Plaquette Order Research Articles

Global phase diagram for the spin-1 antiferromagnet with uniaxial anisotropy on the kagome lattice

The phase diagram of the XXZ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $J_z$ coupling is antiferromagnetic. In the zero magnetic field case, the six previously introduced phases, found using various methods, are: the nondegenerate gapped photon phase which breaks no space symmetry or spin symmetry; the six-fold degenerate phase with plaquette order, which breaks both time reversal symmetry and translational symmetry; the "superfluid" (ferromagnetic) phase with an in-plane global U(1) symmetry broken, when $J_{xy} < 0$; the $\sqrt{3}\times\sqrt{3}$ order when $J_{xy} > 0$; the nematic phase when $D < 0$ and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the six-fold plaquette phases. The transition between nematic phase and the six-fold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\hat{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the six-fold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, BF$_4$ salts, or optical lattices of atoms with three degenerate states on every site.

Exotic Mott phases of the extendedt−Jmodel on the checkerboard lattice at commensurate densities

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 $1∕8$ and $3∕8$ 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 $n=1∕2$ (quarter filling), the system at $n=1∕4$ or $n=3∕4$ 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 $n=1∕2$) and a fourfold degenerate ground state. Interestingly, qualitative differences with the bosonic analog (e.g., known to exhibit columnar order at $n=1∕4$) emphasize the important role of the spin degrees of freedom in, e.g., stabilizing plaquette phases with respect to rotational symmetry-breaking phases.

Phase diagram of the spin-orbital model on the square lattice

We study the phase diagram of the spin-orbital model in both weak and strong limits of the quartic spin-orbital exchange interaction. This allows us to study quantum phase transitions in the model and to approach from both sides the most interesting intermediate-coupling regime and in particular the SU(4)-symmetric point of the Hamiltonian. It was suggested earlier by Li et al. [Phys. Rev. Lett. $81,$ 3527 (1999)] that at this point the ground state of the system is a plaquette spin-orbital liquid. We argue that the state is more complex. There is plaquette order, but it is anisotropic: bonds in one direction are stronger than those in the perpendicular direction. This order is somewhat similar to that found recently in the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg spin model.

Quantum phases of the Shastry-Sutherland antiferromagnet: Application toSrCu2(BO3)2

We study possible paramagnetic phases of antiferromagnets on the Shastry-Sutherland lattice by a gauge-theoretic analysis of fluctuations in a theory with $\mathrm{Sp}(2N)$ symmetry. In addition to the familiar dimer phase, we find a confining phase with plaquette order and a topologically ordered phase with deconfined $S=1/2$ spinons and helical spin correlations. The deconfined phase is contiguous to the dimer phase and in a regime of couplings close to those found in the insulator ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}.$ We suggest that a superconductor obtained by doping this insulator with mobile charge carriers will be an attractive candidate for observing the anomalous magnetic flux properties associated with topological order.

Planar Pyrochlore, Quantum Ice and Sliding Ice

We study quantum antiferromagnetism on the highly frustrated planar pyrochlore lattice, also known as the square lattice with crossings. The quantum Heisenberg antiferromagnet on this lattice is of interest as a two-dimensional analogue of the pyrochlore lattice magnet. By combining several approaches we conclude that this system is most likely ordered for all values of spin, S, with a two-fold degenerate valence-bond solid being the ground state for small S. We show next that the Ising antiferromagnet with a weak four-spin exchange, equivalent to square ice with the leading quantum dynamics, exhibits analogous plaquette order. As a byproduct of this analysis we obtain, in the system of weakly coupled ice planes, a sliding phase with XY symmetry; at intermediate couplings, long range “anti-ferroelectric” order is stabilized.

