Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid Helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a non-trivial sign structure. Even though the RBM state is by far more effective for Néel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly-correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.

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 J1 and next-nearest-neighbor J2 super-exchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models. For J2 = 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 π-flux threading half of the triangular plaquettes. When increasing the frustrating ratio J2/J1, 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. arXiv:1903.05691v3 [cond-mat.str-el]

We employ a variational Monte Carlo approach to efficiently obtain the dynamical structure factor for the spin-1/2 J1 − J2 Heisenberg model on the square lattice. Upon increasing the frustrating ratio J2/J1, the ground state undergoes a continuous transition from a Néel antiferromagnet to a Z2 gapless spin liquid. We identify the characteristic spectral features in both phases and highlight the existence of a broad continuum of excitations in the proximity of the spin-liquid phase. The magnon branch, which dominates the spectrum of the unfrustrated Heisenberg model, gradually loses its spectral weight, thus releasing nearly-deconfined spinons, whose signatures are visible even in the magnetically ordered state. Our results provide an important example on how magnons fractionalize into deconfined spinons across a quantum critical point.

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor J1 and second-neighbor J2 antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. Néel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio J2/J1, we find strong evidence for a continuous transition to a nonmagnetic phase at J2/J1 ≈ 0.23. Close to the transition point, the Gutzwiller-projected uniform resonating valence bond state gives an excellent approximation to the exact ground-state energy. For 0.23 J2/J1 0.36, a gapless Z2 spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the Néel order melts, this ordered state becomes clearly favored only for J2/J1 0.3. Finally, for 0.36 J2/J1 ≤ 0.5, a valencebond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the Néel antiferromagnet from the plaquette valence-bond solid.

The dynamical spin structure factor is computed within a variational framework to study the onedimensional J1 −J2 Heisenberg model. Starting from Gutzwiller-projected fermionic wave functions, the low-energy spectrum is constructed from two-spinon excitations. The direct comparison with Lanczos calculations on small clusters demonstrates the excellent description of both gapless and gapped (dimerized) phases, including incommensurate structures for J2/J1 > 0.5. Calculations on large clusters show how the intensity evolves when increasing the frustrating ratio and give an unprecedented accurate characterization of the dynamical properties of (non integrable) frustrated spin models.