Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems

Abstract

Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement. arXiv:1712.03557v2 [cond-mat.stat-mech] 21 Dec 2017 Entanglement hamiltonian and contour in inhomogeneous 1D critical systems 2

On entanglement Hamiltonians of an interval in massless harmonic chains

“…These sums can be organised in different ways. For instance, by writing the symmetric matrices M and N as sums of a diagonal matrix, an upper triangular matrix and a lower triangular matrix, it is straightforward to obtain (14)…”

“…The quadratic operators H M and H N can be decomposed in different ways, as discussed in §2 and in Appendix A. For the sake of simplicity, in the following we describe the continuum limit for the decomposition given by (14) and (15), but the procedure can be easily adapted to the ones given by (16) and (17) or by (18) and (19). In Appendix A we discuss another decomposition, inspired by the numerical analysis performed in [22].…”

“…-0.0002 0.0000 Figure 2: Diagonals of the matrix M (see (20)) when the subsystem is an interval made by L sites in the infinite line and ωL = 10 −500 . The red solid curve is the parabola andp i+k in (14) and (15) lead to fields whose argument is properly shifted. By employing (29) and the Taylor expansion as a → 0, in the continuum limit it is straightforward to obtain thatq…”

“…)dx in the continuum limit and the divergent factor L provides the factor . From (29), (30) and (31), for the operators (14) and (15) it is straightforward to obtain H M −→ H M and H N −→ H N respectively in the continuum limit, where…”

“…As for the combinations (36) and (46), we find that they lead to the function that vanishes identically in the interval, as expected from CFT. The range of the index i in these combinations do not allow to capture the curve predicted by CFT on the entire interval and this fact motivated us to employ the decompositions (18) and (19) instead of (14) and (15).…”