Valence Bond Crystal Research Articles

Bipartite Rokhsar–Kivelson points and Cantor deconfinement

Quantum dimer models on bipartite lattices exhibit Rokhsar--Kivelson points with exactly known critical ground states and deconfined spinons. We examine generic, weak perturbations around these points. In $d=2+1$ we find a first-order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact $U(1)$ gauge theory in $d=2+1$! In $d=3+1$, we find a continuous transition between the $U(1)$ resonating valence bond phase and a deconfined staggered valence bond crystal. In an appendix, we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in $d=2+1$.

Spin-charge separation in two-dimensional frustrated quantum magnets.

The dynamics of a mobile hole in two-dimensional frustrated quantum magnets is investigated by exact diagonalization techniques. Our results provide evidence for spin-charge separation upon doping the kagome lattice, a prototype of a spin liquid. In contrast, in the checkerboard lattice, a symmetry broken valence bond crystal, a small quasiparticle peak is seen for some crystal momenta, a finding interpreted as a restoration of weak holon-spinon confinement.

QUANTUM PHASES IN TWO-DIMENSIONAL FRUSTRATED SPIN-1/2 ANTIFERROMAGNETS

We describe four phases found in two-dimensional quantum antiferromagnets. Two of them display long range order at T = 0: the Néel state and the Valence Bond Crystal. The last two are Spin-Liquids. Properties of these different states are shortly described and likely conditions of their occurrence outlined.

Bond order from disorder in the planar pyrochlore magnet

We study magnetic order in the Heisenberg antiferromagnet on the checkerboard lattice, a two-dimensional version of the pyrochlore network with strong geometric frustration. By employing the semiclassical (1/S) expansion we find that quantum fluctuations of spins induce a long-range order that breaks the four-fold rotational symmetry of the lattice. The ordered phase is a valence-bond crystal. We discuss similarities and differences with the extreme quantum case S = 1/2 and find a useful phenomenology to describe the bond-ordered phases.

Planar pyrochlore: A valence-bond crystal

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.

One-dimensional behavior and sliding Luttinger liquid phase in a frustrated spin-12crossed chain model: Contribution of exact diagonalizations

Exact diagonalizations indicate that the effective 1-dimensional behavior (sliding Luttinger liquid phase) of the frustrated spin-1/2 crossed chain model, predicted by Starykh, Singh and Levine [Phys. Rev. Lett. 88, 167203 (2002)], persists for a wide range of transverse couplings. The extension of the other phases (plaquette valence bond crystal and N\'eel long range order) is precised. No clear indication of a coexistence of these two phases is found, at variance with a suggestion of Sachdev and Park (cond-mat/0108214).

Generalized Shastry-Sutherland models in three and higher dimensions

We construct Heisenberg antiferromagnetic models in arbitrary dimensions that have isotropic valence-bond crystals (VBC's) as their exact ground states. The $d=2$ model is the Shastry-Sutherland model. In the three-dimensional case we show that it is possible to have a lattice structure, analogous to that of ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2},$ where the stronger bonds are associated with shorter bond lengths. A dimer mean-field theory becomes exact at $\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}$ and a systematic $1/d$ expansion can be developed about it. We study the N\'eel-VBC transition at large d and find that the transition is first order in even but second order in odd dimensions.

Phase diagram of the quadrumerized Shastry-Sutherland model

We determine the phase diagram of a generalized Shastry-Sutherland model, using a combination of dimer- and quadrumer-boson methods and numerical exact diagonalization techniques. Along special lines in the parameter space the model reduces to the standard Shastry-Sutherland model, the 1/5th depleted square lattice and the two-dimensional plaquette square lattice model. We study the evolution of the ordered phases found in the latter two unfrustrated models under the effect of frustration. Furthermore we present exact diagonalization results for the Shastry-Sutherland model on clusters with up to 32 sites, supporting the existence of an intermediate gapped valence bond crystal phase with plaquette long-range order.

Planar pyrochlore: A strong-coupling analysis

Recent investigations of the two-dimensional spin-1/2 checkerboard lattice favor a valence bond crystal with long range quadrumer order [J.-B. Fouet et al., preprint cond-mat/0108070]. Starting from the limit of isolated quadrumers, we perform a complementary analysis of the evolution of the spectrum as a function of the inter quadrumer 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, (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 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.

Exact diagonalization studies of two-dimensional frustrated anti-ferromagnet models

We describe the four kinds of behavior found in two-dimensional isotropic quantum anti-ferromagnets. Two of them display long-range order at T = 0: the Néel state and the valence bond crystal. The last two are spin liquids. Properties of these different states are briefly described and open questions are underlined. PACS Nos.: 75.10Jm, 75.50Ee, 75.40-s

Exact diagonalization studies of two-dimensional frustrated anti-ferromagnet models

We describe the four kinds of behavior found in two-dimensional isotropic quantum antiferromagnets. Two of them display long range order at T=0: the N\'eel state and the Valence Bond Crystal. The last two are Spin-Liquids. Properties of these different states are shortly described and open questions are underlined.

Valence-Bond Crystal and Anisotropic Excitation Spectrum on 3-Dimensionally Frustrated Pyrochlore

The valence-bond crystal is proposed as a ground state of spin-1/2 Heisenberg antiferromagnet on 3-dimensional fully frustrated pyrochlore by the bond-operator theory. This crystal, formed as a set of RVB-like tetrahedral singlet states, has lower energy than the state consisting of two dimers in a tetrahedron. The spectra of triplet excitation have a finite gap and a symmetry lower than that of the lattice. The comparison with Y(Sc)Mn 2 is discussed.