Yang F, Wu M W
Hefei National Laboratory for Physical Sciences at Microscale, Department of Physics, and CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China.
J Phys Condens Matter. 2024 Jul 19;36(42). doi: 10.1088/1361-648X/ad61ae.
Both macroscopic Ginzburg-Landau Lagrangian and microscopic gauge-invariant kinetic equation suggest a finite Higgs-mode generation in the second-order optical response of superconductors at clean limit, whereas the previous derivations through the path-integral approach and Eilenberger equation within the Matsubara formalism failed to give such generation. The crucial treatment leading to this controversy lies at an artificial scheme that whether the external optical frequency is taken as continuous variable or bosonic Matsubara frequency to handle the gap dynamics within the Matsubara formalism. To resolve this issue, we derive the effective action of the superconducting gap nearin the presence of the vector potential through the path-integral approach, to fill in the long missing gap of the microscopic derivation of the Ginzburg-Landau Lagrangian in superconductors. It is shown that only by taking optical frequency as continuous variable within the Matsubara formalism, can one achieve the fundamental Ginzburg-Landau Lagrangian, and in particular, the finite Ginzburg-Landau kinetic term leads to a finite Higgs-mode generation at clean limit. To further eliminate the confusion of the Matsubara frequency through a separate framework, we apply the Eilenberger equation within the Keldysh formalism, which is irrelevant to the Matsubara space. By calculating the gap dynamics in the second-order response, it is analytically proved that the involved optical frequency is a continuous variable rather than bosonic Matsubara frequency, causing a finite Higgs-mode generation at clean limit.
宏观的金兹堡 - 朗道拉格朗日量和微观的规范不变动力学方程均表明，在清洁极限下超导体的二阶光学响应中会产生有限的希格斯模式，而此前通过松原形式体系内的路径积分方法和艾伦伯格方程进行的推导未能得出这种产生。导致这一争议的关键处理在于一个人为设定，即在松原形式体系内处理能隙动力学时，是将外部光学频率视为连续变量还是玻色松原频率。为解决这个问题，我们通过路径积分方法推导了在矢量势存在下超导能隙附近的有效作用量，以填补超导体中金兹堡 - 朗道拉格朗日量微观推导中长久缺失的空白。结果表明，只有在松原形式体系内将光学频率视为连续变量，才能得到基本的金兹堡 - 朗道拉格朗日量，特别是有限的金兹堡 - 朗道动力学项会在清洁极限下导致有限的希格斯模式产生。为了通过一个单独的框架进一步消除松原频率带来的混淆，我们在凯尔迪什形式体系内应用艾伦伯格方程，该方程与松原空间无关。通过计算二阶响应中的能隙动力学，解析证明了所涉及的光学频率是连续变量而非玻色松原频率，这在清洁极限下导致了有限的希格斯模式产生。