Critical dynamics of singlet excitations in a frustrated spin system

We construct and analyze a frustrated quantum spin model with plaquette order, in which the low-energy dynamics is controlled by spin singlets. At a critical value of frustration the singlet spectrum becomes gapless, indicating a quantum transition to a phase with dimer order. The magnetic susceptibility has an activated form throughout the phase diagram, whereas the specific heat exhibits a rich structure and a power-law dependence on temperature at the quantum critical point. This generic behavior can be relevant to quantum antiferromagnets on Kagom\'e and pyrochlore lattices where (almost) all low-energy excitations are spin singlets, as well as to the ${\mathrm{CaV}}_{4}{\mathrm{O}}_{9}$ lattice and the strongly frustrated antiferromagnets ${\mathrm{Cu}}_{2}{\mathrm{Te}}_{2}{\mathrm{O}}_{5}(\mathrm{Cl},\mathrm{Br}{)}_{2}$ and ${\mathrm{Li}}_{2}\mathrm{VO}(\mathrm{Si},\mathrm{Ge}){\mathrm{O}}_{4}.$

Dimer order with striped correlations in theJ1−J2Heisenberg model

Ground-state energies for plaquette and dimer order in the ${J}_{1}{\ensuremath{-}J}_{2}$ square-lattice spin-half Heisenberg model are compared using series expansion methods. We find that these energies are remarkably close to each other at intermediate values of ${J}_{2}{/J}_{1},$ where the model is believed to have a quantum disordered ground state. They join smoothly with those obtained from the Ising expansions for the two-sublattice N\'eel state at ${J}_{2}{/J}_{1}\ensuremath{\approx}0.4,$ suggesting a second-order transition from a N\'eel state to a quantum disordered state, whereas they cross the energy for the four-sublattice ordered state at ${J}_{2}{/J}_{1}\ensuremath{\approx}0.6$ at a large angle, implying a first-order transition to the four-sublattice magnetic state. The strongest evidence that the plaquette phase is not realized in this model comes from the analysis of the series for the singlet and triplet excitation spectra, which suggest an instability in the plaquette phase. Thus our study supports the recent work of Kotov et al., which presents a strong picture for columnar dimer order in this model. We also discuss the striped nature of spin correlations in this phase, with substantial resonance all along columns of dimers.

Presence of midgap states inCaV4O9

Using exact diagonalizations of finite clusters with up to 32 sites, we study the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model on the 1/5 depleted square lattice. Spin-spin correlation functions are consistent with plaquette order in the spin gap phase which exists for intermediate values of ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$. Besides, we show that singlet states will be present in the singlet-triplet gap if ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$ is not too small (${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\gtrsim}0.47). We argue that this property should play a central role in determining the exchange integrals in ${\mathrm{CaV}}_{4}$${\mathrm{O}}_{9}$. \textcopyright{} 1996 The American Physical Society.

Spin gap in CaV4O9: A large-S approach.

We study the stability of the four-site plaquette order in a model recently introduced by Katoh and Imada to explain the spin gap observed in ${\mathrm{Ca}{\mathrm{V}}_{4})}_{9}$. We show that all relevant types of order (four-site plaquette, dimerized, N\'eel) can be described within the framework of Schwinger-boson mean-field theory, and we make predictions regarding their relative stability. The results are compared to exact diagonalization of finite clusters.

Cluster study of the t-J model: Chiral fluctuations.

Uniform and staggered chiral order are investigated in the two-dimensional {ital t}-{ital J} model at arbitrary doping fractions. Equal-time and dynamical correlations are calculated on a 4{times}4 cluster by using a Lanczos algorithm. We found that static uniform correlations are suppressed with increasing doping and {ital t}/{ital J} while static staggered correlations seem enhanced, in agreement with previous mean-field studies. The plaquette-plaquette correlation function falls off rapidly with distance, and, thus, we obtain an upper bound on the plaquette order parameter, which is very small. However, our numerical results show that doping for {ital t}/{ital J}{ge}1 induces large chiral fluctuations at low energy, which may become important even in the absence of long-range order. The staggered order parameter exhibits low-energy fluctuations of energy {similar to}2{ital J}, while for the uniform case fluctuations also occur at a higher energy {similar to}2{ital t}.