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arXiv:2206.04087 · Physical review. B./Physical review. B

Phase diagram of a bilinear-biquadratic spin-1 model on the triangular lattice from density matrix renormalization group simulations

Aaron Szasz, Chong Wang, Yin-Chen He

Published Sep 6, 2022• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.106.115103

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We investigate a highly frustrated spin-1 model on the triangular lattice, with nearest- and next-nearest-neighbor antiferromagnetic S\ifmmode⋅\else\textperiodcentered\fiSS\ifmmode\cdot\else\textperiodcentered\fi{}SS\ifmmode⋅\else\textperiodcentered\fiS interactions and nearest-neighbor (S\ifmmode⋅\else\textperiodcentered\fiS)2{(S\ifmmode\cdot\else\textperiodcentered\fi{}S)}^{2}(S\ifmmode⋅\else\textperiodcentered\fiS)2 interactions. Using the density matrix renormalization group technique, we find three magnetically ordered phases, namely 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ 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∘{120}^{\ensuremath{\circ}}120\ensuremath∘ spiral and antiferroquadrupolar orders, the more likely scenario is a direct continuous transition between these two orders.
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.
We study the spin-1/21/21/2 Heisenberg model on the triangular lattice with the nearest-neighbor J1>0{J}_{1}>0J1​>0, the next-nearest-neighobr J2>0{J}_{2}>0J2​>0 Heisenberg interactions, and the additional scalar chiral interaction ${J}_{\ensuremath{\chi}}({\stackrel{P\vec}{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\stackrel{P\vec}{S}}_{j})\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{S}}_{k}$ for the three spins in all the triangles using large-scale density matrix renormalization group calculation on cylinder geometry. With increasing J2{J}_{2}J2​ (${J}_{2}/{J}_{1}\ensuremath{\le}0.3$) and ${J}_{\ensuremath{\chi}}$ (${J}_{\ensuremath{\chi}}/{J}_{1}\ensuremath{\le}1.0$) interactions, we establish a quantum phase diagram with the magnetically ordered ${120}^{\ensuremath{\circ}}$, stripe, and noncoplanar tetrahedral phase. In between these magnetic order phases, we find a chiral spin liquid (CSL) phase, which is identified as a $\ensuremath{\nu}=1/2$ bosonic fractional quantum Hall state with possible spontaneous rotational symmetry breaking. By switching on the chiral interaction, we find that the previously identified spin liquid in the ${J}_{1}\ensuremath{-}{J}_{2}$ triangular model ($0.08\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.15$) shows a phase transition to the CSL phase at very small ${J}_{\ensuremath{\chi}}$. We also compute the spin triplet gap in both spin liquid phases, and our finite-size results suggest a large gap in the odd topological sector but a small or vanishing gap in the even sector. We discuss the implications of our results on the nature of the spin liquid phases.
The antiferromagnetic Heisenberg model on the triangular lattice is perhaps the best known example of frustrated magnets, but it orders at low temperatures. Recent density matrix renormalization group (DMRG) calculations find that the next nearest neighbor interaction J_{2} enhances the frustration, and it leads to a spin liquid for J_{2}/J_{1}∈(0.08,0.15). In addition, a DMRG study of a dipolar Heisenberg model with longer range interactions gives evidence for a spin liquid at a small dipole tilting angle θ∈[0,10°). In both cases, the putative spin liquid region appears to be small. Here, we show that for the triangular lattice dipolar Heisenberg model, a robust quantum paramagnetic phase exists in a surprisingly wide region, θ∈[0,54°), for dipoles tilted along the lattice diagonal direction. We obtain the phase diagram of the model by functional renormalization group (RG), which treats all magnetic instabilities on equal footing. The quantum paramagnetic phase is characterized by a smooth continuous flow of vertex functions and spin susceptibility down to the lowest RG scale, in contrast to the apparent breakdown of RG flow in phases with stripe or spiral order. Our finding points to a promising direction to search for quantum spin liquids in ultracold dipolar molecules.
Spin systems with frustrated anisotropic interactions are of significant interest due to possible exotic ground states. We have explored their phase diagram on a nearest-neighbor triangular lattice using the density-matrix renormalization group and mapped out the topography of the region that can harbor a spin liquid. We find that this spin-liquid phase is continuously connected to a previously discovered spin-liquid phase of the isotropic J_{1}-J_{2} model. The two limits show nearly identical spin correlations, making the case that their respective spin liquids are isomorphic to each other.
Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-1 model on the square lattice with the nearest-neighbor bilinear (J1{J}_{1}J1​) and biquadratic (K1{K}_{1}K1​) interactions. We provide strong evidence for a nematic spin liquid phase in a range of K1/J1{K}_{1}/{J}_{1}K1​/J1​ near the SU(3)-symmetric point (J1=K1{J}_{1}={K}_{1}J1​=K1​), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks C4{C}_{4}C4​ lattice rotational symmetry, and possesses fluctuations peaked at the wave vector $(\ensuremath{\pi},2\ensuremath{\pi}/3)$. The spin excitation gap drops rapidly with system size and appears to be gapless, and the nematic order is attributed to the dominant $(\ensuremath{\pi},2\ensuremath{\pi}/3)$ fluctuations. Our results provide a novel mechanism for electronic nematic order and, more generally, open up a new avenue to explore frustration-induced exotic ground states.
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.