  In the d-wave superconductors the electronic entropy associated with an
isolated vortex diverges logarithmically with the size of the system even at
low temperatures. In the vortex array the entropy per vortex per layer, $S_V$,
is much larger than $k_B$ and depends on the distribution of the velocity field
$v_s$ around the vortex. If there is a first order transition with the change
of the velocity distribution, then there will be a big entropy jump $\Delta S_V
\sim 1 k_B$ at the transition. This entropy jump comes from the electronic
degrees of freedom on the vortex background, which is modified by the vortex
transition. This can explain the big jump in the entropy observed in the
so-called vortex-melting transition [A. Junod, et. al., Physica C 27, 245
(1997)], in which the vortex array and thus the velocity field are
redistributed. The possibility of the Berezinskii- Kosterlitz-Thouless
transition in the 3-dimensional d-wave superconductor due to the fermionic
bound states in the vortex background is discussed.
