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arXiv:2406.11031 · Physical review. D/Physical review. D.

Entanglement in ( 1 + 1 )-dimensional free scalar field theory: Tiptoeing between continuum and discrete formulations

Dimitrios Katsinis, Georgios Pastras

Published Oct 28, 2024• 8 citations
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Abstract

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

DOI
10.1103/physrevd.110.085015

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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
We find the entanglement entropy of a subregion of the vacuum state in scalar quantum electrodynamics, working perturbatively to the two-loop level. Doing so leads us to derive the Maxwell-Proca propagator in conical Euclidean space. The area law of entanglement entropy is recovered in both the massive and massless limits of the theory, as is expected. These results yield the renormalization group flow of entanglement entropy, and we find that loop contributions suppress entanglement entropy. We highlight these results in the light of the renormalization group flow of couplings and correlators, which are increased in scalar quantum electrodynamics, so that the potential tension between the increase in correlations between two points of spacetime and the decrease in entanglement entropy between two regions of spacetime with energy is discussed. We indeed show that the vacuum of a subregion of spacetime purifies with energy in scalar quantum electrodynamics, which is related to the concept of screening. Published by the American Physical Society 2024
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.
An exact result for the reduced density matrix on a finite interval for a 1+11+11+1 dimensional free real scalar field in the ground state is presented. In the massless case, the Williamson decomposition of the appearing kernels is explicitly performed, which allows to reproduce the known result for the entanglement (modular) Hamiltonian and, for a small mass MMM of the field, find the leading ∼1/ln⁡M\sim1/\ln M∼1/lnM non-local corrections. In the opposite M→∞M\to\inftyM→∞ case, it is argued that, up to terms O(M−1/2)O(M^{-1/2})O(M−1/2), the entanglement Hamiltonian is local with the density being the Hamiltonian density spatially modulated by a triangular-shape function.
A bstract We study the modular Hamiltonian and the entanglement entropy of the BMS-invariant free fermion model. Starting from the modular Hamiltonian on a half-line interval, we calculate the modular Hamiltonian for a region consisting of two disjoint intervals using the uniformization map and a background field method. The resulting modular Hamiltonian contains a local term which generates a geometrical flow, and a bi-local term which mixes operators between the two intervals. We further derive a relation between Wightman functions and the modular Hamiltonian using both diagonalization and a coherent state method. This enables us to compute the entanglement entropy for multi-disjoint intervals. Our explicit results in the BMS free fermion model are compatible with earlier ones based on symmetry and holography.
We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2 + 1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum ( 2 + 1 )-dimensional SU(2) gauge theory does not have such scarred states. Published by the American Physical Society 2024
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.