Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Chiral ordering in a frustrated quantum spin system’ ’’

Johannes Richter, Claudius Gros, Werner Weber

Published May 1, 1992• 3 citations

Abstract

We discuss different definitions of chiral order parameters for spin systems. For the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ frustrated Heisenberg model on a finite square lattice, we calculate the vector chirality in the ground state. Unlike the ``spiral'' order parameter, the vector chiral order parameter shows a significant enhancement for strong frustration ${\mathit{J}}_{2}$\ensuremath{\approxeq}0.5${\mathit{J}}_{1}$ in comparison to its corresponding high-temperature expectation value. That indicates different finite-size contributions for both types of order parameters. Thus, we argue that we found some evidence for enhanced vector chiral correlations in the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model.

DOI: 10.1103/physrevb.45.10113

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1 The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins

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Jan 1, 200991 citations

Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Chiral ordering in a frustrated quantum spin system’ ’’

Johannes Richter, Claudius Gros, Werner Weber

Published May 1, 1992• 3 citations

Publisher

Abstract

We discuss different definitions of chiral order parameters for spin systems. For the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

frustrated Heisenberg model on a finite square lattice, we calculate the vector chirality in the ground state. Unlike the ``spiral'' order parameter, the vector chiral order parameter shows a significant enhancement for strong frustration

{\mathit{J}}_{2}

\ensuremath{\approxeq}0.5

{\mathit{J}}_{1}

in comparison to its corresponding high-temperature expectation value. That indicates different finite-size contributions for both types of order parameters. Thus, we argue that we found some evidence for enhanced vector chiral correlations in the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

model.

DOI: 10.1103/physrevb.45.10113

Related papers

Comment on ‘‘Chiral ordering in a frustrated quantum spin system’’

Didier Poilblanc, Elbio R. Dagotto

The symmetry properties of an operator recently introduced by Richter, Gros, and Weber in the context of the ${\mathit{J}}_{1}$ - ${\mathit{J}}_{2}$ frustrated Heisenberg model in two dimensions are studied. Richter et al. claimed that the enhancement of this operator with ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ indicates the existence of strong ``chiral'' correlations in the model. However, we show that this operator transforms identically as a previously analyzed ``spiral'' operator and thus their physical content, including energy gaps in the spectrum, are the same. Thus, we argue that there are no clear numerical indications of chiral order in the ${\mathit{J}}_{1}$ - ${\mathit{J}}_{2}$ model.The symmetry properties of an operator recently introduced by Richter, Gros, and Weber in the context of the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ frustrated Heisenberg model in two dimensions are studied. Richter et al. claimed that the enhancement of this operator with ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$ indicates the existence of strong ``chiral'' correlations in the model. However, we show that this operator transforms identically as a previously analyzed ``spiral'' operator and thus their physical content, including energy gaps in the spectrum, are the same. Thus, we argue that there are no clear numerical indications of chiral order in the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model.

Physical review. B, Condensed matterMay 1, 19922 citations

Chiral ordering in a frustrated quantum spin system

Johannes Richter, Claudius Gros, Werner Weber

We examine the ${\mathit{J}}_{1}$ - ${\mathit{J}}_{2}$ , spin-1/2 Heisenberg model on a square lattice with 16 and 20 sites. We evaluate the ground-state correlations of an operator, measuring the handiness of plaquettes. We find a strong, physically relevant enhancement in these chiral correlations for intermediate values of ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ . We compare with the known results for correlations between column-wise-ordered singlets. We find both types of correlations to be viable candidates for the ground-state correlations in the thermodynamic limit.We examine the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$, spin-1/2 Heisenberg model on a square lattice with 16 and 20 sites. We evaluate the ground-state correlations of an operator, measuring the handiness of plaquettes. We find a strong, physically relevant enhancement in these chiral correlations for intermediate values of ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$. We compare with the known results for correlations between column-wise-ordered singlets. We find both types of correlations to be viable candidates for the ground-state correlations in the thermodynamic limit.

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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).

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

Wen-Yuan Liu, Shaojun Dong, Chao Wang +4 more

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Static and dynamical correlations in a spin-1/2 frustrated antiferromagnet

