Weyl 半金屬的模型可以被預測在有自旋軌道耦合的系統中產生在。我們目的是在凝態系統中尋找高能 Weyl 理論中的手性磁效應(Chiral Magnetic Effect)。其中要求Weyl 半金屬中兩個Weyl點有能量差。那在凝態系統中有效尋找有能量差的Weyl 點的一種方法便是利用非平衡系統。在本篇文章中我們利用前人建立的Weyl 半金屬模型，並由Floquet 理論引入外加場，來在時間週期上驅動Weyl 半金屬產生有能量差的Weyl 點。但利用Floquet 理論所帶來的影響是會持續加熱系統，因此額外我們也引入冷庫去汲取熱能。在以上的系統設置完成後我們應用外加磁場，並看出非零的電流與磁場間平行的關係，以及電流-電流響應。因此我們提供了一個簡明的系統來實現手性磁效應。

The chiral magnetic effect (CME) has been proposed to exist for a condensed matter system with two Weyl nodes separated in different energies. In the condensed matter systems with spin-orbit coupling that has time-reversal symmetry and the needed Weyl nodes, it is realized that the static CME can not exist and the key factor to realize it is to create a non-equilibrium situation with Weyl nodes. We establish a non-equilibrium electronic systems with Weyl nodes by resorting to a time-periodic driven field on a Weyl semi-metal. The driven field induces band folding creates an effective Floquet Hamiltonian. Under appropriate condition, we show that the effective Floquet Hamiltonian gives rise a band structure with Weyl cones. The driven field plays both roles of modifying the band structure but at the same it makes the system non-equilibrium. As the the driven field pumps energy into the system, we couple a reservoir to extract energy out so that the electronic system can be maintained at some stationary state. We apply the Kelydsh formalism to calculate expectation value of current and the linear response of the current under magnetic field. The CME current in this system is verified to be non-vanishing. Our work thus establishes the existence of dynamical chiral magnetic field in Floquet Hamiltonian with Weyl nodes.

Acknowledgements
摘要i
Abstract ii
1 Introduction 1
2 Floquet Theory3
2.1 Interacting Picture 3
2.2 Floquet Theory 4
3 Weyl Semimetal 7
3.1 Weyl Equation 7
3.2 Topology of Weyl equation 9
3.3 Quantum Anomaly and Chiral Magnetic Effect 12
4 Kelydsh Formalism 17
4.1 Kelydsh method 17
4.2 Linear Response 20
4.3 Current Current Correlation 21
4.4 System Couple to bath 24
5 Linear Response 31
5.1 Retarded Current­Current Correlation 32
6 Result 35
6.1 Model 35
6.1.1 Hamiltonian 35
6.1.2 Effective Hamiltonian 38
6.1.3 Topology 40
6.2 Linear Response 41
6.2.1 Current Expression 41
6.3 Retarded Current Current Correlation 46
6.3.1 DC limit 46
7 Conclusion 53
References 55

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