Entangled Particles Show Charge Distribution Changes with System Size

Symmetry-resolved entanglement in non-relativistic quantum field theories, including Lifshitz scalar chains and Lifshitz fermionic models, is now understood with greater clarity. M. Reza Mohammadi Mozaffar and Ali Mollabashi at University of Guilan in collaboration with Institute for Research in Fundamental Sciences computed symmetry-resolved Renyi and von Neumann entropies using charged moments and the correlator method. These calculations reveal how entanglement characteristics depend on factors like subsystem size and dynamical exponent z. The study shows that approximate equipartition emerges in large-z Lifshitz scalar theories, while genuine equipartition in Lifshitz fermionic models is limited to the relativistic case, offering key insights for experimental platforms like cold atom setups capable of particle-number-resolved measurements.

Scalar versus fermionic entanglement reveals differing equipartition behaviours

Entanglement measures now reveal approximate equipartition among charge sectors in Lifshitz scalar theories at large dynamical exponent values, z, a phenomenon previously unobservable in non-relativistic quantum field theories. This equipartition refers to the near-equal distribution of entanglement across different charge sectors within the system. The Lifshitz scalar model, characterised by its anisotropic scaling, exhibits this behaviour due to the specific dispersion relation of its constituent particles. The dynamical exponent, z, dictates the scaling of space and time, and a larger z value effectively slows down the dynamics, promoting this approximate equipartition. Contrasting sharply, Lifshitz fermionic models only achieve genuine equipartition, an even distribution of entanglement across charge states, when operating within the relativistic limit, and discerning such distinctions proved impossible until now. Fermions, governed by the Pauli exclusion principle, present a fundamentally different entanglement structure compared to bosons described by scalar fields. The relativistic limit, where the speed of light is effectively infinite within the model, alters the fermionic behaviour, allowing for genuine equipartition. Symmetry-resolved Renyi and von Neumann entropies, used to quantify uncertainty within quantum systems, were computed to map these entanglement features, with configurational entropy dominating in scalar theories and fluctuation entropy prevailing in fermionic models. Renyi entropy, a generalisation of von Neumann entropy, provides a more robust measure of entanglement, particularly in the presence of noise. The dominance of configurational entropy in scalar theories suggests that entanglement is primarily determined by the spatial arrangement of particles, while fluctuation entropy in fermionic models indicates that particle number fluctuations play a more significant role.

Genuine equipartition in Lifshitz fermionic models occurs only when the system behaves relativistically, clearly distinguishing how entanglement distributes across charge states depending on the theory type. This relativistic behaviour arises from the altered energy-momentum relation, allowing for a more uniform distribution of entanglement. Calculations utilising charged moments and the correlator method confirmed that subsystem size, charge, mass, and the value of z all influence symmetry-resolved entanglement. The charged moments provide information about the charge distribution within the entangled state, while the correlator method allows for the calculation of entanglement entropies by examining the correlations between different subsystems. Specifically, increasing the subsystem size generally reduces the entanglement entropy, as the system becomes more classical. The charge and mass parameters influence the entanglement structure by modifying the interactions between particles. These findings are particularly relevant to experimental setups involving cold atoms and mesoscopic systems, where particle numbers can be precisely measured, though translating these theoretical insights into practical quantum technologies still requires a deeper understanding of entanglement dynamics in time-dependent scenarios. Cold atom experiments, utilising trapped ions or neutral atoms, offer a promising platform for simulating these non-relativistic quantum field theories. The ability to control and measure particle numbers with high precision is crucial for verifying the theoretical predictions regarding symmetry-resolved entanglement.

Entanglement distribution benchmarks for non-relativistic quantum field theories

Modelling diverse physical scenarios, from complex materials to ultracold atoms, requires understanding how quantum entanglement behaves in systems lacking relativistic symmetry. Many-body physics, condensed matter physics, and the study of strongly correlated systems all benefit from a detailed understanding of entanglement in non-relativistic regimes. Relativistic effects, while important in high-energy physics, are often negligible in these systems. This careful mapping of entanglement distribution within these non-relativistic quantum field theories reveals subtle differences between Lifshitz scalar and fermionic models. The Lifshitz models themselves are important as they represent a broader class of non-relativistic field theories exhibiting anisotropic scaling, which is relevant to describing systems with emergent low-energy behaviour different from standard relativistic quantum field theories. However, the techniques employed rely on analysing specific Lifshitz models, raising whether these findings generalise to all such systems. Further research is needed to explore the extent to which these results hold for other non-relativistic models with different interactions and symmetries.

Establishing a key benchmark for future research, the detailed mapping of entanglement distribution, a measure of how linked particles are, across these models is now complete. This benchmark allows for a direct comparison of different theoretical approaches and experimental results. Accurate modelling of complex materials and ultracold atoms relies on understanding these nuances, offering insights into systems where precise particle-number measurements are achievable, and providing a foundation for exploring entanglement in real-world experiments. For example, understanding entanglement in strongly correlated materials could lead to the development of novel quantum materials with tailored properties. A detailed connection between conserved quantities and the distribution of quantum entanglement within non-relativistic systems is now established, with symmetry-resolved entropies demonstrating that Lifshitz scalar theories exhibit approximate entanglement equipartition at high dynamical exponents, while Lifshitz fermionic models require relativistic conditions for genuine equipartition. These distinctions, revealed through charged moments and the correlator method, highlight how system dynamics influence entanglement structure. The conservation of charge, a fundamental principle in physics, directly impacts the entanglement distribution, as the total charge within the system remains constant. The symmetry-resolved entropies provide a quantitative measure of how entanglement is distributed among different charge sectors, revealing the underlying structure of entanglement in these non-relativistic systems. Further investigation into the time evolution of these entanglement patterns could unlock new avenues for controlling and manipulating quantum systems.

The research demonstrated distinctions in how entanglement distributes itself within two types of non-relativistic quantum systems: Lifshitz scalar and Lifshitz fermionic models. These models showed differing degrees of entanglement equipartition, a balanced distribution across charge sectors, dependent on the dynamical exponent and whether the system was relativistic. This understanding of symmetry-resolved entanglement, measured using charged moments and the correlator method, is relevant to experimental platforms like cold atom setups where particle-number resolution is possible. The authors suggest further research is needed to determine if these results apply to a wider range of non-relativistic models.