Equipartition of the entanglement entropy

Abstract

The entanglement in a quantum system that possess an internal symmetry, characterized by the S z -magnetization or U (1)-charge, is distributed among different sectors. The aim of this letter is to gain a deeper understanding of the contribution to the entanglement entropy in each of those sectors for the ground state of conformal invariant critical one dimensional systems. We find surprisingly that the entanglement entropy is equally distributed among the different magnetization sectors. Its value is given by the standard area law violating logarithmic term, that depends on the central charge c, minus a double logarithmic correction related to the zero temperature susceptibility. This result provides a new method to estimate simultaneously the central charge c and the critical exponents of U (1)-symmetric quantum chains. The method is numerically simple and gives precise results for the spin-1 2 quantum XXZ chain. We also compute the probability distribution of the magnetization in contiguous sublattices.

Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach

“…they exactly satisfy the equipartition of entanglement for any value of ∆. In the critical case, only the leading terms satisfy such equipartition [30,33].…”

“…First of all, there is a very important difference compared to the conformal gapless case [28], i.e. the absence of equipartition of entanglement [30]: the Rényi entropies (63) depend explicitly on q. This dependence is explicitly reported in Figure 2 (a) where, in order to show its variation, we plot it as a continuous function of q, although only integer values are physical.…”

Symmetry resolved entanglement: exact results in 1D and beyond

“…which means that ρ A is block-diagonal with respect to the eigenbasis ofQ A . In such a representation, each block (charge sector) corresponds to an eigenvalue Q A ofQ A , and we can therefore denote this block by ρ (Q A ) A , and define for each such eigenvalue [19,20,21,22,23] S n (Q A ) = Tr ρ…”

“…It is evident that these quantities satisfy S = Q A S (Q A ) and S n = Q A S n (Q A ). Note that some works normalize each block by each trace [21,22,23] before calculating the entropies, which thus quantify the entanglement after a projective charge measurement. We prefer not to do so and instead use (6), following [19,20], because the resulting entropies are not only more accessible to calculations, but are also directly experimentally measurable, using either the replica trick [20,24,25], or random time evolution which conserves the charge [26,27].…”

“…The flux-resolved and charge-resolved REE have previously been approximately calculated for 1D many-body systems using conformal field theory (CFT) and numerical techniques [19,20,21,22,23]. The flux-resolved REE has an analog for discrete symmetries, i.e., when the quantity conserved is Q mod p where p is some natural number (e.g., fermion parity for p = 2) [20].…”