Abstract
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials. However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering. In this work, we investigate the low-energy properties of the $S=1/2$ Heisenberg model on the triangular lattice, including both nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ super-exchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models. For $J_2=0$, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at $K$ points (i.e., at the corners of the Brillouin zone). The strong renormalization of the magnon branch (also including roton-like minima around the $M$ points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a non-trivial magnetic $\pi$-flux threading half of the triangular plaquettes. When increasing the frustrating ratio $J_2/J_1$, we detect a progessive softening of the magnon branch at $M$, which eventually becomes gapless within the spin-liquid phase. This feature is captured by the band structure of the unprojected wave function (with $2$ Dirac points for each spin component). In addition, we observe an intense signal at low energies around the $K$ points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
Highlights
The antiferromagnetic Heisenberg model for S 1⁄4 1=2 spins interacting on the triangular lattice represents the simplest example in which quantum fluctuations give rise to strong modifications of the classical picture, where the minimum energy configuration shows 120° order
We observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids
We present the numerical calculations for the dynamical structure factor Sðq; ωÞ obtained by the variational approach described in the previous section
Summary
The antiferromagnetic Heisenberg model for S 1⁄4 1=2 spins interacting on the triangular lattice represents the simplest example in which quantum fluctuations give rise to strong modifications of the classical picture, where the minimum energy configuration shows 120° order. The spectrum shows gapless excitations at M points; in addition, a strong signal at low energies is present at the pffiffi corners of the Brillouin zone, i.e., K 1⁄4 ð2π=3; 2π= 3Þ and K0 1⁄4 ð4π=3; 0Þ While the former aspect can be understood by inspecting the noninteracting spinon band structure, the latter one is a genuine feature that emerges from the Gutzwiller projector, which includes interactions between spinons and gauge fields. While the noninteracting wave function corresponds to a mean-field approximation, in which gauge fields are completely frozen, the Gutzwiller projection has the effect of inserting back the temporal fluctuations of those fields [38] In this respect, it is worth mentioning that a recent field-theoretical analysis indicated the existence of low-energy (triplet) monopole excitations at the zone corners, which are expected to contribute to the dynamical structure factor [39]
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