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Pursuing fractionalized particles that do not bear properties of conventionalmeasurable objects, exemplified by bare particles in the vacuum such aselectrons and elementary excitations such as magnons, is a challenge inphysics. Here we show that a machine-learning method for quantum many-bodysystems that has achieved state-of-the-art accuracy reveals the existence of aquantum spin liquid (QSL) phase in the region $0.49\lesssimJ_2/J_1\lesssim0.54$ convincingly in spin-1/2 frustrated Heisenberg model withthe nearest and next-nearest neighbor exchanges, $J_1$ and $J_2$, respectively,on the square lattice. This is achieved by combining with the cutting-edgecomputational schemes known as the correlation ratio and level spectroscopymethods to mitigate the finite-size effects. The quantitative one-to-onecorrespondence between the correlations in the ground state and the excitationspectra enables the reliable identification and estimation of the QSL and itsnature. The spin excitation spectra containing both singlet and triplet gaplessDirac-like dispersions signal the emergence of gapless fractionalized spin-1/2Dirac-type spinons in the distinctive QSL phase. Unexplored critical behaviorwith coexisting and dual power-law decays of N\'{e}el antiferromagnetic anddimer correlations is revealed. The power-law decay exponents of the twocorrelations differently vary with $J_2/J_1$ in the QSL phase and thus havedifferent values except for a single point satisfying the symmetry of the twocorrelations. The isomorph of excitations with the cuprate $d$-wavesuperconductors implies a tight connection between the present QSL andsuperconductivity. This achievement demonstrates that the quantum-staterepresentation using machine learning techniques, which had mostly been limitedto benchmarks, is a promising tool for investigating grand challenges inquantum many-body physics.

physical review. x

Dirac-type nodal spin liquid revealed by refined quantum many-body  solver using neural-network wave function, correlation ratio, and level  spectroscopy