Dynamical Structure Factor of the $J_1-J_2$ Heisenberg Model on the Triangular Lattice: Magnons, Spinons, and Gauge Fields

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 $J_2$ superexchanges, 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 rotonlike 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 nontrivial magnetic $\pi$ flux threading half of the triangular plaquettes. When increasing the frustrating ratio $J_2/J_1$, we detect a progressive 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 two 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.

Popular 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.

Figure 1

Upper left-hand panel: The classical spin configuration (in the $XY$ plane) that is determined by the fictitious magnetic field $h$ in the Hamiltonian Eq. (13) with $\mathbf{Q}=(2\pi /3,2\pi /\sqrt{3})$. Upper right-hand panel: Pattern for the sign structure of the nearest-neighbor hopping $s_{i,j}$ of Eq. (13), $s_{i,j}=+1$ ($-1$) for solid (dashed) lines; notice the amplitude for the kinetic terms is chosen to be $t>0$. Lower left-hand panel: The path in the Brillouin zone that is used to plot the results of the dynamical structure factor of the $30\times 30$ triangular lattice (blue arrows); see Figs. 2, 3, 4, 7, and 8. Lower right-hand panel: The path in the Brillouin zone that is used to plot the dynamical structure factor of the $84\times 6$ cylinder (blue arrows); see Fig. 6. In both lower panels the orange shaded area corresponds to the region of the Brillouin zone in which magnon decay is predicted by the spin-wave approximation [16, 17], and the dashed line delimits the magnetic Brillouin zone.

Figure 2

Dynamical structure factor of the nearest-neighbor Heisenberg model on the triangular lattice obtained by using the variational wave function of Eqs. (12) and (13) with $t=0$ on the $30\times 30$ cluster. The path along the Brillouin zone is shown in Fig. 1. A Gaussian broadening of the spectrum has been applied ($\sigma =0.02J_1$). The spin-wave energies of the magnon branch (${\epsilon }_q$), on the same cluster size, are represented by the white dots connected with a solid line. The dashed line corresponds to the bottom of the continuum within linear spin waves, i.e., $E_q={\min }_k\{{\epsilon }_{q-k}+{\epsilon }_k\}$. Notice that $E_q<{\epsilon }_q$ in most of the Brillouin zone, as obtained in Refs. [16, 17].

Figure 3

The same as Fig. 2 but for the optimal variational wave function with both hopping $t$ and fictitious magnetic field $h$. The path along the Brillouin zone is shown in Fig. 1. The dotted line denotes the bottom of the continuum $E_q={\min }_k\{E_0^{q-k}+E_0^k\}$, where $E_0^q$ is the lowest energy for a given momentum $\mathbf{q}$ obtained within our variational approach. Since the spectrum is gapless at the $\mathrm{\Gamma }$ point, we exclude the cases $\mathbf{k}=(0,0)$ and $\mathbf{k}=\mathbf{q}$ in the search of the minimum, because the resulting $E_q$ would simply coincide with the energy of the magnon branch $E_0^q$ all over the Brillouin zone. The purpose of this kinematic analysis is to show that no magnon decay can yield an energy $E_q$ which is lower than the one of the magnon branch $E_0^q$ (in contrast to spin-wave results).

Figure 4

Energies of the magnon branch for the nearest-neighbor Heisenberg model on the triangular lattice obtained with different methods. The path along the Brillouin zone is shown in Fig. 1. The black line corresponds to linear spin wave, the blue squares to series expansion [21], and the orange circles to our variational results (on the $30\times 30$ cluster).

Figure 5

Dispersion relation of the magnon branch (i.e., the lowest-energy excitation) as obtained within our variational approach (on the $30\times 30$ cluster). The linear spin-wave results are also reported for comparison. Dashed lines represent the edges of the magnetic Brillouin zone. The presence of the roton minima at the $M$ and $Y_1$ points in the variational spectrum is evident.

Figure 6

The dynamical structure factor for the nearest-neighbor Heisenberg model on a cylindrical geometry ($84\times 6$), to make a close comparison with DMRG calculations by Verresen et al. [32]. We apply a Gaussian broadening to the spectrum, which is equivalent to the one of the aforementioned DMRG result ($\sigma =0.077J_1$). The path in the Brillouin zone is shown in the inset and in Fig. 1 [point $A$ lies at $1/4$ of the $\mathrm{\Gamma }-K^{\prime \prime }$ line, where $K^{\prime \prime }=(-2\pi /3,2\pi /\sqrt{3})$, and point $B$ lies at $1/4$ of the $K-K^{\prime }$ line]. The dashed line denotes the bottom of the continuum, which is evaluated by taking $E_q=\min \{E_0^{q-K}+E_0^K,E_0^{q+K}+E_0^{-K}\}$, where $E_0^q$ is the lowest energy for a given momentum $q$ obtained within our variational approach and $K=(2\pi /3,2\pi /\sqrt{3})$.

Figure 7

The dynamical structure factor for the $J_1-J_2$ Heisenberg model on the $30\times 30$ cluster with $J_2/J_1=0.07$ (top) and $J_2/J_1=0.09$ (bottom). The path along the Brillouin zone is shown in Fig. 1, and a Gaussian broadening of the spectrum has been applied ($\sigma =0.02J_1$).

Figure 8

The dynamical structure factor for the $J_1-J_2$ Heisenberg model on the $30\times 30$ cluster with $J_2/J_1=0.125$. The variational results (left) are compared to the ones obtained from the unprojected Abrikosov fermion Hamiltonian ${\mathcal{H}}_0$ of Eq. (13) with $t=1$ and $h=0$ (right). The path along the Brillouin zone is shown in Fig. 1. We applied a Gaussian broadening of $\sigma =0.02J_1$ to the variational results. Notice that, for the unprojected data, the energy scale is given by the hopping amplitude $t$ of the unprojected Hamiltonian Eq. (13), instead of $J_1$. In addition, the broadening has been rescaled in order to account for the larger bandwidth of the spectrum.

Figure 9

The dynamical structure factor for the $J_1-J_2$ Heisenberg model on the square lattice ($22\times 22$) with $J_2/J_1=0.55$. The variational results (left) are compared to the ones obtained from the unprojected Abrikosov fermion Hamiltonian ${\mathcal{H}}_0$ (right), which contains a flux-phase hopping (of strength $t$) and a $d_{xy}$ pairing (see Ref. [46] for details). We applied a Gaussian broadening of $\sigma =0.02J_1$ to the variational results. Notice that, for the unprojected data, the energy scale is given by the hopping amplitude $t$ of the unprojected Hamiltonian of Ref. [46], instead of $J_1$. In addition, the broadening has been rescaled in order to account for the larger bandwidth of the spectrum.