Luttinger Liquids Exhibit Finite-Size Quasi-Symmetry Breaking with Sub-Extensive Magnetization in Odd-Sized Chains

The peculiar behaviour of magnetism in one-dimensional systems challenges conventional wisdom, and recent work by Filippo Caleca, Saverio Bocini, Fabio Mezzacapo, and Tommaso Roscilde from ENS de Lyon and Université Lyon 1 demonstrates a surprising new effect in these materials. The team discovered that chains of interacting spins, behaving as ‘Luttinger liquids’, can retain a measurable magnetization even after the external field that initially aligned them is removed, a phenomenon known as finite-size quasi-spontaneous symmetry breaking. This is particularly remarkable because such behaviour is usually associated with larger, more complex systems, and the researchers further show that these chains exhibit ‘spin squeezing’, a state where quantum fluctuations are reduced, that becomes more pronounced as the system grows larger. This achievement reveals that scalable quantum correlations, previously thought to require high-dimensional materials, can emerge naturally within the critical physics of one-dimensional Luttinger liquids, opening new avenues for exploring quantum phenomena in simplified systems.

Number Parity and Finite Quantum Systems

Researchers investigate the behaviour of quantum many-body systems, particularly focusing on spin models and the subtle effects of number parity symmetry breaking. This symmetry, concerning the conservation of particle number, can unexpectedly break down in finite-size quantum systems, leading to observable consequences. The team highlights a significant enhancement of this symmetry breaking, termed a giant number parity effect, which becomes particularly pronounced in certain spin models and influences quantum phase transitions. The research demonstrates that number parity symmetry breaking can act as an indicator of phase transitions, functioning as an order parameter that changes value as the system transitions between different states.

Scientists employ advanced computational techniques, including Quantum Monte Carlo, Dynamical Mean Field Theory, and Matrix Product States, to explore these phenomena. A key finding is that the expectation value of the number parity operator can effectively signal a change in the system’s quantum state. The study suggests a potential link between number parity symmetry breaking and the emergence of topological phases of matter, exotic states with unique properties. Researchers explore the universality of these effects across different spin models and system parameters, seeking to understand if the observed behaviour is consistent regardless of specific details. They also identify a dynamic signature of the symmetry breaking, a measurable change in the system’s behaviour over time, and connect their findings to experimental systems like trapped ions and Rydberg atom arrays, suggesting potential avenues for observation.

Luttinger Liquids and Spontaneous Magnetization Retention

Researchers have discovered that finite-size spin systems can surprisingly retain magnetization even after the removal of an applied field, a phenomenon known as spontaneous symmetry breaking. Extending previous work on systems with an odd number of spin sites, this study demonstrates the same effect in spin chains exhibiting Luttinger-liquid behaviour, a characteristic of gapless one-dimensional systems. Scientists employed time-dependent Hamiltonian simulations, utilizing matrix-product states to model the systems and observe their evolution. The experimental protocol involved applying a time-varying field coupled to the magnetization, and then slowly reducing it to observe the system’s behaviour.

Researchers used exponential field ramps, carefully controlling the ramp time to promote adiabaticity, ensuring the system prepared in a magnetized ground state with conserved parity. By simulating the time evolution, scientists analysed the resulting magnetization to determine if it retained a finite value after the field was removed. Simulations on systems ranging from 30 to 31 spins, across three different models exhibiting Luttinger-liquid phases, revealed a clear distinction. Odd-sized systems retained a finite magnetization, while even-sized systems exhibited oscillating behaviour around zero. Further analysis using the Density Matrix Renormalization Group (DMRG) studied the scaling of residual magnetization with system size, ranging from 25 to 71 spins, confirming quasi-spontaneous symmetry breaking and scalable correlations.

Finite-Size Magnetization and Scaling in Spin Chains

Scientists have discovered that odd-sized spin chains can retain a net magnetization even after the symmetry-breaking field is removed, a phenomenon of finite-size quasi-spontaneous symmetry breaking. The magnitude of this retained magnetization scales with the number of sites and the Luttinger exponent, a value characterizing the system’s low-energy physics. For the XX chain, the residual magnetization is calculated as a constant multiplied by the inverse fourth root of the number of sites, confirming a precise scaling relationship between magnetization and system size. Beyond retaining magnetization, the research reveals that these systems exhibit scalable spin squeezing, meaning the degree of entanglement between spins increases with system size.

Measurements confirm that the minimum variance of the collective spin component scales inversely with system size, indicating an increasing entanglement depth as the number of entangled spins grows. For example, in the XXZ chain, the team observed a decreasing squeezing parameter with system size, confirming the predicted scaling. The team extended this analysis to even-sized systems, discovering that applying a small external field enables scalable spin squeezing, consistent with the findings for odd-sized chains. Numerical simulations, performed on systems ranging from 25 to 71 spins, demonstrate that the theoretical predictions accurately match the observed behaviour across different models, establishing a clear connection between the critical nature of Luttinger liquids and the emergence of scalable correlations and entanglement in one-dimensional quantum spin chains.

Persistent Magnetization and Scalable Spin Squeezing

This research demonstrates that systems of half-integer spins, specifically spin chains, can exhibit a surprising degree of persistent magnetization even after a symmetry-breaking field is removed. Systems with an odd number of sites retain a finite magnetization, a phenomenon termed finite-size quasi-spontaneous symmetry breaking. The magnitude of this retained magnetization scales with the number of sites and the Luttinger parameter, a value characterizing the system’s low-energy physics. Importantly, the team found that these systems also display scalable spin squeezing, meaning the degree of entanglement between spins increases with system size, regardless of whether the number of sites is odd or even.

The strength of this squeezing is again dictated by the Luttinger parameter, highlighting its central role in governing the system’s properties. This finding is significant because scalable correlations are typically associated with higher-dimensional systems, yet this work demonstrates their presence in one-dimensional, gapless systems like Luttinger liquids. Future research could investigate the robustness of these findings to disorder or external perturbations, and explore potential applications of this enhanced entanglement in quantum technologies.