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High-order topological insulator (HOTI) occupies an important position intopological band theory due to its exotic bulk-edge correspondence. Recently,it has been predicted that external magnetic field can introduce rich physicsinto two-dimensional (2D) HOTIs. However, up to now the theoretical descriptionis still incomplete and the experimental realization is still lacking. Here weinvestigate the influence of continuously varying magnetic field on 2DSu-Schriffer-Heeger lattice, which is one of the most celebrated HOTI models,and proposed a corresponding circuit quantum electrodynamics (cQED) simulator.Our numerical calculation shows that the zero energy corner modes (ZECMs),which can serve as evidence of the high order topology of the lattice, exhibitexotic and rich dependence on the imposed magnetic field and the inhomogeneoushopping strength. Moreover, by exploiting the parametric conversion method, wecan establish time- and site-resolved tunable hopping constants in the proposedcQED simulator, thus providing an ideal platform for simulating the magneticfield induced topological phase transitions in 2D HOTIs. Since the high-ordertopological phases of the proposed model can be characterized by the existenceof the ZECMs on the lattice, we further investigate the corner site excitationof the lattice in the steady state limit. Our numerical results imply that thepredicted topological phase transitions can be unambiguously identified by thesteady-state photon number measurement of the corner sites and their fewneighbors. Requiring only current level of technology, our scheme can bereadily tested in experiment and may pave an alternative way towards the futureinvestigation of HOTIs in the presence of magnetic field, disorder, and strongcorrelation.

Circuit QED simulator of two-dimensional Su-Schrieffer-Hegger model:  magnetic field induced topological phase transition in high-order topological  insulators