From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

Abstract

We derive exact formulas for the expectation value of local observables in a one-dimensional gas of bosons with point-wise repulsive interactions (Lieb-Liniger model). Starting from a recently conjectured expression for the expectation value of vertex operators in the sinh-Gordon field theory, we derive explicit analytic expressions for the one-point K-body correlation functions (Ψ † ) K (Ψ) K in the Lieb-Liniger gas, for arbitrary integer K. These are valid for all excited states in the thermodynamic limit, including thermal states, generalized Gibbs ensembles and non-equilibrium steady states arising in transport settings. Our formulas display several physically interesting applications: most prominently, they allow us to compute the full counting statistics for the particle-number fluctuations in a short interval. Furthermore, combining our findings with the recently introduced generalized hydrodynamics, we are able to study multi-point correlation functions at the Eulerian scale in non-homogeneous settings. Our results complement previous studies in the literature and provide a full solution to the problem of computing one-point functions in the Lieb-Liniger model. I. INTRODUCTIONCorrelation functions encode all of the information which can be experimentally extracted from a many-body quantum system. At the same time, the problem of their computation is extremely complicated from the theoretical point of view, restricting us, in general, to rely uniquely on perturbative or purely numerical methods.An outstanding exception to this picture are integrable systems [1], characterized by the existence of an extensive number of local conservation laws, which provide an ideal theoretical laboratory to deepen our knowledge of many-body physics. This is especially true due to the possibility of obtaining exact, unambiguous predictions for several quantities of interest, allowing us, for instance, to test the validity of approximate or numerical methods which can be applied to more general cases. While integrability directly provides the tools for diagonalizing the Hamiltonian, the computation of correlation functions constitute a remarkable challenge, which has attracted a constant theoretical effort over the past fifty years [2][3][4][5]. Classical studies have in particular focused on ground-state and thermal correlations, and joint efforts have led to spectacular results, for example in the case of prototypical interacting spin models such as the well-known Heisenberg chain [6][7][8][9][10][11][12].More recently, new energy has been pumped into the study of integrable models, also due to the new experimental possibilities offered by cold-atom physics. Nearly ideal integrable systems can now be realized in cold-atom experiments both in and out equilibrium [13][14][15], elevating the relevance of existing works beyond the purely theoretical interest, and motivating further advances in the framework of non-equilibrium physics (see [16] for a collection of recent reviews on this topic).From the experimental point of vie...

Transport fluctuations in integrable models out of equilibrium

“…Despite many efforts, only a few results exist: free-fermions with the celebrated Lesovik-Levitov formula [50][51][52], harmonic chains [53] and free field theory [54,55], particular integrable impurity models [56], and one-dimensional critical systems [57,58]; see the review [59]. Some results also exist for fluctuation statistics of other quantities, not related to transport, in certain integrable models, see for instance [60][61][62][63]. A full grasp of counting statistics for transport in interacting many-body systems, especially where integrability and ballistic processes dominate, remains an open problem.…”

Introducing iFluid: a numerical framework for solving hydrodynamical equations in integrable models

“…Thus, an infinite set of advection equations emerge, which through the thermodynamic Bethe ansatz, can be formulated as a single Euler-scale equation for a quasiparticle distribution. Since the inception of GHD several applications have been added to the framework, such as calculations of entanglement spreading [16][17][18][19], correlation functions [20][21][22], diffusive corrections [23,24], and many others [25][26][27][28]. Recently it has also been demonstrated to capture the dynamics of a cold Bose gas trapped on an atom-chip [29].…”

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“…Being a very natural concept FCS has been studied for a long time in different communities. It has been studied in the context of charge fluctuations 45,46 , Bose gases [47][48][49][50][51] , particle number fluctuations [52][53][54][55][56] , quantum spin chains [57][58][59][60][61][62][63][64] and out of equilibrium quantum systems 51,65,66 .…”