arXiv:2212.14223 · Physical review. B./Physical review. B

Global quantum phase diagram and non-Abelian chiral spin liquid in a spin- 32 square-lattice antiferromagnet

Weiwei Luo, Yixuan Huang, D. N. Sheng, W. Zhu

Published Jul 17, 2023• 4 citations

Abstract

Since strong quantum fluctuations are essential for the emergence of quantum spin liquids, there have been extensive exploration and identification of spin liquid candidates in spin-$\frac{1}{2}$ systems, while such activities are rare in higher spin systems. Here we report an example of non-Abelian chiral spin liquid emerging in a spin-$\frac{3}{2}$ Heisenberg model on a square lattice. By tuning Heisenberg exchange interaction and scalar chirality interaction, we map out a quantum phase diagram enclosing three conventional magnetic orders and a chiral spin liquid based on density-matrix renormalization group studies. The nature of the spin liquid is identified as a long-sought bosonic version of the Read-Rezayi state that supports non-Abelian Fibonacci anyonic statistics, identified by the ground state entanglement spectrum. Significantly, we establish that the non-Abelian chiral spin liquid emerges through the enlarged local degrees of freedom and enhanced quantum fluctuations near the classical phase boundaries of competing magnetic orders. Our numerical discovery of an exotic quantum spin liquid in a spin-$\frac{3}{2}$ system suggests a route for discovering fractionalized quantum phases in frustrated higher spin magnetic compounds.

DOI: 10.1103/physrevb.108.035130

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arXiv:2212.14223 · Physical review. B./Physical review. B

Global quantum phase diagram and non-Abelian chiral spin liquid in a spin- 32 square-lattice antiferromagnet

Weiwei Luo, Yixuan Huang, D. N. Sheng, W. Zhu

Published Jul 17, 2023• 4 citations

arXiv HTML PDF

Abstract

Since strong quantum fluctuations are essential for the emergence of quantum spin liquids, there have been extensive exploration and identification of spin liquid candidates in spin-

\frac{1}{2}

systems, while such activities are rare in higher spin systems. Here we report an example of non-Abelian chiral spin liquid emerging in a spin-

\frac{3}{2}

Heisenberg model on a square lattice. By tuning Heisenberg exchange interaction and scalar chirality interaction, we map out a quantum phase diagram enclosing three conventional magnetic orders and a chiral spin liquid based on density-matrix renormalization group studies. The nature of the spin liquid is identified as a long-sought bosonic version of the Read-Rezayi state that supports non-Abelian Fibonacci anyonic statistics, identified by the ground state entanglement spectrum. Significantly, we establish that the non-Abelian chiral spin liquid emerges through the enlarged local degrees of freedom and enhanced quantum fluctuations near the classical phase boundaries of competing magnetic orders. Our numerical discovery of an exotic quantum spin liquid in a spin-

\frac{3}{2}

system suggests a route for discovering fractionalized quantum phases in frustrated higher spin magnetic compounds.

DOI: 10.1103/physrevb.108.035130

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Jianwei Yang, Zhao Liu, Ling Wang

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Physical review. B./Physical review. BDec 2, 20242 citations

Since strong quantum fluctuations are essential for the emergence of quantum spin liquids, there have been extensive exploration and identification of spin liquid candidates in spin-

\frac{1}{2}

systems, while such activities are rare in higher spin systems. Here we report an example of non-Abelian chiral spin liquid emerging in a spin-

\frac{3}{2}

\frac{3}{2}

system suggests a route for discovering fractionalized quantum phases in frustrated higher spin magnetic compounds.

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.

1

square-lattice model with the first- and second-neighbor bilinear (

J_1,J_2

) and biquadratic interactions (

K_1,K_2

), as well as the three-spin scalar chiral coupling

J_χ

. Inspired by the evidence of the Moore-Read state in exact diagonalization and infinite Projected Entangled Pair States calculations, we fix

J_1=1.0

J_2/J_1=0.623

K_1/J_1=-0.176

, and obtain a quantum phase diagram for

0 \leq K_2/J_1 \leq 0.4

and

0.3 \leq J_χ/J_1 \leq 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.

\frac{1}{2}

XY model with nearest-neighbor

({J}_{1})

, and next-nearest-neighbor

({J}_{2})

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.

\text{SU}{(2)}_{1}

chiral spin liquid emerges in the highly frustrated region, which is unambiguously identified by two topologically degenerate ground states, a modular matrix, and characteristic level counting in an entanglement spectrum, featuring the same topological order of the $\ensuremath{\nu}=1/2$ bosonic Laughlin state. The phase boundaries among the different orders are determined by the energy level crossing analysis and wave-function fidelity susceptibility.

We study the spin-

1/2

Heisenberg model on the triangular lattice with the nearest-neighbor

{J}_{1}>0

, the next-nearest-neighobr

{J}_{2}>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

{J}_{2}

(${J}_{2}/{J}_{1}\ensuremath{\le}0.3$) 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.

\frac{1}{2}

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

{J}_{2}

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

\mathcal{S}

matrix of minimally entangled states on a torus and the entanglement spectra of even and odd sectors on a cylinder, employing ED and

\mathrm{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.