continuum mechanics of quantum many-body systems
DESCRIPTION
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 PresentationTRANSCRIPT
Continuum Mechanics of Quantum Continuum Mechanics of Quantum Many-Body SystemsMany-Body Systems
J. Tao1,2, X. Gao1,3, G. Vignale2, I. V. Tokatly4
New Approaches in Many-Electron TheoryMainz, September 20-24, 2010
1. Los Alamos National Lab2. University of Missouri-Columbia3. Zhejiang Normal University, China4. Universidad del Pais Vasco
Continuum Mechanics: what is it?
€
ρ0
∂ 2u(r, t)
∂t 2=
r ∇ ⋅
t σ (r, t)
Stress tensor1 2 3
Internal Force1 2 4 3 4
+ F(r, t)External Volume force
1 2 3
F
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”.
€
σ ij (r r , t) = B
Bulkmodulus
{r
∇ ⋅r u δij + S
Shearmodulus
{∂ui
∂rj
+∂u j
∂ri
−2
3
r ∇ ⋅
r u δ ij
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
€
ρ0
∂ 2u(r, t)
∂t 2= B + 1−
2
3
⎛
⎝ ⎜
⎞
⎠ ⎟S
⎡
⎣ ⎢
⎤
⎦ ⎥∇ ∇ ⋅u( ) + S∇ 2u
Internal forces1 2 4 4 4 4 4 3 4 4 4 4 4
+ F(r, t)Volume force1 2 3
Elasticity
€
u(r, t)
ρ0
displacement field
Can continuum mechanics be applied to quantum mechanical systems?
In principle, yes!
Heisenberg Equations of Motion:
€
∂n(r, t)
∂tDerivative of particle density
1 2 3 = −∇ ⋅ j(r, t)
Current density
1 2 3
€
∂j(r, t)
∂t= −∇ ⋅
t P (r, t)
Stresstensor
1 2 3 - n(r, t) ∇ V0(r) + V1(r, t)[ ]
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.
€
ˆ H (t) = ˆ T Kinetic Energy
{ + ˆ W Interaction Energy
{ + ˆ V 0 Externalstatic potential
{ + dr∫ V1(r, t) Externaltime-dependentpotential (small)
1 2 3 ˆ n (r, t)Hamiltonian:
Local conservation of particle number
Local conservation of momentum
€
−nr
∇V0 +r
∇VH( )Volume force density F
1 2 4 4 3 4 4 =
r ∇ ⋅
t P (r)
Stress tensor{
Equilibrium of Quantum Mechanical SystemsEquilibrium of Quantum Mechanical Systems
€
Pij (r) = Ps,ij (r) Non -interacting kinetic part (Kohn-Sham)
1 2 3 + Pxc,ij (r)
Exchange-correlation part1 2 3
The two components of the stress tensor
1s
2s
+
€
Ps,ij =1
2m∂iψ lσ
* ∂ jψ lσ + ∂ jψ lσ* ∂iψ lσ −
1
2∇ 2nδij
⎛
⎝ ⎜
⎞
⎠ ⎟
lσ
∑Kohn-Sham
orbitalsDensity
F
J. Tao, GV, I.V. Tokatly,PRL 100, 206405 (2008)
Pressure and shear forces in atomsPressure and shear forces in atoms
€
Ps,ij (r) = ps(r)δ ij + π s(r)rirj
r2−
1
3δ ij
⎡
⎣ ⎢
⎤
⎦ ⎥
pressure shear
€
π s(r) = ∂rψ lσ
2−
∂θψ lσ
2
r2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
lσ
∑
€
ps(r) =1
3∇ψ lσ
2
lσ
∑ −∇ 2n(r)
4
€
Fp = −∇ps
€
Fπ = −∇π s −2π s
r
Continuum mechanics in the linear response regime
0, 0
€
⋅⋅⋅
n, n
2, 2
1, 1
“Linear response regime” means that we are in a non-stationary state that is “close” to the ground-state, e.g.
