Physical review. B, Condensed matter

Chiral ordering in a frustrated quantum spin system

Johannes Richter, Claudius Gros, Werner Weber

Published Jul 1, 1991• 18 citations

Abstract

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.

DOI: 10.1103/physrevb.44.906

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

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

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

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Physical review. B, Condensed matter

Chiral ordering in a frustrated quantum spin system

Johannes Richter, Claudius Gros, Werner Weber

Published Jul 1, 1991• 18 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.44.906

Related papers

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

Phase diagram of the frustrated spin-1/2 Heisenberg antiferromagnet in 2 dimensions

Elbio R. Dagotto, Adriana Moreo

Using a Lanczos technique we study the frustrated spin-(1/2 Heisenberg model on square lattices of 16 and 20 sites. Frustration is introduced by an interaction along the diagonals of the plaquettes with coupling ${J}_{2}$ \ensuremath{\ge}0. For large ${J}_{2}$ we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of ${J}_{2}$ , the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.Using a Lanczos technique we study the frustrated spin-(1/2 Heisenberg model on square lattices of 16 and 20 sites. Frustration is introduced by an interaction along the diagonals of the plaquettes with coupling ${J}_{2}$\ensuremath{\ge}0. For large ${J}_{2}$ we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of ${J}_{2}$, the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.

Physical Review LettersNov 6, 1989329 citations

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

Johannes Richter, Claudius Gros, Werner Weber

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.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.

Physical review. B, Condensed matterMay 1, 19923 citations

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

Spin- 12 J1 - J2 Heisenberg antiferromagnet on a square lattice: A plaquette renormalized tensor network study

Ji-Feng Yu, Ying-Jer Kao

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$ - ${J}_{2}$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with $L$ $=$ 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different ${J}_{2}/{J}_{1}$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high ${J}_{2}/{J}_{1}$ , we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$-${J}_{2}$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with $L$ $=$ 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different ${J}_{2}/{J}_{1}$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high ${J}_{2}/{J}_{1}$, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.

arXiv:1108.3393Physical Review BMar 5, 201270 citations

Quantum phase diagram of the spin-1 J1−J2 Heisenberg model on the honeycomb lattice

Shou-Shu Gong, Wei Kang Zhu, D. N. Sheng

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.

arXiv:1508.00515Physical Review BNov 6, 201544 citations

We examine the

{\mathit{J}}_{1}

{\mathit{J}}_{2}

{\mathit{J}}_{2}

{\mathit{J}}_{1}

{\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.

{J}_{2}

\ensuremath{\ge}0. For large

{J}_{2}

we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of

{J}_{2}

, the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.

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.

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 apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg

{J}_{1}

{J}_{2}

model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with

L

=

8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different

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

values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high

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

, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.

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.