arXiv:cond-mat/0007329 · Physical review. B, Condensed matter

Quantum phase transitions in the two-dimensional J1−J2 model

Oleg P. Sushkov, J. Oitmaa, Zheng Weihong

Published Feb 20, 2001• 151 citations

Abstract

We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the ${J}_{1}{\ensuremath{-}J}_{2}$ model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the N\'eel state to the spin liquid state at ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38,$ and the first-order transition from the spin liquid state to the collinear state at ${(J}_{2}{/J}_{1}{)}_{c4}\ensuremath{\approx}0.60.$ We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at ${(J}_{2}{/J}_{1}{)}_{c3}=0.50\ifmmode\pm\else\textpm\fi{}0.02,$ and the transition from the simple N\'eel state to the N\'eel state with spin columnar dimerization at ${(J}_{2}{/J}_{1}{)}_{c1}=0.34\ifmmode\pm\else\textpm\fi{}0.04.$ We also present an independent calculation of ${(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38$ using another approach.

DOI: 10.1103/physrevb.63.104420

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arXiv:cond-mat/0007329 · Physical review. B, Condensed matter

Quantum phase transitions in the two-dimensional J1−J2 model

Oleg P. Sushkov, J. Oitmaa, Zheng Weihong

Published Feb 20, 2001• 151 citations

arXiv HTML PDF

Abstract

We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the

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

model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the N'eel state to the spin liquid state at

{(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38,

and the first-order transition from the spin liquid state to the collinear state at

{(J}_{2}{/J}_{1}{)}_{c4}\ensuremath{\approx}0.60.

We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at

{(J}_{2}{/J}_{1}{)}_{c3}=0.50\ifmmode\pm\else\textpm\fi{}0.02,

and the transition from the simple N'eel state to the N'eel state with spin columnar dimerization at

{(J}_{2}{/J}_{1}{)}_{c1}=0.34\ifmmode\pm\else\textpm\fi{}0.04.

We also present an independent calculation of

{(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38

using another approach.

DOI: 10.1103/physrevb.63.104420

Related papers

J1−J2 frustrated two-dimensional Heisenberg model: Random phase approximation and functional renormalization group

Johannes Reuther, P. Wölfle

We study the ground-state properties of the two-dimensional spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ 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 random-phase approximation, demonstrating the appearance of a paramagnetic phase between the N\'eel-ordered and Collinear-ordered phases, at sufficiently large pseudofermion damping. Second 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 N\'eel order at $g={J}_{2}/{J}_{1}\ensuremath{\lesssim}{g}_{\text{c}1}\ensuremath{\approx}0.4\dots{}0.45$ and Collinear order at $g\ensuremath{\gtrsim}{g}_{\text{c}2}\ensuremath{\approx}0.66\dots{}0.68$, 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 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ 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 random-phase approximation, demonstrating the appearance of a paramagnetic phase between the N\'eel-ordered and Collinear-ordered phases, at sufficiently large pseudofermion damping. Second 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 N\'eel order at $g={J}_{2}/{J}_{1}\ensuremath{\lesssim}{g}_{\text{c}1}\ensuremath{\approx}0.4\dots{}0.45$ and Collinear order at $g\ensuremath{\gtrsim}{g}_{\text{c}2}\ensuremath{\approx}0.66\dots{}0.68$, 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.

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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.

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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.

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Low-energy singlet and triplet excitations in the spin-liquid phase of the two-dimensional J1−J2 model

Valeri N. Kotov, J. Oitmaa, Oleg P. Sushkov +1 more

We analyze the stability of the spontaneously dimerized phase of the frustrated Heisenberg antiferromagnet---the ${J}_{1}{\ensuremath{-}J}_{2}$ model---in two dimensions. The lowest triplet excitation, corresponding to breaking of a singlet bond, is found to be stable in the region $0.38\ensuremath{\lesssim}{J}_{2}{/J}_{1}\ensuremath{\lesssim}0.62.$ In addition we find a stable low-energy collective singlet mode, which reflects the spontaneous violation of the discrete symmetry. The spontaneous dimerization vanishes at the point of second-order quantum transition into the N\'eel ordered phase $[{(J}_{2}{/J}_{1}{)}_{c1}\ensuremath{\approx}0.38].$ We argue that the disappearance of dimer order is related to the vanishing of the singlet energy gap at the transition point.We analyze the stability of the spontaneously dimerized phase of the frustrated Heisenberg antiferromagnet---the ${J}_{1}{\ensuremath{-}J}_{2}$ model---in two dimensions. The lowest triplet excitation, corresponding to breaking of a singlet bond, is found to be stable in the region $0.38\ensuremath{\lesssim}{J}_{2}{/J}_{1}\ensuremath{\lesssim}0.62.$ In addition we find a stable low-energy collective singlet mode, which reflects the spontaneous violation of the discrete symmetry. The spontaneous dimerization vanishes at the point of second-order quantum transition into the N\'eel ordered phase $[{(J}_{2}{/J}_{1}{)}_{c1}\ensuremath{\approx}0.38].$ We argue that the disappearance of dimer order is related to the vanishing of the singlet energy gap at the transition point.

