arXiv:2310.06054 · Journal of High Energy Physics

Perturbative method for mutual information and thermal entropy of scalar quantum fields

Joseph Bramante, Andrew Buchanan

Published Apr 4, 2024• 2 citations

Abstract

A bstract A new approach is presented to compute entropy for massless scalar quantum fields. By perturbing a skewed correlation matrix composed of field operator correlation functions, the mutual information is obtained for disjoint spherical regions of size r at separation R , including an expansion to all orders in r / R . This approach also permits a perturbative expansion for the thermal field entropy difference in the small temperature limit ( T ≪ 1/ r ).

DOI: 10.1007/jhep04(2024)019

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arXiv:2310.06054 · Journal of High Energy Physics

Perturbative method for mutual information and thermal entropy of scalar quantum fields

Joseph Bramante, Andrew Buchanan

Published Apr 4, 2024• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep04(2024)019

Related papers

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An inverse mass expansion for entanglement entropy in free massive scalar field theory

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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 $1+1$ 1 + 1 , $2+1$ 2 + 1 and $3+1$ 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 $2+1$ 2 + 1 and $3+1$ 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.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 $$1+1$$ 1 + 1 , $$2+1$$ 2 + 1 and $$3+1$$ 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 $$2+1$$ 2 + 1 and $$3+1$$ 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.

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Markus Schröfl, Stefan Floerchinger

Abstract Relative entropy is a powerful measure of the dissimilarity between two statistical field theories in the continuum. In this work, we study the relative entropy between Gaussian scalar field theories in a finite volume with different masses and boundary conditions. We show that the relative entropy depends crucially on d , the dimension of Euclidean space. Furthermore, we demonstrate that the mutual information between two disjoint regions in $\mathbb {R}^d$ R d is finite if the two regions are separated by a finite distance and satisfies an area law. We then construct an example of “touching” regions between which the mutual information is infinite. We argue that the properties of mutual information in scalar field theories can be explained by the Markov property of these theories.Abstract Relative entropy is a powerful measure of the dissimilarity between two statistical field theories in the continuum. In this work, we study the relative entropy between Gaussian scalar field theories in a finite volume with different masses and boundary conditions. We show that the relative entropy depends crucially on d , the dimension of Euclidean space. Furthermore, we demonstrate that the mutual information between two disjoint regions in $$\mathbb {R}^d$$ R d is finite if the two regions are separated by a finite distance and satisfies an area law. We then construct an example of “touching” regions between which the mutual information is infinite. We argue that the properties of mutual information in scalar field theories can be explained by the Markov property of these theories.

arXiv:2307.15548Annales Henri PoincaréDec 17, 20242 citations

Entanglement entropy of disjoint regions in excited states: an operator method

Noburo Shiba

We develop the computational method of entanglement entropy based on the idea that $\mathrm{T}\mathrm{r}{\rho}_{{}^{\varOmega}}^n$ is written as the expectation value of the local operator, where ρ Ω is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I (n)(A, B) = S () + S () − S ∪ () of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we compute I (n)(A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, I (n)(A, B) − I (n)(A, B)| r → ∞ = C () /r α(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute C (2) systematically.We develop the computational method of entanglement entropy based on the idea that $$ \mathrm{T}\mathrm{r}{\rho}_{{}^{\varOmega}}^n $$ is written as the expectation value of the local operator, where ρ Ω is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I (n)(A, B) = S () + S () − S ∪ () of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we compute I (n)(A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, I (n)(A, B) − I (n)(A, B)| r → ∞ = C () /r α(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute C (2) systematically.

arXiv:1408.0637Journal of High Energy PhysicsDec 1, 201419 citations

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+1

1 + 1 ,

2+1

2 + 1 and

3+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+1

2 + 1 and

3+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.

\mathbb {R}^d

R d is finite if the two regions are separated by a finite distance and satisfies an area law. We then construct an example of “touching” regions between which the mutual information is infinite. We argue that the properties of mutual information in scalar field theories can be explained by the Markov property of these theories.

We develop the computational method of entanglement entropy based on the idea that

\mathrm{T}\mathrm{r}{\rho}_{{}^{\varOmega}}^n

is written as the expectation value of the local operator, where ρ Ω is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I (n)(A, B) = S () + S () − S ∪ () of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we compute I (n)(A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, I (n)(A, B) − I (n)(A, B)| r → ∞ = C () /r α(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute C (2) systematically.