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Casimir effect mechanism of pairing between fermions in the vicinity of a magnetic quantum critical point

We consider two immobile spin $1/2$ fermions in a two-dimensional magnetic system that is close to the $ O(3)$ magnetic quantum critical point (QCP) which separates magnetically ordered and disordered phases. Focusing on the disordered phase in the vicinity of the QCP, we demonstrate that the criticality results in a strong long range attraction between the fermions, with potential $V(r) \propto -1/r^ν$, $ν\approx 0.75$, where $r$ is separation between the fermions. The mechanism of the enhanced attraction is similar to Casimir effect and corresponds to multi-magnon exchange processes between the fermions. While we consider a model system, the problem is originally motivated by recent establishment of magnetic QCP in hole doped cuprates under the superconducting dome at doping of about 10\%. We suggest the mechanism of magnetic critical enhancement of pairing in cuprates.