Quantum impurity entanglement

Entanglement in J₁–J₂, S = 1/2 quantum spin chains with an impurity is studied using analytic methods as well as large scale numerical density matrix renormalization group methods. The entanglement is investigated in terms of the von Neumann entropy, S = −Trρ_Alogρ_A, for a subsystem A of size r of the chain. The impurity contribution to the uniform part of the entanglement entropy, S_imp, is defined and analysed in detail in both the gapless, J₂≤J₂^c, as well as the dimerized phase, J₂>J₂^c, of the model. This quantum impurity model is in the universality class of the single channel Kondo model and it is shown that in a quite universal way the presence of the impurity in the gapless phase, J₂≤J₂^c, gives rise to a large length scale, ξ_K, associated with the screening of the impurity, the size of the Kondo screening cloud. The universality of Kondo physics then implies scaling of the form S_imp(r/ξ_K,r/R) for a system of size R. Numerical results are presented clearly demonstrating this scaling. At the critical point, J₂^c, an analytic approach based on a Fermi liquid picture, valid at distances and energy scales , is developed and analytic results at T = 0 are obtained showing S_imp = πξ_K[1+π(1−r/R)cot(πr/R)]/(12R) for finite R. For T>0, in the thermodynamic limit, we find S_imp = [π²ξ_KT/(6v)]coth(2πrT/v), with v the spin-wave velocity. In the dimerized phase an appealing picture of the entanglement is developed in terms of a thin soliton (TS) ansatz and the notions of impurity valence bonds (IVB) and single particle entanglement (SPE) are introduced. The TS-ansatz permits a variational calculation of the complete entanglement in the dimerized phase that appears to be exact in the thermodynamic limit at the Majumdar–Ghosh point, J₂ = J₁/2, and surprisingly precise even close to the critical point J₂^c. In the appendices the TS-ansatz is further used to calculate and with high precision at the Majumdar–Ghosh point and the relation between the finite temperature entanglement entropy, S(T), and the thermal entropy, S_th(T), is discussed. Finally, the alternating part of S_imp is discussed, together with its relation to the boundary induced dimerization.