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The power of machine learning algorithms to automatically classify differentphases of matter and detect quantum phase transitions without necessity tocharacterize phases by various quantities like local order parameters ortopological invariants as in conventional approaches defined machine learningphases of matter as a new research frontier and basic research tool incondensed matter and statistical physics. We study quantum criticality in thespin-1/2 square-lattice J1-J2 model with additional plaquette structure bycombination of reinforcement and supervised machine learning techniques. In ourcalculations the ground-state spin-spin correlation matrices for several systemsizes are first found by restricted Boltzmann machine based variational MonteCarlo method, equivalent to reinforcement learning, and then used as a trainingdata for convolutional neural network based supervised machine learningalgorithm for phases classification. The model exhibits a quantum phasetransition from paramagnetic plaquette resonating valence bond state to anantiferromagnetic state and has been a topic of great interest because of itsclose connection to cuprate superconductors and possibility of realization incold atoms experiments. We consider both frustrated and unfrustrated regimesand compare our results with the results obtained previously with othermethods. We find that our results are in good agreement with the resultsobtained by coupled cluster and real space renormalization group methods forboth frustrated and unfrustrated regimes. The quantum Monte Carlo andfinite-size scaling result for the unfrustrated case however slightly differsfrom our result for the critical value of the inter-plaquette couplingstrength, although the results are still in reasonable agreement.

Machine learning quantum criticality in the spin-1/2 quantum antiferromagnets on the square lattice with plaquette structure