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arXiv:0707.1195 · Physical Review B

Spontaneous trimerization in a bilinear-biquadratic S=1 zig-zag chain

Philippe Corboz, Andreas Martin Läuchli, Keisuke Totsuka, Hirokazu Tsunetsugu

Published Dec 12, 2007• 32 citations
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Abstract

Recent theoretical studies raised the possibility of a realization of spin nematic states in the $S=1$ triangular lattice compound ${\mathrm{NiGa}}_{2}{\mathrm{S}}_{4}$. We study the bilinear-biquadratic spin 1 chain in a zig-zag geometry by means of the density matrix renormalization group method and exact diagonalization. We present the phase diagram focusing on antiferromagnetic interactions. Adjacent to the known Haldane--double Haldane and the extended critical phase with dominant spin nematic correlations we find a trimerized phase with a nonvanishing energy gap. We discuss results for different order parameters, energy gaps, correlation functions, and the central charge, and make connection to field theoretical predictions for the phase diagram.

DOI
10.1103/physrevb.76.220404

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Recent theoretical studies raised the possibility of a realization of spin nematic states in the S=1S=1S=1 triangular lattice compound NiGa2S4{\mathrm{NiGa}}_{2}{\mathrm{S}}_{4}NiGa2​S4​. We study the bilinear-biquadratic spin 1 chain in a zig-zag geometry by means of the density matrix renormalization group method and exact diagonalization. We present the phase diagram focusing on antiferromagnetic interactions. Adjacent to the known Haldane--double Haldane and the extended critical phase with dominant spin nematic correlations we find a trimerized phase with a nonvanishing energy gap. We discuss results for different order parameters, energy gaps, correlation functions, and the central charge, and make connection to field theoretical predictions for the phase diagram.
We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as exact diagonalization, density matrix renormalization group, and strong coupling series expansions. For the single chain, we clarify the dominant correlation function in the enigmatic gapless period-three phase for $\ensuremath{\theta}∊(\ensuremath{\pi}∕4,\ensuremath{\pi}∕2)$, which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for $\ensuremath{\theta}∊(\ensuremath{-}3\ensuremath{\pi},4,\ensuremath{-}\ensuremath{\pi}∕2)$ and---surprisingly---seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg S=1S=1S=1 ladder and therefore also with the Haldane phase of isolated chains.
We study the phase diagram of the spin-1 quantum bilinear-biquadratic antiferromagnet on the kagome lattice using exact diagonalization and the density matrix renormalization group algorithm. The SU(3)\mathrm{SU}(3)SU(3)-symmetric point of this model Hamiltonian is a spontaneously trimerized state whose qualitative nature persists even at the Heisenberg point, a finding that contrasts previous proposals. We report the ground state energy per site of the Heisenberg model to be $\ensuremath{-}1.410(2)$ and establish the presence of a spin gap.
The magnetization process of the S = 1 antiferromagnetic chain with the biquadratic interaction is investigated using the numerical diagonalization. As a result, it is found that the quantum spin nematic liquid phase appears below the saturation magnetization for sufficiently large negative biquadratic interaction. The ground state phase diagram is also presented. This result is consistent with the previous density matrix renormalization group study. In addition the present study confirms that the two magnon excitation is gapless, while the single magnon one is gapped in the spin nematic liquid phase.
We investigate a highly frustrated spin-1 model on the triangular lattice, with nearest- and next-nearest-neighbor antiferromagnetic $S\ifmmode\cdot\else\textperiodcentered\fi{}S$ interactions and nearest-neighbor ${(S\ifmmode\cdot\else\textperiodcentered\fi{}S)}^{2}$ interactions. Using the density matrix renormalization group technique, we find three magnetically ordered phases, namely ${120}^{\ensuremath{\circ}}$ spiral order, stripe order, and tetrahedral order, as well as two spin nematic phases: ferroquadrupolar and antiferroquadrupolar. While our data could be consistent with a spin liquid phase between the ${120}^{\ensuremath{\circ}}$ spiral and antiferroquadrupolar orders, the more likely scenario is a direct continuous transition between these two orders.
Nematic order is an exotic property observed in several strongly correlated systems, such as iron-based superconductors. Using large-scale density matrix renormalization group (DMRG) techniques, we study at zero temperature the nematic spin liquid that competes with spin dipolar and quadrupolar orders. We use these nematic orders to characterize different quantum phases and quantum phase transitions. More specifically, we study a spin-1 bilinear-biquadratic Heisenberg model on the square lattice with couplings beyond nearest neighbors. We focus on parameter regions around the highly symmetric SU(3)SU(3)SU(3) point where the bilinear and biquadratic interactions are equal. With growing further-neighbor biquadratic interactions, we identify different spin dipolar and quadrupolar orders. We find that the DMRG results for cylindrical geometries correctly detect nematicity in different quantum states and accurately characterize the quantum phase transitions among them. Therefore, spin-driven nematicity, here defined as the spontaneous breaking of the lattice invariance under a ${90}^{\ensuremath{\circ}}$ rotation, is an order parameter which can be studied directly in DMRG calculations in two dimensions in different quantum states.
We investigate the effect of a single-ion anisotropy on the bilinear-biquadratic spin-1 J1-J2 chain, focusing on the quantum phase transitions out of the trimerized phase. Using large-scale density matrix renormalization group simulations, we uncover a rich phase diagram comprising five gapped phases and, remarkably, two critical floating phases. These incommensurate Luttinger liquid phases emerge from the proliferation of non-magnetic domains - 0-states and dimers - within a trimerized background and are confined to the zero magnetization sector, while magnetic excitations remain gapped. We show that the transition between the topological Haldane phase and the floating phases are governed by a composite critical line with central charge c=2, consistent with a coexistence of magnetic and non-magnetic critical modes. Our results shed new light on the long-standing problem of the Haldane-trimerized transition.