2.4 Entropy of a system of finite size at finite temperature

In a more general situation one starts with a system of finite size L in a mixed thermal state at temperature T. This system is divided by the entangling surface Σ in two sub-systems of characteristic size l. Then, the entanglement entropy is a function of several parameters (if the field in question is massive then mass m should be added to the parameters on which the entropy should depend)
S = S(T, L, l, 𝜖), (8 )
where 𝜖 is a UV cut-off. Clearly, the entanglement entropy in this general case is due to a combination of different factors: the entanglement between two sub-systems and the thermal nature of the initial mixed state. In d dimensions even for simple geometries this function of 4 variables is not known explicitly. However, in two spacetime dimensions, in some particular cases, the explicit form of this function is known.
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