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arXiv:1502.04831 · Physical Review B

Spin liquid phase of the S = 1 2 J 1 − J 2 Heisenberg model on the triangular lattice

Zhenyue Zhu, Steven R. White

Published Jul 7, 2015• 275 citations
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Abstract

We study the $S=1/2$ Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor interactions ${J}_{1}$ and ${J}_{2}$ with the density matrix renormalization group, on long open cylinders with widths up to nine lattice spacings. In an intermediate ${J}_{2}$ region $0.06\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.17$, we find evidence for a spin liquid (SL) state with short range spin-spin, bond-bond, and chiral correlation lengths, bordered by a classical ${120}^{\ensuremath{\circ}}$ N\'eel ordered state at small ${J}_{2}$ and by a two sublattice collinear magnetically ordered state at larger ${J}_{2}$. Focusing on ${J}_{2}/{J}_{1}=0.1$, we find a number of signatures of a gapped SL phase.

DOI
10.1103/physrevb.92.041105

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We study the S=1/2S=1/2S=1/2 Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor interactions J1{J}_{1}J1​ and J2{J}_{2}J2​ with the density matrix renormalization group, on long open cylinders with widths up to nine lattice spacings. In an intermediate J2{J}_{2}J2​ region 0.06\ensuremath≲J2/J1\ensuremath≲0.170.06\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.170.06\ensuremath≲J2​/J1​\ensuremath≲0.17, we find evidence for a spin liquid (SL) state with short range spin-spin, bond-bond, and chiral correlation lengths, bordered by a classical 120\ensuremath∘{120}^{\ensuremath{\circ}}120\ensuremath∘ N'eel ordered state at small J2{J}_{2}J2​ and by a two sublattice collinear magnetically ordered state at larger J2{J}_{2}J2​. Focusing on J2/J1=0.1{J}_{2}/{J}_{1}=0.1J2​/J1​=0.1, we find a number of signatures of a gapped SL phase.
We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the triangular lattice with nearest- (J1)({J}_{1})(J1​), next-nearest- (J2)({J}_{2})(J2​), and third-neighbor (J3)({J}_{3})(J3​) interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with SU(2)\mathrm{SU}(2)SU(2) spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to Lx=48{L}_{x}=48Lx​=48 and circumference Ly=6−−12{L}_{y}=6--12Ly​=6−−12. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of J3{J}_{3}J3​ interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.
We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.
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.
Topological Spin Liquid Phase in the Spin-1/21/21/2 Triangular Heisenberg Model
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