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.
SciPost Phys. 4, 037 (2018) ·
published 25 June 2018

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Motivated by the calculation of correlation functions in inhomogeneous
onedimensional (1d) quantum systems, the 2d Inhomogeneous Gaussian Free Field
(IGFF) is studied and solved. The IGFF is defined in a domain $\Omega \subset
\mathbb{R}^2$ equipped with a conformal class of metrics $[{\rm g}]$ and with a
real positive coupling constant $K: \Omega \rightarrow \mathbb{R}_{>0}$ by the
action $\mathcal{S}[h] = \frac{1}{8\pi } \int_\Omega \frac{\sqrt{{\rm g}} d^2
{\rm x}}{K({\rm x})} \, {\rm g}^{i j} (\partial_i h)(\partial_j h)$. All
correlations functions of the IGFF are expressible in terms of the Green's
functions of generalized Poisson operators that are familiar from 2d
electrostatics in media with spatially varying dielectric constants.
This formalism is then applied to the study of ground state correlations of
the LiebLiniger gas trapped in an external potential $V(x)$. Relations with
previous works on inhomogeneous Luttinger liquids are discussed. The main
innovation here is in the identification of local observables $\hat{O} (x)$ in
the microscopic model with their field theory counterparts $\partial_x h, e^{i
h(x)}, e^{i h(x)}$, etc., which involve nonuniversal coefficients that
themselves depend on position  a fact that, to the best of our knowledge,
was overlooked in previous works on correlation functions of inhomogeneous
Luttinger liquids , and that can be calculated thanks to Bethe Ansatz form
factors formulae available for the homogeneous LiebLiniger model. Combining
those positiondependent coefficients with the correlation functions of the
IGFF, ground state correlation functions of the trapped gas are obtained.
Numerical checks from DMRG are provided for densitydensity correlations and
for the oneparticle density matrix, showing excellent agreement.
SciPost Phys. 2, 012 (2017) ·
published 4 April 2017

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The oneparticle density matrix of the onedimensional TonksGirardeau gas
with inhomogeneous density profile is calculated, thanks to a recent
observation that relates this system to a twodimensional 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. [Phys. Rev. A 67, 043607 (2003)] from a
different method.
Mr Brun: "We thank the referee for their..."
in Submissions  report on Conformal field theory on top of a breathing onedimensional gas of hard core bosons