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arXiv:1907.00646 · Journal of High Energy Physics

Correlation function of modular Hamiltonians

Jiang Long

Published Nov 1, 2019• 13 citations
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Abstract

Abstract We investigate varies correlation functions of modular Hamiltonians defined with respect to spatial regions in quantum field theory. These correlation functions are divergent in general. We extract finite correlators by removing divergent terms for two dimensional massless free scalar theory. We reproduce the same correlators in general two dimensional conformal field theories.

DOI
10.1007/jhep11(2019)163

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Abstract We investigate varies correlation functions of modular Hamiltonians defined with respect to spatial regions in quantum field theory. These correlation functions are divergent in general. We extract finite correlators by removing divergent terms for two dimensional massless free scalar theory. We reproduce the same correlators in general two dimensional conformal field theories.
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We study (m)(m)(m)-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find logarithmic divergence for these correlation functions. We justify the logarithmic behaviour from three different approaches: massless free scalar theory, Selberg integral and conformal block. Cutoff independent coefficients are obtained from analytic continuation of conformal blocks. A UV/IR relation has been found in connected correlation functions. We could derive a formal ``first law of thermodynamics'' for a subsystem using deformed reduced density matrix. Area law of connected correlation function in higher dimensions is also discussed briefly.
We study the (m)-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find a logarithmic divergence for these correlation functions. We justify the logarithmic behaviour from two different approaches: massless free scalar theory and conformal block. Cutoff independent coefficients are obtained from analytic continuation of conformal blocks. A UV/IR relation has been found in the connected correlation functions. We could derive a formal “first law of thermodynamics” for a subsystem using the deformed reduced density matrix. An area law of the connected correlation function in higher dimensions is also discussed briefly.
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Abstract We study the two point correlation function of a local operator on an n-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable w, dual to the Renyi parameter n. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform f (w) has bounded behavior at infinity in the complex w plane, the discontinuity of the Renyi transform disc f (w) provides the unique analytic continuation in n of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large N holographic CFTd in dimensions d > 2. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in 2d conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in 2d, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.