Quantum entanglement in condensed matter systems

Abstract: This review focuses on the field of quantum entanglement applied to condensed matter physics systems with strong correlations, a domain which has rapidly grown over the last decade. By tracing out part of the degrees of freedom of correlated quantum systems, useful and non-trivial informations can be obtained through the study of the reduced density matrix, whose eigenvalue spectrum (the entanglement spectrum) and the associated R\'enyi entropies are now well recognized to contains key features. In particular, the celebrated area law for the entanglement entropy of ground-states will be discussed from the perspective of its subleading corrections which encode universal details of various quantum states of matter, e.g. symmetry breaking states or topological order. Going beyond entropies, the study of the low-lying part of the entanglement spectrum also allows to diagnose topological properties or give a direct access to the excitation spectrum of the edges, and may also raise significant questions about the underlying entanglement Hamiltonian. All these powerful tools can be further applied to shed some light on disordered quantum systems where impurity/disorder can conspire with quantum fluctuations to induce non-trivial effects. Disordered quantum spin systems, the Kondo effect, or the many-body localization problem, which have all been successfully (re)visited through the prism of quantum entanglement, will be discussed in details. Finally, the issue of experimental access to entanglement measurement will be addressed, together with its most recent developments.

• The paper elucidates the area law and entanglement entropy in strongly correlated systems by highlighting subleading corrections that indicate distinct quantum phases.
• It demonstrates that the entanglement spectrum acts as a diagnostic tool, linking low-lying excitations to topological orders and critical phenomena.
• The paper also addresses experimental approaches and challenges in measuring quantum entanglement, particularly in disordered and many-body localized systems.

Quantum Entanglement in Condensed Matter Systems

This essay provides an expert overview of Nicolas Laflorencie's review on the role of quantum entanglement in condensed matter physics, particularly in systems with strong correlations. The analysis centers on the insights derived from the reduced density matrix (RDM) and the entanglement spectrum, offering a deeper understanding of different states of matter and the effects of disorders.

Key Insights from the Review

1. Area Law and Entanglement Entropy: The paper elaborates on the area law for ground-state entanglement entropy, widely applicable in one-dimensional (1D) and higher-dimensional systems. The study highlights the importance of subleading corrections that encode universal features of symmetry-breaking states and topological orders. For example, the area law in 1D systems includes universal corrections related to the Luttinger liquid parameter.
2. Entanglement Spectrum (ES): Beyond the entropy, the ES provides a powerful diagnostic tool for understanding quantum phases. In systems like quantum spin chains and ladders, the low-lying part of the ES corresponds to the energy spectrum, revealing universal characteristics relevant in diagnosing topological properties and critical phenomena.
3. Disordered Systems and Infinite Randomness: The review details how quantum entanglement serves as a lens to understand disordered systems, such as quantum spin systems with impurity and many-body localization (MBL) issues. For instance, in the random singlet phase, the disorder-average von Neumann entropy showcases a logarithmic growth associated with infinite randomness fixed points.
4. Experimental Implications: Laflorencie emphasizes the challenge and progress in measuring quantum entanglement in experimental settings, such as quasi-2D quantum systems. Techniques leveraging quantum noise, bipartite fluctuations, and newly developed protocols in ultra-cold atomic systems are promising avenues for practical entanglement detection.
5. Future Directions and Open Questions: The essay concludes by identifying open questions, such as the universality of entanglement spectra and understanding entanglement in non-Fermi liquid phases. Additionally, prospects in computational techniques and the study of ergodicity breaking in high-energy states of interacting systems are highlighted as potential future developments.

In summary, this review anchors the quantum entanglement landscape as a fundamental construct in understanding condensed matter systems, providing robust theoretical and numerical techniques that unravel the complexities of correlations, disorder, and experimental realizations.