Continuum mechanics for quantum many-body systems: Linear response regime

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 integrodifferential 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.

Figure 1

(a) Longitudinal and (b) transverse modes for a homogeneous electron gas at $r_s=1,3,5$. Wave vector $q$ is in units of $k_F$ and frequency is in units of $2E_F$. The curves labeled ${\omega }_{+}\left(q\right)$ and ${\omega }_{-}\left(q\right)$ are the boundaries of the electron-hole continuum (Ref. 38). Single-particle excitations exist for ${\omega }_{-}\left(q\right)<\omega <{\omega }_{+}\left(q\right)$ whereas multiparticle excitations are distributed all over the plane. The elastic approximation replaces the exact spectrum by the two branches ${\omega }_L\left(q\right)$ and ${\omega }_T\left(q\right)$, which carry the entire spectral weight. Notice that the $r_s$ dependence is barely discernible on a large $q$ scale but becomes clearly visible at smaller $q$ as shown in the insets.

Figure 2

(Color online) Evolution of the excitation energies for two electrons in a one-dimensional harmonic trap. Solid lines denote exact excitation energies, labeled by $\left(n,m\right)$ as explained in the text. Solid dots indicate the calculated eigenvalues of the QCM equation of motion. Crosses on the right denote the strong-correlation limit of the eigenvalues, given by Eq. (113).

Figure 3

(Color online) Top panel: the displacement field $u_{nm}\left(x\right)$ for $\left(n,m\right)=\left(0,1\right)$, (0,2), and (0,3) in the strong correlation limit. Bottom panel: the same for $\left(n,m\right)=\left(1,1\right)$, (1,2), and (2,1). The thin solid lines represent the density profile. The large value of the displacement field for $x\sim 0$ does not have a physical significance since the density is exponentially small in that region.

Figure 4

(Color online) Evolution of the displacement fields of excitations (1,1) and (3,0) (even), and (2,0) and (2,1) (odd) as a function of correlation strength $\chi ={\omega }_0^{-1/2}$, as shown in the top left panel. Notice the variation in the number of nodes as $\chi$ increases from the weakly correlated with the strongly correlated limit.