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arXiv:1805.00354 · Physical review. B./Physical review. B

Ground-state study of the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice using tensor networks

Ido Niesen, Philippe Corboz

Published Jun 27, 2018• 38 citations
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Abstract

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.

DOI
10.1103/physrevb.97.245146

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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∘{120}^{\ensuremath{\circ}}120\ensuremath∘ 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∘{120}^{\ensuremath{\circ}}120\ensuremath∘ 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 \ensuremathθ=\ensuremathπ/2\ensuremath{\theta}=\ensuremath{\pi}/2\ensuremathθ=\ensuremathπ/2, and we show that, also contrary to the square lattice case, this phase is not present on the triangular lattice.
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.
Infinite projected entangled pair states simulations of the S=1S=1S=1 bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled S=1S=1S=1 chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1{J}_{1}J1​ and the next-nearest-neighbor J2{J}_{2}J2​ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly=10{L}_{y}=10Ly​=10, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.
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.
We study the quantum phase diagram of the spin-1/21/21/2 nearest-neighbor Heisenberg model on the maple-leaf lattice using infinite projected entangled pair states (iPEPS) combined with a corner transfer matrix renormalization group scheme adapted to C3C_3C3​-symmetric lattices. Focusing on the fully antiferromagnetic JJJ-JdJ_dJd​ model with Jh=Jt:=JJ_h = J_t := JJh​=Jt​:=J, we map out the ground-state phase diagram as a function of the dimer coupling JdJ_dJd​. Our results show that the system hosts only two phases: a magnetically ordered canted-120∘120^\circ120∘ phase and an exact dimer singlet product phase. We identify a first-order transition between these two phases at Jd/J≈1.45J_d/J \approx 1.45Jd​/J≈1.45. Within the magnetically ordered phase, we observe small but finite magnetic moments. We also resolve the quantum renormalization of the canting angle, which deviates from the classical prediction over almost the entire magnetically ordered phase.