arXiv:2406.17417 · Physical review. B./Physical review. B

Plaquette-type valence bond solid state in the J 1 − J 2 square-lattice Heisenberg model

Jiale Huang, Xiangjian Qian, Mingpu Qin

Published Nov 4, 2024• 5 citations

Abstract

We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy ${\mathrm{\ensuremath{\Delta}}}_{y}$ in the nearest-neighboring coupling in the $y$ direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as $D=25\phantom{\rule{0.16em}{0ex}}000$ in FAMPS, simulating systems at a maximum size of $14\ifmmode\times\else\texttimes\fi{}14$. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite ${\mathrm{\ensuremath{\Delta}}}_{y}$ scaling analysis of the VBS dimer order parameters, we identify the VBS phase as a PVBS type, meaning there is no spontaneous rotational symmetry breaking in the VBS phase. This paper not only supports the presence of PVBS order in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.

DOI: 10.1103/physrevb.110.195111

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

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Evidence of valence bond solid and quantum paramagnetism in the $J_1$--$J_2$--$J_3$ Heisenberg model from pseudofermion functional renormalization group

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arXiv:2406.17417 · Physical review. B./Physical review. B

Plaquette-type valence bond solid state in the J 1 − J 2 square-lattice Heisenberg model

Jiale Huang, Xiangjian Qian, Mingpu Qin

Published Nov 4, 2024• 5 citations

arXiv HTML PDF

Abstract

We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the

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

square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy

{\mathrm{\ensuremath{\Delta}}}_{y}

in the nearest-neighboring coupling in the

y

direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as

D=25\phantom{\rule{0.16em}{0ex}}000

in FAMPS, simulating systems at a maximum size of

14\ifmmode\times\else\texttimes\fi{}14

. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite

{\mathrm{\ensuremath{\Delta}}}_{y}

scaling analysis of the VBS dimer order parameters, we identify the VBS phase as a PVBS type, meaning there is no spontaneous rotational symmetry breaking in the VBS phase. This paper not only supports the presence of PVBS order in the

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

square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.

DOI: 10.1103/physrevb.110.195111

Related papers

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

Plaquette valence bond state in the spin- 12 J1−J2 XY model on the square lattice

Yang‐Hao Chan, Hong‐Chen Jiang, Ying-Cheng Chen

We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at ${J}_{2}/{J}_{1}<0.50$ and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at ${J}_{2}/{J}_{1}<0.50$ and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.

arXiv:2212.04863Physical review. B./Physical review. BJun 1, 20233 citations

Phase diagram and magnetic excitations of J1–J3 Heisenberg model on the square lattice

Muwei Wu, Shou-Shu Gong, Dao‐Xin Yao +1 more

We study the phase diagram and the dynamical spin structure factor of the spin-1/2 ${J}_{1}--{J}_{3}$ Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the $X$ and $M$ points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the ${J}_{1}--{J}_{3}$ Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.We study the phase diagram and the dynamical spin structure factor of the spin-1/2 ${J}_{1}--{J}_{3}$ Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the $X$ and $M$ points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the ${J}_{1}--{J}_{3}$ Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.

arXiv:2205.07910Physical review. B./Physical review. BSep 16, 20227 citations

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Jun Takahashi, Anders W. Sandvik

