Multilocal Fermionization

Rehren, Karl-Henning; Tedesco, Gennaro

doi:10.1007/s11005-012-0582-5

Cited by 21 publications

(34 citation statements)

References 15 publications

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“…For the fermion field and any number of intervals, or the current field in the single interval case, this same phase factor determines completely the dependence of all eigenvectors in s. In this case the modular flow has a simple geometrical picture as a translation in the variable ω, ω → ω + 2πτ [22]. Perhaps a reason for the fermion to be special is the multilocal symmetries described by Rehren and Tedesco [46].…”

Section: Final Remarksmentioning

confidence: 99%

Entropy and modular Hamiltonian for a free chiral scalar in two intervals

et al. 2018

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We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current j(x) = ∂φ(x) corresponding to a chiral free scalar φ in d = 2. We also compute explicitly the mutual information between the intervals. This model shows a failure of Haag duality for two intervals that translates into a loss of a symmetry property for the mutual information usually associated with modular invariance. Contrary to the case of a free massless fermion, the modular Hamiltonian turns out to be completely non local. The calculation is done diagonalizing the density matrix by computing the eigensystem of a correlator kernel operator. These eigenvectors are obtained by a novel method that involves solving an equivalent problem for an holomorphic function in the complex plane where multiplicative boundary conditions are imposed on the intervals. Using the same technique we also re-derive the free fermion modular Hamiltonian in a more transparent way. arXiv:1809.00026v2 [hep-th]

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Section: Final Remarksmentioning

confidence: 99%

Entropy and modular Hamiltonian for a free chiral scalar in two intervals

et al. 2018

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“…We will find that the multi-interval modular flow can be obtained by gluing together single interval modular flows. In the operator algebra language, this gluing is implemented by a non-local isomorphism between the single interval algebra of n free fermions, and the n-interval algebra of a single fermion, as was first observed in [1]. Our work provides a path integral interpretation of this isomorphism.…”

Section: Introductionmentioning

confidence: 80%

“…Undoing the diagonalization then expresses this coupling in terms of an antisymmetric matrix gauge field [1]:…”

Section: Generalization To Many Intervals and Finite Sizementioning

confidence: 99%

Gluing together modular flows with free fermions

Wong

2019

J. High Energ. Phys.

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We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the modular action. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it's non-local conformal symmetry and U (1) Kacs-Moody symmetry.

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“…Now let K = L 2 (I 1 ) ⊕ L 2 (−I 1 ). Our goal is to have an explicit formula forĵ K : this is based on the formula for the modular operator for disjoint intervals in the free fermion case in [22], [29] and [7].…”

Section: Explicit Formula In a Special Casementioning

confidence: 99%

“…We will start with the following linear isomorphism from H ⊕ H to H which is inspired by Prop. 3 in [29] …”

Section: Explicit Formula In a Special Casementioning

confidence: 99%

Von Neumann Entropy in QFT

Longo

2020

Commun. Math. Phys.

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In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones O ⊂ O of the spacetime, where the closure of O is contained in O. Given a QFT net A of local von Neumann algebras A(O), we consider the von Neumann entropy S A (O, O) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O) ⊂ A( O) (split property). We show that this canonical entanglement entropy S A (O, O ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy S A (O, O), the infimum of the vacuum von Neumann entropy of F , where F here runs over all the intermediate, discrete type I von Neumann algebras. We prove that S A (O, O) is finite for the local chiral conformal net generated by finitely many commuting U (1)-currents.Von Neumann entropy is the basic concept in quantum information and extends the classical Shannon's information entropy notion to the non commutative setting.As is well known, a state ω on a matrix algebra M is given by a density matrix ρ, namely ω(T ) = Tr(ρT ), T ∈ M . The von Neumann entropy of ω is given byEntanglement entropy is a measure of quantum information by the degree of quantum entanglement of a quantum state.Let's consider a bipartite, finite-dimensional quantum system M = A ⊗ B, where A and B are matrix algebras. Given a pure state ω on the matrix algebra M , let ρ A and ρ B the density matrices associated with the restrictions of ω to A and B respectively on A and B. The entanglement entropy of ω is defined as(1)The above definition directly extends to the case M = k A k ⊗ B k is a direct sum, where A k , B k are matrix algebras and the restrictions ω| A k ⊗B k are pure (not normalised). It also extends to the infinite-dimensional case where A k , B k are type I factors; in this last case, however, the entanglement entropy may be infinite. Entanglement is certainly one of the main feature of quantum physics and there is a famous, long standing debate on its interpretation (EPR paradox, Bell's inequalities, etc.). An overview of the matter lies beyond the purpose of this introduction.The role of entanglement in Quantum Field Theory is more recent and increasingly important; it represents a piece of the quantum information framework in this subject. It appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc.

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Multilocal Fermionization

Cited by 21 publications

References 15 publications

Entropy and modular Hamiltonian for a free chiral scalar in two intervals

Entropy and modular Hamiltonian for a free chiral scalar in two intervals

Gluing together modular flows with free fermions

Von Neumann Entropy in QFT

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