Spontaneous symmetry breaking in finite size systems due to parity effects in U(1) - symmetric quantum spin models

Spontaneous symmetry breaking (SSB) is traditionally understood as a phenomenon that arises due to the non-invertibility of two limits: the thermodynamic limit and the external symmetry-breaking field going to zero. Typically, SSB is observed only when the number of degrees of freedom N becomes large, as fluctuations in finite size systems tend to restore the symmetry. However, we demonstrate that a parity effect in U(1)-symmetric quantum spin models leads to SSB even at finite sizes. Specifically, due to a doubly-degenerate ground state emerging for odd number of spins, symmetry is not restored during the real-time evolution of a closed quantum system: starting from a fully polarized state with a high external field and slowly reducing the field to zero, the magnetization remains non-zero, defying the expectation that symmetry should be restored. We investigate this phenomenon in both two-dimensional systems, which exhibit long-range order (LRO), and one-dimensional systems, which display quasi-long-range order (QLRO). Using a variety of numerical and analytical techniques, we show that while the residual magnetization persists in 2d systems as the thermodynamic limit is approached, as expected, it vanishes in 1d systems, in line with the behavior of systems with QLRO. Finally, we demonstrate that this mechanism enables scalable spin squeezing in both one-dimensional and two-dimensional quantum spin systems, achieving Heisenberg-limited scaling in systems with ground states that exhibit long-range order.