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

TL;DR

This work demonstrates that one-dimensional spin-1/2 chains in a Luttinger-liquid phase exhibit a giant number-parity effect: odd N yields a subextensive residual magnetization due to Kramers degeneracy, enabling quasi-spontaneous symmetry breaking at finite size. By quasi-adiabatically ramping a symmetry-breaking field and exploiting parity conservation, the authors prepare states with scalable spin squeezing, with the squeezing scaling governed by the Luttinger parameter K as ξ_R^2 ∝ N^{-1+1/(2K)}. The results, supported by DMRG and TDVP simulations across several short-range models, reveal that gapless 1D systems can host metrologically useful multipartite entanglement without true long-range order. The findings extend the landscape of scalable quantum correlations to gapless many-body systems and provide a concrete route to harnessing LL criticality for quantum-enhanced metrology in realistic platforms.

Abstract

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, $ξ_R^2 \sim N^{-1+1/(2K)}$. 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.

Figures (6)

• Figure 1: Giant number-parity effect in Luttinger liquids. (a)-(c) Evolution of the magnetization along an exponential field ramp $\Omega_0 e^{-t/\tau}$ for $N=30,\ 31$. Symbols correspond to the dynamical results, while the dashed line are the ground-state values for positive parity. In all panels the ramp parameters are $\tau=N/10$ and $\Omega_0=20\mathcal{J}$.
• Figure 2: Giant number-parity effect in Luttinger liquids. (a)-(c) Residual magnetization in odd-sized lattices as a function of the system size; shaded regions correspond to the upper and lower bounds defined by Eq. \ref{['eq:Jx_bound']}.
• Figure 3: Scalable spin squeezing in Luttinger liquids, with different panels corresponding to the XXZ chain with $\Delta = -0.5$ (a), the XX chain (b) and the dipolar XX model (c). Triangles correspond to the spin squeezing parameter in the $|+\rangle$ state (with odd $N$) calculated by means of DMRG after fixing the parity sector. We compare it with the minimal spin squeezing obtained in the ground state at finite $\Omega$ field, with circles (crosses) corresponding to odd (even) $N$. In all panels, solid lines correspond to the predictions of Eq. \ref{['eq:xi_vs_N']} with the $c$ prefactor adjusted so as to match the squeezing parameter for the smallest system size.
• Figure 4: Scalings of (a)-(c) magnetization $\langle J^x\rangle/N$, (d)-(f) transverse variance $\text{Var}(J^z)/N$ and (g)-(i) spin squeezing parameter $\xi_R^2$ against the applied field $\Omega$. Different colors indicate different models, with the same color code used in the previous figures. Continuous lines indicate numerical results as obtained by DMRG for system sizes $N\in[20,\dots,37]$. Dotted lines represent the prediction from the theory of critical phenomena (see text).
• Figure 5: Scalings of (a)-(c) magnetization $\langle J^x\rangle/N$, (d)-(f) transverse variance $\text{Var}(J^z)/N$ and (g)-(i) spin squeezing parameter $\xi_R^2$ against the $\Omega$ field as obtained along the field ramps discussed in Sec. \ref{['sec:dynamic_preparation']}. Different colors indicate different models, with the same color code used in the previous figures. Continuous lines indicate numerical results for system sizes $N\in[25,\dots,41]$. Dotted lines represent the prediction from the theory of critical phenomena.