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The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases
by Yannis Brun, Jérôme Dubail
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Submission summary
Ontological classification 
Academic field: 
Physics 
Specialties: 
 Mathematical Physics
 Atomic, Molecular and Optical Physics  Theory
 Condensed Matter Physics  Theory
 Quantum Physics
 Statistical and Soft Matter Physics

Approach: 
Theoretical 
Abstract
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.
Author comments upon resubmission
We thank the editor for taking in charge the submission and refereeing of our manuscript. We here resubmit a new version of the manuscript with minor modifications, see answer to report 1.