SciPost Phys. 6, 051 (2019) ·
published 30 April 2019

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The recent results of [J. Dubail, J.M. St\'ephan, J. Viti, P. Calabrese,
Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale
correlation functions of inhomogeneous critical onedimensional quantum systems
 e.g. a gas of hard core bosons in a trapping potential  are extended to a
dynamical situation: a breathing gas in a timedependent harmonic trap. Hard
core bosons in a timedependent harmonic potential are well known to be exactly
solvable, and can thus be used as a benchmark for the approach. An extensive
discussion of the approach and of its relation with classical and quantum
hydrodynamics in one dimension is given, and new formulas for correlation
functions, not easily obtainable by other methods, are derived. In particular,
a remarkable formula for the large scale asymptotics of the bosonic
$n$particle function $\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger
(x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>$ is obtained. Numerical
checks of the approach are carried out for the fermionic twopoint function 
easier to access numerically in the microscopic model than the bosonic one 
with perfect agreement.
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