Mean-field theory of interacting triplons in a two-dimensional valence-bond solid

R. L. Doretto

We study a system of $\overline{N}$ 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 $\overline{N}$ 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 ${J}_{2}=0.48{J}_{1}$ and $0.52{J}_{1}$, 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 $\overline{N}$. 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 $\overline{N}$. Moreover, only for ${J}_{2}=0.48{J}_{1}$, the spin correlations indicate that the many-triplon states with large triplon number $\overline{N}$ 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 $\overline{N}$ and gapped spin-liquid states.

Tetsuro Saso, 哲郎 佐宗

Dai S. Hirashima, 平島 大

Bohm‐Jung Yang, Arun Paramekanti, Yong Baek Kim

We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two-dimensional star lattice which exhibits geometric frustration. Using slave particle mean-field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-$N$ $\text{SU}(N)$ theory as well as numerical exact diagonalization studies, we also examine various valence-bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest-energy spin liquid candidate for this model. We also find from a study of energetics using Gutzwiller projected wave functions and bond operator approaches that this spin liquid is unstable toward two different VBS states---a VBS state which respects all the Hamiltonian symmetries and a VBS state which exhibits $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order---depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.

Marie-Bernadette Lepetit, Alain Gellé

Lucas S. Buzo, R. L. Doretto

We calculate the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$ for the columnar valence bond solid phase of a two-dimensional frustrated antiferromagnet. In particular, we consider the square lattice spin-$1/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model with an additional Dzyaloshinskii-Moriya interaction between the spins and in the presence of an external magnetic field. We concentrate on the intermediate parameter region of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, where a quantum paramagnetic phase is stable, and consider a Dzyaloshinskii-Moriya vector pattern associated with the couplings between the spins in the ${\mathrm{CuO}}_{2}$ planes of the ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6}$ compound. We describe the columnar valence bond solid phase within the bond-operator formalism, which allows us to map the Heisenberg model into an effective interacting boson model written in terms of triplet operators. The effective boson model is studied within the harmonic approximation, and the triplon excitation bands of the columnar valence bond solid phase are determined. We then calculate the Berry curvature and the Chern numbers of the triplon excitation bands and, finally, determine the thermal Hall conductivity due to triplons as a function of the temperature. We find that the Dzyaloshinskii-Moriya interaction yields a finite Berry curvature for the triplon bands, but the corresponding Chern numbers vanish. Although the triplon excitations are topologically trivial, the thermal Hall conductivity of the columnar valence bond solid phase in the square lattice antiferromagnet is finite at low temperatures. Our results complement a previous study by Samajdar et al., Phys. Rev. B 99, 165126 (2019) concerning the thermal Hall effect due to spinons of a spin-liquid phase on a square lattice. We also comment on the relations of our results with a no-go condition for a thermal Hall effect previously derived for ordered magnets by Katsura et al., Phys. Rev. Lett. 104, 066403 (2010).