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arXiv:1802.10281 · Physical review. B./Physical review. B

Spin-one bilinear-biquadratic model on a star lattice

Hyun-Yong Lee, Naoki Kawashima

Published May 21, 2018• 15 citations
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Abstract

We study the ground-state phase diagram of the $S=1$ bilinear-biquadratic model (BLBQ) on the star lattice with the state-of-art tensor network algorithms. The system has four phases: the ferromagnetic, antiferromagnetic, ferroquadrupolar, and spin-liquid phases. The phases and their phase boundaries are determined by examining various local observables, correlation functions, and transfer matrices exhaustively. The spin-liquid phase, which is the first quantum disordered phase found in the two-dimensional BLBQ model, is gapped and devoid of any conventional long-range order. It is also characterized by fixed-parity virtual bonds in the tensor network formalism, analogous to the Haldane phase, while the parity varies depending on the location of the bond.

DOI
10.1103/physrevb.97.205123

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We study the ground-state phase diagram of the S=1S=1S=1 bilinear-biquadratic model (BLBQ) on the star lattice with the state-of-art tensor network algorithms. The system has four phases: the ferromagnetic, antiferromagnetic, ferroquadrupolar, and spin-liquid phases. The phases and their phase boundaries are determined by examining various local observables, correlation functions, and transfer matrices exhaustively. The spin-liquid phase, which is the first quantum disordered phase found in the two-dimensional BLBQ model, is gapped and devoid of any conventional long-range order. It is also characterized by fixed-parity virtual bonds in the tensor network formalism, analogous to the Haldane phase, while the parity varies depending on the location of the bond.
Making use of infinite projected entangled pair states, we investigate the ground-state phase diagram of the nearest-neighbor spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. In agreement with previous studies, we find the ferromagnetic, ${120}^{\ensuremath{\circ}}$ magnetically ordered, ferroquadrupolar and antiferroquadrupolar phases, and confirm that all corresponding phase transitions are first order. Moreover, we provide an accurate estimate of the location of the ferroquadrupolar to ${120}^{\ensuremath{\circ}}$ magnetically ordered phase transition, thereby fully establishing the phase diagram. Also, we do not encounter any signs of the existence of a quantum paramagnetic phase. In particular, contrary to the equivalent square lattice model, we demonstrate that on the triangular lattice the one-dimensional Haldane phase does not reach all the way up to the two-dimensional limit. Finally, we investigate the possibility of an intermediate partially magnetic partially quadrupolar phase close to $\ensuremath{\theta}=\ensuremath{\pi}/2$, and we show that, also contrary to the square lattice case, this phase is not present on the triangular lattice.
Frustrated quantum antiferromagnetism is an important physical phenomena leading to many of the key properties of quantum matter, such as quantum spin liquids and topological order. Here, the authors employ novel techniques based on tensor network methods for the triangle-honeycomb lattice in order to compute ground-state properties of a frustrated quantum antiferromagnet on the star lattice, with high accuracy, in the thermodynamic limit. Importantly, their findings contradict some of the existing results in the literature on this model. In particular, they find a new resonating valence-bond solid (VBS) phase that had not been observed before, as well as a different VBS structure at the isotropic point than the one previously conjectured by other methods. Moreover, by using tensor network techniques, they also determine the existence of a continuous quantum phase transition in the system between different VBS orders.
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 have precisely determined the ground state phase diagram of the quantum spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice using the tensor renormalization group method. We find that the ferromagnetic, ferroquadrupolar, and a large part of the antiferromagnetic phases are stable against quantum fluctuations. However, around the phase where the ground state is antiferroquadrupolar ordered in the classical limit, quantum fluctuations suppress completely all magnetic orders, leading to a plaquette order phase which breaks the lattice symmetry but preserves the spin SU(2) symmetry. On the evidence of our numerical results, the quantum phase transition between the antiferromagnetic phase and the plaquette phase is found to be either a direct second order or a very weak first order transition.
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.