Predicting the behaviour of in the vortex state is far
more complicated. This is because, compared to the Meissner state,
the problem must be solved in two dimensions,
rather than one. A second complication is that the magnitude
of the order parameter is spatially inhomogeneous due to the presence
of vortices.
Very recently, some progress has been made in understanding
the nature of the field
dependence of in the vortex state of a *d*_{x2-y2}-wave
superconductor [40].
The nonlinear effects which were discussed in the previous section
also affect the supercurrent response in the vortex state. The mechanism
for these nonlinear effects is identical to that in the Meissner phase.
In a *d*_{x2-y2}-wave superconductor, the
supercurrent response is also nonlocal due to the nodes on the Fermi surface.
Near the nodes ,which was discussed earlier in connection with Eq. (2.37).
There it was noted that these
nonlocal effects will affect the temperature dependence of at very low *T*.

Nonlocal effects will not affect the measured field dependence of
in the Meissner state. This can be realized, for instance, by
incorporating a -dependence into the London kernel
. The terms in *Q*_{L} (*k*) which contain *k* will
affect the precise way the field decays into the superconductor.
However, since *k* itself does not depend on the magnetic field,
this will not create a field-dependence for .

The situation is quite different in the vortex state. Since the
theory is nontrivial and the
predicted behaviour of in Ref. [40]
is given as a numerical result, the discussion here will be qualitative only.
The authors of Ref. [40] introduce an appropriate
-dependent kernel into the London model
to account for the nonlocal effects arising at the nodes.
The higher-order *k* terms are more important for large values of *k*.
Since large *k* values correspond to small values of *r* in real space,
the nonlocal effects are most important near the vortex
cores. At fields just above *H*_{c1} where the vortex cores are
isolated from each other, the
measured penetration depth is virtually unaffected. However, with
increasing applied field, the vortex cores move closer together
and the nonlocal regions overlap. This in turn leads to significant
changes in the distribution of magnetic field between the vortices
and a field dependence for the corresponding effective
penetration depth.
In particular, the effective is found to increase with
increasing magnetic field. At fields in which the vortices begin to
interact, this field dependence is very strong. The strength of the
field dependence weakens somewhat at higher magnetic fields.
In the same study, the nonlinear corrections are found to
have a small effect on the field dependence of .However, it should be noted that the calculations in Ref. [40]
were performed assuming that the size of the vortex cores are
small, which is not always the case.