Universality of Entropy Scaling in One Dimensional Gapless Models

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval $(0,x)$. At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.