Nicolas Laflorencie

Quantum entanglement in condensed matter systems

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 Renyi 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.

Boundary Effects in the Critical Scaling of Entanglement Entropy in 1D Systems

We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.

Entanglement entropy in quantum impurity systems and systems with boundaries

We review research on a number of situations where a quantum impurity or a physical boundary has an interesting effect on entanglement entropy. Our focus is mainly on impurity entanglement as it occurs in one-dimensional systems with a single impurity or a boundary, in particular quantum spin models, but generalizations to higher dimensions are also reviewed. Recent advances in the understanding of impurity entanglement as it occurs in the spin-boson and Kondo impurity models are discussed along with the influence of boundaries. Particular attention is paid to (1 + 1)-dimensional models where analytical results can be obtained for the case of conformally invariant boundary conditions and a connection to topological entanglement entropy is made. New results for the entanglement in systems with mixed boundary conditions are presented. Analytical results for the entanglement entropy obtained from Fermi liquid theory are also discussed as well as several different recent definitions of the impurity contribution to the entanglement entropy.

Extended slow dynamical regime close to the many-body localization transition

Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L=28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth

Bipartite fluctuations as a probe of many-body entanglement

S(t)propto t^{1/z}

Quantum impurity entanglement

with a disorder-dependent exponent

Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

zge1

Scaling of entanglement entropy in the random singlet phase

, in contrast with the pure ballistic growth

Impurity entanglement entropy and the Kondo screening cloud

z=1

Simulations of pure and doped low-dimensional spin-1/2 gapped systems

of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance