Physical review. B, Condensed matter

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

Didier Poilblanc, Elbio R. Dagotto

Published May 1, 1992• 2 citations

Abstract

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.

DOI: 10.1103/physrevb.45.10111

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

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

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Chiral ordering in a frustrated quantum spin system

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J_1-J_2 frustrated 2d Heisenberg model: RPA and Functional RG

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Dec 4, 20090 citations

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Shu Zhang, Gerhard H. Muller

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Physical review. B, Condensed matter

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

Didier Poilblanc, Elbio R. Dagotto

Published May 1, 1992• 2 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.45.10111

Related papers

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

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

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.

Physical review. B, Condensed matterJul 1, 199118 citations

Dimerization and incommensurate spiral spin correlations in the zigzag spin chain: Analogies to the Kondo lattice

Steven R. White, Ian K. Affleck

Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling ${\mathit{J}}_{1}$ and frustrating second-neighbor coupling ${\mathit{J}}_{2}$ , particularly in the limit ${\mathit{J}}_{2}$ \ensuremath{\gg} ${\mathit{J}}_{1}$ . This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling ${\mathit{J}}_{1}$ and frustrating second-neighbor coupling ${\mathit{J}}_{2}$, particularly in the limit ${\mathit{J}}_{2}$\ensuremath{\gg}${\mathit{J}}_{1}$. This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9602126Physical review. B, Condensed matterOct 1, 1996548 citations

J_1-J_2 frustrated 2d Heisenberg model: RPA and Functional RG

Johannes Reuther, P. Wölfle

We study the ground state properties of the two-dimensional spin-1/2 J_1-J_2-Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary fermion formulation with exact projection. In a first approximation we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities and the spin correlation length within RPA, demonstrating the appearance of a paramagnetic phase between the Neel ordered and Collinear ordered phases, at sufficiently large pseudo fermion damping. Secondly we use a Functional Renormalization Group formulation. We find that the conventional truncation scheme omitting three-particle and higher order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find Neel order at g = J_2/J_1 0.66 ... 0.68, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase we find enhanced columnar dimer and plaquette fluctuations of equal strength.We study the ground state properties of the two-dimensional spin-1/2 J_1-J_2-Heisenberg model on a square lattice, within diagrammatic approximations using an auxiliary fermion formulation with exact projection. In a first approximation we assume a phenomenological width of the pseudofermion spectral function to calculate the magnetization, susceptibilities and the spin correlation length within RPA, demonstrating the appearance of a paramagnetic phase between the Neel ordered and Collinear ordered phases, at sufficiently large pseudo fermion damping. Secondly we use a Functional Renormalization Group formulation. We find that the conventional truncation scheme omitting three-particle and higher order vertices is not sufficient. We therefore include self-energy renormalizations in the single-scale propagator as recently proposed by Katanin, to preserve Ward identities in a better way. We find Neel order at g = J_2/J_1 0.66 ... 0.68, which is in good agreement with results obtained by numerical studies. In the intervening quantum paramagnetic phase we find enhanced columnar dimer and plaquette fluctuations of equal strength.

Dec 4, 20090 citations

Order-parameter fluctuations in the frustrated Heisenberg model on the square lattice

Shu Zhang, Gerhard H. Muller

The T=0 dynamics of the two-dimensional s=1/2 Heisenberg model with competing nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions is explored via the recursion method, specifically the frequency-dependent fluctuations of the order parameters associated with some of the known or suspected ordering tendencies in this system, i.e., Néel, collinear, dimer, and chiral order. The results for the dynamic structure factors of the respective fluctuation operators show a strong indication of collinear order at J2/J1≳0.6 and a potential for dimer order at 0.5≲J2/J1≲0.6, whereas the chiral ordering tendency is observed to be considerably weaker.The T=0 dynamics of the two-dimensional s=1/2 Heisenberg model with competing nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions is explored via the recursion method, specifically the frequency-dependent fluctuations of the order parameters associated with some of the known or suspected ordering tendencies in this system, i.e., Néel, collinear, dimer, and chiral order. The results for the dynamic structure factors of the respective fluctuation operators show a strong indication of collinear order at J2/J1≳0.6 and a potential for dimer order at 0.5≲J2/J1≲0.6, whereas the chiral ordering tendency is observed to be considerably weaker.

arXiv:cond-mat/9508145Journal of Applied PhysicsApr 15, 19960 citations

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}

{\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.

{\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 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.

Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling

{\mathit{J}}_{1}

and frustrating second-neighbor coupling

{\mathit{J}}_{2}

, particularly in the limit

{\mathit{J}}_{2}

\ensuremath{\gg}

{\mathit{J}}_{1}

. This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.