Topological Spin Liquid Phase in the Spin-$1/2$ Triangular Heisenberg Model

Wen-Jun Hu, Shou-Shu Gong, Wei Kang Zhu, D. N. Sheng

Published Apr 2, 2015• 1 citation

Abstract

We study the spin-1/2 Heisenberg model on the triangular lattice with the antiferromagnetic first (J1) and second nearest-neighbor (J2) interactions using density matrix renormalization group. By studying the spin correlation function, we find a 120◦ magnetic order phase for J2 . 0.08J1 and a stripe antiferromagnetic phase for J2 & 0.16J2. Between these two phases, we identify a gapped spin liquid phase characterized by the exponential decaying spin and dimer correlations, as well as finite spin singlet and triplet excitation gaps. We also find two topological degenerating ground states in two topological sectors. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. While the sector with spinon is found to respect the time reveral symmetry, the even sector without a spinon may break such a symmetry. We discuss the implications of numerical results for the nature of the topological spin liquid phase.

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Topological Spin Liquid Phase in the Spin- $1/2$ Triangular Heisenberg ModelTopological Spin Liquid Phase in the Spin-$1/2$ Triangular Heisenberg Model

Wen-Jun Hu, Shou-Shu Gong, Wei Kang Zhu, D. N. Sheng

Published Apr 2, 2015• 1 citation

OpenAlex

Abstract

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Topological Spin Liquid Phase in the Spin-

1/2

Triangular Heisenberg Model

We study the spin-

\frac{1}{2}

Heisenberg model on the triangular lattice with antiferromagnetic first-

({J}_{1})

and second-

({J}_{2})

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

{Z}_{2}

spin liquids.

{Z}_{2}

topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.

We perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-

1/2

antiferromagnetic square lattice Heisenberg

{J}_{1}

{J}_{2}

model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.62$, separating conventional N\'eel and striped antiferromagnetic states for smaller and larger

{J}_{2}/{J}_{1}

, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\ensuremath{\gamma}=0.70\ifmmode\pm\else\textpm\fi{}0.02$, extremely close to $\ensuremath{\gamma}=\mathrm{ln}(2)\ensuremath{\approx}0.69$ (expected for a

{Z}_{2}

quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.

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.

We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-

1/2

antiferromagnetic Heisenberg model on the triangular lattice with nearest-

({J}_{1})

, next-nearest-

({J}_{2})

, and third-neighbor

({J}_{3})

interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with

\mathrm{SU}(2)

spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to

{L}_{x}=48

and circumference

{L}_{y}=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

{J}_{3}

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 ground-state properties of a spin-1 Heisenberg model on a square lattice with the first- and second-nearest-neighbor antiferromagnetic couplings

{J}_{1}

and

{J}_{2}

and a three-spin scalar chirality term ${J}_{\ensuremath{\chi}}$. Using the density matrix renormalization group calculation, we map out a global phase diagram including various magnetic order phases and an emergent quantum spin liquid phase. The nature of the spin liquid is identified as a bosonic non-Abelian Moore-Read state from the fingerprint of the entanglement spectra and identification of a full set of topological sectors. We further unveil a stripe magnetic order coexisting with this spin liquid. Our results not only establish a rare example of non-Abelian spin liquids in simple spin systems but also demonstrate the coexistence of fractionalized excitations and magnetic order beyond mean-field descriptions.