Abstract
Conformal field theory (CFT) has been extremely successful in describing large-scale universal effects in one-dimensional (1D) systems at quantum critical points. Unfortunately, its applicability in condensed matter physics has been limited to situations in which the bulk is uniform because CFT describes low-energy excitations around some energy scale, taken to be constant throughout the system. However, in many experimental contexts, such as quantum gases in trapping potentials and in several out-of-equilibrium situations, systems are strongly inhomogeneous. We show here that the powerful CFT methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases. The system's inhomogeneity enters the field theory action through parameters that vary with position; in particular, the metric itself varies, resulting in a CFT in curved space. This approach allows us to derive exact formulas for entanglement entropies which were not known by other means.
Highlights
We show here that the powerful Conformal field theory (CFT) methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases
We focus on the example of the free Fermi gas, in a few illustrative in- and out-of-equilibrium inhomogeneous situations
Inhomogeneous 1D quantum systems are difficult to tackle and this is motivating an enormous activity in order to provide exact results in some regimes, as for example the recently developed integrable hydrodynamics [49,50] which may have important ramifications into transport in 1D systems [51,52] such as quantum wires
Summary
Low-dimensional quantum systems are a formidable arena for the study of many-body physics: in one or two spatial dimensions (1D or 2D), the effects of strong correlations and interactions are enhanced and lead to dramatic effects. There is a caveat in the CFT approach to 1D physics though: since it describes low-energy excitations around some fixed energy scale (e.g. the Fermi energy), CFT does not accommodate strong variations of that scale throughout the system This rules out a priori the possibility of tackling inhomogeneous systems, in which the relevant energy scale varies. It is clear that unravelling the global theory for the inhomogeneous system is a problem of geometric nature: it is about understanding the global geometric data (e.g. metric tensor, coupling constants, gauge fields, etc.) that enter the action This is the program we illustrate with the few simple examples below. We demonstrate the power of this formalism by providing new exact asymptotic formulas for entanglement entropies
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