2.13 Entanglement entropy in non-Lorentz invariant theories

Non-Lorentz invariant theories are characterized by a modified dispersion relation, 2 2 ω + F (p ) = 0, between the energy ω and the 3-momentum p. These theories can be described by a wave operator of the following type
š’Ÿ = − ∂2t + F (− Δx ) , (45 )
where Δ = ∑d− 1∂2 x i i is the spatial Laplace operator. Clearly, the symmetry with respect to the Lorentz boosts is broken in operator (45View Equation) if F(q) ā„= q.

As in the Lorentz invariant case to compute the entanglement entropy associated with a surface Σ we choose (d − 1) spatial coordinates {xi, i = 1,..,d − 1} = {x, za, a = 1,..,d − 2}, where x is the coordinate orthogonal to the surface Σ and za are the coordinates on the surface Σ. Then, after going to Euclidean time τ = it, we switch to the polar coordinates, τ = r sin(Ļ• ), x = rcos(Ļ•). In the Lorentz invariant case the conical space, which is needed for calculation of the entanglement entropy, is obtained by making the angular coordinate Ļ• periodic with period 2πα by applying the Sommerfeld formula (22View Equation) to the heat kernel. If Lorentz invariance is broken, as it is for the operator (45View Equation), there are certain difficulties in applying the method of the conical singularity when one computes the entanglement entropy. The difficulties come from the fact that the wave operator š’Ÿ, if written in terms of the polar coordinates r and Ļ•, becomes an explicit function of the angular coordinate Ļ•. As a result of this, the operator š’Ÿ is not invariant under shifts of Ļ• to arbitrary Ļ• + w. Only shifts with w = 2πn, where n is an integer are allowed. Thus, in this case one cannot apply the Sommerfeld formula since it explicitly uses the symmetry of the differential operator under shifts of angle Ļ•. On the other hand, a conical space with angle deficit 2π(1 − n) is exactly what we need to compute Tr ρn for the reduced density matrix. In [184Jump To The Next Citation Point], by using some scaling arguments it was shown that the trace of the heat kernel K (s) = e−sš’Ÿ on a conical space with 2πn periodicity, is

1 TrKn (s) = nTrKn=1 (s) + (4π-)dāˆ•22πn C2 (n)A (Σ )Pd−2(s), (46 )
where nTrKn=1 (s) is the bulk contribution. By the arguments presented in Section 2.7 there is a unique analytic extension of this formula to non-integer n. A simple comparison with the surface term in the heat kernel of the Lorentz invariant operator, which was obtained in Section 2.12, shows that the surface terms of the two kernels are identical. Thus, we conclude that the entanglement entropy is given by the same formula
∫ ∞ S = -----A-(Σ-)---- ds-Pd−2(s), (47 ) 12 ⋅ (4π )(d−2)āˆ•2 šœ–2 s
where P (s) n is defined in Eq. (40View Equation), as in the Lorentz invariant case (41View Equation). A similar property of the entanglement entropy was observed for a non-relativistic theory described by the Schrödinger operator [205Jump To The Next Citation Point] (see also [59] for a holographic derivation). For polynomial operators, k F (q) ∼ q, some scaling arguments can be used [205Jump To The Next Citation Point] to get the form of the entropy that follows from Eq. (47View Equation).

In the rest of the review we shall mostly focus on the study of Lorentz invariant theories, with field operator quadratic in derivatives, of the Laplace type, 2 š’Ÿ = − (∇ + X ).

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