* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Vignale - www2.mpip
Perturbation theory wikipedia , lookup
Quantum field theory wikipedia , lookup
Coupled cluster wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Renormalization wikipedia , lookup
Schrödinger equation wikipedia , lookup
Wave function wikipedia , lookup
Probability amplitude wikipedia , lookup
Coherent states wikipedia , lookup
Density functional theory wikipedia , lookup
Renormalization group wikipedia , lookup
Particle in a box wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Hydrogen atom wikipedia , lookup
EPR paradox wikipedia , lookup
Matter wave wikipedia , lookup
Wave–particle duality wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Noether's theorem wikipedia , lookup
Scalar field theory wikipedia , lookup
Dirac equation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Density matrix wikipedia , lookup
Hidden variable theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Canonical quantization wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
New Approaches in Many-Electron Theory Mainz, September 20-24, 2010 Continuum Mechanics of Quantum Many-Body Systems J. Tao1,2, X. Gao1,3, G. Vignale2, I. V. Tokatly4 1. Los Alamos National Lab 2. University of Missouri-Columbia 3. Zhejiang Normal University, China 4. 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 0 2u(r,t) r t 0 (2r,t) + F(r,t) 2 1 3 123 t Stress tensor External Volume force displacement field u(r,t) 1 4 2 43 Internal Force u u 2 j i ij (r ,t) B uij S r r 3 uij Bulk Shear j i modulus modulus F 2u(r,t) 2 2 0 B 1 S u S u + F(r,t) t 2 3 Volume force Internal forces Can continuum mechanics be applied to quantum mechanical systems? In principle, yes! Hamiltonian: ˆ Hˆ (t) T {ˆ { W Kinetic Energy Interaction Energy ˆ V {0 External static potential dr ˆ V (r,t) 112 3 n(r,t) External time -dependent potential (small) Heisenberg Equations of Motion: Local conservation of particle number Local conservation of momentum n(r,t) 1 j(r,t) 23 1 2t 3 Current Derivative of particle density density t j(r,t) P 1(r,t) 2 3 - n(r,t) V0 (r) V1(r, t) t Stress tensor The Runge-Gross theorem asserts that P(r,t) is a unique functionalof the current density (and of the initial quantum state) -- thus closing the equations of motion. Equilibrium of Quantum Mechanical Systems J. Tao, GV, I.V. Tokatly,PRL 100, 206405 (2008) 2s r r r t n V0 VH P (r) { 1 4 4 2 4 43 Stress tensor 1s + Volume force density F F The two components of the stress tensor Pij (r) Ps,ij (r) 123 Non -interacting kinetic part (Kohn -Sham) Pxc,ij (r) 123 Exchange-correlation part * 1 1 2 * Ps,ij i l j l jl i l nij 2m l 2 Kohn-Sham orbitals Density Pressure and shear forces in atoms r r 1 Ps,ij (r) ps (r)ij s (r) i 2j ij 3 r pressure shear l 2 s (r) r l r2 l 2 F s s r 1 n(r) 2 ps (r) l 3 l 4 2 Fp ps 2 Continuum mechanics in the linear response regime Yn, En Y2, E2 Y1, E1 “Linear response regime” means that we are in a non-stationary state that is “close” to the ground-state, e.g. Yn 0 (t) Y0 eiE0 t Yn eiEn t 1 Y0, E0 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 Yn ˆj(r) Y0 un 0 (r,t) eiE n E 0 t c.c iE n E 0 n 0 (r) Continuum mechanics in the linear response regime - continued Excitation energies in linear continuum mechanics are obtained by Fourier analyzing the displacement field Yn ˆj(r) Y0 un 0 (r,t) eiE n E 0 t c.c iE n E 0 n 0 (r) However, the correspondence between excited states and displacement fields can be many-to-one. Different excitations can have the same displacement Excitations fields (up to a constant). This Displacement fields implies that the equation for the displacement field, while linear, cannot be rigorously cast as a conventional eigenvalue Y0, E0 problem. Continuum Mechanics – Lagrangian formulation I. V. Tokatly, PRB 71, 165104 & 165105 (2005); PRB 75, 125105 (2007) 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. r,t u(,t) 123 ds g d d Displacement Field ds dr dr n r Euclidean space u( ,t) j( ,t) , t n( ,t) Wave function in Lagrangian frame n' n r( ) Curved space Hamiltonian in Lagrangian frame 2 u ( ,t) V0 1 ˜ H˜ [u] ˜ V1 m u ( t) ( t) t 2 n0 ( ) u 1 Generalized force Continuum Mechanics: the Elastic Approximation 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... The elastic equation of motion Ý Ý(,t) F[u(,t)] V1(,t) mu F[u(,t)] 1 Y0[u] Tˆ Wˆ Vˆ0 Y0 [u] 2 n0 u(,t) 1 4 4 4 4 2 4 4 4 4 3 E 2 [u ], energy of deformed state to second order in u. Y0[u] is the deformed ground state wave function: r1,..., rN Y0[u] Y0 (r1 u(r1),..., rN u(rN ))g1/ 4 (r1)... g1/ 4 (rN ) 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 derivation Start from the equation for the linear response of the current: j() n0A1 () K() A1 () and go to the high frequency limit: K() j; j M 2 M = Y0 [[ Hˆ , j], j] Y0 First spectral moment : - Inverting