Topological Phase Transition And Detectable Edge State In A Quasi-three-dimensional Circuit Quantum Electrodynamic Lattice

In recent years,the discovery of topological insulators has attracted great interest and strong attention of researchers in the topological states of quantum matter.As a novel discovery of quantum matter,topological insulators behave as insulators in the bulk,but contain gapless boundary states.Topological states can be characterized by the topological invariants rather than Landau’s symmetry breaking theory.If the energy gap is not closed,topological invariants do not change their value,so topological states are insensitive to disorder.Topology as a branch of mathematics,which is concerned with the properties that a geometric figure or space can remain unchanged after changing its shape continuously.With a large number of discoveries related to the Quantum Hall effect,the link between topology and the band structure of matter has been gradually established.Topological insulator like ordinary insulators has band gap in the bulk,but there are topologically protected electronic states on the edge or surface.The dimension of the electronic state is one dimension lower than the interior of the bulk,such as a three-dimensional topological insulator,whose surface electronic state is two-dimensional and has many novel properties.These properties have the potential to be widely applied and related topological materials have been theoretically predicted and experimentally observed in various systems.On the other hand,it provides great convenience for the introduction of synthetic dimensions due to the flexibility and diversity of circuit quantum electrodynamics lattice system.The constraints of geometric physical dimensions on exploring high-dimensional physical systems can be broken by introducing synthetic dimensions,and new opportunities are provided to realize exotic topological models and explore and exploit topological effects in new ways.In this paper,we study the properties of high dimensional topological insulators by mapping of one dimensional circuits quantum electrodynamics lattice.We propose a simple and feasible method to realize the mapping from a onedimensional to a quasi-three-dimensional topological system by introducing two periodic spatial dimensions in a circuit quantum electrodynamics(QED)lattice system.We investigate the topological phase transition and the edge states in a quasi-three-dimensional topological system mapped by a one-dimensional circuit QED lattice system.It is found that with the increasing of the periodically modulated on-site potential strength,the system undergoes a topological phase transition,corresponding to the change of the number of Weyl points under the periodic boundary condition.Under the open boundary condition,the phase transition is reflected by the energy band separation and the appearance of new edge states.Interestingly,the system holds two pairs of crossed edge states in the energy gaps when the periodic parameters take certain values.Furthermore,we show that,benefiting from the Bose statistical properties of the circuit quantum electrodynamic,the edge states of the system can be directly detected by measuring the average photon number of the cavity field in the steady state.