Entanglement negativity in the harmonic chain out of equilibrium

Abstract

We study the entanglement in a chain of harmonic oscillators driven out of equilibrium by preparing the two sides of the system at different temperatures, and subsequently joining them together. The steady state is constructed explicitly and the logarithmic negativity is calculated between two adjacent segments of the chain. We find that, for low temperatures, the steady-state entanglement is a sum of contributions pertaining to left-and right-moving excitations emitted from the two reservoirs. In turn, the steady-state entanglement is a simple average of the Gibbs-state values and thus its scaling can be obtained from conformal field theory. A similar averaging behaviour is observed during the entire time evolution. As a particular case, we also discuss a local quench where both sides of the chain are initialized in their respective ground states.

Entanglement scrambling in 2d conformal field theory

“…Similarly, we expect our observations on light cone singularities to be important in the computation of the entanglement negativity of two disjoint intervals A, B after a global or local quench [23,[74][75][76], which in [23,76] was found to be exactly 3/4 of the quasiparticle mutual information I(A, B). …”

Entanglement negativity after a local quantum quench in conformal field theories

“…The entanglement negativity for a harmonic chain has been numerically studied in several works [9,22,26,27,39]. Here we follow the method developed in these works, and apply it to the local quench problem.…”

“…At t = 0, two disconnected chains are joined together as one harmonic chain with number of sites 2N under the Dirichlet boundary condition (similar to Ref. [27]). …”

“…Most recently, finite temperature effects on the entanglement negativity in conformal field theories were studied [41]. In particular, finite temperature effects on a local quench of the entanglement negativity were studied numerically based on a critical harmonic chain [27]. It is interesting to generalize our method to the finite temperature case, and have an analytical picture of the finite temperature effects on a local quench of the negativity.…”

Topological entanglement negativity in Chern-Simons theories

“…Recently, the entanglement negativity has been extensively studied in conformal field theories [11][12][13], quantum spin chain systems [14][15][16][17][18][19][20], coupled harmonic oscillators in one and two dimensions [21][22][23][24][25][26][27], free fermion systems [28][29][30][31][32], topological ordered systems [33][34][35], and holographic entanglement [36][37][38]. Furthermore, the entanglement negativity has JHEP09(2016)012 also been studied in the non-equilibrium case [39][40][41][42] as well as the finite temperature case [39,43,44].…”