Scaling dimension of Cooper pair operator from the black hole interior

TL;DR

This paper presents a holographic approach to determine the scaling dimension of the Cooper pair operator by imposing finiteness conditions inside and outside a black hole, offering a novel way to derive this quantity.

Contribution

It introduces a method to compute the scaling dimension of order parameters in holographic superconductors based on finiteness conditions inside black holes, avoiding manual setting of scalar mass.

Findings

The scaling dimension is quantized under the finiteness condition.

The method applies to any order parameter in holographic models.

Provides a new perspective on operator dimensions in holography.

Abstract

We have shown that in holographic superconductivity theory for 3+1 dimensional system, the scaling dimension of Cooper pair operator can be obtained as a quantized value if we request that the the scalar function describing the order parameter is finite inside the black hole as well as outside. This should be contrasted to the usual situation where we set the mass squared of the scalar by hand. Our method can be applied to any order parameters.