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arXiv:2603.24158

Entanglement Entropy of Massive Scalar Fields: Mass Suppression, Violation of Universal mR Scaling, and Implications for Black Hole Thermodynamics

Stefano Bellucci A, M. Shatnev, L. G. Zazunov

Published Mar 25, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the entanglement entropy of a massive scalar field using the spherical shell lattice model introduced by Das and Shankaranarayanan. A systematic numerical analysis is performed to study the dependence of the entropy on the field mass and on the size of the entangling region for both ground and excited states. For the ground state, we find that the entanglement entropy is exponentially suppressed by the field mass, reflecting the presence of a finite correlation length, while the geometric area-law scaling remains robust for all masses. For localized excited states, however, we uncover a qualitatively different behavior. The excess entropy does not exhibit universal scaling in the dimensionless variable mR. Instead, numerical results show that data points with identical mR but different (m,R) pairs do not collapse onto a single curve, demonstrating a clear violation of simple scaling. This breakdown is traced to the presence of an additional length scale associated with the finite width of the wave-packet excitation. This result identifies the coexistence of multiple infrared scales as a key feature of excited-state entanglement in massive quantum field theories. Mutual information provides an additional finite diagnostic of correlations in the chosen nested geometry. The numerical results show a strong dependence on the field mass, although the detailed behavior is sensitive to the geometric setup used in the calculation. These findings clarify how particle mass and excitation structure jointly determine entanglement properties, and suggest that the matter contribution to the generalized entropy in semiclassical gravity may depend on independent infrared parameters rather than on a single correlation scale. Implications for black hole entropy and the island formula are briefly discussed.

DOI
10.48550/arxiv.2603.24158

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We investigate the entanglement entropy of a massive scalar field using the spherical shell lattice model introduced by Das and Shankaranarayanan. A systematic numerical analysis is performed to study the dependence of the entropy on the field mass and on the size of the entangling region for both ground and excited states. For the ground state, we find that the entanglement entropy is exponentially suppressed by the field mass, reflecting the presence of a finite correlation length, while the geometric area-law scaling remains robust for all masses. For localized excited states, however, we uncover a qualitatively different behavior. The excess entropy does not exhibit universal scaling in the dimensionless variable mR. Instead, numerical results show that data points with identical mR but different (m,R) pairs do not collapse onto a single curve, demonstrating a clear violation of simple scaling. This breakdown is traced to the presence of an additional length scale associated with the finite width of the wave-packet excitation. This result identifies the coexistence of multiple infrared scales as a key feature of excited-state entanglement in massive quantum field theories. Mutual information provides an additional finite diagnostic of correlations in the chosen nested geometry. The numerical results show a strong dependence on the field mass, although the detailed behavior is sensitive to the geometric setup used in the calculation. These findings clarify how particle mass and excitation structure jointly determine entanglement properties, and suggest that the matter contribution to the generalized entropy in semiclassical gravity may depend on independent infrared parameters rather than on a single correlation scale. Implications for black hole entropy and the island formula are briefly discussed.
The entanglement entropy correlates two quantum subsystems which are part of a larger system. A logarithmic divergence term present in the entanglement entropy is universal in nature and directly proportional to the conformal anomaly. We study this logarithmic divergence term of entropy for massive scalar field in [Formula: see text] dimensions by applying numerical techniques to entanglement entropy approach. This (2+1)-dimensional massive theory can be obtained from the (3+1)-dimensional massless scalar field via dimensional reduction. We also calculated mass corrections to entanglement entropy for scalar field. Finally, we observe that the area law contribution to the entanglement entropy is not affected by this mass term and the universal quantities depend upon the basic properties of the system.
Abstract We evaluate the entanglement entropy of a single connected region in excited states of one-dimensional massive free theories with finite numbers of particles, in the limit of large volume and region length. For this purpose, we use finite-volume form factor expansions of branch-point twist field two-point functions. We find that the additive contribution to the entanglement due to the presence of particles has a simple “qubit” interpretation, and is largely independent of momenta: it only depends on the numbers of groups of particles with equal momenta. We conjecture that at large momenta, the same result holds for any volume and region lengths, including at small scales. We provide accurate numerical verifications.
We study analytically the corrections to the leading terms in the R\'enyi entropy of a massive lattice theory, showing significant deviations from naive expectations. In particular, we show that finite size and finite mass effects give rise to different contributions (with different exponents) and thus violate a simple scaling argument. In the specific, we look at the entanglement entropy of a bipartite XYZXYZXYZ spin-1/21/21/2 chain in its ground state. When the system is divided into two semi-infinite half-chains, we have an analytical expression of the R\'enyi entropy as a function of a single mass parameter. In the scaling limit, we show that the entropy as a function of the correlation length formally coincides with that of a bulk Ising model. This should be compared with the fact that, at criticality, the model is described by a c=1c=1c=1 conformal field theory and the corrections to the entropy due to finite size effects show exponents depending on the compactification radius of the theory. We will argue that there is no contradiction between these statements. If the lattice spacing is retained finite, the relation between the mass parameter and the correlation length generates new subleading terms in the entropy, whose form is path dependent in phase space and whose interpretation within a field theory is not available yet. These contributions arise as a consequence of the existence of stable bound states and are thus a distinctive feature of truly interacting theories, such as the XYZXYZXYZ chain.
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 extend the entanglement entropy calculation performed in the seminal paper by Srednicki (Phys Rev Lett 71:666, 1993) for free real massive scalar field theories in 1+11+11+1 1 + 1 , 2+12+12+1 2 + 1 and 3+13+13+1 3 + 1 dimensions. We show that the inverse of the scalar field mass can be used as an expansion parameter for a perturbative calculation of the entanglement entropy. We perform the calculation for the ground state of the system and for a spherical entangling surface at third order in the inverse mass expansion. The calculated entanglement entropy contains a leading area law term, as well as subleading terms that depend on the regularization scheme, as expected. Universal terms are non-perturbative effects in this approach. Interestingly, this perturbative expansion can be used to approximate the coefficient of the area law term, even in the case of a massless scalar field in 2+12+12+1 2 + 1 and 3+13+13+1 3 + 1 dimensions. The presented method provides the spectrum of the reduced density matrix as an intermediate result, which is an important advantage in comparison to the replica trick approach. Our perturbative expansion underlines the relation between the area law and the locality of the underlying field theory.
In this thesis, we study the entanglement entropy of a chain of coupled harmonic oscillators, which is used to model a one-dimensional bosonic scalar field. Initially, a model for a massless scalar field is discussed. When calculating the entanglement entropy from the reduced density matrix we run into a problem due to the presence of a zero mode, and we are only able to compute the entanglement entropy of one particular case. These difficulties are avoided when we consider a massive scalar field instead, which we accomplish by adding a mass term to the dispersion relation. We numerically study the entanglement entropy of the massive scalar field model as a function of the number of coordinates and the mass parameter and compare with analytical results. By fitting our numerical data we construct a single cross-over function, which describes the entanglement entropy as a function of both the number of coordinates and the mass parameter. The cross-over function agrees well with the numerical data, especially for small values of the mass parameter. Taking then the massless limit, the entanglement entropy of a massless scalar field can nevertheless be determined and reproduces the prediction of conformal field theory.