Master Equation For Open Two-band Systems And Its Applications To Hall Conductance

Quantum mechanics and the theory of relativity are considered as two foundations of modern physics.Quantum mechanics is the science describing the microscopic material and its interactions and promotes the progress of physics dramatically.In most realistic conditions,the quantum mechanical system which is different from the isolated system is always coupled with the environment whose degree is typically large.The coupling leads the exchange of the energy and information between the system and environment.Quantum mechanical systems are always regarded as the open systems in practice and open quantum theory has been extended to many aspects,such as quantum optics,condensed matter physics and quantum transport.We can apply the master equation to study open quantum systems in general.In this thesis,we derive a master equation and the Hall conductance of the open two-band system.There are many researches about the Markov master equations,but the decay of the master equations is almost induced phenomenologically,which is different from ours.In this thesis,we start from the concrete model which is different from most researches.The Hamiltonian of the two-band system influenced by the environment is constructed firstly.With the noise electromagnetic field along x axis,the crystal momentum changes which causes the variation of the Hamiltonian.The total Hamiltonian is rewritten in the Hilbert space spanned by the eigenstates of the system Hamiltonian and the master equation is deduced by using the Born approximation,Markovian approximation and the rotating approximation.The master equation of the two-band system influenced by the environment depends on the Bloch vectors and the temperature,which has distinctions with the master equation induced phenomenologically.Based on the perturbation theory,we can calculate the quantum Hall conductance using Kubo formula,which is the linear response of the electric current density.We apply the perturbation theory to solve the steady state of the master equation and apply it to the calculation of the Hall conductance which can be expanded in powers of the decay rates.As a result,the zero-order Hall conductance can go back to the Hall conductivity predicted by the Kubo formula while others can not.Finally,we apply the Kubo formula to the tight-binding electrons in two-dimemsional lattice and the dependence of the decay rates on the Bloch vector is given.We find the phase transition points are robust against the environment.