Quantum critical scaling of altermagnetism

The term altermagnetism has recently been introduced to describe the N\'eel order of a class of materials whose magnetic sublattices are neither related by translation nor inversion. While these materials arguably have large technological potential, little effort has been devoted to studying the universal distinction of this phase of matter compared to collinear antiferromagnetism. Employing a recently proposed minimal microscopic model, we explicitly derive a nonlinear sigma model describing long-wavelength fluctuations of the staggered magnetization in this system, including quantum effects to leading order. The term that distinguishes the altermagnetic nonlinear sigma model from its antiferromagnetic counterpart is an interaction term that derives directly from the Berry phase of the microscopic spin degrees of freedom. Its effects on the one-loop renormalization group flow in $d=2+\epsilon$ dimensions are examined. Extending the theory to describe the fermionic excitations of the metallic altermagnet, we find an effective low-energy model of $d$-wave spin-split Dirac fermions interacting with the magnetic fluctuations. Using a Dyson-Schwinger approach, we derive the many-body effects on the dynamical critical scaling due to the competition between the long-range Coulomb interaction and the fluctuations of the staggered magnetization.