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arXiv:2301.00294 · Journal of High Energy Physics

Modular Hamiltonian of the scalar in the semi infinite line: dimensional reduction for spherically symmetric regions

Marina Huerta, Guido van der Velde

Published Jun 16, 2023• 12 citations
arXivHTMLPDF

Abstract

Abstract We focus our attention on the one dimensional scalar theories that result from dimensionally reducing the free scalar field theory in arbitrary d dimensions. As is well known, after integrating out the angular coordinates, the free scalar theory can be expressed as an infinite sum of theories living in the semi-infinite line, labeled by the angular modes $$ \left\{\ell, \overrightarrow{m}\right\} $$ ℓ m → . We show that their modular Hamiltonian in an interval attached to the origin is, in turn, the one obtained from the dimensional reduction of the modular Hamiltonian of the conformal parent theory in a sphere. Remarkably, this is a local expression in the energy density, as happens in the conformal case, although the resulting one-dimensional theories are clearly not conformal. We support this result by analyzing the symmetries of these theories, which turn out to be a portion of the original conformal group, and proving that the reduced modular Hamiltonian is in fact the operator generating the modular flow in the interval. By studying the spectrum of these modular Hamiltonians, we also provide an analytic expression for the associated entanglement entropy. Finally, extending the radial regularization scheme originally introduced by Srednicki, we sum over the angular modes to successfully recover the conformal anomaly in the entropy logarithmic coefficient in even dimensions, as well as the universal constant F term in d = 3.

DOI
10.1007/jhep06(2023)097

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Abstract We focus our attention on the one dimensional scalar theories that result from dimensionally reducing the free scalar field theory in arbitrary d dimensions. As is well known, after integrating out the angular coordinates, the free scalar theory can be expressed as an infinite sum of theories living in the semi-infinite line, labeled by the angular modes {ℓ,m→}\left\{\ell, \overrightarrow{m}\right\}{ℓ,m} ℓ m → . We show that their modular Hamiltonian in an interval attached to the origin is, in turn, the one obtained from the dimensional reduction of the modular Hamiltonian of the conformal parent theory in a sphere. Remarkably, this is a local expression in the energy density, as happens in the conformal case, although the resulting one-dimensional theories are clearly not conformal. We support this result by analyzing the symmetries of these theories, which turn out to be a portion of the original conformal group, and proving that the reduced modular Hamiltonian is in fact the operator generating the modular flow in the interval. By studying the spectrum of these modular Hamiltonians, we also provide an analytic expression for the associated entanglement entropy. Finally, extending the radial regularization scheme originally introduced by Srednicki, we sum over the angular modes to successfully recover the conformal anomaly in the entropy logarithmic coefficient in even dimensions, as well as the universal constant F term in d = 3.
In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in ddd dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to 1/r1/r1/r, the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group. Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant FFF term in d=3d=3d=3, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.
A bstract In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in d dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to 1/ r , the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group. Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant F term in d = 3, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.
We review some classic works on ground state entanglement entropy in ( 1 + 1 )-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length L . Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuum limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge. Published by the American Physical Society 2024
Abstract We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular Hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement Hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O (1) terms in the entanglement entropy.
We extend previous work on the perturbative expansion of the Rényi entropy, S q , around q = 1 for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Rényi entropies. On the other hand, the perturbative results agree with known Rényi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Rényi entropy of N=4 \mathcal{N}=4 N=4 super-Yang-Mills near q = 1. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.
Abstract Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1 + 1 space–time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval. We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors. Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.