Didier Poilblanc, Eduardo R. Gagliano, Silvia Bacci +1 more

We study the frustrated spin-1/2 antiferromagnetic Heisenberg model on a square lattice of 16 sites, using a Lanczos algorithm and group-theoretical techniques. We investigate the competition between collinear, dimer, chiral, and spiral orders by calculating the dynamical and spatial correlation functions. Our numerical results strongly suggest the presence of long-range collinear order in the interval 0.55\ensuremath{\le} ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ \ensuremath{\le}0.9 (we cannot show numerically that this order exists for larger values of ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ ). Dimer- and chiral-order parameters both show increasing low-energy fluctuations and a longer range of the correlations in space around ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ \ensuremath{\sim}0.55. We propose an order parameter for chiral order exhibiting the full symmetry of the cluster and hence being less sensitive to finite-size effects. The mean value of the square of this plaquette chiral-order parameter is enhanced for ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ \ensuremath{\sim}0.55, where the characteristic chiral spin-fluctuation frequency is reduced, but no indications of long-range chiral order were found. Although the correlation function of the twist-order parameter is not a simple monotonically decreasing function of distance, little change is induced by frustration, suggesting that spiral order is not strongly favored. The energy gap corresponding to dimer order is smaller than those of spin waves, spiral order, and chiral order; thus, this state is the best candidate for an intermediate-( ${\mathit{J}}_{2}$ / ${\mathit{J}}_{1}$ ) phase in the ${\mathit{J}}_{1}$ - ${\mathit{J}}_{2}$ model.We study the frustrated spin-1/2 antiferromagnetic Heisenberg model on a square lattice of 16 sites, using a Lanczos algorithm and group-theoretical techniques. We investigate the competition between collinear, dimer, chiral, and spiral orders by calculating the dynamical and spatial correlation functions. Our numerical results strongly suggest the presence of long-range collinear order in the interval 0.55\ensuremath{\le}${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\le}0.9 (we cannot show numerically that this order exists for larger values of ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$). Dimer- and chiral-order parameters both show increasing low-energy fluctuations and a longer range of the correlations in space around ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\sim}0.55. We propose an order parameter for chiral order exhibiting the full symmetry of the cluster and hence being less sensitive to finite-size effects. The mean value of the square of this plaquette chiral-order parameter is enhanced for ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\sim}0.55, where the characteristic chiral spin-fluctuation frequency is reduced, but no indications of long-range chiral order were found. Although the correlation function of the twist-order parameter is not a simple monotonically decreasing function of distance, little change is induced by frustration, suggesting that spiral order is not strongly favored. The energy gap corresponding to dimer order is smaller than those of spin waves, spiral order, and chiral order; thus, this state is the best candidate for an intermediate-(${\mathit{J}}_{2}$/${\mathit{J}}_{1}$) phase in the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model.

Physical review. B, Condensed matterMay 1, 199143 citations

1 The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N = 40 spins

Johannes Richter, J. Schulenburg

We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1

Jan 1, 200991 citations

We discuss different definitions of chiral order parameters for spin systems. For the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

{\mathit{J}}_{2}

\ensuremath{\approxeq}0.5

{\mathit{J}}_{1}

{\mathit{J}}_{1}

{\mathit{J}}_{2}

model.

The symmetry properties of an operator recently introduced by Richter, Gros, and Weber in the context of the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

frustrated Heisenberg model in two dimensions are studied. Richter et al. claimed that the enhancement of this operator with

{\mathit{J}}_{2}

{\mathit{J}}_{1}

indicates the existence of strong ``chiral'' correlations in the model. However, we show that this operator transforms identically as a previously analyzed ``spiral'' operator and thus their physical content, including energy gaps in the spectrum, are the same. Thus, we argue that there are no clear numerical indications of chiral order in the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

model.

We examine the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

, spin-1/2 Heisenberg model on a square lattice with 16 and 20 sites. We evaluate the ground-state correlations of an operator, measuring the handiness of plaquettes. We find a strong, physically relevant enhancement in these chiral correlations for intermediate values of

{\mathit{J}}_{2}

{\mathit{J}}_{1}

. We compare with the known results for correlations between column-wise-ordered singlets. We find both types of correlations to be viable candidates for the ground-state correlations in the thermodynamic limit.

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).

J_1

J_2

model on the open boundary conditions (OBC). In this reply, we show that their argument does not apply to our case.

{\mathit{J}}_{2}

{\mathit{J}}_{1}

\ensuremath{\le}0.9 (we cannot show numerically that this order exists for larger values of

{\mathit{J}}_{2}

{\mathit{J}}_{1}

). Dimer- and chiral-order parameters both show increasing low-energy fluctuations and a longer range of the correlations in space around

{\mathit{J}}_{2}

{\mathit{J}}_{1}

\ensuremath{\sim}0.55. We propose an order parameter for chiral order exhibiting the full symmetry of the cluster and hence being less sensitive to finite-size effects. The mean value of the square of this plaquette chiral-order parameter is enhanced for

{\mathit{J}}_{2}

{\mathit{J}}_{1}

\ensuremath{\sim}0.55, where the characteristic chiral spin-fluctuation frequency is reduced, but no indications of long-range chiral order were found. Although the correlation function of the twist-order parameter is not a simple monotonically decreasing function of distance, little change is induced by frustration, suggesting that spiral order is not strongly favored. The energy gap corresponding to dimer order is smaller than those of spin waves, spiral order, and chiral order; thus, this state is the best candidate for an intermediate-(

{\mathit{J}}_{2}

{\mathit{J}}_{1}

) phase in the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

model.