arXiv:1711.06523 · Physical review. B./Physical review. B

Nematic spin liquid phase in a frustrated spin-1 system on the square lattice

Wen-Jun Hu, Shou-Shu Gong, Hsin-Hua Lai, Haoyu Hu, Qimiao Si A, Andriy H. Nevidomskyy

Published Oct 28, 2019• 16 citations

Abstract

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 (${J}_{1}$) and biquadratic (${K}_{1}$) interactions. We provide strong evidence for a nematic spin liquid phase in a range of ${K}_{1}/{J}_{1}$ near the SU(3)-symmetric point (${J}_{1}={K}_{1}$), 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 ${C}_{4}$ 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.

DOI: 10.1103/physrevb.100.165142

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Ground state phase diagram and the exotic phases in spin- 1 2 square lattice J 1 − J 2 − J χ model

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Ground state phase diagram and the exotic phases in the spin-1/2 square lattice J1-J2-Jx model

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Density matrix renormalization group study of nematicity in two dimensions: Application to a spin-1 bilinear-biquadratic model on the square lattice

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

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Quantum Spin Liquid in Spin 1/2J1–J2Heisenberg Model on Square Lattice: Many-Variable Variational Monte Carlo Study Combined with Quantum-Number Projections

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arXiv:1711.06523 · Physical review. B./Physical review. B

Nematic spin liquid phase in a frustrated spin-1 system on the square lattice

Wen-Jun Hu, Shou-Shu Gong, Hsin-Hua Lai, Haoyu Hu, Qimiao Si A, Andriy H. Nevidomskyy

Published Oct 28, 2019• 16 citations

arXiv HTML PDF

Abstract

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 (

{J}_{1}

) and biquadratic (

{K}_{1}

) interactions. We provide strong evidence for a nematic spin liquid phase in a range of

{K}_{1}/{J}_{1}

near the SU(3)-symmetric point (

{J}_{1}={K}_{1}

), 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

{C}_{4}

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.

DOI: 10.1103/physrevb.100.165142

Related papers

Ground state phase diagram and the exotic phases in spin- 1 2 square lattice J 1 − J 2 − J χ model

Jianwei Yang, Zhao Liu, Ling Wang

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- $\frac{1}{2}$ 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.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-$\frac{1}{2}$ 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.

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

Ground state phase diagram and the exotic phases in the spin-1/2 square lattice J1-J2-Jx model

Jianwei Yang, Zhao Liu, Ling Wang

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.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.

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Density matrix renormalization group study of nematicity in two dimensions: Application to a spin-1 bilinear-biquadratic model on the square lattice

Wen-Jun Hu, Shou-Shu Gong, Hsin-Hua Lai +2 more

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 $SU(3)$ 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.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 $SU(3)$ 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.

arXiv:2001.05377Physical review. B./Physical review. BJan 15, 202012 citations

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

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.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.

arXiv:2206.04087Physical review. B./Physical review. BSep 6, 20228 citations

Fragility of the nematic spin liquid induced by diagonal couplings in the square-lattice SU(3) model

Xiaotian Zhang, Wen-Jun Hu, Elbio R. Dagotto +1 more

We present a large-scale density matrix renormalization group (DMRG) study of the spin-1 SU(3) bilinear-biquadratic model on the square lattice, which was suggested to host a nematic spin liquid state in recent DMRG calculations. We report that this spin liquid appears to strongly compete with a three-sublattice magnetic order. To further study the competition between the two states, and the reason of the emergent nematic spin liquid, we included an additional next-nearest-neighbor SU(3) symmetric interactions along one of the two plaquette diagonal directions. This allows to tune the square-lattice model to the triangular-lattice model. By computing spin correlation functions and various order parameters, we find that the three-sublattice order may develop at infinitesimal additional new coupling, at least within the precision of our study. Compared with the previous findings that the nematic spin liquid is stable in extended parameter regions with additional couplings that respect the lattice symmetries of the square lattice, we argue that here the diagonal couplings, which frustrate the bipartite-lattice structure, rapidly suppress the two-sublattice fluctuations and the three-sublattice order thus wins. This numerical result is consistent with the conjecture that the nematic spin liquid emerges from the competition between two- and three-sublattice fluctuations.We present a large-scale density matrix renormalization group (DMRG) study of the spin-1 SU(3) bilinear-biquadratic model on the square lattice, which was suggested to host a nematic spin liquid state in recent DMRG calculations. We report that this spin liquid appears to strongly compete with a three-sublattice magnetic order. To further study the competition between the two states, and the reason of the emergent nematic spin liquid, we included an additional next-nearest-neighbor SU(3) symmetric interactions along one of the two plaquette diagonal directions. This allows to tune the square-lattice model to the triangular-lattice model. By computing spin correlation functions and various order parameters, we find that the three-sublattice order may develop at infinitesimal additional new coupling, at least within the precision of our study. Compared with the previous findings that the nematic spin liquid is stable in extended parameter regions with additional couplings that respect the lattice symmetries of the square lattice, we argue that here the diagonal couplings, which frustrate the bipartite-lattice structure, rapidly suppress the two-sublattice fluctuations and the three-sublattice order thus wins. This numerical result is consistent with the conjecture that the nematic spin liquid emerges from the competition between two- and three-sublattice fluctuations.

arXiv:2108.12747Physical review. B./Physical review. BNov 19, 20213 citations

Quantum Spin Liquid in Spin 1/2J1–J2Heisenberg Model on Square Lattice: Many-Variable Variational Monte Carlo Study Combined with Quantum-Number Projections

Satoshi Morita, Ryui Kaneko, Masatoshi Imada

The nature of quantum spin liquids is studied for the spin- $1/2$ antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, $J_1$ , and those between next-nearest-neighbor sites, $J_2$ . We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$ , while the gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le 0.6$ . The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle \propto 1/r^{z+η}$ at a long distance $r$ appears to increase from $z+η\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC phase at $J_1/J_1\sim0.5$ . Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.The nature of quantum spin liquids is studied for the spin-$1/2$ antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, $J_1$, and those between next-nearest-neighbor sites, $J_2$. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$, while the gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le 0.6$. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle \propto 1/r^{z+η}$ at a long distance $r$ appears to increase from $z+η\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC phase at $J_1/J_1\sim0.5$. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.

arXiv:1410.7557Journal of the Physical Society of JapanJan 29, 2015122 citations

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 (

{J}_{1}

) and biquadratic (

{K}_{1}

) interactions. We provide strong evidence for a nematic spin liquid phase in a range of

{K}_{1}/{J}_{1}

near the SU(3)-symmetric point (

{J}_{1}={K}_{1}

{C}_{4}

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.

\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.

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.

The nature of quantum spin liquids is studied for the spin-

1/2

antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites,

J_1

, and those between next-nearest-neighbor sites,

J_2

. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region

0.4< J_2/J_1\le 0.6

sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of

0.4 < J_2/J_1 \le 0.5

, while the gapped valence-bond-crystal (VBC) phase is stabilized for

0.5 < J_2/J_1 \le 0.6

. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the

C_{4v}

symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation

\langle S(0)S(r)\rangle \propto 1/r^{z+η}

at a long distance

r

appears to increase from

z+η\sim 1

J_2/J_1\sim0.4

toward the continuous transition to the VBC phase at

J_1/J_1\sim0.5

. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.