arXiv:1205.0681 · Physical Review B

Ground-state phase diagram of the spin- 12 square-lattice J1 - J2 model with plaquette structure

Oliver Götze, Sven E. Krüger, F. Fleck, J. Schulenburg, Johannes Richter

Published Jun 21, 2012• 16 citations

Abstract

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 $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.

DOI: 10.1103/physrevb.85.224424

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Spin- 12 J1 - J2 Heisenberg model on a cross-striped square lattice

R. F. Bishop, P. H. Y. Li, C. E. Campbell

arXiv:1308.4573Physical Review BDec 19, 201310 citations

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Fabio Mezzacapo

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arXiv:1205.0681 · Physical Review B

Ground-state phase diagram of the spin- 12 square-lattice J1 - J2 model with plaquette structure

Oliver Götze, Sven E. Krüger, F. Fleck, J. Schulenburg, Johannes Richter

Published Jun 21, 2012• 16 citations

arXiv HTML PDF

Abstract

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

DOI: 10.1103/physrevb.85.224424

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Spin- 12 J1 - J2 Heisenberg model on a cross-striped square lattice

R. F. Bishop, P. H. Y. Li, C. E. Campbell

Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ( $s=\frac{1}{2}$ ) ${J}_{1}$ - ${J}_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength ${J}_{1}$ and two next-nearest-neighbor (diagonal) bonds of strength ${J}_{2}$ . The ${J}_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no ${J}_{2}$ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when ${J}_{1}$ and ${J}_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum $s=\frac{1}{2}$ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the $s=\frac{1}{2}$ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the $s=\frac{1}{2}$ model are of continuous deconfined type.Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ($s=\frac{1}{2}$) ${J}_{1}$-${J}_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength ${J}_{1}$ and two next-nearest-neighbor (diagonal) bonds of strength ${J}_{2}$. The ${J}_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no ${J}_{2}$ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when ${J}_{1}$ and ${J}_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum $s=\frac{1}{2}$ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the $s=\frac{1}{2}$ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the $s=\frac{1}{2}$ model are of continuous deconfined type.

arXiv:1308.4573Physical Review BDec 19, 201310 citations

Ground-state phases of the frustrated spin- 12 J1 – J2 – J3 Heisenberg ferromagnet ( J1<0 ) on the honeycomb lattice with J3=J2>0

P. H. Y. Li, R. F. Bishop, DAMIAN J. J. FARNELL +2 more

We study the ground-state (GS) properties of the frustrated spin- $\frac{1}{2}$ ${J}_{1}$ -- ${J}_{2}$ -- ${J}_{3}$ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor ( ${J}_{2}>0$ ) and next-next-nearest-neighbor ( ${J}_{3}>0$ ) exchanges, for the case ${J}_{3}={J}_{2}$ . We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region $\frac{1}{10}<{J}_{2}<\frac{1}{5}$ . Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with ${J}_{1}=+1$ ). 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Unlike in the antiferromagnetic model (with ${J}_{1}=+1$ ), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$ , with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<{J}_{2}<\frac{1}{5}$ .We study the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$--${J}_{2}$--${J}_{3}$ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (${J}_{2}>0$) and next-next-nearest-neighbor (${J}_{3}>0$) exchanges, for the case ${J}_{3}={J}_{2}$. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region $\frac{1}{10}<{J}_{2}<\frac{1}{5}$. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with ${J}_{1}=+1$). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter ${J}_{2}$. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with ${J}_{1}=+1$) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with ${J}_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<{J}_{2}<\frac{1}{5}$.

arXiv:1109.6229Physical Review BFeb 21, 201231 citations

Phase diagram of a frustrated Heisenberg antiferromagnet on the honeycomb lattice: The J1 - J2 - J3 model

P. H. Y. Li, R. F. Bishop, DAMIAN J. J. FARNELL +1 more

We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 ${J}_{1}$ - ${J}_{2}$ - ${J}_{3}$ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values ${J}_{3}<0$ . Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 ${J}_{1}$-${J}_{2}$-${J}_{3}$ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values ${J}_{3}<0$. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.

arXiv:1204.2390Physical Review BOct 9, 201299 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

Complete phase diagram of the spin- 12 J1 - J2 - J3 model (with J3 = J2 ) on the honeycomb lattice

R. F. Bishop, P. H. Y. Li

We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin- $\frac{1}{2}$ ${J}_{1}$ - ${J}_{2}$ - ${J}_{3}$ model on the honeycomb lattice, with nearest-neighbor exchange coupling ${J}_{1}$ plus next-nearest-neighbor ( ${J}_{2}$ ) and next-next-nearest-neighbor ( ${J}_{3}$ ) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when ${J}_{1}>0$ , compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$-${J}_{2}$-${J}_{3}$ model on the honeycomb lattice, with nearest-neighbor exchange coupling ${J}_{1}$ plus next-nearest-neighbor (${J}_{2}$) and next-next-nearest-neighbor (${J}_{3}$) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when ${J}_{1}>0$, compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.

arXiv:1202.6249Physical Review BApr 20, 201227 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

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

\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 (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (

s=\frac{1}{2}

)

{J}_{1}

{J}_{2}

Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength

{J}_{1}

and two next-nearest-neighbor (diagonal) bonds of strength

{J}_{2}

. The

{J}_{2}

bonds are arranged so that the basic square plaquettes in alternating columns have either both or no

{J}_{2}

bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when

{J}_{1}

and

{J}_{2}

can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum

s=\frac{1}{2}

model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the

s=\frac{1}{2}

model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the

s=\frac{1}{2}

model are of continuous deconfined type.

We study the ground-state (GS) properties of the frustrated spin-

\frac{1}{2}

{J}_{1}

{J}_{2}

{J}_{3}

Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (

{J}_{2}>0

) and next-next-nearest-neighbor (

{J}_{3}>0

) exchanges, for the case

{J}_{3}={J}_{2}

. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at

{J}_{2}^{\mathrm{cl}}=\frac{1}{7}

($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region

\frac{1}{10}<{J}_{2}<\frac{1}{5}

. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with

{J}_{1}=+1

). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter

{J}_{2}

. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with

{J}_{1}=+1

) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with

{J}_{1}=+1

), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for

{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}

, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range

\frac{1}{10}<{J}_{2}<\frac{1}{5}

We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2

{J}_{1}

{J}_{2}

{J}_{3}

model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values

{J}_{3}<0

. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.

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 coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin-

\frac{1}{2}

{J}_{1}

{J}_{2}

{J}_{3}

model on the honeycomb lattice, with nearest-neighbor exchange coupling

{J}_{1}

plus next-nearest-neighbor (

{J}_{2}

) and next-next-nearest-neighbor (

{J}_{3}

) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when

{J}_{1}>0

, compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.

{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.