Abstract
We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal O(1) term γ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.
Highlights
Most of the above developments concern entanglement between spatially separated subsystems
The present work proposes a characterisation of the Renormalisation Group (RG) flow in 1+1 dimensional quantum field theories through a quantification of the entanglement between left and right moving excitations, which are decoupled in the ultraviolet limit and become entangled during the flow to the infrared
In relativistic quantum field theories, scale invariance is generally promoted to conformal symmetry [24, 25], which is extended to an infinite dimensional symmetry in the 1+1 dimensional case [26]. 1+1 dimensional Conformal Field Theories (CFTs) obey holomorphic factorisation: the excitation spectrum consists of left (l) and right (r) movers which transform under a separate chiral symmetry algebra and do not interact
Summary
The QFT Hamiltonian on the cylinder associated to the Euclidean action in eq (1.1) is. Where the relevant scalar field φ has conformal dimension ∆φ ≡ 2hφ at the UV fixed point (see eq (A.2)). The IR regime is reached for mL 1, which is equivalent to either increasing the volume L or the coupling constant λ. The UV fixed point is a CFT with central charge c and a torus partition function which is assumed to correspond to a diagonal modular invariant. In finite volume any eigenstate of the QFT Hamiltonian can be expanded in the eigenstates of HCFT which form the so-called conformal basis (cf appendix A)
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