Mutual Information and Boson Radius in ac=1Critical System in One Dimension

Abstract

We study the generic scaling properties of the mutual information between two disjoint intervals, in a class of one-dimensional quantum critical systems described by the c = 1 bosonic field theory. A numerical analysis of a spin-chain model reveals that the mutual information is scale-invariant and depends directly on the boson radius. We interpret the results in terms of correlation functions of branch-point twist fields. The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge. Given a microscopic model, an important and often nontrivial issue is how to obtain the effective field theory controlling its long-distance behavior. The notion of quantum entanglement, or more specifically, the entanglement entropy, has been extensively applied as a new way to address this basic matter. From a quantum ground state |Ψ , one constructs the reduced density matrix ρ A := TrĀ |Ψ Ψ| on a subsystem A by tracing out the exteriorĀ. The entanglement entropy is defined as S A := −Tr ρ A log ρ A . In 1D quantum critical systems, the entanglement entropy for an interval A = [x 1 , x 2 ] embedded in a chain exhibits a universal scaling [6,7,8,9,10,11]:where c is the central charge of the CFT and s 1 is a nonuniversal constant related to the ultra-violet (UV) cutoff. This scaling allows to determine the universal number c as a representative of the ground state structure, without having to worry about the precise correspondence between the microscopic model and the field theory.As it is well known, the central charge is not the only important number specifying a CFT. In the bosonic field theory with c = 1, the boson compactification radius R (or equivalently, the TLL parameter K = 1/(4πR 2 )) is a dimensionless parameter which changes continuously in a phase and controls the power-law behavior of various physical quantities. It is natural to ask how to identify the boson radius as a generic structure of the ground state. In this Letter, we demonstrate that the entanglement entropy can achieve this task if we consider two disjoint intervals, A = [x 1 , x 2 ] and B = [x 3 , x 4 ]. We analyze the scaling of the mutual information defined asThis measures the amount of information shared by two subsystems [12,13]. A numerical analysis of a spin-chain model reveals a robust relation between I A:B and R, irrespective of microscopic details. We compare the result with the general prediction of Calabrese and Cardy (CC) [9], and find a relevant correction to their result. Roughly speaking, the mutual information (2) may be regarded as a region-region correlator. It is known that I A:B is non-negative, and becomes zero iff ρ A∪B = ρ A ⊗ ρ B , i.e., in a situation of no correlation [14]. A motivation to consider I A:B comes from that microscopic details at short-range scales, which are often obstacles when analyzing point-point correlators, can be smoothed out over regions. As we enlar...

Negativity spectrum in 1D gapped phases of matter

“…To proceed, it is convenient to first derive the asymptotic behaviour of the combinations q + ± q − . From (14), one has that the generating function of (q + + q − )/2 is G 2 (x 2 )G 2 (x), On the other hand, one has that the generating function for (q + − q − )/2 is which is the result reported in the main text (cf. (34)).…”

“…The sum is easily done using (14). Using also the explicit value of ζ 0 (17), the logarithmic negativity of two adjacent intervals, in the thermodynamic limit, is…”

“…The terms G 2 (x 2 ) and G 2 (x) in (14) correspond to µ i µ j and √ µ k µ l in (8), respectively. The second term in the square bracket in (14) properly counts the diagonal term with k = l in (8). A nontrivial consequence of (14) is that q + (n) = q − (n) for n odd, as obvious from Figure 4.…”

Bipartite fluctuations as a probe of many-body entanglement

“…It was originally predicted in Ref. [7] that for a free boson the entanglement entropy is given by [103,104] that this is not quite correct, due to the fact that for disjoint intervals the topology of the Riemann surface used in the derivation is highly non-trivial, i.e., non-local. It is interesting that, as we show below, the particle number fluctuations within LL theory, i.e., for a free boson, retains the simpler form of Eq.…”

“…For the entanglement entropy, there has been growing interest in the case where the subsystem A is composed of disjoint intervals [7,103,104]. In particular, consider the case where A extends from x 1 to x 2 and also x 3 to x 4 (we consider PBCs once more).…”