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arXiv:cond-mat/9701058 · Physical review. B, Condensed matter

Exact ground and excited states of frustrated antiferromagnets on the CaV4O9 lattice

Indrani Bose, Asimkumar Ghosh

Published Aug 1, 1997• 21 citations
arXivHTMLPDF

Abstract

The experimental observation of a spin gap in the antiferromagnetic spin-$\frac{1}{2}$ compound CaV${}_{4}$O${}_{9}$ has been attributed to the formation of a plaquette-resonating-valence-bond (PRVB) state in the underlying 1/5-depleted square lattice. We construct a spin-$\frac{1}{2}$ model on the 1/5-depleted lattice for which the PRVB state can be shown to be the exact ground state in a particular parameter regime. In a subspace of this parameter regime, the first excited state can be calculated exactly. In a different parameter regime, the dimer state is shown to be the exact ground state.

DOI
10.1103/physrevb.56.3149

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The experimental observation of a spin gap in the antiferromagnetic spin-12\frac{1}{2}21​ compound CaV4{}_{4}4​O9{}_{9}9​ has been attributed to the formation of a plaquette-resonating-valence-bond (PRVB) state in the underlying 1/5-depleted square lattice. We construct a spin-12\frac{1}{2}21​ model on the 1/5-depleted lattice for which the PRVB state can be shown to be the exact ground state in a particular parameter regime. In a subspace of this parameter regime, the first excited state can be calculated exactly. In a different parameter regime, the dimer state is shown to be the exact ground state.
A theoretical model is presented to explain the spin gap observed for CaV4O9\mathrm{CaV}{}_{4}\mathrm{O}{}_{9}CaV4​O9​. The underlying lattice of the 1/5-depleted square lattice favors a formation of the plaquette resonating-valence-bond state. Inclusion of the frustrating second neighbor interaction enhances this tendency, leading to a quantum disordered state of a two-dimensional spin-1/2 Heisenberg model with a sufficiently big spin gap compatible with experiments.
We examine the ground state and the excitations of an one-dimensional Heisenberg spin 1/2 antiferromagnet with alternating dimers and four-spin plaquettes (dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact-numerical and perturbational results are presented. We find that the system is gapped for all parameter values. For the latter one we find an effective S=1 Haldane like behaviour in the limit of dominating dimer bonds. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system CaV4O9CaV_4O_9CaV4​O9​.
We study the proposed plaquette resonating valence bond (pRVB) state in the honeycomb lattice ${J}_{1}\ensuremath{-}{J}_{2}$ model with frustration arising from next-nearest-neighbor interactions. Starting with the limit of decoupled hexagons, we develop a plaquette operator approach to describe the pRVB state and its low-energy excitations. Our calculation clarifies that the putative pRVB state necessarily has fff-wave symmetry: the plaquette wave function is an antisymmetric combination of the Kekul\'e structures. We estimate the plaquette ordering amplitude, ground-state energy, and spin gap as a function of J2/J1{J}_{2}/{J}_{1}J2​/J1​. The pRVB state is most stable around ${J}_{2}/{J}_{1}\ensuremath{\sim}0.25$. We identify the wave vectors of the lowest triplet excitations, which can be verified using exact diagonalization or density-matrix renormalization group (DMRG) studies. When J2{J}_{2}J2​ is reduced, we can have either a deconfined quantum phase transition (QPT) or a first-order transition into a N\'eel state. When J2{J}_{2}J2​ is increased, we surmise that the system will undergo a first-order phase transition into a state which breaks lattice rotational symmetry.
We investigate the nonmagnetic phase of the spin-half frustrated Heisenberg antiferromagnet on the square lattice using exact diagonalization (up to 36 sites) and quantum Monte Carlo techniques (up to 144 sites). The spin gap and the susceptibilities for the most important crystal symmetry breaking operators are computed. A genuine and somehow unexpected "plaquette resonating valence bond," with spontaneously broken translation symmetry and no broken rotation symmetry, comes out from our numerical simulations as the most plausible ground state for J(2)/J(1) approximately 0.5.
Using both exact diagonalizations and diagonalizations in a subset of short-range valence-bond singlets, we address the nature of the groundstate of the Heisenberg spin-1∕21∕21∕2 antiferromagnet on the square lattice with competing next-nearest and next-next-nearest neighbor antiferromagnetic couplings (${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ model). A detailed comparison of the two approaches reveals a region along the line (J2+J3)∕J1=1∕2({J}_{2}+{J}_{3})∕{J}_{1}=1∕2(J2​+J3​)∕J1​=1∕2, where the description in terms of nearest-neighbor singlet coverings is excellent, therefore providing evidence for a magnetically disordered region. Furthermore, a careful analysis of dimer-dimer correlation functions, dimer structure factors and plaquette-plaquette correlation functions provides striking evidence for the presence of a plaquette valence bond crystal order in part of the magnetically disordered region.
We study the S>1/2S>1/2S>1/2 antiferromagnetic Heisenberg model on the 1/5-depleted square lattice as a function of the ratio of the intraplaquette coupling to the interplaquette coupling. Using stochastic series expansion quantum Monte Carlo simulations, we numerically identify three quantum phases, including the dimer phase, N\'eel phase, and plaquette valence-bond-solid phase. We also obtain the accurate quantum critical points that belong to the OOO(3) universality class using the large-scale finite-size scaling. Most importantly, we study the dynamic spin structure factors of different phases, which can be measured by inelastic neutron scattering experiments. The low-energy excitations can be explained as triplons in the dimer phase and plaquette valence-bond-solid phase, while in the N\'eel phase, the more prominent magnon mode can be found as the spin magnitude increases. Furthermore, we find a broader continuum at smaller SSS, which may be the dynamical signature of nearly deconfined spinon excitations.