# Effective Potential for Emergent Majorana Fermions in Superconductor Systems

## TL;DR

This paper calculates the effective potential for emergent Majorana fermions in superconductor systems, revealing how superconducting parameters depend on temperature and interaction strength, with implications for topological insulators and high-Tc superconductivity.

## Contribution

It introduces a novel calculation of the effective potential for Majorana fermions in various superconductor models, including topological insulators, using functional integration techniques.

## Key findings

- Superconductor parameters vary with temperature in the Kitaev chain.
- Derived a gap equation for surface superconductivity in topological insulators.
- Identified a critical interaction strength for induced superconductivity in p-wave systems.

## Abstract

Majorana fermions cannot be found in nature as a free fundamental particle. Nevertheless, in condensed matter systems, they can emerge as a collective excitation. In this work, using functional integration techniques, we calculated the effective potential for emergent Majorana fermions in the Kitaev chain. In this case, we have shown how the superconductor parameter behaves as a function of temperature. Furthermore, we considered surface-induced superconductivity in a Topological Insulator and calculated the effective potential for emergent Majorana fermions in this system. In the case of an s-wave superconductor, we obtained a gap equation equivalent to that one appearing in a quasi-two-dimensional Dirac electronic system, a candidate to explain high-Tc superconductivity. Finally, for the p-wave superconductor, we have obtained a critical value of the electronelectron interaction in the surface of the Topological Insulator, determining the existence or not of induced superconductivity, a remarkable result to guide experiments