  Upon application of an external tuning parameter, a magnetic state can be
driven to a normal metal state at zero temperature. This phenomenon is known as
quantum criticality and leads to fascinating responses in thermodynamics and
transport of the compound. In the standard picture, a single quantum critical
point occurs at zero temperature, which results in a nontrivial critical
behaviour in its vicinity. Here we show that in two dimensions the scenario is
considerably more complex due to the enormous amount of quantum fluctuations.
Instead of the single point separating the antiferromagnet from the normal
metal, we have discovered a broad region between these two phases where the
magnetic order is destroyed but certain areas of the Fermi surface are closed
by a large gap. This gap reflects the formation of a novel quantum state
characterised by a superposition of d-wave superconductivity and a
quadrupole-density wave, id est a state in which an electron quadrupole density
spatially oscillates with a period drastically different from the one of the
original spin-density wave. At moderate temperatures both orders co-exist at
short distances but thermal fluctuations destroy the long-range order. Below a
critical temperature the fluctuations are less essential and superconductivity
becomes stable. This new phenomenon may shed some light on the origin of the
mysterious pseudogap state and of the high-temperature transition into the
superconducting state in cuprates. Our results demonstrate that quantum phase
transitions between antiferromagnets and normal metals in layered materials may
be the proper playground for search of new high temperature superconductors.
