2.5 Entropy in (1+1)-dimensional spacetime

The state of a quantum field in two dimensions is defined on a union of intersecting intervals A ∪ B. The 2-point correlation functions behave logarithmically in the limit of coincident points. Correspondingly, the leading UV divergence of the entanglement entropy in two dimensions is logarithmic. For example, for a 2D massless conformal field theory, characterized by a central charge c, the entropy is [208, 33Jump To The Next Citation Point, 133Jump To The Next Citation Point]
S = cn-ln lA-+ s(l ∕l ), (9 ) 2d 6 𝜖 A B
where n is the number of intersections of intervals A and B, where the sub-systems are defined, l A (lB) is the length of the interval A (B). The second term in Eq. 9View Equation) is a UV finite term. In some cases the conformal symmetry in two dimensions can be used to calculate not only the UV divergent term in the entanglement entropy but also the UV finite term, thus obtaining the complete answer for the entropy, as was shown by Holzhey, Larsen and Wilczek [133] (see [161, 29] for more recent developments). There are two different limiting cases when the conformal symmetry is helpful. In the first case, one considers a pure state of the conformal field theory on a circle of circumference L, the subsystem is defined on a segment of size l of the circle. In the second situation, the system is defined on an infinite line, the subsystem lives on an interval of length l of the line and the global system is in a thermal mixed state with temperature T. In Euclidean signature both geometries represent a cylinder. For a thermal state the compact direction on the cylinder corresponds to Euclidean time τ compactified to form a circle of circumference β = 1∕T. In both cases the cylinder can be further conformally mapped to a plane. The invariance of the entanglement entropy under conformal transformation can be used to obtain
( ) S = c-ln L--sin( πl- (10 ) 3 π𝜖 L
in the case of a pure state on a circle and
c ( β πl) S = --ln ---sinh(--- (11 ) 3 π 𝜖 β
for a thermal mixed state on an infinite line. In the limit of large l the entropy (11View Equation) approaches
c c l c β S = 3-πlT + 3-ln(π𝜖) + 3-ln l-, (12 )
where the first term represents the entropy of the thermal gas (7View Equation) in a cavity of size l, while the second term represent the purely entanglement contribution (note that the intersection of A and B contains two points in this case so that n = 2). The third term is an intermediate term due to the interaction of both factors, thermality and entanglement. This example clearly shows that for a generic thermal state the entanglement entropy is due to the combination of two factors: the entanglement between two subsystems and the thermal nature of the mixed state of the global system.
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