  Many physical properties of high-temperature (high-Tc) superconductors are
two-dimensional phenomena derived from their square planar CuO2 building
blocks. This is especially true of the magnetism from the copper ions. As
mobile charge carriers enter the CuO2 layers, the antiferromagnetism of the
parent insulators, where each copper spin is antiparallel to its nearest
neighbours1, evolves into a fluctuating state where the spins show tendencies
towards magnetic order of a longer periodicity. For certain charge carrier
densities, quantum fluctuations are sufficiently suppressed to yield static
long-period order2,3,4,5,6, and external magnetic fields also induce such
order7,8,9,10,11,12. Here we show that in contrast to the chemically-controlled
order in superconducting samples, the field-induced order in these same samples
is actually three-dimensional, implying significant magnetic linkage between
the CuO2 planes. The results are important because they show that there are
three-dimensional magnetic couplings which survive into the superconducting
state, and coexist with the crucial inter-layer couplings responsible for
three-dimensional superconductivity. Both types of coupling will straighten the
vortex lines, implying that we have finally established a direct link between
technical superconductivity, which requires zero electrical resistance in an
applied magnetic field and depends on vortex dynamics, and the underlying
antiferromagnetism of the cuprates.
