Spin and pair density waves in 2D altermagnetic metals

Altermagnetism, a recently proposed and experimentally confirmed class of magnetic order, features collinear compensated magnetism with unconventional d-, g-, or i-wave spin order. Here, we show that in a metallic 2D d-wave altermagnet with combined two-fold spin and four-fold lattice rotational symmetry $[C_2||C_4]$, secondary instabilities can arise. Using an unbiased functional renormalization group approach, we analyze the weak-coupling instabilities of a 2D Hubbard model with a preexisting altermagnetic order inspired by our ab initio electronic structure calculations of realistic material candidates from V$_2$X$_2$O (X = Te, Se) family. We identify two distinct spin density wave (SDW) states that break the underlying altermagnetic $[C_2||C_4]$ symmetry. Additionally, we find spin-fluctuation-induced instabilities leading to a singlet d-wave superconducting state and an unconventional commensurate pair density wave (PDW) state with extended s-wave and spin-triplet symmetry. We establish a general criterion for the unusual exchange statistics for these pair density waves and characterize their excitation spectrum, which exhibits Bogoliubov Fermi surfaces or nodal points depending on the gap size.