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