We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally nonlocal and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how generalized hydrodynamics emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that generalized hydrodynamics in the weak interaction regime is robust under nonintegrable perturbations.
We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally nonlocal and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how generalized hydrodynamics emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that generalized hydrodynamics in the weak interaction regime is robust under nonintegrable perturbations.
We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. Starting from an ansatz for the state at time t , which we call the correlated fluid-cell ensemble, we show that the dynamics of local observables at macroscopic times and length scales can be cast into a generalized Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy formulated in terms of the quasiparticle densities of the underlying integrable model and their correlations. We derive this hierarchy and validate these predictions against microscopic molecular-dynamics simulations, finding perfect agreement. At late times, the one-particle distribution relaxes via a Boltzmann-type scattering integral encoding the interplay between integrable contact processes and long-range collisions, whereas higher-point correlations remain strongly nonthermal on thermalization timescales, indicative of a form of incomplete or generalized thermalization. Focusing on long-range dipolar quantum gases, where the relevant matrix elements can be obtained explicitly, we show that our collision integral reduces exactly to the Fermi golden rule result and provide a complete theoretical account of the experimental observations of Tang []. More broadly, our framework extends the BBGKY program to regimes with strong local interactions and applies to a wide class of experimentally relevant systems, from one-dimensional dipolar cold-atom gases to Lennard-Jones molecular fluids.
In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.
We consider the dynamics of |N| bosons in three dimensions. We assume that the pair interaction is given by |N3β−1V(Nβ⋅)|. By studying an associated many-body wave operator, we introduce a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the |N|-body wave function, for |$\beta \in ( 0,1] ∣,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣N\rightarrow \infty ∣limit,all∣k∣−bodyscatteringprocessesvanishif∣k\geqslant 3∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣\beta \in ( 0,1) ∣andweaklyfor∣\beta =1∣.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣\beta <1∣,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣\beta \in (0,1)$|.
The rigorous evolution equations for the one-body Green's functions (g>,g<) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, N (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (g>,g<).
The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. In this Letter, we show that it supersedes the widely used conventional hydrodynamics (CHD) of one-dimensional Bose gases. We illustrate this by studying "nonlinear sound waves" emanating from initial density accumulations in the Lieb-Liniger model. We show that, at zero temperature and in the absence of shocks, GHD reduces to CHD, thus for the first time justifying its use from purely hydrodynamic principles. We show that sharp profiles, which appear in finite times in CHD, immediately dissolve into a higher hierarchy of reductions of GHD, with no sustained shock. CHD thereon fails to capture the correct hydrodynamics. We establish the correct hydrodynamic equations, which are finite-dimensional reductions of GHD characterized by multiple, disjoint Fermi seas. We further verify that at nonzero temperature, CHD fails at all nonzero times. Finally, we numerically confirm the emergence of hydrodynamics at zero temperature by comparing its predictions with a full quantum simulation performed using the NRG-TSA-abacus algorithm. The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility, and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two minimally invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer timescale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimally invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.