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
J1 and
J2 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.
We study the spin-
1/2 Heisenberg model on the triangular lattice with the nearest-neighbor
J1>0, the next-nearest-neighobr
J2>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 (${J}_{2}/{J}_{1}\ensuremath{\le}0.3$) 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.
Non-Abelian chiral spin liquids (CSLs) host non-Abelian anyon excitations and are promising platforms for topological quantum computation. However, realizing non-Abelian CSLs in short-range interacting spin systems remains elusive. Here, we study a spin-
1 square-lattice model with the first- and second-neighbor bilinear (
J1,J2) and biquadratic interactions (
K1,K2), as well as the three-spin scalar chiral coupling
Jχ. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix
J1=1.0,
J2/J1=0.623,
K1/J1=−0.176, and obtain a quantum phase diagram for
0≤K2/J1≤0.4 and
0.3≤Jχ/J1≤0.6 by using the density matrix renormalization group (DMRG) calculations. We identify a stripe antiferromagnetic phase, a Néel antiferromagnetic phase, a magnetically ordered chiral spin state phase, and a non-Abelian CSL phase emerging near the boundaries of the ordered phases. The unbiased DMRG results provide strong evidence for the Moore-Read state, including the three topological sectors, the quantized spin Chern number
C=1, and the level counting of low-lying entanglement spectra which is consistent with the chiral SU(2)
2 conformal field theory. Our results may shed new light on searching for non-Abelian CSLs in other higher-spin (
S>1/2) systems with short-range couplings.
The frustrated XY model on the honeycomb lattice has drawn lots of attention because of the potential emergence of chiral spin liquid (CSL) with the increasing of frustrations or competing interactions. In this paper, we study the extended spin-
21 XY model with nearest-neighbor
(J1), and next-nearest-neighbor
(J2) interactions in the presence of a three-spins chiral $({J}_{\ensuremath{\chi}})$ term using density matrix renormalization group methods. We obtain a quantum phase diagram with both conventionally ordered and topologically ordered phases. In particular, the long-sought Kalmeyer-Laughlin CSL is shown to emerge under a small ${J}_{\ensuremath{\chi}}$ perturbation due to the interplay of the magnetic frustration and chiral interactions. The CSL, which is a nonmagnetic phase, is identified by the scalar chiral order, the finite spin gap on a torus, and the chiral entanglement spectrum described by chiral
SU(2)1 conformal field theory.
The intricate interplay between frustration and spin chirality, induced by the Dzyaloshinskii-Moriya interaction, holds promise for unveiling novel phases in frustrated quantum magnets. This study investigates the ground state phase diagram of the spin-
21 square lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model upon incorporation of the chiral interaction ${J}_{\ensuremath{\chi}}$. Employing exact diagonalization (ED) with full lattice symmetries, we analyze the phase evolution as a function of
J2 at fixed ${J}_{\ensuremath{\chi}}$, utilizing highly symmetric energy levels. Critical level crossings and ground state fidelity susceptibility (FS) techniques aid in pinpointing phase boundaries: Magnetic to nonmagnetic phase transitions are identified through critical level crossings between gapless magnetic excitations and quasidegenerate ground states of nonmagnetic phases. Direct transitions between two nonmagnetic phases are characterized by FS peaks due to avoided ground state level crossing. Based on these observations, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase within a significant range of ${J}_{\ensuremath{\chi}}$, demarcated by a nearly vertical boundary line at ${J}_{2}\ensuremath{\approx}0.65$. This critical line terminates at the lower boundary in ${J}_{\ensuremath{\chi}}$ of a magnetic ordered chiral spin solid phase, which gains prominence with increasing ${J}_{\ensuremath{\chi}}$ from both CSL and NSL phases. The topological nature of the CSL is confirmed using the modular
S matrix of minimally entangled states on a torus and the entanglement spectra of even and odd sectors on a cylinder, employing ED and
SU(2)-symmetric density matrix renormalization group method, respectively. Furthermore, a comprehensive discussion on the nature of the NSL is provided, exploring aspects such as ground state degeneracy, local bond energy landscape, and singlet and triplet gaps on various tori, offering substantial evidence supporting the nematic nature of the NSL.
The intricate interplay between frustration and spin chirality has the potential to give rise to unprecedented phases in frustrated quantum magnets. We examine the ground state phase diagram of the spin-1/2 square lattice J1-J2-Jx model by employing critical level crossings and ground state fidelity susceptibility (FS) using exact diagonalization (ED) with full lattice symmetries. Our analysis reveals the evolution of highly symmetric energy levels as a function of J2 at fixed Jx. During a magnetic to non-magnetic phase transition, the precise identification of the phase boundary is achieved through critical level crossings between the gapless excitation of a magnetic phase and the quasi-degenerate ground state of a non-magnet phase. Conversely, a direct transition between two non-magnetic phases is characterized by a FS peak accompanied by an avoided ground state level crossing, serving as a distinctive signal. Within a substantial range of Jx, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase with a degeneracy of two on a cylinder. These two phases are demarcated by a nearly vertical boundary line at J2 = 0.65. This critical line terminates at the lower boundary of a magnetic ordered chiral spin solid (CSS) phase. We validate the topological nature of the CSL using the modular S matrix of the minimum entangled states (MES) on a torus, along with the entanglement spectra (ES) of even and odd sectors on a cylinder, employing an SU(2)-symmetric density matrix renormalization group (DMRG) method. Furthermore, we delve into a comprehensive discussion on the nature of the NSL, exploring aspects such as ground state degeneracy, the local bond energy landscape, and the singlet and triplet gaps on various tori. These analysis provide substantial evidence supporting the nematic nature of the NSL.