arXiv:cond-mat/9903154Physical review. B, Condensed matterDec 1, 199983 citations

Tensor network simulation of the phase diagram of the frustrated J1 - J2 Heisenberg model on a checkerboard lattice

Yang‐Hao Chan, Yong‐Jian Han, L.-M. Duan

We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic ${J}_{1}$ - ${J}_{2}$ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when ${J}_{2}<0.88{J}_{1}$ , a plaquette valence bond solid state when $0.88<{J}_{2}/{J}_{1}<1.11$ , and a stripe phase when ${J}_{2}>1.11{J}_{1}$ , with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic ${J}_{1}$-${J}_{2}$ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when ${J}_{2}<0.88{J}_{1}$, a plaquette valence bond solid state when $0.88<{J}_{2}/{J}_{1}<1.11$, and a stripe phase when ${J}_{2}>1.11{J}_{1}$, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.

arXiv:1108.1595Physical Review BDec 12, 201116 citations

J 1 − J 2 model: First-order phase transition versus deconfinement of spinons

Jesko Sirker, Zheng Weihong, Oleg P. Sushkov +1 more

We revisit the phase transition from the N\'eel ordered to a valence bond solid (VBS) state in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model. In the first part we address the question whether or not this transition could be an example of a second-order phase transition due to a deconfinement of spinons. We give arguments based on series expansion and spin-wave theory that this is not the case and the transition is most likely first order. The method proposed here to detect first-order phase transitions seems to be very sensitive and might be useful in other models as well. In the second part we analyze possible VBS patterns in the magnetically disordered phase based on numerical data for different susceptibilities, obtained in the ordered phase, which test the breaking of lattice symmetries. We conclude that a columnar dimerization pattern is the most likely candidate.We revisit the phase transition from the N\'eel ordered to a valence bond solid (VBS) state in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model. In the first part we address the question whether or not this transition could be an example of a second-order phase transition due to a deconfinement of spinons. We give arguments based on series expansion and spin-wave theory that this is not the case and the transition is most likely first order. The method proposed here to detect first-order phase transitions seems to be very sensitive and might be useful in other models as well. In the second part we analyze possible VBS patterns in the magnetically disordered phase based on numerical data for different susceptibilities, obtained in the ordered phase, which test the breaking of lattice symmetries. We conclude that a columnar dimerization pattern is the most likely candidate.

arXiv:cond-mat/0601183Physical Review BMay 17, 2006138 citations

We analyze the phase diagram of the frustrated Heisenberg antiferromagnet, the

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

model, in two dimensions. Two quantum phase transitions in the model are already known: the second-order transition from the N'eel state to the spin liquid state at

{(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38,

and the first-order transition from the spin liquid state to the collinear state at

{(J}_{2}{/J}_{1}{)}_{c4}\ensuremath{\approx}0.60.

We have found evidence for two second-order phase transitions: the transition from the spin columnar dimerized state to the state with plaquette-type modulation at

{(J}_{2}{/J}_{1}{)}_{c3}=0.50\ifmmode\pm\else\textpm\fi{}0.02,

and the transition from the simple N'eel state to the N'eel state with spin columnar dimerization at

{(J}_{2}{/J}_{1}{)}_{c1}=0.34\ifmmode\pm\else\textpm\fi{}0.04.

We also present an independent calculation of

{(J}_{2}{/J}_{1}{)}_{c2}\ensuremath{\approx}0.38

using another approach.

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.

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 use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic

{J}_{1}

{J}_{2}

Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when

{J}_{2}<0.88{J}_{1}

, a plaquette valence bond solid state when

0.88<{J}_{2}/{J}_{1}<1.11

, and a stripe phase when

{J}_{2}>1.11{J}_{1}

, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.