We introduce a quantum spin-1/2 model with many-body correlated Heisenberg-type interactions on the two-dimensional square lattice, designed so that the system can host a fourfold degenerate plaquette valencebond solid (PVBS) ground state that spontaneously breaks Z 4 symmetry.The system is sign-problem free and amenable to large-scale quantum Monte Carlo simulations, thus allowing us to carry out a detailed study of the quantum phase transition between the standard Néel antiferromagnetic (AFM) and PVBS states.We find a first-order transition, in contrast to previously studied continuous transitions from the AFM phase into a columnar valence-bond solid (CVBS) phase.The theory of deconfined quantum criticality predicts generic continuous AFM-CVBS and AFM-PVBS transitions, and, in one version of the theory, the two critical order parameters transform under SO(5) symmetry.Emergent SO(5) symmetry has indeed been observed in studies of the AFM-CVBS transition, and here we show that the first-order AFM-PVBS transition also exhibits SO(5) symmetry at the transition point.Such unexpected symmetry of the coexistence state, which implies a lack of energy barriers between the coexisting phases, has recently been observed at other first-order transitions, but the case presented here is the first example with SO(5) symmetry.The extended symmetry may indicate that the transition is connected to a deconfined critical point.We also discuss the first-order transition in the context of a recent proposal of spinons with fracton properties in the PVBS state, concluding that the fracton scenario is unlikely.Furthermore, we discover a novel type of eightfold degenerate VBS phase, arising when the PVBS state breaks a remaining Z 2 symmetry.This second phase transition, which is continuous, implies that the PVBS phase can be regarded as an intermediate "vestigial" phase, a concept recently introduced to describe multistage phase transitions involving a continuous symmetry.Here we construct a six-dimensional order parameter and also introduce a general graph-theoretic approach to describe the two-stage discrete symmetry breaking.We discuss different ways of breaking the symmetries in one or two stages at zero and finite temperatures.In the latter case, we observe fluctuation-induced first-order transitions, which are hallmarks of vestigial phase transitions.We also mention possible connections of the AFM-PVBS transition to the SO(5) theory of high-T c superconductivity.We introduce a quantum spin-1/2 model with many-body correlated Heisenberg-type interactions on the two-dimensional square lattice, designed so that the system can host a fourfold degenerate plaquette valencebond solid (PVBS) ground state that spontaneously breaks Z 4 symmetry.The system is sign-problem free and amenable to large-scale quantum Monte Carlo simulations, thus allowing us to carry out a detailed study of the quantum phase transition between the standard Néel antiferromagnetic (AFM) and PVBS states.We find a first-order transition, in contrast to previously studied continuous transitions from the AFM phase into a columnar valence-bond solid (CVBS) phase.The theory of deconfined quantum criticality predicts generic continuous AFM-CVBS and AFM-PVBS transitions, and, in one version of the theory, the two critical order parameters transform under SO(5) symmetry.Emergent SO(5) symmetry has indeed been observed in studies of the AFM-CVBS transition, and here we show that the first-order AFM-PVBS transition also exhibits SO(5) symmetry at the transition point.Such unexpected symmetry of the coexistence state, which implies a lack of energy barriers between the coexisting phases, has recently been observed at other first-order transitions, but the case presented here is the first example with SO(5) symmetry.The extended symmetry may indicate that the transition is connected to a deconfined critical point.We also discuss the first-order transition in the context of a recent proposal of spinons with fracton properties in the PVBS state, concluding that the fracton scenario is unlikely.Furthermore, we discover a novel type of eightfold degenerate VBS phase, arising when the PVBS state breaks a remaining Z 2 symmetry.This second phase transition, which is continuous, implies that the PVBS phase can be regarded as an intermediate "vestigial" phase, a concept recently introduced to describe multistage phase transitions involving a continuous symmetry.Here we construct a six-dimensional order parameter and also introduce a general graph-theoretic approach to describe the two-stage discrete symmetry breaking.We discuss different ways of breaking the symmetries in one or two stages at zero and finite temperatures.In the latter case, we observe fluctuation-induced first-order transitions, which are hallmarks of vestigial phase transitions.We also mention possible connections of the AFM-PVBS transition to the SO(5) theory of high-T c superconductivity.

arXiv:2001.10045Physical Review ResearchSep 22, 202040 citations

Evidence of valence bond solid and quantum paramagnetism in the $J_1$ -- $J_2$ -- $J_3$ Heisenberg model from pseudofermion functional renormalization groupEvidence of valence bond solid and quantum paramagnetism in the $J_1$--$J_2$--$J_3$ Heisenberg model from pseudofermion functional renormalization group

GÜN GÜNAL, Erhai Zhao, Ahmet Keleş

Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the $J_1$ -- $J_2$ -- $J_3$ Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in $J_1$ , i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the $J_1$--$J_2$--$J_3$ Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in $J_1$, i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.

arXiv:2610.26135Oct 19, 20260 citations

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

We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the

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

square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy

{\mathrm{\ensuremath{\Delta}}}_{y}

in the nearest-neighboring coupling in the

y

direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as

D=25\phantom{\rule{0.16em}{0ex}}000

in FAMPS, simulating systems at a maximum size of

14\ifmmode\times\else\texttimes\fi{}14

. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite

{\mathrm{\ensuremath{\Delta}}}_{y}

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

square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.

{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 studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor

{J}_{1}

and second-neighbor

{J}_{2}

antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at

{J}_{2}/{J}_{1}<0.50

and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.

We study the phase diagram and the dynamical spin structure factor of the spin-1/2

{J}_{1}--{J}_{3}

Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the

X

and

M

points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the

{J}_{1}--{J}_{3}

Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.

Evidence of valence bond solid and quantum paramagnetism in the

J_1

J_2

J_3

Heisenberg model from pseudofermion functional renormalization group

Motivated by recent tensor network studies reporting a quantum spin liquid (QSL) within the quantum paramagnetic (QP) regime, we investigate numerically the zero-temperature phase diagram of the

J_1

J_2

J_3

Heisenberg model on the square lattice using the pseudofermion functional renormalization group (pf-fRG). Thanks to improved frequency resolution and larger lattice truncations, we are able to explore a significantly larger parameter space than previous pf-fRG work to map out the phase boundary of an extended QP region. We further resolve possible bond order within the QP regime by measuring the dimer response to small deformations in

J_1

, i.e. bond anisotropy that breaks the point-group symmetry of the Hamiltonian. The functional renormalization group flow of the dimer susceptibility suggests that the QP window splits into two: a region where it grows continuously down to the lowest energy scale to signal valence bond solid (VBS) order, and another region where it reaches a peak at a finite scale and is subsequently suppressed to indicate a QSL.

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.