We construct and study the two frustrated quantum spin-
1∕2 models on square lattice, which are like the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model with some additional four-spin exchange interactions. These models admit an exactly solvable case in which the ground state consists of four degenerate columnar-dimer singlet (CDS) configurations. Away from the exact case, we employ bond-operator mean-field theory to investigate the evolution of the ground state by varying the interaction parameters. The mean-field calculation reveals a quantum phase diagram in which the CDS phase undergoes a continuous phase transition to either N\'eel or collinear ordered antiferromagnetic phases.
We study a system of
N interacting triplons (the elementary excitations of a valence-bond solid) described by an effective interacting boson model derived within the bond operator formalism in order to determine the stability and the properties of many-triplon states. In particular, we consider the square lattice spin-$1/2\phantom{\rule{4pt}{0ex}}{J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg model, focus on the intermediate parameter region where a quantum paramagnetic phase sets in, and consider the columnar valence-bond solid as a reference state. Within the bond operator theory, the Heisenberg model is mapped into an effective boson model in terms of triplet operators
t. The effective boson model is studied at the harmonic approximation and the energy of the triplons and the expansion of the triplon operators
b in terms of the triplet operators
t are determined. Such an expansion allows us to perform a second mapping and, therefore, determine an effective interacting boson model in terms of the triplon operators
b. We then consider systems with a fixed number
N of triplons and study the stability of many-triplon states within a mean-field approximation. We show that many-triplon states are stable, the lowest-energy ones are constituted by a small number of triplons, and the excitation gaps are finite. For
J2=0.48J1 and
0.52J1, we also calculate spin-spin and dimer-dimer correlation functions, dimer order parameters, and the bipartite von Neumann entanglement entropy within our mean-field formalism in order to determine the properties of the many-triplon state as a function of the triplon number
N. We find that the spin and the dimer correlations decay exponentially and that the entanglement entropy obeys an area law, regardless of the triplon number
N. Moreover, only for
J2=0.48J1, the spin correlations indicate that the many-triplon states with large triplon number
N might display a more homogeneous singlet pattern than the columnar valence-bond solid. We also comment on possible relations between the many-triplon states with large triplon number
N and gapped spin-liquid states.
We introduce a new representation of S=1/2 quantum spins in terms of bond operators. The bond operators create and annihilate singlet and triplet bonds between a pair of spins. The representation is useful in describing the transition between dimerized and magnetically ordered phases of quantum antiferromagnets. It is used to obtain a mean-field theory of the two-dimensional frustrated quantum Heisenberg antiferromagnets considered recently by Gelfand, Singh, and Huse. The method should also be useful in the analysis of random quantum antiferromagnets.
In a recent work [Phys. Rev. B {\bf 77}, 014419 (2008)], a quantum spin-1/2 model having an exact fourfold degenerate Shastry-Sutherland ground state was constructed, and studied using finite-size numerical exact diagonalization. There, a schematic quantum phase diagram suggesting many competing phases was also proposed. In the present work, an extended version of this model is investigated using dimer and plaquette triplon mean-field theories. The quantum phase diagram of the extended model (in the thermodynamic limit) is found to have many interesting phases: commensurate and incommensurate magnetically ordered phases, and columnar dimer, sublattice columnar dimer, Shastry-Sutherland and plaquette spin-gapped phases. Among themselves, these phases undergo a continuous or a level-crossing transition.
We look into the quantum phase diagram of a spin- antiferromagnet on the square lattice with degenerate Shastry–Sutherland ground states, for which only a schematic phase diagram is known so far. Many exotic phases were proposed in the schematic phase diagram by the use of exact diagonalization on very small system sizes. In our present work, an important extension of this antiferromagnet is introduced and investigated in the thermodynamic limit using triplon mean-field theory. Remarkably, this antiferromagnet shows a stable plaquette spin-gapped phase like the original Shastry–Sutherland antiferromagnet, although both of these antiferromagnets differ in the Hamiltonian construction and ground state degeneracy. We propose a sublattice columnar dimer phase which is stabilized by the second and third neighbor antiferromagnetic Heisenberg exchange interactions. There are also some commensurate and incommensurate magnetically ordered phases, and other spin-gapped phases which find their places in the quantum phase diagram. Mean-field results suggest that there is always a level-crossing phase transition between two spin-gapped phases, whereas in other situations, either a level-crossing or a continuous phase transition happens.
Aiming to describe frustrated quantum magnets with nonmagnetic singlet ground states, we have extended the Rokhsar-Kivelson (RK) loop expansion to derive a generalized quantum dimer model containing only connected terms up to arbitrary order. For the square-lattice frustrated Heisenberg antiferromagnet (${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ model), an expansion up to eighth order shows that the leading correction to the original RK model comes from dimer moves on length-6 loops. This model free of the original sign problem is treated by advanced numerical techniques. The results suggest that a rotationally anisotropic plaquette phase [A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett. 100, 037201 (2008)] is the ground state of the Heisenberg model in the parameter region of maximum frustration.