Eight-fold classification of superconducting orders

Abstract

We present a symmetry-based classification for superconducting pairing states, organized by the exchange properties of the anomalous correlation function rather than by a specific microscopic pairing mechanism. The classification is built from the pairwise permutation of spin, orbital, spatial, and temporal indices, leading to the fermionic constraint ${\cal S} {\cal P}^\ast {\cal O} {\cal T}^\ast = -1$, and is further organized by separating relative and center-of-mass space-time coordinates. This construction defines what we call the Berezinskii--Abrahams hypercube, in which conventional Bardeen--Cooper--Schrieffer superconductivity, unconventional $p$- and $d$-wave pairing, odd-frequency superconductivity, Fulde--Ferrell--Larkin--Ovchinnikov states, pair-density-wave states, and time-modulated superconducting orders appear as different sectors of a unified framework. Beyond organizing known phases, the Berezinskii--Abrahams hypercube identifies symmetry-allowed hybrid orders that have received comparatively little attention, including odd-frequency Fulde--Ferrell--Larkin--Ovchinnikov or pair-density-wave states and odd-frequency time-modulated superconducting states. We discuss microscopic routes, candidate platforms, experimental signatures, and stability constraints for these sectors, emphasizing the distinction between symmetry allowance and physical realizability. We also present the proximity induced odd-frequency pair-density-wave state, and its driven analog as the two new examples of the states that naturally emerge in the Berezinskii--Abrahams hypercube. The resulting framework provides a guide for connecting established superconducting phenomena with unexplored symmetry-allowed forms of order.