Excited state entanglement in one dimensional quantum critical systems: Extensivity and the role of microscopic details

Abstract

We study entanglement via the subsystem purity relative to bipartitions of arbitrary excited states in (1+1)-dimensional conformal field theory, equivalent to the scaling limit of one dimensional quantum critical systems. We compute the exact subpurity as a function of the relative subsystem size for numerous excited states in the Ising and three-state Potts models. We find that it decays exponentially when the system and the subsystem sizes are comparable until a saturation limit is reached near half-partitioning, signaling that excited states are maximally entangled. The exponential behavior translates into extensivity for the second Rényi entropy. Since the coefficient of this linear law depends only on the excitation energy, this result shows an interesting, new relationship between energy and quantum information and elucidates the role of microscopic details.

Figures (2)

• Figure 1: (Logarithmic) plots of the subsystem purities of the first few spin-zero excited states in the Ising universality class ($\Psi=\psi\bar{\psi}$, the legend shows only the chiral generators $\psi$). We organized the plots according to the descendance level of the excited states (and the primaries are not shown). The exponents of the purity in the (first) exponential domain depend on the excitation energy in a nonlinear way. The Rényi entropy $S_{2}=-\log P$ can also be read off these plots. For the level one and two states we also show data from the critical Ising spin chain marked on the plots by points. The agreement is very convincing and serves as a check of the present framework. Note, that the scale is arbitrary depending on the regularization (e.g. system size) and was set so that the exponential decay would begin around $P=1$.
• Figure 2: (Logarithmic) plots of the subsystem purities of the first few spin-zero excited states from the identity and energy towers in the Potts universality class.