Eq. (1) to first order we get 2 d ImK( ) 0 1 1 M 1 A 1 () j( ) j(r', ) 2 n0 n0 n0 Finally, using j() in0u() V1 ( ) A 1 ( ) i Ý Ý(r,t) n0 (r)u dr'M(r,r') u(r',t) n (r)V (r, t) 0 F[u] 1 E 2 [u] u(r,t) Elastic equation of motion: a variational derivation The variational Ansatz density operator (t) e current density operator i nˆ (r) (r,t )dr im phase e ˆj(r)u(r,t )dr displacement 0 groundstate The Lagrangian The Euler-Lagrange equations of motion Ý, uÝ) = (t) it Hˆ (t) L(,u, uÝ= 1 E 2[u] Ý u Ý Ý= n 0 u This approach is easily generalizable to include static magnetic fields. The elastic equation of motion: discussion 1. The linear functional F[u] is calculable from the exact oneand two body density matrices of the ground-state. The latter can be obtained from Quantum Monte Carlo calculations. 2. The eigenvalue problem is hermitian and yields a complete set of orthonormal eigenfunction. Orthonormality defined with respect to a modified scalar product with weight n0(r). u (r) u (r)n (r)dr ' 0 ' 3. The positivity of the eigenvalues (=excitation energies) is guaranteed by the stability of the ground-state 4. All the excitations of one-particle systems are exactly reproduced. The sum rule Let u(r) be a solution of the elastic eigenvalue problem with eigenvalue 2.The following relation exists between 2and the exact excitation energies: 2 f n E n E 0 2 n Oscillator strength fn rule f-sum dr u (r) j 0n n Elastic QCM (r) n 1 j 0n En E0 f Exact excitation energies 2 2 (r) Y0 ˆj(r) Yn rigorously satisfied in 1D systems A group of levels may collapse into one but the spectral weight is preserved within each group! Example 1: Homogeneous electron gas LONGITUDINAL TRANSVERSAL uLq r qˆ e iq r uTq r tˆ q e iq r L2 q 2p 2t(n)q 2 2p n 4 q 4 2dq' qˆ qˆ ' Sq q' Sq' 2 3 static structure 2t(n) 2 factor T2 q q 3 2 dq' 2 ˆ ˆ p 3 q q' S q q' S q' n 2 /EF 1 particle excitations Multiparticle excitations Similar, but not identical to Bijl-Feynman theory: L Yq nˆ q Y0 Plasmon frequency T Multiparticle excitations p q/kF q2 L (q) 2S(q) Example 2 Elastic equation of motion for 1-dimensional systems (3T0u) (n0u) Ý= uV0 muÝ n0 4n0 dxK(x, x')u(x) u(x') a fourth-order integro-differential equation 1 n 0(x) T0 (x) = xx(x, x ) x x' 1 4 2 43 2 4 Oneparticle density matrix K(x, x') = x ) w''(x 1 4 2 4x') 3 142 (x, 2 43 Twoparticle Second derivative density matrix of interaction From Quantum Monte Carlo A. Linear Harmonic Oscillator 1 d 4u d 3u d 2u du 2 2 x 3 (x 2) 2 3x 1 2 u 0 4 4 dx dx dx dx 0 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). Eigenvalues: Eigenfunctions: n n 0 un (x) Hn1 (x) B. Hydrogen atom (l=0) 1 d 4 u r 1 d 3u r 2 1 d 2u r 3 du r 2 2 1 3 1 2 2 2 3 4 u r 0 4 r dr r r dr 4 dr r dr r Z Eigenvalues: Eigenfunctions: Z 2 1 n 1 2 2 n 2r unr (r) L2n2 n C. Two interacting particles in a 1D harmonic potential – Spin singlet 6 Center 4 7of Mass 4 8 6 4 4 Relative 4 7 Motion 4 4 4 8 P 2 02 2 02 2 1 2 H X p x 2 2 4 2 8 x a 1 4 2 43 Ynm (X, x) n (X)m (x) Parabolic trap n,m non-negative integers WEAK CORRELATION 0>>1 E nm Soft Coulomb repulsion STRONG CORRELATION 0<<1 0 n 2m E nm 0 n m 3 n0(x) n0(x) Evolution of exact excitation energies E/0 Breathing mode Kohn’s mode WEAK CORRELATION 2 0 STRONG CORRELATION (5,0) (3,1) (1,2) (4,0) (2,1) (0,2) (3,0) (1,1) (2,0) (0,1) (1,0) Exact excitation energies (lines) vs QCM energies (dots) E/0 (5,0) (3,1) (1,2) (4,0) (2,1) (0,2) (3,0) (1,1) (2,0) (0,1) (1,0) WEAK CORRELATION 2 0 STRONG CORRELATION 3.94 2.63 Strong Correlation Limit even odd States with the same n+m and the same parity of m have identical displacement fields. At the QCM level they collapse into a single mode with energy 2 3 3k 6k(k 1) 2 3 (1) m 2 3 k k n m 1 (1,2) (3,0) nm 4.46 3 3.94 0 (0,2) 3.46 2.63 (2,0) 2 even (1,0) (1,1) (3,0),(1,2) odd (2,1)(0,3) Other planned applications Periodic system: Replace bands of single –particle excitations by bands of collective modes Luttinger liquid in a harmonic trap a g1D 2 2h a3D ma2 3D scattering length radius of tube Conclusions and speculations I 1. Our 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 - The theory of dispersive Van derWaals forces, especially in complex geometries - Possible nonlocal refinement of the plasmon pole approximation in GW calculations - Studying dynamics in the strongly correlated regime, which is dominated by a collective response (e.g., collective modes in the quantum Hall regime) Conclusions and speculations II 2. As a byproduct we got an explicit analytic representation of the exact xc kernel in the high-frequency (anti-adiabatic) limit -This kernel should help us to study an importance of the space and time nonlocalities in the KS formulation of timedependent CDFT. -It is interesting to try to interpolate between the adiabatic and anti-adiabatic extremes to construct a reasonable frequencydependent functional