arXiv:1806.07031 · Physical review. B./Physical review. B

Gapless spin liquid ground state of the spin- 12 J1−J2 Heisenberg model on square lattices

Wen-Yuan Liu, Shaojun Dong, Chao Wang, Yong‐Jian Han, Hong An, Guang‐Can Guo, Lixin He

Published Dec 17, 2018• 71 citations

Abstract

The spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on square lattices is investigated via the finite projected entangled pair states (PEPS) method. Using the recently developed gradient optimization method combined with Monte Carlo sampling techniques, we are able to obtain the ground state energies that are competitive with the best results. The calculations show that there is no N\'eel order, dimer order, or plaquette order in the region of $0.42\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$, suggesting a single spin liquid phase in the intermediate region. Furthermore, the calculated staggered spin, dimer, and plaquette correlation functions all have power law decay behaviors, which provide strong evidence that the intermediate nonmagnetic phase is a single gapless spin liquid state.

DOI: 10.1103/physrevb.98.241109

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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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Tensor-product state approach to spin- 12 square J1−J2 antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality

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Gapless spin liquid and valence-bond solid in the J1 - J2 Heisenberg model on the square lattice: Insights from singlet and triplet excitations

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Phase diagram of the spin- 12 J1 - J2 Heisenberg model on a honeycomb lattice

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arXiv:1306.6067Physical Review BOct 29, 2013113 citations

Quantum phase diagram of spin-1 J1−J2 Heisenberg model on the square lattice: An infinite projected entangled-pair state and density matrix renormalization group study

Reza Haghshenas, Wang-Wei Lan, Shou-Shu Gong +1 more

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arXiv:1806.07031 · Physical review. B./Physical review. B

Gapless spin liquid ground state of the spin- 12 J1−J2 Heisenberg model on square lattices

Wen-Yuan Liu, Shaojun Dong, Chao Wang, Yong‐Jian Han, Hong An, Guang‐Can Guo, Lixin He

Published Dec 17, 2018• 71 citations

arXiv HTML PDF

Abstract

The spin-

\frac{1}{2}

{J}_{1}\ensuremath{-}{J}_{2}

Heisenberg model on square lattices is investigated via the finite projected entangled pair states (PEPS) method. Using the recently developed gradient optimization method combined with Monte Carlo sampling techniques, we are able to obtain the ground state energies that are competitive with the best results. The calculations show that there is no N'eel order, dimer order, or plaquette order in the region of

0.42\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6

, suggesting a single spin liquid phase in the intermediate region. Furthermore, the calculated staggered spin, dimer, and plaquette correlation functions all have power law decay behaviors, which provide strong evidence that the intermediate nonmagnetic phase is a single gapless spin liquid state.

DOI: 10.1103/physrevb.98.241109

Related papers

Comment on “Gapless spin liquid ground state of the spin- 12 J1−J2 Heisenberg model on square lattices”

Bowen Zhao, Jun Takahashi, Anders W. Sandvik

Liu et al. [Phys. Rev. B 98, 241109 (2018)] used Monte Carlo sampling of the physical degrees of freedom of a projected entangled pair state type wave function for the $S=1/2$ frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice and found a nonmagnetic state argued to be a gapless spin liquid when the coupling ratio $g={J}_{2}/{J}_{1}$ is in the range $g\ensuremath{\in}[0.42,0.6]$. Here we show that their definition of the order parameter for another candidate ground state within this coupling window---a spontaneously dimerized state---is problematic. The order parameter as defined will not detect dimer order when lattice symmeties are broken due to open boundaries or asymmetries originating from the calculation itself. Thus, a dimerized phase for some range of $g$ cannot be excluded (and is likely based on several other recent works).Liu et al. [Phys. Rev. B 98, 241109 (2018)] used Monte Carlo sampling of the physical degrees of freedom of a projected entangled pair state type wave function for the $S=1/2$ frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice and found a nonmagnetic state argued to be a gapless spin liquid when the coupling ratio $g={J}_{2}/{J}_{1}$ is in the range $g\ensuremath{\in}[0.42,0.6]$. Here we show that their definition of the order parameter for another candidate ground state within this coupling window---a spontaneously dimerized state---is problematic. The order parameter as defined will not detect dimer order when lattice symmeties are broken due to open boundaries or asymmetries originating from the calculation itself. Thus, a dimerized phase for some range of $g$ cannot be excluded (and is likely based on several other recent works).

arXiv:1909.12788Physical review. B./Physical review. BApr 9, 20208 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

Tensor-product state approach to spin- 12 square J1−J2 antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality

Ling Wang, Zheng‐Cheng Gu, Frank Verstraete +1 more

The ground state phase of a spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $({J}_{2}\ensuremath{\sim}0.5{J}_{1})$ has been debated for decades. Here we study this model using the cluster update algorithm for tensor-product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point ${J}_{2}^{{c}_{1}}=0.572(5){J}_{1}$ and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at ${J}_{2}/{J}_{1}=0.6$ . These results are consistent with a potential gapless $U(1)$ spin-liquid phase. However, since the $U(1)$ spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at ${J}_{2}^{{c}_{1}}$ . Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.The ground state phase of a spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $({J}_{2}\ensuremath{\sim}0.5{J}_{1})$ has been debated for decades. Here we study this model using the cluster update algorithm for tensor-product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point ${J}_{2}^{{c}_{1}}=0.572(5){J}_{1}$ and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at ${J}_{2}/{J}_{1}=0.6$. These results are consistent with a potential gapless $U(1)$ spin-liquid phase. However, since the $U(1)$ spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at ${J}_{2}^{{c}_{1}}$. Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.

