arXiv:1308.4573 · Physical Review B

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

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

Published Dec 19, 2013• 10 citations

Abstract

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.

DOI: 10.1103/physrevb.88.214418

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arXiv:1308.4573 · Physical Review B

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

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

Published Dec 19, 2013• 10 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.88.214418

Related papers

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

Quasiclassical magnetic order and its loss in a spin- 1 2 Heisenberg antiferromagnet on a triangular lattice with competing bonds

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

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin- $\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin- $\frac{1}{2}$ model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter, $M=0.198(5)$ (in units where the classical value is ${M}^{\mathrm{cl}}=\frac{1}{2}$ ), which are among the best available. For the spin- $\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin-$\frac{1}{2}$ model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter, $M=0.198(5)$ (in units where the classical value is ${M}^{\mathrm{cl}}=\frac{1}{2}$), which are among the best available. For the spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.

arXiv:1410.6003Physical Review BJan 22, 201586 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 ${J}_{1}=+1$ ). 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 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

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

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

Ground state phases of the spin-1/2 J1–J2 Heisenberg antiferromagnet on the square lattice: A high-order coupled cluster treatment

R. Darradi, Oleg Derzhko, Ronald Zinke +3 more

Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 ${J}_{1}--{J}_{2}$ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 ${J}_{1}--{J}_{2}$ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.

arXiv:0806.3825Physical Review BDec 11, 2008183 citations

Ground-state phases of the spin- 1 J 1 − J 2 Heisenberg antiferromagnet on the honeycomb lattice

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

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.

arXiv:1602.08915Physical review. B./Physical review. BJun 28, 201621 citations

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}

s=\frac{1}{2}

s=\frac{1}{2}

s=\frac{1}{2}

model are of continuous deconfined type.

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 (CCM) to study the zero-temperature ground-state (GS) properties of a spin-

\frac{1}{2}

${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings

{J}_{1}>0

and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin-

\frac{1}{2}

model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter,

M=0.198(5)

(in units where the classical value is

{M}^{\mathrm{cl}}=\frac{1}{2}

), which are among the best available. For the spin-

\frac{1}{2}

${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.

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}

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 complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2

{J}_{1}--{J}_{2}

frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling

{J}_{1}>0

and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.