Eliashberg Theory in the weak-coupling limit: Results on the real frequency axis

• Author / Creator

  Sepideh Mirabi

• One of the theoretical mechanisms for accurately describing superconductivity is the electron-phonon interaction. This attractive interaction occurs between two electrons near the Fermi surface and leads to gapped quasi-particle states. The equations describing the mean-field properties are called the Eliashberg equations. By accounting for the frequency of phonons, it follows that the interaction within Eliashberg theory is retarded in time and local in space. In the BCS theory of superconductivity, however, the nature of the interaction is instantaneous in time.

  The dynamical nature of the order parameter in Eliashberg theory is the foremost difference with the order parameter in the BCS counterpart. The motivation for this thesis is to explore this disparity, with a particular emphasis on the nature of the weak-coupling limits of the two variant theories.
  In this thesis, both the numerical solution and analytical approximation for the Eliashberg theory on the real frequency axis are evaluated in the weak electron-phonon coupling limit.
  To first order in the weak-coupling limit, the analytical solution is shown to agree well with the numerical result. Previous literature has shown that the weak-coupling limit of the BCS and Eliashberg theories do not converge to the same result, and moreover there exists a difference of $1/\sqrt{e}$ between the critical temperatures of the two theories.
  Here, we have extended this result further to investigate the behaviour of the zero-temperature gap edge on the real frequency axis. This quantity is analytically solved on the real axis under certain approximations, and it is found that there exists the same pre-factor of $1/\sqrt{e}$ in the zero-temperature gap edge as observed in the critical temperature in the Eliashberg theory. Remarkably, in the weak-coupling limit, the ratio of these two quantities therefore approaches the same universal value as obtained in the BCS theory.

  In chapter one, a brief development of the history of the BCS and Eliashberg theories is given.
  In chapter two, a full derivation of the Eliashberg equations is included, based on solving the time development of the Green's functions. The Green's function can be evaluated on either the imaginary or real frequency axes. It is much more convenient to use the Matsubara formalism, especially when performing the numerical calculations. However, most of the dynamical and physical properties are determined on the real axis. By implementing the correct analytical continuation procedure on the real axis, we recover the results on the imaginary frequency axis.

  In chapter three, the imaginary axis Eliashberg equations are considered, first in the case where the renormalization factor is included, and then in the case where it is neglected. The critical temperature is calculated using the linearized gap function. Also, the analytical solution using digamma expressions for the gap function is provided both on the real and imaginary frequency axes.

  In chapter four, the numerical solutions on the real and imaginary axes in the zero-temperature, finite-temperature, and critical-temperature limits are studied.

  Finally, in chapter five the thermodynamic properties of superconductors are studied. The free energy, heat capacity and the critical magnetic field are considered using both BCS and Eliashberg theories. It is shown that the heat capacity ratio at the critical temperature of both theories are identical, and in the zero-temperature limit in the Eliashberg theory the free energy has a correction factor of $(1/\sqrt{e})^{2}$ compared to its BCS counterpart.

• Subjects / Keywords

  • real frequency axis
  • Eliashberg Theory
  • weak-coupling limit
  • superconductivity

• Graduation date

  Fall 2020

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/r3-6frk-xf06

• License

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