arXiv:1112.3331Physical review. B./Physical review. BAug 19, 2016102 citations

Gapless spin liquid and valence-bond solid in the J1 - J2 Heisenberg model on the square lattice: Insights from singlet and triplet excitations

Francesco Ferrari, Federico Becca

The spin- $1/2$ ${J}_{1}$ - ${J}_{2}$ Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.The spin-$1/2$ ${J}_{1}$-${J}_{2}$ Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.

arXiv:2005.12941Physical review. B./Physical review. BJul 10, 202079 citations

Phase diagram of the spin- 12 J1 - J2 Heisenberg model on a honeycomb lattice

Shou-Shu Gong, D. N. Sheng, Olexei I. Motrunich +1 more

We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin- $\frac{1}{2}$ Heisenberg model on a honeycomb lattice with first ( ${J}_{1}$ ) and second ( ${J}_{2}$ ) neighbor antiferromagnetic interactions, where a ${Z}_{2}$ spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size $N=2\ifmmode\times\else\texttimes\fi{}6\ifmmode\times\else\texttimes\fi{}6$. With increasing ${J}_{2}$ , we find a N\'eel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment ${m}_{s}^{2}$ on finite-size cylinders to the thermodynamic limit, we find the N\'eel order vanishing at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.22$. For $0.25<{J}_{2}/{J}_{1}\ensuremath{\le}0.35$, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For $0.22<{J}_{2}/{J}_{1}<0.25$ , both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-$\frac{1}{2}$ Heisenberg model on a honeycomb lattice with first (${J}_{1}$) and second (${J}_{2}$) neighbor antiferromagnetic interactions, where a ${Z}_{2}$ spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size $N=2\ifmmode\times\else\texttimes\fi{}6\ifmmode\times\else\texttimes\fi{}6$. With increasing ${J}_{2}$, we find a N\'eel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment ${m}_{s}^{2}$ on finite-size cylinders to the thermodynamic limit, we find the N\'eel order vanishing at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.22$. For $0.25<{J}_{2}/{J}_{1}\ensuremath{\le}0.35$, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For $0.22<{J}_{2}/{J}_{1}<0.25$, both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.

arXiv:1306.6067Physical Review BOct 29, 2013113 citations

Quantum phase diagram of spin-1 J1−J2 Heisenberg model on the square lattice: An infinite projected entangled-pair state and density matrix renormalization group study

Reza Haghshenas, Wang-Wei Lan, Shou-Shu Gong +1 more

We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor ${J}_{1}$ and the next-nearest-neighbor ${J}_{2}$ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width ${L}_{y}=10$ , we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor ${J}_{1}$ and the next-nearest-neighbor ${J}_{2}$ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width ${L}_{y}=10$, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.

arXiv:1802.00874Physical review. B./Physical review. BMay 31, 201821 citations

The spin-

\frac{1}{2}

{J}_{1}\ensuremath{-}{J}_{2}

0.42\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6

Liu et al. [Phys. Rev. B 98, 241109 (2018)] used Monte Carlo sampling of the physical degrees of freedom of a projected entangled pair state type wave function for the

S=1/2

frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice and found a nonmagnetic state argued to be a gapless spin liquid when the coupling ratio

g={J}_{2}/{J}_{1}

is in the range $g\ensuremath{\in}[0.42,0.6]$. Here we show that their definition of the order parameter for another candidate ground state within this coupling window---a spontaneously dimerized state---is problematic. The order parameter as defined will not detect dimer order when lattice symmeties are broken due to open boundaries or asymmetries originating from the calculation itself. Thus, a dimerized phase for some range of

g

cannot be excluded (and is likely based on several other recent works).

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.

{J}_{2}^{{c}_{1}}=0.572(5){J}_{1}

and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at

{J}_{2}/{J}_{1}=0.6

. These results are consistent with a potential gapless

U(1)

spin-liquid phase. However, since the

U(1)

spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at

{J}_{2}^{{c}_{1}}

. Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.

The spin-

1/2

{J}_{1}

{J}_{2}

Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.

We use the density matrix renormalization group (DMRG) algorithm to study the phase diagram of the spin-

\frac{1}{2}

Heisenberg model on a honeycomb lattice with first (

{J}_{1}

) and second (

{J}_{2}

) neighbor antiferromagnetic interactions, where a

{Z}_{2}

spin liquid region has been proposed. By implementing SU(2) symmetry in the DMRG code, we are able to obtain accurate results for long cylinders with a width slightly over 15 lattice spacings and a torus up to the size $N=2\ifmmode\times\else\texttimes\fi{}6\ifmmode\times\else\texttimes\fi{}6$. With increasing

{J}_{2}

, we find a N\'eel phase with a vanishing spin gap and a plaquette valence-bond (PVB) phase with a nonzero spin gap. By extrapolating the square of the staggered magnetic moment

{m}_{s}^{2}

on finite-size cylinders to the thermodynamic limit, we find the N\'eel order vanishing at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.22$. For $0.25<{J}_{2}/{J}_{1}\ensuremath{\le}0.35$, we find a possible PVB order, which shows a fast growing PVB decay length with increasing system width. For

0.22<{J}_{2}/{J}_{1}<0.25

, both spin and dimer orders vanish in the thermodynamic limit, which is consistent with a possible spin liquid phase. We present calculations of the topological entanglement entropy, compare the DMRG results with the variational Monte Carlo, and discuss possible scenarios in the thermodynamic limit for this region.

We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor

{J}_{1}

and the next-nearest-neighbor

{J}_{2}

couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width

{L}_{y}=10

, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.