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Mesoscopic correlations in Tb2Ti2O7 spin liquid

We have studied the spin correlations with $\bf{k}$= ($\frac12$, $\frac12$, $\frac12$) propagation vector which appear below 0.4\, K in \tbti\ spin liquid by combining powder neutron diffraction and specific heat on Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ samples with $x$=0, 0.01, -0.01. The $\bf{k}$= ($\frac12$, $\frac12$, $\frac12$) order clearly appears on all neutron patterns by subtracting a pattern at 1.2(1)\,K. Refining the subtracted patterns at 0.07\,K yields two possible spin structures, with spin-ice-like and monopole-like correlations respectively. Mesoscopic correlations involve Tb moments of 1 to 2 \mub\ ordered on a length scale of about 20 Å. In addition, long range order involving a small spin component of 0.1 to 0.2 \mub\ is detected for the $x$= 0 and 0.01 samples showing a peak in the specific heat. Comparison with previous single crystals data suggests that the ($\frac12$, $\frac12$, $\frac12$) order settles in through nanometric spin textures with dominant spin ice character and correlated orientations, analogous to nanomagnetic twins.