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Conformal field theory (CFT) has been extremely successful in describinglarge-scale universal effects in one-dimensional (1D) systems at quantumcritical points. Unfortunately, its applicability in condensed matter physicshas been limited to situations in which the bulk is uniform because CFTdescribes low-energy excitations around some energy scale, taken to be constantthroughout the system. However, in many experimental contexts, such as quantumgases in trapping potentials and in several out-of-equilibrium situations,systems are strongly inhomogeneous. We show here that the powerful CFT methodscan be extended to deal with such 1D situations, providing a few concreteexamples for non-interacting Fermi gases. The system's inhomogeneity enters thefield theory action through parameters that vary with position; in particular,the metric itself varies, resulting in a CFT in curved space. This approachallows us to derive exact formulas for entanglement entropies which were notknown by other means.

Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases