ensuremath{-}{J}_{\ensuremath{\perp}}|/{J}_{\ensuremath{\parallel}}\ensuremath{\sim}0.283$ in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.
Uniform‐field effects for frustrated odd‐leg integer‐spin tubes (cylinder type spin systems) are investigated in the weak interchain‐coupling regime. We predict that, as the field exceeds the spin gap, a two‐component Tomonaga‐Luttinger liquid (TLL) appears due to the condensation of the doubly degenerate lowest magnons. Furthermore, it is argued that when the uniform field is so strong that the second lowest magnons are also condensed, the two‐component TLL is destroyed and a new one‐component TLL emerges. This quantum phase transition may be detected as a magnetization cusp.