Thermal stability of pair density wave in a $d$-wave altermagnetic superconductor

We study finite-momentum superconductivity in a two-dimensional $d$-wave altermagnetic superconductor using a non-perturbative Monte Carlo approach beyond mean-field theory. We show that altermagnetism stabilizes a pair density wave (PDW) state without external magnetic fields and enables its survival at finite temperatures with robust phase coherence. Our results establish altermagnetism as a promising route to realizing thermally stable PDW superconductivity and identify clear thermodynamic and spectroscopic signatures.