Journal of Physical Studies 23(3), Article 3709 [6 pages] (2019)
DOI: https://doi.org/10.30970/jps.23.3709

THE FUNCTIONAL INTEGRATION METHOD IN THE TWO BAND SUPERCONDUCTIVITY THEORY

A. Shutovskyi, A. Svidzinskyi, V. Sakhnyuk, O. Pastukh

Lesya Ukrainka Eastern European National University, 13, Voli Ave., Lutsk, UA-43000, Ukraine

Applying the thermodynamic perturbation theory, an evolution operator of a superconducting system can mathematically be expressed as a product of two multipliers. The first multiplier can physically be interpreted as an evolution operator describing a system of free electrons. The second multiplier is the so called ordered exponential containing the operator of the pairing interaction. In the theory of superconductivity, an ordered exponential is usually considered as a product of exponentials. This fact helps to consider a functional integral as a way to parameterize complex operator functions. To represent an ordered exponential in the form of a functional integral, the generalized Poisson parametrization was used. By substituting the obtained parametrization of an ordered exponential into a definition of the partition function, the partition function representation of a superconductor with two energy gaps was successfully constructed. During the construction of the mentioned representation, the operators in the momentum representation were taken. To calculate the partition function of a two band superconductor, the mean field theory was formulated. The thermodynamic potential in the mean field approximation was also introduced. The mean field is a name of two complex functions dependent on the so called band indices, spatial coordinates and the so called imaginary time. Each of the mentioned complex functions is also introduced as a solution of an integral equation.

PACS number(s): 74.70.Ad, 03.65.Db

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