Critical Density-Wave Vestigial Phases of Commensurate Pair Density Wave

The pair-density-wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U(1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with n-times (n ≥ 3) expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg-Landau theory, we get the low energy effective model Hamiltonian from typical PDW ground state configurations. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For nt ≤ 4 (t = 1 for odd n and t = 1/2 for even n), besides the charge-4e/2e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for nt ≥ 5 (specifically, for odd n, n ≥ 5, and for even n, n ≥ 10), the restore of the translational symmetry with increasing temperature is realized through two successive Berezinskii-Kosterlitz-Thouless transitions. Such a two-step process leads into two critical vestigial phases, i.e. the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading into a power-law decaying density-density correlation even at 2D. Our work appeals for experimental verifications.