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arXiv:2208.14118 · Journal of High Energy Physics

Rényi entropy and negativity for massless complex boson at conformal interfaces and junctions

Luca Capizzi, Sara Murciano, Pasquale Calabrese

Published Nov 18, 2022• 11 citations
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Abstract

Abstract We consider the ground state of a theory composed byMspecies of massless complex bosons in one dimension coupled together via a conformal interface. We compute both the Rényi entropy and the negativity of a generic partition of wires, generalizing the approach developed in a recent work for free fermions. These entanglement measures show a logarithmic growth with the system size, and the universal prefactor depends both on the details of the interface and the bipartition. We test our analytical predictions against exact numerical results for the harmonic chain.

DOI
10.1007/jhep11(2022)105

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Abstract We consider the ground state of a theory composed byMspecies of massless complex bosons in one dimension coupled together via a conformal interface. We compute both the Rényi entropy and the negativity of a generic partition of wires, generalizing the approach developed in a recent work for free fermions. These entanglement measures show a logarithmic growth with the system size, and the universal prefactor depends both on the details of the interface and the bipartition. We test our analytical predictions against exact numerical results for the harmonic chain.
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We investigate the second Rényi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second Rényi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second Rényi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual Rényi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
We investigate the second R\'enyi entropy of two intervals in the massless Thirring model describing a self-interacting Dirac fermion in two dimensions. Boson-fermion duality relating this model to a free compact boson theory enables us to simplify the calculation of the second R\'enyi entropy, reducing it to the evaluation of the partition functions of the bosonic theory on a torus. We derive exact results on the second R\'enyi entropy, and examine the dependence on the sizes of the intervals and the coupling constant of the model both analytically and numerically. We also explore the mutual R\'enyi information, a measure quantifying the correlation between the two intervals, and find that it generally increases as the coupling constant of the Thirring model becomes larger.
Abstract In this paper, we consider the computation of charged moments of the reduced density matrix of two disjoint intervals in the 1+1 dimensional free compactified boson conformal field theory (CFT) by studying the four-point function of the fluxed twist fields. We obtained the exact scaling function of this four-point function and discussed its decompactification limit. This scaling function was used to obtain the charged moments of the partial transpose which we refer as charged Rényi negativity. These charged moments and the charged moments of the partial transpose are essential for the problem of symmetry decomposition of the corresponding entanglement measures. We test our analytic formula against exact numerical computation in the complex harmonic chain, finding perfect agreements.
Abstract We consider the ground state of two species of one-dimensional critical free theories coupled together via a conformal interface. They have an internalU(1) global symmetry and we investigate the quantum fluctuations of the total charge on one side of the interface, giving analytical predictions for the full counting statistics, the charged moments of the reduced density matrix and the symmetry resolved Rényi entropies. Our approach is based on the relation between the geometry with the defect and the homogeneous one, and it provides a way to characterize the spectral properties of the correlation functions restricted to one of the two species. Our analytical predictions are tested numerically, finding a perfect agreement.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.