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arXiv:2201.03595 · Journal of High Energy Physics

Euclidean time approach to entanglement entropy on lattices and fuzzy spaces

A. Allouche, D. Dou

Published Apr 8, 2022• 1 citation
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Abstract

Abstract In a recent letter, Phys. Lett. B792 (2019) 60, we developed a novel Euclidean time approach to compute Rényi entanglement entropy on lattices and fuzzy spaces based on Green’s function. The present work is devoted in part to the explicit proof of the Green’s matrix function formula which was quoted and used in the previous letter, and on the other part to some applications of this formalism. We focus on scalar theory on 1+1 lattice. We also use the developed approach to systematically go beyond the Gaussian case by considering interacting models, in particular our results confirm earlier expectations concerning the correction to the entanglement at first order. We finally outline how this approach can be used to compute the entanglement entropy on fuzzy spaces for free and interacting scalar theories.

DOI
10.1007/jhep04(2022)051

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Abstract In a recent letter, Phys. Lett. B792 (2019) 60, we developed a novel Euclidean time approach to compute Rényi entanglement entropy on lattices and fuzzy spaces based on Green’s function. The present work is devoted in part to the explicit proof of the Green’s matrix function formula which was quoted and used in the previous letter, and on the other part to some applications of this formalism. We focus on scalar theory on 1+1 lattice. We also use the developed approach to systematically go beyond the Gaussian case by considering interacting models, in particular our results confirm earlier expectations concerning the correction to the entanglement at first order. We finally outline how this approach can be used to compute the entanglement entropy on fuzzy spaces for free and interacting scalar theories.
We develop a Green's function approach to compute Rényi entanglement entropy on lattices and fuzzy spaces. The Rényi entropy resulting from tracing out an arbitrary collection of subsets of coupled harmonic oscillators is written as zero temperature partition function generated by an Euclidean action with n-fold step potential. The associated Green's function is explicitly constructed and an alternative new formula for the Rényi entropy is obtained. Finally it is outlined how this approach can be used to go beyond the gaussian state and include interaction by writing a perturbative expansion for the entanglement entropy.
We investigate the impact of self-interactions on the Rényi and entanglement entropies of a scalar field on (2+1)(2+1)(2+1)-dimensional spacetimes, whose spatial sections are modeled by fuzzy spaces, specifically the fuzzy sphere and the fuzzy disc. We compute the first-order perturbative correction induced by a λφ4λφ^4λφ4 interaction using the Green's function approach. In contrast to the free theory, where the entanglement entropy is dominated by degrees of freedom near the entangling boundary and obeys an area law, we find that the interaction correction has an extensive bulk contribution, receiving significant contributions from degrees of freedom throughout the fuzzy space. For the fuzzy sphere, the correction exhibits strong infrared sensitivity associated with the zero mode. We isolate and resolve this zero-mode IR divergence by projecting out the zero mode, thereby obtaining a physically meaningful quantity. In the commutative continuum limit, the interaction correction has the same degree of UV divergence as the free entropy but does not obey a pure area law. Furthermore, we analyze the Moyal plane limit, where the interaction correction exhibits a distinct IR divergence. We discuss the physical origin of these extensive bulk features and examine their possible connection to the celebrated UV/IR mixing phenomenon in noncommutative quantum field theories.
We study entanglement entropy on the fuzzy sphere. We calculate it in a scalar field theory on the fuzzy sphere, which is given by a matrix model. We use a method that is based on the replica method and applicable to interacting fields as well as free fields. For free fields, we obtain results consistent with the previous study, which serves as a test of the validity of the method. For interacting fields, we perform Monte Carlo simulations at strong coupling and see a novel behavior of entanglement entropy.
Abstract Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.
Understanding the dependence of entanglement entropy on the renormalized mass in quantum field theories can provide insight into phenomena such as quantum phase transitions, since the mass varies in a singular way near the transition. Here we perturbatively calculate the entanglement entropy in interacting scalar field theory, focusing on the dependence on the field’s mass. We study λϕ 4 and g ϕ 3 theories in their ground state. By tracing over a half space, using the replica trick and position space Green’s functions on the cone, we show that spacetime volume divergences cancel and renormalization can be consistently performed in this conical geometry. We establish finite contributions to the entanglement entropy up to two-loop order, involving a finite area law. The resulting entropy is simple and intuitive: the free theory result in d = 3 (that we included in an earlier publication) Δ S ∼ A m 2 ln ( m 2 ) is altered, to leading order, by replacing the bare mass m by the renormalized mass m r evaluated at the renormalization scale of zero momentum.
We investigate entanglement entropy in a scalar field theory on the fuzzy sphere. The theory is realized by a matrix model. In our previous study, we confirmed that entanglement entropy in the free case is proportional to the square of the boundary area of a focused region. Here we argue that this behavior of entanglement entropy can be understood by the fact that the theory is regularized by matrices, and further examine the dependence of entanglement entropy on the matrix size. In the interacting case, by performing Monte Carlo simulations, we observe a transition from a generalized volume law, which is obtained by integrating the square of area law, to the square of area law.