Continuum Mechanics for Quantum Many-Body Systems: The Linear Response Regime Xianlong Gao Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang Province, 321004, China Jianmin Tao Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 G. Vignale Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211 I. V. Tokatly IKERBASQUE, Basque Foundation for Science, E-48011, Bilbao, Spain ETSF Scientific Development Centre, Dpto. F´ısica de Materiales, Universidad del Pa´ıs Vasco, Centro de F´ısica de Materiales CSIC-UPV/EHU-MPC, Av. Tolosa 72, E-20018 San Sebasti´ an, Spain and Moscow Institute of Electronic Technology, Zelenograd, 124498 Russia (Dated: November 1, 2018) We derive a closed equation of motion for the current density of an inhomogeneous quantum many- body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integro-differential equation, whose only inputs are the one-particle density matrix and the pair correlation function of the ground-state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle, and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems. The dynamics of quantum many-particle systems poses a major challenge to computational physicists and chemists. In the study of ground-state properties one can rely on a variational principle, which enables a vari- ety of powerful statistical methods (in addition to exact diagonalization) such as the quantum variational Monte Carlo method and the diffusion Monte Carlo method. 1 In time-dependent situations, the absence of a practi- cal variational principle has greatly hindered the devel- opment of equally powerful methods. Yet it is hard to overestimate the importance of developing effective tech- niques to tackle the quantum dynamical problem. Such a technique could allow, for example, to follow in real time the evolution of chemical reactions, ionization and collision processes. One of the most successful computational meth- ods developed to date is the time-dependent density functional theory (TDDFT), or its more recent ver- sion – time-dependent current density functional theory (TDCDFT). 2 In this approach, the interacting electronic An alternative approach, which actually dates back to the early days of the quantum theory, attempts to calculate the collective variables of interest, density and current, without appealing to the underlying wave function. 7–9 This approach we call “quantum continuum mechanics” (QCM), because in analogy with classical theories of continuous media (elasticity and hydrodynam- ics), it attempts to describe the quantum many-body sys- tem without explicit reference to the individual particles system is treated as a noninteracting electronic system subjected to an effective scalar potential (a vector poten- tial in TDCDFT) which is self-consistently determined by the electronic density (or by the current density). 3,4 Thus, one avoids the formidable problem of solving the time-dependent Schr¨ odinger equation for the many-body wave function. Even this simplified problem, however, is quite complex, since it involves the determination of N time-dependent single particle orbitals – one for each particle. Furthermore, there are features such as multi- particle excitations 5 and dispersion forces 6 that are very difficult to treat within the conventional approximation schemes. I. INTRODUCTION arXiv:1001.0616v1 [cond-mat.other] 5 Jan 2010