arXiv:0809.4035 · Physical Review B

Hierarchical mean-field approach to the J1−J2 Heisenberg model on a square lattice

Леонид Исаев, Gerardo Ortíz, J. Dukelsky

Published Jan 9, 2009• 97 citations

Abstract

We study the quantum phase diagram and excitation spectrum of the frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ spin-1/2 Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea of identifying relevant degrees of freedom, is developed. Thus, by performing educated, manifestly symmetry-preserving mean-field approximations, we unveil fundamental properties of the system. We then compare various coverings of the square lattice with plaquettes, dimers, and other degrees of freedom, and show that only the symmetric plaquette covering, which reproduces the original Bravais lattice, leads to the known phase diagram. The intermediate quantum paramagnetic phase is shown to be a (singlet) plaquette crystal, connected with the neighboring N\'eel phase by a continuous phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and demonstrate that in the paramagnetic phase the ground and first excited states are separated by a finite gap, which closes in the N\'eel and columnar phases. Our results suggest that the quantum phase transition between N\'eel and paramagnetic phases can be properly described within the Ginzburg-Landau-Wilson paradigm.

DOI: 10.1103/physrevb.79.024409

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arXiv:0809.4035 · Physical Review B

Hierarchical mean-field approach to the J1−J2 Heisenberg model on a square lattice

Леонид Исаев, Gerardo Ortíz, J. Dukelsky

Published Jan 9, 2009• 97 citations

arXiv HTML PDF

Abstract

We study the quantum phase diagram and excitation spectrum of the frustrated

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

spin-1/2 Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea of identifying relevant degrees of freedom, is developed. Thus, by performing educated, manifestly symmetry-preserving mean-field approximations, we unveil fundamental properties of the system. We then compare various coverings of the square lattice with plaquettes, dimers, and other degrees of freedom, and show that only the symmetric plaquette covering, which reproduces the original Bravais lattice, leads to the known phase diagram. The intermediate quantum paramagnetic phase is shown to be a (singlet) plaquette crystal, connected with the neighboring N'eel phase by a continuous phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and demonstrate that in the paramagnetic phase the ground and first excited states are separated by a finite gap, which closes in the N'eel and columnar phases. Our results suggest that the quantum phase transition between N'eel and paramagnetic phases can be properly described within the Ginzburg-Landau-Wilson paradigm.

DOI: 10.1103/physrevb.79.024409

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The phase diagram of the Heisenberg antiferromagnet with four-spin interactions

Леонид Исаев, Gerardo Ortíz, J. Dukelsky

We study the quantum phase diagram of the Heisenberg planar antiferromagnet with a subset of four-spin ring exchange interactions, using the recently proposed hierarchical mean-field approach. By identifying relevant degrees of freedom, we are able to use a single variational ansatz to map the entire phase diagram of the model and uncover the nature of its various phases. It is shown that there exists a transition between a Néel state and a quantum paramagnetic phase, characterized by broken translational invariance. The non-magnetic phase preserves the lattice rotational symmetry, and has a correlated plaquette nature. Our results also suggest that this phase transition can be properly described within the Landau paradigm.We study the quantum phase diagram of the Heisenberg planar antiferromagnet with a subset of four-spin ring exchange interactions, using the recently proposed hierarchical mean-field approach. By identifying relevant degrees of freedom, we are able to use a single variational ansatz to map the entire phase diagram of the model and uncover the nature of its various phases. It is shown that there exists a transition between a Néel state and a quantum paramagnetic phase, characterized by broken translational invariance. The non-magnetic phase preserves the lattice rotational symmetry, and has a correlated plaquette nature. Our results also suggest that this phase transition can be properly described within the Landau paradigm.

