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New Approaches in Many-Electron Theory Mainz, September 20-24, 2010. Continuum Mechanics of Quantum Many-Body Systems. J. Tao 1,2 , X. Gao 1,3 , G. Vignale 2 , I. V. Tokatly 4. Los Alamos National Lab 2. University of Missouri-Columbia 3. Zhejiang Normal University, China - PowerPoint PPT Presentation

Continuum Mechanics of Quantum Many-Body SystemsJ. Tao1,2, X. Gao1,3, G. Vignale2, I. V. Tokatly4 New Approaches in Many-Electron TheoryMainz, September 20-24, 2010Los Alamos National Lab2. University of Missouri-Columbia3. Zhejiang Normal University, China4. Universidad del Pais Vasco

Continuum Mechanics: what is it?An attempt to describe a complex many-body system in terms of a few collective variables -- density and current -- without reference to the underlying atomic structure. Classical examples are Hydrodynamics and Elasticity.Elasticity

Can continuum mechanics be applied to quantum mechanical systems?In principle, yes!Heisenberg Equations of Motion:The Runge-Gross theorem asserts that P(r,t) is a unique functional of the current density (and of the initial quantum state) -- thus closing the equations of motion.Local conservation of particle numberLocal conservation of momentum

Equilibrium of Quantum Mechanical SystemsThe two components of the stress tensorKohn-Sham orbitalsDensityJ. Tao, GV, I.V. Tokatly,PRL 100, 206405 (2008)

Pressure and shear forces in atoms

Continuum mechanics in the linear response regimeLinear response regime means that we are in a non-stationary state that is close to the ground-state, e.g. The displacement field associated with this excitation is the expectation value of the current in Yn0 divided by the ground-state density n0 and integrated over time

Continuum mechanics in the linear response regime - continuedExcitation energies in linear continuum mechanics are obtained by Fourier analyzing the displacement field

Make a change of coordinates to the comoving frame -- an accelerated reference frame that moves with the electron liquid so that the density is constant and the current density is zero everywhere. Continuum Mechanics Lagrangian formulationI. V. Tokatly, PRB 71, 165104 & 165105 (2005); PRB 75, 125105 (2007)

Assume that the wave function in the Lagrangian frame is time-independent - the time evolution of the system being entirely governed by the changing metric. We call this assumption the elastic approximation. This gives...Continuum Mechanics: the Elastic ApproximationThe elastic equation of motionY0[u] is the deformed ground state wave function:The elastic approximation is expected to work best in strongly correlated systems, and is fully justified for (1) High-frequency limit (2) One-electron systems. Notice that this is the opposite of an adiabatic approximation.

Elastic equation of motion: an elementary derivationand go to the high frequency limit:

Elastic equation of motion: a variational derivationThe LagrangianThe Euler-Lagrange equations of motionThis approach is easily generalizable to include static magnetic fields.

The elastic equation of motion: discussion3. The positivity of the eigenvalues (=excitation energies) is guaranteed by the stability of the ground-state4. All the excitations of one-particle systems are exactly reproduced.1. The linear functional F[u] is calculable from the exact one- and two body density matrices of the ground-state. The latter can be obtained from Quantum Monte Carlo calculations.

The sum rule Let ul(r) be a solution of the elastic eigenvalue problem with eigenvalue wl2. The following relation exists between wl2 and the exact excitation energies: Oscillator strengthf-sum rule

Example 1: Homogeneous electron gas Plasmon frequency

Example 2 Elastic equation of motion for 1-dimensional systemsa fourth-order integro-differential equation

A. Linear Harmonic Oscillator This equation can be solved analytically by expanding u(x) in a power series of x and requiring that the series terminates after a finite number of terms (thus ensuring zero current at infinity).B. Hydrogen atom (l=0)

- C. Two interacting particles in a 1D harmonic potential Spin singletWEAK CORRELATION w0>>1STRONG CORRELATION w0
Evolution of exact excitation energiesE/w0WEAK CORRELATIONSTRONG CORRELATIONKohns modeBreathing mode

Exact excitation energies (lines) vs QCM energies (dots)E/w0WEAK CORRELATIONSTRONG CORRELATION(1,0)(0,1)(2,0)(1,1)(3,0)(0,2)(2,1)(4,0)(1,2)(3,1)(5,0)

Strong Correlation Limit

(1,0)(1,1)(3,0),(1,2)

(2,1)(0,3)

Other planned applicationsPeriodic system: Replace bands of single particle excitations by bands of collective modesLuttinger liquid in a harmonic trap

Conclusions and speculations IOur Quantum Continuum Mechanics is a direct extension of the collective approximation (Bijl-Feynman) for the homogeneous electron gas to inhomogeneous quantum systems. We expect it to be useful for

Conclusions and speculations IIThis kernel should help us to study an importance of the space and time nonlocalities in the KS formulation of time-dependent CDFT.As a byproduct we got an explicit analytic representation of the exact xc kernel in the high-frequency (anti-adiabatic) limit It is interesting to try to interpolate between the adiabatic and anti-adiabatic extremes to construct a reasonable frequency-dependent functional

**Quantum mechanical probles as equilibrium problems - The quantum force and its components.*Quantum mechanical probles as equilibrium problems - The quantum force and its components.*Define wq.*Quantum mechanical probles as equilibrium problems - The quantum force and its components.*Quantum mechanical probles as equilibrium problems - The quantum force and its components.****