€
Ψn0(t) = Ψ0 e−iE0t + λ Ψn e−iEnt
λ <<1
The displacement field associated with this excitation is the expectation value of the current in Ψn0 divided by the ground-state density n0 and integrated over time
€
un0(r, t) = λΨn
ˆ j (r) Ψ0
i En − E0( )n0(r)e−i En −E0( )t + c.c
Continuum mechanics in the linear Continuum mechanics in the linear response regime - continuedresponse regime - continued
Excitation energies in linear continuum mechanics are obtained by Fourier analyzing the displacement field
€
un0(r, t) =Ψn
ˆ j (r) Ψ0
i En − E0( )n0(r)e−i En −E0( )t + c.c
However, the correspondence between excited states and displacement fields can be many-to-one. Different excitations
€
⋅⋅⋅
0, 0
ExcitationsDisplacement
fields
can have the same displacement fields (up to a constant). This implies that the equation for the displacement field, while linear, cannot be rigorously cast as a conventional eigenvalue problem.
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 formulation
€
r ξ, t( ) = ξ + u(ξ, t)Displacement Field
1 2 3
Euclidean space
€
ds = dr ⋅dr
€
n r( )
Curved space
€
ds = gμν dξ μ dξ ν
€
n' ξ( ) ≡ n r(ξ )( )
€
∂u(ξ , t)
∂t=
j(ξ , t)
n(ξ , t),
I. V. Tokatly, PRB 71, 165104 & 165105 (2005); PRB 75, 125105 (2007)
Hamiltonian in Lagrangian frame
€
m∂ 2uμ (ξ , t)
∂t 2+ u ⋅∇
∂V0
∂ξ μ
= −1
n0(ξ )˜ ψ (t)
δ ˜ H [u]
δuμ
˜ ψ (t)1
−∂V1
∂ξ μ
Generalized force
Wave function in Lagrangian frame
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 Approximation
The elastic equation of motion
€
m˙ ̇ u (ξ , t) = F[u(ξ , t)]−∇V1(ξ , t)
€
F[u(ξ , t)] = −1
n0
δ
δu(ξ , t)Ψ0[u] ˆ T + ˆ W + ˆ V 0 Ψ0[u]
2
= E2 [u ], energy of deformed stateto second order in u .
1 2 4 4 4 4 3 4 4 4 4
€
r1,..., rN Ψ0[u] = Ψ0(r1 − u(r1),..., rN − u(rN ))g−1/ 4 (r1)... g−1/ 4 (rN )
0[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 Notice that this is the opposite opposite of an of an adiabatic adiabatic approximation.approximation.
Elastic equation of motion: an elementary derivation
Start from the equation for the linear response of the current:
€
j(ω) = n0A1 (ω) + K(ω) ⋅A1 (ω)
and go to the high frequency limit:
Inverting Eq. (1) to first order we get
€
A1 (ω) =1
n0
j(ω) +1
n0
M
ω2
1
n0
⋅ j(r',ω)
€
n0(r)˙ ̇ u (r, t) = dr'∫ M(r, r') ⋅u(r', t) − n0(r)∇V1(r, t)
Finally, using
€
A1 (ω) = −∇V1(ω)
iω
€
j(ω) = −iωn0u(ω)
€
F[u] =δE2[u]
δu(r, t)
€
K(ω) = j;jω ω →∞
⏐ → ⏐ ⏐ M
ω2
€
M = − Ψ0 [[ ˆ H , j], j] Ψ0
€
First spectral moment : -2
πdω
0
∞
∫ ω ImK(ω)
Elastic equation of motion: a variational derivation
The Lagrangian
€
L(ϕ ,u, ˙ ϕ , ˙ u ) = ψ ( t) i∂t − ˆ H ψ (t)
The Euler-Lagrange equations of motion
€
˙ u = ∇ϕ
∇ ˙ ϕ = ˙ ̇ u = −1
n0
δE2[u]
δu
⎧ ⎨ ⎪
⎩ ⎪
This approach is easily generalizable to include static magnetic fields.
