arXiv:1504.08173 · Journal of Physics Condensed Matter

Ground-state phase structure of the spin- anisotropic planar pyrochlore

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

Published Sep 8, 2015• 5 citations

Abstract

We study the zero-temperature ground-state (GS) properties of the spin-1/2 anisotropic planar pyrochlore, using the coupled cluster method (CCM) implemented to high orders of approximation. The system comprises a J 1-J 2 model on the checkerboard lattice, with isotropic Heisenberg interactions of strength J 1 between all nearest-neighbour pairs of spins on the square lattice, and of strength J 2 between half of the next-nearest-neighbour pairs (in the checkerboard pattern). We calculate results for the GS energy and average local GS on-site magnetization, using various antiferromagnetic classical ground states as CCM model states. We also give results for the susceptibility of one of these states against the formation of crossed-dimer valence-bond crystalline (CDVBC) ordering. The complete GS phase diagram is presented for arbitrary values of the frustration parameter k≡J2/J1, and when each of the exchange couplings can take either sign.

DOI: 10.1088/0953-8984/27/38/386002

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arXiv:1504.08173 · Journal of Physics Condensed Matter

Ground-state phase structure of the spin- anisotropic planar pyrochlore

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

Published Sep 8, 2015• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0953-8984/27/38/386002

Related papers

A frustrated spin-1J1-J2Heisenberg antiferromagnet: An anisotropic planar pyrochlore model

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

The zero-temperature ground-state (GS) properties and phase diagram of a frustrated spin-1 J 1 - J 2 Heisenberg model on the checkerboard square lattice are studied, using the coupled cluster method. We consider the case where the nearest-neighbour exchange bonds have strength J 1 > 0 and the next-nearest-neighbour exchange bonds present (viz., in the checkerboard pattern of the planar pyrochlore) have strength J 2 = κ J 1 > 0. We find significant differences from both the spin-1/2 and classical versions of the model. We find that the spin-1 model has a first phase transition at κ C1 ≈ 1.00 ± 0.01 (as does the classical model at κ cl = 1) between two antiferromagnetic phases, viz., a quasiclassical Néel phase (for κ < κ C1 ) and one of the infinitely degenerate family of quasiclassical phases (for k > κ C1 ) that exists in the classical model for κ > κ c1 , which is now chosen by the order by disorder mechanism as (probably) the "doubled Néel" (or Néel*) state. By contrast, none of this family survives quantum fluctuations to form a stable GS phase in the spin-1/2 case. We also find evidence for a second quantum critical point at κ C2 ≈ 2.0 ± 0.5 in the spin-1 model, such that for κ > κ C2 the quasiclassical (Néel*) ordering melts and a nonclassical phase appears, which, on the basis of preliminary evidence, appears unlikely to have crossed-dimer valence-bond crystalline (CDVBC) ordering, as in the spin-1/2 case. Unlike in the spin-1/2 case, where the Néel and CDVBC phases are separated by a phase with plaquette valence-bond crystalline (PVBC) ordering, we find very preliminary evidence for such a PVBC state in the spin-1 model for all κ > κ C2 .The zero-temperature ground-state (GS) properties and phase diagram of a frustrated spin-1 J 1 - J 2 Heisenberg model on the checkerboard square lattice are studied, using the coupled cluster method. We consider the case where the nearest-neighbour exchange bonds have strength J 1 > 0 and the next-nearest-neighbour exchange bonds present (viz., in the checkerboard pattern of the planar pyrochlore) have strength J 2 = κ J 1 > 0. We find significant differences from both the spin-1/2 and classical versions of the model. We find that the spin-1 model has a first phase transition at κ C1 ≈ 1.00 ± 0.01 (as does the classical model at κ cl = 1) between two antiferromagnetic phases, viz., a quasiclassical Néel phase (for κ < κ C1 ) and one of the infinitely degenerate family of quasiclassical phases (for k > κ C1 ) that exists in the classical model for κ > κ c1 , which is now chosen by the order by disorder mechanism as (probably) the "doubled Néel" (or Néel*) state. By contrast, none of this family survives quantum fluctuations to form a stable GS phase in the spin-1/2 case. We also find evidence for a second quantum critical point at κ C2 ≈ 2.0 ± 0.5 in the spin-1 model, such that for κ > κ C2 the quasiclassical (Néel*) ordering melts and a nonclassical phase appears, which, on the basis of preliminary evidence, appears unlikely to have crossed-dimer valence-bond crystalline (CDVBC) ordering, as in the spin-1/2 case. Unlike in the spin-1/2 case, where the Néel and CDVBC phases are separated by a phase with plaquette valence-bond crystalline (PVBC) ordering, we find very preliminary evidence for such a PVBC state in the spin-1 model for all κ > κ C2 .

arXiv:1312.2420Journal of Physics Conference SeriesAug 14, 20141 citation

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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.

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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.

