One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance

Abstract

The one-particle density matrix of the one-dimensional Tonks-Girardeau gas with inhomogeneous density profile is calculated, thanks to a recent observation that relates this system to a two-dimensional conformal field theory in curved space. The result is asymptotically exact in the limit of large particle density and small density variation, and holds for arbitrary trapping potentials. In the particular case of a harmonic trap, we recover a formula obtained by Forrester et al.[23] from a different method.

The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases

“…The content of this subsection is very similar to arguments given in Refs. [17,18]; we repeat those here only for completeness.…”

“…While Luttinger liquids have been studied extensively in homogeneous, translation invariant, situations, the present paper follows on from the recent series [13][14][15][16][17][18][19][20][21] that aims at extending the free boson CFT, or GFF, to inhomogeneous situations. [Troughout this work, inhomogeneity is understood as spatial dependence of physical quantities and parameters.]…”

“…In that case, the inhomogeneous Luttinger liquid is nothing but a 2d CFT in a curved metric, a fact that can be exploited to easily get nice exact analytic formulae in a variety of interesting physical situations, see Refs. [13][14][15][16][17][18][19][20][21].…”

Entanglement entropies of inhomogeneous Luttinger liquids

“…Above, I a/α and M are defined in Eq. (21). In principle, I a/α (and thus M) still bear a non-trivial dependence on x 2 , but the resulting expression must be x 2 −independent.…”

Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas

“…Our results can also be used as starting point for the determination of correlation functions in inhomogeneous setting, e.g. on an Eulerian scale [91], or in the low energy regime using conformal field theories in curved backgrounds [92,93].…”