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Entanglement entropy of a one-dimensional scalar field

K.R. de Ruiter

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O(N)\mathrm{O}(N)O(N) symmetry in the limit of large NNN for general entanglement geometries. We show that the leading large-NNN result can be obtained from the entanglement entropy of NNN Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of NNN massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=\ensuremath{\infty}$. Notably, for a semi-infinite cylindrical region it scales as N0{N}^{0}N0 in the Wilson-Fisher theory, in stark contrast to the NNN-linear result of the Gaussian fixed point.
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.
Abstract We construct a contour function for the entanglement entropies in generic harmonic lattices. In one spatial dimension, numerical analysis are performed by considering harmonic chains with either periodic or Dirichlet boundary conditions. In the massless regime and for some configurations where the subsystem is a single interval, the numerical results for the contour function are compared to the inverse of the local weight function which multiplies the energy-momentum tensor in the corresponding entanglement hamiltonian, found through conformal field theory methods, and a good agreement is observed. A numerical analysis of the contour function for the entanglement entropy is performed also in a massless harmonic chain for a subsystem made by two disjoint intervals.