Complex phase diagram of Ba$_{1-x}$Na$_{x}$Fe$_{2}$As$_{2}$: a multitude of phases striving for the electronic entropy

The low-temperature electronic phase diagram of Ba$_{1-x}$Na$_{x}$Fe$_{2}$As$_{2}$, obtained using high-resolution thermal-expansion and specific-heat measurements, is shown to be considerably more complex than previously reported, containing nine different phases. Besides the magnetic $C_{2}$ and reentrant $C_{4}$ phases, we find evidence for an additional, presumably magnetic, phase below the usual SDW transition, as well as a possible incommensurate magnetic phase. All these phases coexist and compete with superconductivity, which is particularily strongly suppressed by the $C_{4}$-magnetic phase due to a strong reduction of the electronic entropy available for pairing in this phase.