Hall Conductance For Open Two-band System Beyond Rotating-wave Approximation

The discovery of integer quantum Hall effect is a milestone in the history of condensed matter physics.The topology mechanism of integer quantum Hall effect was discussed by Thouless,Kohmoto,Nightingale and Nijs.And Chern number was introduced to describe the topological property of quantum Hall conductance.Quantum Hall state could be described by the topological property of the ground state wave function without any symmetry breaking,and the topological invariant would exist in the quantum Hall system.In contrast to ordinary band insulators,topological insulators whose surface states are protected topologically are new states of quantum matter.Linear response theory is widely used in condensed matter physics and material science.The response of system to an external weakly classical field can be expressed in terms of Kubo formula,while the quantum Hall conductance is the linear response of current to external electric field.In most realistic conditions,the quantum system inevitably interacts with the environment,and the response of the open system to external fields would in general be different from that of a strictly isolated one.In this paper,we systematically study the Hall conductance of a two-band model under the influence of its electromagnetic environment by treating the system and its environment on equal footing.In order to clarify some well-established conclusions about the Hall conductance,we do not use the rotating wave approximation(RWA)in obtaining an effective Hamiltonian for open two-band system.We first derive the ground state of the whole system(the system plus the environment described by multi-mode quantized field)beyond RWA,then calculate an analytical expression for Hall conductance of this open system in the ground state.The Hall conductance for open system could no longer be expressed by a multiple of a Chern number and a constant,or a weighted sum of Chern numbers.In a word,there is no topological invariant for open two-band systems.We apply the expression to the calculation of the concrete model,and demonstrate the influence of the environment on the quantum Hall conductance.The results show that the transition points of Hall conductance are robust against the environment,and the topological phase transition still exists in the open two-band system.