arXiv:cond-mat/0108070

The planar pyrochlore: a Valence Bond Crystal

J.-B. Fouet, Matthieu Mambrini, Philippe Sindzingre, C. Lhuillier

Published Aug 3, 2001• 0 citations

Abstract

Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36 in the full Hilbert space and N=40 in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long range order in 4-spin S=0 plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom{é} antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a Valence Bond Crystal. Some results concerning the 3-dimensionnal spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its 2-dimensional analog.

DOI: 10.48550/arxiv.cond-mat/0108070

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arXiv:cond-mat/0108070

The planar pyrochlore: a Valence Bond Crystal

J.-B. Fouet, Matthieu Mambrini, Philippe Sindzingre, C. Lhuillier

Published Aug 3, 2001• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.cond-mat/0108070

Related papers

Planar pyrochlore: A valence-bond crystal

J.-B. Fouet, Matthieu Mambrini, Philippe Sindzingre +1 more

Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to $N=36$ in the full Hilbert space and $N=40$ in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long-range order in four-spin $S=0$ plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom\'e antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a valence bond crystal. Some results concerning the three-dimensional spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its two-dimensional analog.Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to $N=36$ in the full Hilbert space and $N=40$ in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long-range order in four-spin $S=0$ plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom\'e antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a valence bond crystal. Some results concerning the three-dimensional spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its two-dimensional analog.

Physical review. B, Condensed matterFeb 10, 2003107 citations

Anisotropic pyrochlores and the global phase diagram of the checkerboard antiferromagnet

Oleg A. Starykh, Akira Furusaki, Leon Balents

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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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

Closely competing valence bond crystal orders in the ground state of the spin- $\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational studyClosely competing valence bond crystal orders in the ground state of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study

Cheng, Rong, Tao Li

The spin- $\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin- $\frac{1}{2}$ PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with $2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}$ periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in $\langle \hat{\mathbf{S}}^{2}_{u} \rangle$ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.The spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-$\frac{1}{2}$ PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with $2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}$ periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in $\langle \hat{\mathbf{S}}^{2}_{u} \rangle$ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.

arXiv:2509.13746Sep 17, 20250 citations

Plaquette bond order wave in the quarter-filled extended Hubbard model on the checkerboard lattice

Martin Indergand, Carsten Honerkamp A, Andreas Martin Läuchli +2 more

An extended Hubbard model (including nearest-neighbor repulsion and antiferromagnetic spin exchange) is investigated on the frustrated checkerboard lattice, a two-dimensional analog of the pyrochlore lattice. Combining Gutzwiller renormalized mean-field calculations, exact diagonalization techniques, and a weak-coupling renormalization group analysis we provide strong evidence for a crystalline valence bond plaquette phase at quarter-filling. The ground state is twofold degenerate and breaks translation symmetry. The bond energies show a staggering while the charge distribution remains uniform.An extended Hubbard model (including nearest-neighbor repulsion and antiferromagnetic spin exchange) is investigated on the frustrated checkerboard lattice, a two-dimensional analog of the pyrochlore lattice. Combining Gutzwiller renormalized mean-field calculations, exact diagonalization techniques, and a weak-coupling renormalization group analysis we provide strong evidence for a crystalline valence bond plaquette phase at quarter-filling. The ground state is twofold degenerate and breaks translation symmetry. The bond energies show a staggering while the charge distribution remains uniform.

arXiv:cond-mat/0609129Physical Review BJan 4, 200713 citations

Mixed columnar-plaquette crystal of correlated fermions on the two-dimensional pyrochlore lattice at fractional filling

