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arXiv:0806.4763 · Physical Review B

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

Fabien Trousselet, Didier Poilblanc, Roderich Moessner

Published Nov 3, 2008• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.78.195101

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Columnar dimer and plaquette resonating-valence-bond orders in the quantum dimer model

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We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Nov 1, 1996•72 citations
We study a model of strongly correlated S=1/2S=1/2S=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∘\else\textdegree\fi90\ifmmode^\circ\else\textdegree\fi{}90\ifmmode∘\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.
Coulomb repulsion between electrons moving on a frustrated lattice can give rise, at simple commensurate electronic densities, to exotic insulating phases of matter. Such a phenomenon is illustrated using an extended $t\text{\ensuremath{-}}J$ model on a planar pyrochlore lattice for which the work on the quarter-filled case [D. Poilblanc et al., Phys. Rev. B 75, 220503 (2007)] is complemented and extended to 1∕81∕81∕8 and 3∕83∕83∕8 fillings. The location of the metal-insulator transition as a function of the Coulomb repulsion is shown to depend strongly on the sign of the hopping. Quite generally, the metal-insulator transition is characterized by lattice symmetry breaking but the nature of the insulating Mott state is more complex than a simple charge density wave. Indeed, in the limit of large Coulomb repulsion, the physics can be described in the framework of (extended) quantum fully packed loop or dimer models carrying extra spin degrees of freedom. Various diagonal and off-diagonal plaquette correlation functions are computed and the low-energy spectra are analyzed in detail in order to characterize the nature of the insulating phases. We provide evidence that, as for an electronic density of n=1∕2n=1∕2n=1∕2 (quarter filling), the system at n=1∕4n=1∕4n=1∕4 or n=3∕4n=3∕4n=3∕4 exhibits also plaquette order by forming a (lattice rotationally invariant) resonant singlet pair crystal, although with a quadrupling of the lattice unit cell (instead of a doubling for n=1∕2n=1∕2n=1∕2) and a fourfold degenerate ground state. Interestingly, qualitative differences with the bosonic analog (e.g., known to exhibit columnar order at n=1∕4n=1∕4n=1∕4) emphasize the important role of the spin degrees of freedom in, e.g., stabilizing plaquette phases with respect to rotational symmetry-breaking phases.
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.
We examine the ground-state phase diagram of repulsively interacting hard-core bosons on the quarter-filled checkerboard lattice using quantum Monte Carlo simulations constrained to the canonical ensemble. At a sufficiently large repulsion, the system undergoes a quantum phase transition from a superfluid into a valence-bond-solid regime. We identify this as a weakly first-order quantum melting transition at quarter filling. Furthermore, by examining appropriate order parameters, we exhibit the mixed nature of the valence bond solid, with both columnar and plaquette long-ranged order. Our results demonstrate the realization of the mixed columnar-plaquette phase found recently for the square lattice quantum dimer model in a microscopic bosonic system.
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.
We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.