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We study a system of interacting triplons (the elementary excitations of avalence-bond solid) described by an effective interacting boson model derivedwithin the bond-operator formalism. In particular, we consider the squarelattice spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model, focus on theintermediate parameter region, where a quantum paramagnetic phase sets in, andconsider the columnar valence-bond solid phase. Within the bond-operatortheory, the Heisenberg model is mapped into an effective boson model in termsof triplet operators $t$. The effective boson model is studied at the harmonicapproximation and the energy of the triplons and the expansion of the triplonoperators $b$ in terms of the triplet operators $t$ are determined. Such anexpansion allows us to performed a second mapping, and therefore, determine aneffective interacting boson model in terms of the triplon operators $b$. Wethen consider systems with a fixed number of triplons and determined theground-state energy and the spectrum of elementary excitations within amean-field approximation. We show that many-triplon states are stable, thelowest-energy ones are constituted by a small number of triplons, and theexcitation gaps are finite. For $J_2=0.48 J_1$ and $J_2=0.52 J_1$, we alsocalculate spin-spin and dimer-dimer correlation functions, dimer orderparameters, and the bipartite von Neumann entanglement entropy within ourmean-field formalism in order to determine the properties of the many-triplonstate as a function of the triplon number. We find that the spin and the dimercorrelations decay exponentially and that the entanglement entropy obeys anarea law, regardless the triplon number. Moreover, only for $J_2=0.48 J_1$, thespin correlations indicate that the many-triplon states with large triplonnumber might display a more homogeneous singlet pattern than the columnarvalence-bond solid.

Mean-field theory of interacting triplons in a two-dimensional valence-bond solid: stability and properties of many-triplon states