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We study the phase diagram of two models of spin- $1∕2$ antiferromagnets composed of corner-sharing tetrahedra, the basis of the pyrochlore structure. Primarily, we focus on the Heisenberg antiferromaget on the checkerboard lattice (also called the planar pyrochlore and crossed-chain model). This model has an anisotropic limit, when the dimensionless ratio of two exchange constants ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪡1$, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. Using recently developed techniques combining renormalization group ideas and one-dimensional bosonization and current algebra methods, we show that in this limit the model enters a crossed-dimer state with twofold spontaneous symmetry breaking but no magnetic order. We complement this result by an approximate ``quadrumer triplet boson'' calculation, which qualitatively captures the physics of the ``plaquette valence-bond solid'' state believed to obtain for ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J\ensuremath{\approx}1$. Using these known points in parameter space, the instabilities pointed to by the quadrumer boson calculation, and the simple limit ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪢1$, we construct a few candidate global phase diagrams for the model, and discuss the nature of the quantum phase transitions contained therein. Finally, we apply our quasi-one-dimensional techniques to an anisotropic limit of the three-dimensional pyrochlore antiferromagnet, an approximate model for magnetism in $\mathrm{Ge}{\mathrm{Cu}}_{2}{\mathrm{O}}_{4}$ . A crossed-dimer state is predicted here as well.We study the phase diagram of two models of spin-$1∕2$ antiferromagnets composed of corner-sharing tetrahedra, the basis of the pyrochlore structure. Primarily, we focus on the Heisenberg antiferromaget on the checkerboard lattice (also called the planar pyrochlore and crossed-chain model). This model has an anisotropic limit, when the dimensionless ratio of two exchange constants ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪡1$, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. Using recently developed techniques combining renormalization group ideas and one-dimensional bosonization and current algebra methods, we show that in this limit the model enters a crossed-dimer state with twofold spontaneous symmetry breaking but no magnetic order. We complement this result by an approximate ``quadrumer triplet boson'' calculation, which qualitatively captures the physics of the ``plaquette valence-bond solid'' state believed to obtain for ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J\ensuremath{\approx}1$. Using these known points in parameter space, the instabilities pointed to by the quadrumer boson calculation, and the simple limit ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪢1$, we construct a few candidate global phase diagrams for the model, and discuss the nature of the quantum phase transitions contained therein. Finally, we apply our quasi-one-dimensional techniques to an anisotropic limit of the three-dimensional pyrochlore antiferromagnet, an approximate model for magnetism in $\mathrm{Ge}{\mathrm{Cu}}_{2}{\mathrm{O}}_{4}$. A crossed-dimer state is predicted here as well.

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

We study the zero-temperature ( $T=0$ ) ground-state (GS) properties of a frustrated spin-half ${J}_{1}^{\mathit{X}XZ}$ -- ${J}_{2}^{\mathit{X}XZ}$ model on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, using the coupled cluster method. Both interactions are of the anisotropic XXZ type. We present the $T=0$ GS phase diagram of the model in the ranges $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$ of the spin-space anisotropy parameter and $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$ of the frustration parameter. A possible quantum spin-liquid region is identified.We study the zero-temperature ($T=0$) ground-state (GS) properties of a frustrated spin-half ${J}_{1}^{\mathit{X}XZ}$--${J}_{2}^{\mathit{X}XZ}$ model on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, using the coupled cluster method. Both interactions are of the anisotropic XXZ type. We present the $T=0$ GS phase diagram of the model in the ranges $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$ of the spin-space anisotropy parameter and $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$ of the frustration parameter. A possible quantum spin-liquid region is identified.

arXiv:1404.0194Physical Review BJun 24, 201424 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

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 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 study the phase diagram of two models of spin-

1∕2

antiferromagnets composed of corner-sharing tetrahedra, the basis of the pyrochlore structure. Primarily, we focus on the Heisenberg antiferromaget on the checkerboard lattice (also called the planar pyrochlore and crossed-chain model). This model has an anisotropic limit, when the dimensionless ratio of two exchange constants ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪡1$, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. Using recently developed techniques combining renormalization group ideas and one-dimensional bosonization and current algebra methods, we show that in this limit the model enters a crossed-dimer state with twofold spontaneous symmetry breaking but no magnetic order. We complement this result by an approximate ``quadrumer triplet boson'' calculation, which qualitatively captures the physics of the ``plaquette valence-bond solid'' state believed to obtain for ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J\ensuremath{\approx}1$. Using these known points in parameter space, the instabilities pointed to by the quadrumer boson calculation, and the simple limit ${J}_{\ifmmode\times\else\texttimes\fi{}}∕J⪢1$, we construct a few candidate global phase diagrams for the model, and discuss the nature of the quantum phase transitions contained therein. Finally, we apply our quasi-one-dimensional techniques to an anisotropic limit of the three-dimensional pyrochlore antiferromagnet, an approximate model for magnetism in

\mathrm{Ge}{\mathrm{Cu}}_{2}{\mathrm{O}}_{4}

. A crossed-dimer state is predicted here as well.

We study the zero-temperature (

T=0

) ground-state (GS) properties of a frustrated spin-half

{J}_{1}^{\mathit{X}XZ}

{J}_{2}^{\mathit{X}XZ}

model on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings

{J}_{1}>0

and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, using the coupled cluster method. Both interactions are of the anisotropic XXZ type. We present the

T=0

GS phase diagram of the model in the ranges $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$ of the spin-space anisotropy parameter and $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$ of the frustration parameter. A possible quantum spin-liquid region is identified.

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.