Phase diagram of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>J</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mtext>−</mml:mtext><mml:msub><mml:mi>J</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> Heisenberg second-orde

Competing interactions in quantum magnets lead to a variety of emergent states, including ordered phases, nematic magnets, and quantum spin liquids. Among them, topological quantum magnets represent a promising platform to create topological excitations protected by the bulk many-body excitation gap. Here we establish the phase diagram of a breathing frustrated antiferromagnetic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>J</a:mi><a:mn>1</a:mn></a:msub><a:mtext>−</a:mtext><a:msub><a:mi>J</a:mi><a:mn>2</a:mn></a:msub></a:math>-Heisenberg model, featuring both ordered states and a higher-order topological quantum magnet state. Using variational many-body methods based on neural network quantum states and tensor networks, we determine the existence of a first-order phase transition between stripe order and the topological quantum magnet and the second-order phase transition between the Néel order and quantum magnet phase, further corroborated by calculations of the many-body gap. Using an auxiliary fermion parton formalism, we show the emergence of topological spinon corner modes stemming from the breathing order parameter of the parent Heisenberg model. Our results establish the breathing frustrated square lattice Heisenberg model as a paradigmatic system to engineer topological quantum magnetism, as recently realized in Ti lattices at MgO.