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We present a combined neutron diffraction and high field muon spin rotation ($\mu$SR) study of the magnetically ordered and superconducting phases of the high-temperature superconductor La$_{1.94}$Sr$_{0.06}$CuO$_{4+y}$ ($T_{\rm c} = 37.5(2)$~K) in a magnetic field applied perpendicular to the CuO$_2$ planes. We observe a linear field-dependence of the intensity of the neutron diffraction peak that reflects the modulated antiferromagnetic stripe order. The magnetic volume fraction extracted from $\mu$SR data likewise increases linearly with applied magnetic field. The combination of these two observations allows us to unambiguously conclude that stripe-ordered regions grow in an applied field, whereas the stripe-ordered magnetic moment itself is field-independent. This contrasts with earlier suggestions that the field-induced neutron diffraction intensity in La-based cuprates is due to an increase in the ordered moment. We discuss a microscopic picture that is capable of reconciling these conflicting viewpoints.

Proper characterisation of investigated samples is vital when studying superconductivity as impurities and doping inhomogeneities can affect the physical properties of the measured system. We present a method where a polarised neutron imaging setup utilises the precession of spin-polarised neutrons in the presence of a trapped field in the superconducting sample to spatially map out the critical temperature for the phase transition between superconducting and non-superconducting states. We demonstrate this method on a superconducting crystal of the prototypical high-temperature superconductor (La,Sr)$_2$CuO$_4$. The results, which are backed up by complementary magnetic susceptibility measurements, show that the method is able to resolve minor variations in the transition temperature across the length of the LSCO crystal, caused by inhomogeneities in strontium doping., Comment: 11 pages, 7 figures

We present detailed neutron scattering studies of the static and dynamic stripes in an optimally doped high-temperature superconductor, La$_2$CuO$_{4+y}$. We find that the dynamic stripes do not disperse towards the static stripes in the limit of vanishing energy transfer. We conclude that the dynamic stripes observed in neutron scattering experiments are not the Goldstone modes associated with the broken symmetry of the simultaneously observed static stripes, but rather that the signals originate from different domains in the sample. These domains may be related by structural twinning, or may be entirely different phases, where the static stripes in one phase are pinned versions of the dynamic stripes in the other. Our results explain earlier observations of unusual dispersions in underdoped La$_{2-x}$Sr$_x$CuO$_{4}$ ($x=0.07$) and La$_{2-x}$Ba$_x$CuO$_{4}$ ($x=0.095$). Our findings are relevant for all compounds exhibiting magnetic stripes, and may thus be a vital part in unveiling the nature of high temperature superconductivity.