We present a systematic study of the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model by using the Green's function Monte Carlo (GFMC) technique, implemented within the fixed-node (FN) approximation and a wave function that contains both antiferromagnetism and
d-wave pairing. This enables us to study the interplay between these two kinds of order and compare the GFMC results with the ones obtained by the simple variational approach. By using a generalization of the forward-walking technique, we are able to calculate true FN ground-state expectation values of the pair-pair correlation functions. In the case of ${t}^{\ensuremath{'}}=0$, there is a large region with a coexistence of superconductivity and antiferromagnetism that survives up to ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.10$ for
J∕t=0.2, and ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.13$ for
J∕t=0.4. The presence of a finite ${t}^{\ensuremath{'}}∕t<0$ induces a strong suppression of both magnetic (with ${\ensuremath{\delta}}_{c}\ensuremath{\lesssim}0.03$ for
J∕t=0.2 and ${t}^{\ensuremath{'}}∕t=\ensuremath{-}0.2$) and pairing correlations. In particular, the latter ones are depressed both in the low-doping regime and around $\ensuremath{\delta}\ensuremath{\sim}0.25$, where strong size effects are present.
We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site
E0/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be
Zc==c/
cs=1.14(5).
Finding high-quality trial wave functions for quantum Monte Carlo calculations of light nuclei requires a strong intuition for modeling the interparticle correlations as well as large computational resources for exploring the space of variational parameters. Moreover, for systems with three-body interactions, the wave function should account for many-body effects beyond simple pairwise correlations. In this Letter, we design neural networks that efficiently incorporate these factors to generate expressive wave function Ansätze for light nuclei using variational Monte Carlo. Our neural-network approach for A=3 nuclei can capture, already at the level of variational Monte Carlo, the overwhelming majority of the ground-state energy estimated by Green's function Monte Carlo (GFMC). It achieves a ground-state energy within 0.45% of the GFMC result for ^{3}H using the softest chiral interaction, representing a substantial improvement over standard variational Monte Carlo, which exhibits a 3.7% deviation. The result indicates the potential of neural networks to construct effective trial wave functions for quantum Monte Carlo calculations.