Giant number-parity effect and scalable spin squeezing in Luttinger liquids

Finite-size quantum spin systems can be magnetized by the application of a symmetry-breaking field, but in general their symmetry is expected to be restored once the field is turned off adiabatically. Recently (F. Caleca et al., arXiv:2412.15493) we have shown that systems of half-integer spins with an odd number of sites and a parity-preserving Hamiltonian can retain a finite magnetization, hence exhibiting spontaneous symmetry breaking (SSB) at finite size. Here we generalize this phenomenon to spin chains whose low-energy physics (in zero field) realizes a Luttinger-liquid phase. We observe that odd-sized chains can exhibit a phenomenon of finite-size quasi-SSB, in which a net sub-extensive magnetization, $M \sim N^{1-1/(4K)}$ is retained, where $N$ is the number of sites and $K$ the Luttinger exponent. Interestingly, the states prepared by turning off the symmetry-breaking field quasi-adiabatically display scalable spin squeezing -- namely stronger the bigger the system -- regardless of the parity of $N$. The scaling of the squeezing parameter is dictated again by the Luttinger exponent, $\xi_R^2 \sim N^{-1+1/(2K)}$. This result shows that scalable quantum correlations with metrological significance, associated typically with high-dimensional systems, can be found as well in gapless one-dimensional ones; and they are a direct consequence of the critical nature of Luttinger liquids.