Dynamical Structure Factor of the J1−J2 Heisenberg Model on the Triangular Lattice: Magnons, Spinons, and Gauge Fields

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 theS=1/2Heisenberg model on the triangular lattice, including both nearest-neighborJ1and next-nearest-neighborJ2superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models. ForJ2=0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations atKpoints (i.e., at the corners of the Brillouin zone). The strong renormalization of the magnon branch (also including rotonlike minima around theMpoints, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magneticπflux threading half of the triangular plaquettes. When increasing the frustrating ratioJ2/J1, we detect a progressive softening of the magnon branch atM, which eventually becomes gapless within the spin-liquid phase. This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component). In addition, we observe an intense signal at low energies around theKpoints, 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.

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Plain Language Summary

Magnetism in materials arises when the magnetic moments of the electrons (i.e., the spins) are arranged in some specific ordered pattern. This usually happens when magnetic materials are cooled down to sufficiently low temperatures such that the spins “freeze.” Frustrated magnets are an exception: The spins resist ordering even at extremely low temperatures. For this reason, frustrated magnets can host an unconventional phase of matter: the spin liquid, in which spins point in random directions as if floating inside a fluid. Here, we characterize the transition from a normal magnet to a spin liquid in a theoretical spin model by numerical simulations.

In actual materials, neutron-scattering experiments can discriminate between magnetic-ordered and spin-liquid phases. In the first case, the spectrum is dominated by magnons, which are collective oscillations of the spins around their preferred orientations; in the second case, the spins break up, releasing fractional particles known as spinons.

Exploiting a novel numerical technique, we characterize the spectrum of a quantum spin model that transitions from a magnetic-ordered phase to a spin liquid. We observe the fractionalization of magnons into spinons across the transition and, most importantly, we detect signatures of a third kind of excitation that is related to the fluctuations of emergent gauge fields.

Our work provides the first measurable evidence of the effect of gauge fluctuations on the spectra of frustrated magnets. Therefore, we hope that the present work will boost future investigations to further clarify the nature of the elementary excitations of these systems.