The area dependence of entanglement entropy of a free scalar field is often understood in terms of coupled harmonic oscillators. In Schrodinger quantization, the Gaussian nature of ground state wave function for these oscillators is sufficient to provide the exact form of the reduced density matrix and its eigenvalues, thus giving the entanglement entropy. However, in polymer quantization the ground state is not Gaussian and the formalism that can provide the exact analytical form of the reduced density matrix is not yet known. In order to address this issue, here we treat the interaction between two coupled harmonic oscillators in the perturbative approach and evaluate the entanglement entropy in Schrodinger and polymer quantization. Contrary to Schrodinger quantization, we show that in the high frequency regime the entanglement entropy decreases for polymer quantization, keeping the ratio of coupling strength to the square of individual oscillator frequency fixed. Furthermore for the free scalar field we validate the area dependence of entanglement entropy in Fock quantization and demonstrate that polymer quantization produces a similar area law.
The area dependence of entanglement entropy of a free scalar field is often understood in terms of coupled harmonic oscillators. In Schrodinger quantization, the Gaussian nature of ground state wave function for these oscillators is sufficient to provide the exact form of the reduced density matrix and its eigenvalues, thus giving the entanglement entropy. However, in polymer quantization the ground state is not Gaussian and the formalism that can provide the exact analytical form of the reduced density matrix is not yet known. In order to address this issue, here we treat the interaction between two coupled harmonic oscillators in the perturbative approach and evaluate the entanglement entropy in Schrodinger and polymer quantization. Contrary to Schrodinger quantization, we show that in the high frequency regime the entanglement entropy decreases for polymer quantization, keeping the ratio of coupling strength to the square of individual oscillator frequency fixed. Furthermore for the free scalar field we validate the area dependence of entanglement entropy in Fock quantization and demonstrate that polymer quantization produces a similar area law.
A bstract The entanglement entropy of a free scalar field in its ground state is dominated by an area law term. It is noteworthy, however, that the study of entanglement in scalar field theory has not advanced far beyond the ground state. In this paper, we extend the study of entanglement of harmonic systems, which include free scalar field theory as a continuum limit, to the case of the most general Gaussian states, namely the squeezed states. We find the eigenstates and the spectrum of the reduced density matrix and we calculate the entanglement entropy. We show that our method is equivalent to the correlation matrix method. Finally, we apply our method to free scalar field theory in 1+1 dimensions and show that, for very squeezed states, the entanglement entropy is dominated by a volume term, unlike the ground-state case. Even though the state of the system is time-dependent in a non-trivial manner, this volume term is time-independent. We expect this behaviour to hold in higher dimensions as well, as it emerges in a large-squeezing expansion of the entanglement entropy for a general harmonic system.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.
Entanglement entropy (EE) is a pivotal quantity in quantum field theory (QFT) that captures the nonlocal correlations between subsystems, offering deep insights into the quantum structure of spacetime, field dynamics, and phase transitions. This work derives EE in QFT using the covariance matrix formalism, establishing its foundations through the von Neumann entropy of reduced density matrices. We analytically compute EE for coupled harmonic oscillators as a prototypical model, extending the results to quantum fields via perturbative expansions. In the context of Maxwell QFT, we analyze entropy corrections induced by external perturbations, revealing their fractal dynamics through the emergence of Julia sets. The study further incorporates noncommutative geometry, where the deformation of spacetime modifies the covariance matrix, field strength tensors, and modular indices. Numerical simulations validate the derived scaling laws, entropy corrections, and noncommutative effects. These findings provide a mathematically rigorous framework to explore the interplay between EE, field theory, and geometric structures in high-energy physics.
It is well-known that entanglement entropy in field theory at its ground state is dominated by an area law term, presenting a similarity to the entropy of black holes. It is interesting to investigate whether this similarity can be extended by showing that gravitational dynamics emerges from the first law of entanglement thermodynamics. Answering this question requires the specification of the modular Hamiltonian. Motivated by the above, we study entanglement in the toy model of harmonic systems lying at any classicalmost state, i.e. any coherent state. We specify explicitly the reduced density matrix and its time-evolution, as well as the modular Hamiltonian. Interestingly, the time evolution is unitary and we specify the effective Hamiltonian which generates it. Our results provide the tools to investigate the similarity between gravity and entanglement in discretized free scalar field theory in the framework of (Phys Rev Lett 71:666, 1993).
In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency w of sqrt(w+ w-) (where w+ and w- are defined later). We find this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2. Only one set w,T for a single oscillator should map into the integrated result.
We investigate interaction-induced corrections to entanglement entropy by mapping a screened Yukawa-type interaction to an effective harmonic oscillator system with controlled anharmonic perturbations. Starting from a one-dimensional interaction V(x)=−g2e−αmx/x, we reformulate the problem in terms of a four-dimensional radial oscillator, where the finite screening length generates a systematic hierarchy of polynomial interactions in the radial coordinate. This mapping enables a controlled Rayleigh-Schrodinger perturbative treatment of the ground-state wavefunction and an explicit spectral analysis of the reduced density matrix. Working in the weak-screening regime, we compute the leading non-Gaussian correction arising from the quartic interaction ρ4, which appears at order α2 in the expansion of the Yukawa-like potential. We obtain closed analytic expressions for the resulting small eigenvalues of the reduced density matrix and evaluate their contribution to the von Neumann entanglement entropy. We show that the entropy receives analytic corrections at order α2, originating both from explicit anharmonic state-mixing effects and from the implicit α dependence of the Gaussian width parameter. Our results clarify the distinct roles of harmonic renormalization and genuinely non-Gaussian interactions in generating entanglement, establish a systematic power-counting and normalization scheme for higher-order ρ2n perturbations, and provide a transparent oscillator-based framework for computing entanglement entropy in weakly interacting low-dimensional and field-theoretic systems.