arXiv:1802.00874 · Physical review. B./Physical review. B

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, D. N. Sheng

Published May 31, 2018• 21 citations

Abstract

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.

DOI: 10.1103/physrevb.97.184436

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arXiv:1802.00874 · Physical review. B./Physical review. B

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, D. N. Sheng

Published May 31, 2018• 21 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.97.184436

Related papers

U(1) -symmetric infinite projected entangled-pair states study of the spin-1/2 square J1−J2 Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$ -symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin- $1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$ -symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite- $D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$ . For $0.53<{J}_{2}/{J}_{1}<0.61$ , a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$ , where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-$1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite-$D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$. For $0.53<{J}_{2}/{J}_{1}<0.61$, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

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Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin $S=1$ Heisenberg model on the honeycomb lattice with the antiferromagnetic first- ( ${J}_{1}$ ) and second-neighbor ( ${J}_{2}$ ) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large ${J}_{2}$ side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin- $1/2$ system.Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin $S=1$ Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (${J}_{1}$) and second-neighbor (${J}_{2}$) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large ${J}_{2}$ side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-$1/2$ system.

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The $U(1)$ -symmetric iPEPS study of the spin- $1/2$ square $J_{1}$ - $J_{2}$ Heisenberg modelThe $U(1)$-symmetric iPEPS study of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$ -symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin- $1/2$ square $J_{1}$ - $J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$ -symmetric iPEPS ansatz (for a fixed bond dimension $D$ ) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large- $D$ limit, we find that a deconfined quantum-critical point occurs at $J^{c_1}_{2}=0.530(5)$ , where the correlation length of the system diverges as $\xi \sim D^{1.2}$ . The estimated critical anomalous exponents $\eta_{s}=0.65(5)$ and $\eta_{d}=1.9(1)$ for spin and dimer correlations respectively show strong deviation from those of the well-known $\mathcal{J}$ - $\mathcal{Q}$ model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with $J_2$ and finally a first-order quantum phase transition takes place at $J^{c_2}_{2}=0.610(3)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-$D$ limit, we find that a deconfined quantum-critical point occurs at $J^{c_1}_{2}=0.530(5)$, where the correlation length of the system diverges as $\xi \sim D^{1.2}$. The estimated critical anomalous exponents $\eta_{s}=0.65(5)$ and $\eta_{d}=1.9(1)$ for spin and dimer correlations respectively show strong deviation from those of the well-known $\mathcal{J}$-$\mathcal{Q}$ model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with $J_2$ and finally a first-order quantum phase transition takes place at $J^{c_2}_{2}=0.610(3)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

Nov 21, 20170 citations

Absence of spin liquid phase in the J1−J2 Heisenberg model on the square lattice

Xiangjian Qian, Mingpu Qin

We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at ${J}_{2}/{J}_{1}=0.535(3)$ , suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at ${J}_{2}/{J}_{1}=0.535(3)$ in the model. From the crossing of the first derivative of the energies with ${J}_{2}$ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at ${J}_{2}/{J}_{1}=0.610(5)$ .We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at ${J}_{2}/{J}_{1}=0.535(3)$, suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at ${J}_{2}/{J}_{1}=0.535(3)$ in the model. From the crossing of the first derivative of the energies with ${J}_{2}$ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at ${J}_{2}/{J}_{1}=0.610(5)$.

arXiv:2309.13630Physical review. B./Physical review. BApr 3, 202425 citations

Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networks

Juraj Hašík, Didier Poilblanc, Federico Becca

The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2 J 1 − J 2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45 J 2 / J 1 ≤ 0.45 . The unrestricted iPEPS simulations reveal an U(1) U ( 1 ) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1) J 2 / J 1 ≈ 0.46 ( 1 ) . The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2 J 1 − J 2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45 J 2 / J 1 ≤ 0.45 . The unrestricted iPEPS simulations reveal an U(1) U ( 1 ) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1) J 2 / J 1 ≈ 0.46 ( 1 ) . The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.

arXiv:2009.02313SciPost PhysicsJan 20, 202166 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

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}

{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

{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 develop an improved variant of

U(1)

-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-

1/2

square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the

U(1)

-symmetric iPEPS ansatz (for a fixed bond dimension

D)

set a better upper bound, improving previous tensor-network-based results. By studying the finite-

D

scaling of the magnetically order parameter, we find a N\'eel phase for

{J}_{2}/{J}_{1}<0.53

. For

0.53<{J}_{2}/{J}_{1}<0.61

, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at

{J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)

, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with

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

and finally a first-order quantum phase transition takes place at

{J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)

to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin

S=1

Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (

{J}_{1}

) and second-neighbor (

{J}_{2}

) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large

{J}_{2}

side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-

1/2

system.

The

U(1)

-symmetric iPEPS study of the spin-

1/2

square

J_{1}

J_{2}

Heisenberg model

We develop an improved variant of

U(1)

-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-

1/2

square

J_{1}

J_{2}

Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the

U(1)

-symmetric iPEPS ansatz (for a fixed bond dimension

D

) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-

D

limit, we find that a deconfined quantum-critical point occurs at

J^{c_1}_{2}=0.530(5)

, where the correlation length of the system diverges as

\xi \sim D^{1.2}

. The estimated critical anomalous exponents

\eta_{s}=0.65(5)

and

\eta_{d}=1.9(1)

for spin and dimer correlations respectively show strong deviation from those of the well-known

\mathcal{J}

\mathcal{Q}

model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with

J_2

and finally a first-order quantum phase transition takes place at

J^{c_2}_{2}=0.610(3)

to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

{J}_{2}/{J}_{1}=0.535(3)

, suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at

{J}_{2}/{J}_{1}=0.535(3)

in the model. From the crossing of the first derivative of the energies with

{J}_{2}

for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at

{J}_{2}/{J}_{1}=0.610(5)

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.