Direct three-dimensional ordering of quasi-one-dimensional quantum dimer system near critical fields

As the temperature lowers in the gapless region, quantum quasi-1D spin systems show one-dimensional (1D) short-range ordering (SRO) corresponding to a Tomonaga-Luttinger liquid (TLL) by the interaction within chains, and then at lower temperatures where the small interchain interaction cannot be ignored, they have a three-dimensional (3D) long- range ordered (LRO, BEC) phase. This generally accepted scenario for the dimensional crossover of quasi-1D spin ordering, however, is debatable around the critical field, because the 1D SRO (TLL) regime has a linear boundary on the $H$-$T$ phase diagram, while the 3D LRO transition temperature has a convex shape. To this dimensionality paradox around the critical field, the authors have given the solution, experimentally by detailed analyses of the specific heat and ac susceptibility for bond-alternating antiferromagnetic chains in pentafluorophenyl nitronyl nitroxide (F${}_{5}$PNN). They reveal the existence of a definite field region near a critical field, where 3D LRO (BEC) directly occurs without passing through 1D SRO region at higher temperatures, even if the interchain interaction is sufficiently small compared to the intrachain one.