Fabien Trousselet, Didier Poilblanc, Roderich Moessner

We study a model of strongly correlated $S=1/2$ fermions on the planar pyrochlore, or checkerboard, lattice, at fractional (1/8) filling. Starting with the extended Hubbard model in the limit of strong Coulomb repulsion, low-energy configurations can be mapped onto hard-core dimer configurations whose dimers carry a spin degree of freedom. An effective Hamiltonian similar to the kinetic term of a quantum dimer model on the square lattice which rotates two parallel dimers (in a spin-singlet configuration) by $90\ifmmode^\circ\else\textdegree\fi{}$ naturally emerges. We also introduce an additional term in the Hamiltonian, a generalized dimer plaquette interaction, in order to realize a closer analogy to the latter model. For a strong dimer plaquette attraction stabilizing a columnar phase, a spontaneous dimerization takes place in the direction of the columns of (spin-carrying) dimers. Using exact diagonalizations of two-dimensional periodic clusters, the analysis of the low-energy spectrum and of several types of correlation functions gives indeed evidence for a new type of lattice symmetry breaking phase, the eightfold degenerate mixed columnar-plaquette crystal, and for a transition from this phase to a resonating singlet-pair crystal (found in previous studies) which restores the rotational symmetry of the lattice. Similar conclusions and phase diagram are also reached from a simple variational approach.We study a model of strongly correlated $S=1/2$ fermions on the planar pyrochlore, or checkerboard, lattice, at fractional (1/8) filling. Starting with the extended Hubbard model in the limit of strong Coulomb repulsion, low-energy configurations can be mapped onto hard-core dimer configurations whose dimers carry a spin degree of freedom. An effective Hamiltonian similar to the kinetic term of a quantum dimer model on the square lattice which rotates two parallel dimers (in a spin-singlet configuration) by $90\ifmmode^\circ\else\textdegree\fi{}$ naturally emerges. We also introduce an additional term in the Hamiltonian, a generalized dimer plaquette interaction, in order to realize a closer analogy to the latter model. For a strong dimer plaquette attraction stabilizing a columnar phase, a spontaneous dimerization takes place in the direction of the columns of (spin-carrying) dimers. Using exact diagonalizations of two-dimensional periodic clusters, the analysis of the low-energy spectrum and of several types of correlation functions gives indeed evidence for a new type of lattice symmetry breaking phase, the eightfold degenerate mixed columnar-plaquette crystal, and for a transition from this phase to a resonating singlet-pair crystal (found in previous studies) which restores the rotational symmetry of the lattice. Similar conclusions and phase diagram are also reached from a simple variational approach.

arXiv:0806.4763Physical Review BNov 3, 200810 citations

Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to

N=36

in the full Hilbert space and

N=40

in the restricted subspace of first neighbor dimers. This antiferromagnet does not break SU(2) symmetry and displays long-range order in four-spin

S=0

plaquettes. Both the symmetry properties of the spectrum and various correlations functions are extensively studied. At variance with the kagom\'e antiferromagnet, the Heisenberg quantum model on a checkerboard lattice is a valence bond crystal. Some results concerning the three-dimensional spin-1/2 pyrochlore magnet (for sizes 16 and 32) are also shown: this system could behave differently from its two-dimensional analog.

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.

Closely competing valence bond crystal orders in the ground state of the spin-

\frac{1}{2}

antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study

The spin-

\frac{1}{2}

antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-

\frac{1}{2}

PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with

2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}

periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in

\langle \hat{\mathbf{S}}^{2}_{u} \rangle

for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.

We study a model of strongly correlated

S=1/2

fermions on the planar pyrochlore, or checkerboard, lattice, at fractional (1/8) filling. Starting with the extended Hubbard model in the limit of strong Coulomb repulsion, low-energy configurations can be mapped onto hard-core dimer configurations whose dimers carry a spin degree of freedom. An effective Hamiltonian similar to the kinetic term of a quantum dimer model on the square lattice which rotates two parallel dimers (in a spin-singlet configuration) by $90\ifmmode^\circ\else\textdegree\fi{}$ naturally emerges. We also introduce an additional term in the Hamiltonian, a generalized dimer plaquette interaction, in order to realize a closer analogy to the latter model. For a strong dimer plaquette attraction stabilizing a columnar phase, a spontaneous dimerization takes place in the direction of the columns of (spin-carrying) dimers. Using exact diagonalizations of two-dimensional periodic clusters, the analysis of the low-energy spectrum and of several types of correlation functions gives indeed evidence for a new type of lattice symmetry breaking phase, the eightfold degenerate mixed columnar-plaquette crystal, and for a transition from this phase to a resonating singlet-pair crystal (found in previous studies) which restores the rotational symmetry of the lattice. Similar conclusions and phase diagram are also reached from a simple variational approach.