Spatially resolved magnetization in the Bose-Einstein condensed state of BaCuSi${}_2$O${}_6$: Evidence for imperfect frustration

In order to understand the nature of the two-dimensional Bose-Einstein condensed (BEC) phase in BaCuSi${}_2$O${}_6$, we performed detailed ${}^{63}$Cu and ${}^{29}$Si NMR above the critical magnetic field, $H_{c1}=23.4$ T. The two different alternating layers present in the system have very different local magnetizations close to $H_{c1}$; one is very weak, and its size and field dependence are highly sensitive to the nature of inter-layer coupling. Its precise value could only be determined by “on-site” ${}^{63}$Cu NMR, and the data are fully reproduced by a model of interacting hard-core bosons in which the perfect frustration associated to tetragonal symmetry is slightly lifted, leading to the conclusion that the population of the less populated layers is not fully incoherent but must be partially condensed.