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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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Oliver Götze, Sven E. Krüger, F. Fleck +2 more

Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin- $\frac{1}{2}$ ${J}_{1}$ - ${J}_{2}$ model on the square lattice with plaquette structure. We consider antiferromagnetic ( ${J}_{1}>0$ ) as well as ferromagnetic ( ${J}_{1}<0$ ) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction ${J}_{2}>0$ . The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform ${J}_{1}$ - ${J}_{2}$ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic ${J}_{1}$ ) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case $s=1/2$ . For the antiferromagnetic case ( ${J}_{1}>0$ ) N\'eel antiferromagnetic long-range order at small ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case ( ${J}_{1}<0$ ) we have the trivial ferromagnetic ground state at small ${J}_{2}/|{J}_{1}|$ . By increasing ${J}_{2}$ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length $s=2$ are antiferromagnetically long-range ordered. Further increasing of ${J}_{2}$ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-$\frac{1}{2}$ ${J}_{1}$-${J}_{2}$ model on the square lattice with plaquette structure. We consider antiferromagnetic (${J}_{1}>0$) as well as ferromagnetic (${J}_{1}<0$) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction ${J}_{2}>0$. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform ${J}_{1}$-${J}_{2}$ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic ${J}_{1}$) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case $s=1/2$. For the antiferromagnetic case (${J}_{1}>0$) N\'eel antiferromagnetic long-range order at small ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (${J}_{1}<0$) we have the trivial ferromagnetic ground state at small ${J}_{2}/|{J}_{1}|$. By increasing ${J}_{2}$ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length $s=2$ are antiferromagnetically long-range ordered. Further increasing of ${J}_{2}$ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.

arXiv:1205.0681Physical Review BJun 21, 201216 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

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

Oleg P. Sushkov, J. Oitmaa, Zheng Weihong

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.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.

arXiv:cond-mat/0007329Physical review. B, Condensed matterFeb 20, 2001151 citations

Ground-state phase diagram of the quantum J1−J2 model on the square lattice

Fabio Mezzacapo

We study the ground-state phase diagram of the quantum ${J}_{1}\ensuremath{-}{J}_{2}$ model on a square lattice by means of an entangled-plaquette variational Ansatz. In the range $0\ensuremath{\leqslant}{J}_{2}/{J}_{1}\ensuremath{\leqslant}1$, we find classical magnetic order of N\'eel and collinear type, for ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.5$ and ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.6$, respectively. For intermediate values of ${J}_{2}/{J}_{1}$ the ground state is a spin liquid (i.e., paramagnetic with no valence-bond crystalline order). Our estimates of the entanglement entropy show that such a spin liquid is topological.We study the ground-state phase diagram of the quantum ${J}_{1}\ensuremath{-}{J}_{2}$ model on a square lattice by means of an entangled-plaquette variational Ansatz. In the range $0\ensuremath{\leqslant}{J}_{2}/{J}_{1}\ensuremath{\leqslant}1$, we find classical magnetic order of N\'eel and collinear type, for ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.5$ and ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.6$, respectively. For intermediate values of ${J}_{2}/{J}_{1}$ the ground state is a spin liquid (i.e., paramagnetic with no valence-bond crystalline order). Our estimates of the entanglement entropy show that such a spin liquid is topological.

arXiv:1203.6381Physical Review BJul 16, 201291 citations

We study the quantum phase diagram and excitation spectrum of the frustrated

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

Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-

\frac{1}{2}

{J}_{1}

{J}_{2}

model on the square lattice with plaquette structure. We consider antiferromagnetic (

{J}_{1}>0

) as well as ferromagnetic (

{J}_{1}<0

) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction

{J}_{2}>0

. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform

{J}_{1}

{J}_{2}

model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic

{J}_{1}

) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case

s=1/2

. For the antiferromagnetic case (

{J}_{1}>0

) N\'eel antiferromagnetic long-range order at small

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

and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large

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

and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (

{J}_{1}<0

) we have the trivial ferromagnetic ground state at small

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

. By increasing

{J}_{2}

this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length

s=2

are antiferromagnetically long-range ordered. Further increasing of

{J}_{2}

then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.

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

the ground state is a spin liquid (i.e., paramagnetic with no valence-bond crystalline order). Our estimates of the entanglement entropy show that such a spin liquid is topological.