Entanglement entropy and quantum field theory

Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A log ρ A corresponding to the reduced density matrix ρ A of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S A ∼ (c/3) log ℓ of Holzhey et al. when A is a finite interval of length ℓ in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals (See note added). For such a system away from its critical point, when the correlation length ξ is large but finite, we show that S A ∼ A(c/6) log ξ, where A is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

Shared information in stationary states at criticality

“…4). The expression ofL = L C (4.3) is the same as in the quantum case and is well understood [1]. The value of γ h (see Table 1) can be inferred from the calculation of Jacobsen and Saleur [19] for the periodic case.…”

Entanglement entropy, trace anomalies and holography

“…In d = 2 CFT universal features of entanglement entropy (EE in the following) were studied in [2]: in particular the dependence of the EE on the scale of the region defining the EE was shown to be related to the trace anomaly of the CFT.…”

“…Following the d = 2 example [2] it was postulated that the EE in flat space is given by the usual partition function when the theory is defined in a metric with a conical singularity supported on a submanifold of codimension 2. The submani-fold is the boundary between the two regions, A and B, defining the EE.…”

“…The paper is organized as follows: In Section 2 we generalize the replica trick of [2] to d = 4 and we deduce an explicit form for the singular metric in the most general case needed for EE.…”

Non-Fermi-liquid d-wave metal phase of strongly interacting electrons

“…We detail these findings in the Supplementary Information, while here we only mention that such Cooper pair correlations have the slowest power law decay among all the discussed observables, including the electron Green's function. This is in stark contrast with the conventional metal and suggests that the d-metal phase has some incipient d-wave superconductivity in two dimensions.As a final piece of "smoking gun" evidence that the realized DMRG phase is in fact the d-metal, we have measured the number of 1D gapless modes, i.e., the effective central charge c, via scaling of the bipartite entanglement entropy 46,47 in the DMRG and VMC 48,49 wave functions. As explained above, we expect c = 2 in the conventional Luttinger liquid and c = 3 in the d-metal.…”