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arXiv:cond-mat/0311082 · Physical Review B

Spin nematics correlations in bilinear-biquadratic S=1 spin chains

Andreas Martin Läuchli, Guido Schmid, Simon Trebst

Published Oct 26, 2006• 180 citations
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Abstract

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 $S=1$ ladder and therefore also with the Haldane phase of isolated chains.

DOI
10.1103/physrevb.74.144426

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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 \ensuremathθ∊(\ensuremathπ∕4,\ensuremathπ∕2)\ensuremath{\theta}∊(\ensuremath{\pi}∕4,\ensuremath{\pi}∕2)\ensuremathθ∊(\ensuremathπ∕4,\ensuremathπ∕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 \ensuremathθ∊(\ensuremath−3\ensuremathπ,4,\ensuremath−\ensuremathπ∕2)\ensuremath{\theta}∊(\ensuremath{-}3\ensuremath{\pi},4,\ensuremath{-}\ensuremath{\pi}∕2)\ensuremathθ∊(\ensuremath−3\ensuremathπ,4,\ensuremath−\ensuremathπ∕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 numerically the ground-state phase diagram of the bilinear-biquadratic spin-1 chain near the ferromagnetic instability point, where the existence of a gapped or gapless nondimerized quantum nematic phase has been suggested. Our results, obtained by a highly accurate density-matrix renormalization-group calculation, are consistent with the view that the order parameter characterizing the dimer phase vanishes only at the point where the system becomes ferromagnetic, although the existence of a gapped or gapless nondimerized phase in a very narrow parameter range between the ferromagnetic and the dimerized regimes cannot be ruled out.
We systematically study the effects of higher-order interactions on the S=32,1S=\frac{3}{2},1S=23​,1 orthogonal dimer chains using exact diagonalization and density matrix renormalization group. Due to frustration and higher spin, there are rich quantum phases, including three Haldane phases, two gapless phases, and several magnetically ordered phases. To characterize these phases and their phase transitions, we study various physical quantities such as energy gap, energy-level crossing, fidelity susceptibility, spin correlation, entanglement spectrum, and central charge. According to our calculations, the biquadratic term can enhance the Haldane phase regions. In particular, we numerically identify that a Haldane phase in S=32S=\frac{3}{2}S=23​ case is adiabatically connected to the exact AKLT (Affleck, Kennedy, Lieb, and Tasaki) point when adding a bicubic term. Our study on the orthogonal dimer model, which is a 1D version of the Shastry-Sutherland model, provides insights into understanding the possible S=32,1S=\frac{3}{2},1S=23​,1 Haldane phases in quasi-1D and 2D frustrated magnetic materials.
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 the Berry phase on quadrupoles that occur, for example, in the low-energy description of spin models. Specifically, we study here the one-dimensional bilinear-biquadratic spin-one model. An open question for many years about this model is whether it has a nondimerized fluctuating nematic phase. The dimerization has recently been proposed to be related to Berry phases of the quantum fluctuations. We use an effective low-energy description to calculate the scaling of the dimerization according to this theory and then verify the predictions using large scale density-matrix renormalization group simulations, giving good evidence that the state is dimerized all the way up to its transition into the ferromagnetic phase. We furthermore discuss the multiplet structure found in the entanglement spectrum of the ground state wave functions.
We study frustrated spin-1/2 dimer systems in two dimensions with a bilayer structure, where spins are ferromagnetically coupled in dimers. Our model includes frustrated two-spin exchange interactions as well as four-spin interactions. We pay particular attention to the spin nematic phase, which does not exhibit any magnetic (spin-dipole) order but has a spin-quadrupolar long-range order. Employing a perturbation calculation, a mean-field approximation, and a numerical many-variable variational Monte Carlo method, we determine ground-state phase diagrams on various two-dimensional lattices. It is found that the model exhibits the spin nematic phase with ferroquadrupolar order in a wide parameter region, in addition to conventional magnetically ordered phases. In particular, it is shown that even when the four-spin interactions are absent, frustrated two-spin exchange interactions can realize the spin nematic phase as a result of strong interdimer correlations. It is also found that the phase transitions between the spin nematic phase and antiferromagnetic phases can be continuous. Furthermore, we present some exact arguments that various phases including the spin nematic phase and the vector chiral (p-type nematic) phase emerge from an SU(4) symmetric point in the model by the addition of appropriate perturbative interactions. The spin nematic phase generated from the SU(4) point is connected with the spin nematic phase found numerically in the system with only two-spin interactions.