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arXiv:2607.04268

Entanglement of excited states after measurements in conformal field theory

Hui-Huang Chen

Published Jul 5, 2026• 0 citations
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Abstract

We study the entanglement of low-energy excited states after a fixed-outcome projective measurement on a spatial interval in a (1+1)-dimensional conformal field theory (CFT). We restrict to post-selected measurement outcomes whose induced boundary condition flows in the infrared to a conformal boundary condition. Such an outcome is represented by a slit carrying that boundary condition, while excited states are introduced by operator insertions in the Euclidean path integral. After mapping the replicated slit geometry to a disk, the excited-state contribution to the post-measurement Rényi entropy is expressed as a normalized boundary-CFT correlation function. We apply this framework to the compact free boson CFT. For the chiral current excitation, the relevant ratios are given by current hafnians. We also study coherent superpositions of $J$ and $\bar J$, and of conjugate compact vertex operators. In the conjugate-vertex case, the ordinary-cylinder second Rényi ratio is independent of the relative phase, whereas the finite-slit post-measurement ratio contains phase-sensitive interference terms. Finally, we describe free-fermion and multi-Slater determinant methods for testing these predictions in the critical XX chain.

DOI
10.48550/arxiv.2607.04268

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The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a 2D conformal field theory~(CFT), the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum, where one of the boundary conditions arise due to the `entanglement cut'. The latter has been argued to be conformal and has been numerically found to be the `free' boundary condition for Ising, Potts and free boson theories. For these models, the `free' boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary minimal models of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the entanglement entropy from the CFT partition functions. These are compared with numerical results.