  High-order topological insulator (HOTI) occupies an important position in
topological band theory due to its exotic bulk-edge correspondence. Recently,
it has been predicted that external magnetic field can introduce rich physics
into two-dimensional (2D) HOTIs. However, up to now the theoretical description
is still incomplete and the experimental realization is still lacking. Here we
investigate the influence of continuously varying magnetic field on 2D
Su-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, exhibit
exotic and rich dependence on the imposed magnetic field and the inhomogeneous
hopping strength. Moreover, by exploiting the parametric conversion method, we
can establish time- and site-resolved tunable hopping constants in the proposed
cQED simulator, thus providing an ideal platform for simulating the magnetic
field induced topological phase transitions in 2D HOTIs. Since the high-order
topological phases of the proposed model can be characterized by the existence
of the ZECMs on the lattice, we further investigate the corner site excitation
of the lattice in the steady state limit. Our numerical results imply that the
predicted topological phase transitions can be unambiguously identified by the
steady-state photon number measurement of the corner sites and their few
neighbors. Requiring only current level of technology, our scheme can be
readily tested in experiment and may pave an alternative way towards the future
investigation of HOTIs in the presence of magnetic field, disorder, and strong
correlation.
