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arXiv:cond-mat/0111405 · Physical review. B, Condensed matter

Planar pyrochlore: A strong-coupling analysis

Wolfram Brenig, A. Honecker

Published Apr 4, 2002• 54 citations
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Abstract

Recent investigations of the two-dimensional spin-1/2 checkerboard lattice favor a valence bond crystal with long-range quadrumer order [J.-B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, cond-mat/0108070 (unpublished)]. Starting from the limit of isolated quadrumers, we perform a complementary analysis of the evolution of the spectrum as a function of the interquadrumer coupling j using both exact diagonalization (ED) and series expansion (SE) by continuous unitary transformation. We compute (i) the ground-state energy, (ii) the elementary triplet excitations, and (iii) singlet excitations on finite systems and find very good agreement between SE and ED. In the thermodynamic limit we find a ground-state energy substantially lower than that documented in the literature. The elementary triplet excitation is shown to be gapped and almost dispersionless, whereas the singlet sector contains strongly dispersive modes. Evidence is presented for the low energy singlet excitations in the spin gap in the vicinity of $j=1$ to result from a large downward renormalization of local high-energy states.

DOI
10.1103/physrevb.65.140407

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Recent investigations of the two-dimensional spin-1/2 checkerboard lattice favor a valence bond crystal with long-range quadrumer order [J.-B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, cond-mat/0108070 (unpublished)]. Starting from the limit of isolated quadrumers, we perform a complementary analysis of the evolution of the spectrum as a function of the interquadrumer coupling j using both exact diagonalization (ED) and series expansion (SE) by continuous unitary transformation. We compute (i) the ground-state energy, (ii) the elementary triplet excitations, and (iii) singlet excitations on finite systems and find very good agreement between SE and ED. In the thermodynamic limit we find a ground-state energy substantially lower than that documented in the literature. The elementary triplet excitation is shown to be gapped and almost dispersionless, whereas the singlet sector contains strongly dispersive modes. Evidence is presented for the low energy singlet excitations in the spin gap in the vicinity of j=1j=1j=1 to result from a large downward renormalization of local high-energy states.
Exact diagonalizations of the spin-1/2 Heisenberg model on the checkerboard lattice have been performed for sizes up to N=36N=36N=36 in the full Hilbert space and N=40N=40N=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=0S=0S=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 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.
Using series expansion based on the flow equation method we study the zero temperature properties of the spin-1∕21∕21∕2 planar pyrochlore antiferromagnet in the limit of strong diagonal coupling. Starting from the limit of decoupled crossed dimers we analyze the evolution of the ground state energy and the elementary triplet excitations in terms of two coupling constants describing the inter dimer exchange. In the limit of weakly coupled spin-1∕21∕21∕2 chains we find that the fully frustrated inter chain coupling is critical, forcing a dimer phase which adiabatically connects to the state of isolated dimers. This result is consistent with findings by O. Starykh, A. Furusaki, and L. Balents [Phys. Rev. B 72, 094416 (2005)] which is inconsistent with a two-dimensional sliding Luttinger liquid phase at finite inter chain coupling.
We study the phase diagram of two models of spin-1∕21∕21∕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 GeCu2O4\mathrm{Ge}{\mathrm{Cu}}_{2}{\mathrm{O}}_{4}GeCu2​O4​. A crossed-dimer state is predicted here as well.
Using a series expansion based on the flow-equation method we study the ground-state energy and the elementary triplet excitations of a generalized model of crossed spin-1/21/21/2 chains starting from the limit of decoupled quadrumers. The triplet dispersion is shown to be very sensitive to the interquadrumer frustration, exhibiting a line of almost complete localization as well as lines of quantum phase transitions limiting the stability of the valence-bond crystal phase. In the vicinity of the checkerboard point, a finite window of exchange couplings is found with a nonzero spin-gap, consistent with known results from exact diagonalization. The ground-state energy is lower than that of the bare quadrumer case for all exchange couplings investigated. In the limiting situation of the fully frustrated checkerboard magnet our results agree with earlier series expansion studies.
We present exact diagonalization calculations for the spin-1/2 nearest-neighbor antiferromagnet on the pyrochlore lattice. We study a section of the lattice in the [111] direction and analyze the Hamiltonian of the breathing pyrochlore system with two coupling constants J1{J}_{1}J1​ and J2{J}_{2}J2​ for tetrahedra of different orientations and investigate the evolution of the system from the limit of disconnected tetrahedra (J2=0{J}_{2}=0J2​=0) to a correlated state at J1=J2{J}_{1}={J}_{2}J1​=J2​. We evaluate the low-energy spectrum, two and four spin correlations, and spin chirality correlations for a system size of up to 36 sites. The model shows a fast decay of spin correlations and we confirm the presence of several singlet excitations below the lowest magnetic excitation. We find chirality correlations near J1=J2{J}_{1}={J}_{2}J1​=J2​ to be small at the length scales available at this system size. Evaluation of dimer-dimer correlations and analysis of the nature of the entanglement of the tetrahedral unit shows that the triplet sector of the tetrahedron contributes significantly to the ground-state entanglement at J1=J2{J}_{1}={J}_{2}J1​=J2​.