An inverse mass expansion for the mutual information in free scalar QFT at finite temperature

Abstract

We study the entanglement entropy and the mutual information in coupled harmonic systems at finite temperature. Interestingly, we find that the mutual information does not vanish at infinite temperature, but it rather reaches a specific finite value, which can be attributed to classical correlations solely. We further obtain high and low temperature expansions for both quantities. Then, we extend the analysis performed in the seminal paper by Srednicki [1] for free real scalar field theories in Minkowski space-time in 3 + 1 dimensions at a thermal state. We find that the mutual information obeys an area law, similar to that obeyed by the entanglement entropy at vanishing temperature. The coefficient of this area law does not vanish at infinite temperature. Then, we calculate this coefficient perturbatively in an 1/µ expansion, where µ is the mass of the scalar field. Finally, we study the high and low temperature behaviour of the area law term.

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

“…But the homogeneous polynomials are a finite dimensional vector space, meaning that the constants A k,a,d n,m,j 1 ,j 2 can be computed through a straightforward finite dimensional linear system of equations. 14 Here we note that for these coefficients, the indices k and a are indices of the Gegenbauer polynomials, the index d corresponds to the spatial dimension, while n, m, j 1 , j 2 label a basis for homogeneous polynomials. There is likely no neat formula for these coefficients in general.…”

“…For instance, Srednicki had to impose an ultraviolet cutoff by discretizing the sphere with a finite lattice. Rather than deal with this divergence directly, some authors choose to study a related quantity, the mutual information of two spatially JHEP04(2024)019 separated regions [12][13][14][15]. Given a bipartite state with density operator ρAB and reduced density operator ρA and ρB , the mutual information can be defined in terms of entropy, I = S(ρ A ) + S(ρ B ) − S(ρ AB ).…”

Entanglement entropy and gravity

“…In [1] we generalize the approach of [42] in a method for the perturbative calculation of the spectrum of the reduced density matrix and entanglement entropy as well. In [6,7] we study free massive scalar QFT at finite temperature. We show that it is the mutual information that obeys an area law and that there is a natural way to separate the contribution of classical and quantum correlations to the mutual information.…”

“…It has been shown that in lattice spin systems the mutual information obeys an area law bound [240]. This Part of the dissertation is based on the publications [1,6,7]. Its structure is as follows: in section 8, we review the derivation of entanglement entropy in systems of coupled harmonic oscillators lying at their ground state and extend the calculation in free scalar field theory including a mass term, closely following [42].…”

“…Στην εργασία [1] γενικεύουμε την προσέγγιση της εργασίας [42] σε μια μέθοδο για τον διαταρακτικό υπολογισμό του φάσματος του ανηγμένου τελεστή πυκνότητας καθώς και της εντροπίας διεμπλοκής. Στις εργασίες [6,7] μελετάμε την ελεύθερη βαθμωτή ΚΘΠ με μάζα σε πεπερασμένη θερμοκρασία. Δείχνουμε ότι σε αυτή την περίπτωση η αμοιβαία πληροφορία υπακούει νόμο εμβαδού και ότι υπάρχει ένας φυσικός τρόπος να διαχωριστούν οι κβαντικές από τις κλασσικές συσχετίσεις στην αμοιβαία πληροφορία.…”