The variational Ansatz
€
ψ(t) = ei ˆ n (r )ϕ (r,t )dr∫ e
−im ˆ j (r )⋅u(r,t )dr∫ ψ 0
ground-state
density operator
current density operator
phase displacement
The elastic equation of motion: discussion
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).
€
uλ∫ (r) ⋅uλ '(r)n0(r)dr = δλλ '
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.
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 The sum rule Let u(r) be a solution of the elastic eigenvalue problem with eigenvalue
The following relation exists between
and the exact excitation energies:
€
2 = fn
λ En − E0( )2
n
∑
€
fnλ =
2 dr uλ (r) ⋅ j0n (r)∫2
En − E0
€
j0n (r) = Ψ0ˆ j (r) Ψn( )Oscillator strength
€
fnλ
n
∑ =1f-sum rule
Exact excitation energies
Elastic QCMA group of levels may collapse into onebut the spectral weight is preservedwithin each group!
rigorously satisfied in 1D systems
Example 1: Homogeneous electron gas Example 1: Homogeneous electron gas
€
uLq r( ) = ˆ q e iq⋅r
€
L2 q( ) = ωp
2 + 2t(n)q2 +q4
4
+ωp
2
n
dq'
2π( )3
ˆ q ⋅ ˆ q '( )2
S q − q'( ) − S q'( )[ ]∫
LONGITUDINAL
Multiparticle excitations
Multiparticle excitations
q/kF
/E
F
1 particle excitations L
T
p
Plasmon frequency
Similar, but not identical to Bijl-Feynman theory:
€
L (q) =q2
2S(q)€
Ψq ∝ ˆ n q Ψ0
€
uTq r( ) = ˆ t qe iq⋅r
€
T2 q( ) =
2t(n)
3q2
+ωp
2
n
dq'
2π( )3
ˆ q × ˆ q '2
S q − q'( ) − S q'( )[ ]∫
TRANSVERSAL
static structure factor
Example 2 Example 2 Elastic equation of motion for 1-dimensional Elastic equation of motion for 1-dimensional
systemssystems
€
m˙ ̇ u = −u ′ ′ V 0 +(3T0 ′ u ′ )
n0
−(n0 ′ ′ u ′ ′ )
4n0
+ d ′ x ∫ K(x, x ') u(x) − u(x ')[ ]
€
T0(x) =1
2∂x∂ ′ x ρ(x, ′ x )
x= x '
One−particledensity matrix
1 2 4 3 4 −
′ ′ n 0(x)
4
€
K(x,x ') = ρ 2(x, ′ x )Two−particledensity matrix
1 2 4 3 4 w' '(x − x')Second derivative of interaction
1 2 4 3 4
a fourth-order integro-differential equation
From Quantum Monte Carlo
A. Linear Harmonic Oscillator A. Linear Harmonic Oscillator
€
1
4
d4u
dx 4− x
d3u
dx 3+ (x 2 − 2)
d2u
dx 2+ 3x
du
dx+ 1−
ω2
ω02
⎛
⎝ ⎜
⎞
⎠ ⎟u = 0
€
n = ±nω0
un (x) = Hn−1(x)
Eigenvalues:
Eigenfunctions:
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 (B. Hydrogen atom (ll=0)=0)
€
1
4
d4ur
dr4− 1−
1
r
⎛
⎝ ⎜
⎞
⎠ ⎟d3ur
dr3+ 1−
2
r−
1
r2
⎛
⎝ ⎜
⎞
⎠ ⎟d2ur
dr2+
3
r2
dur
dr+
2
r3+
ω2
Z 4
⎛
⎝ ⎜
⎞
⎠ ⎟ur = 0
€
n =Z 2
21−
1
n2
⎛
⎝ ⎜
⎞
⎠ ⎟Eigenvalues:
Eigenfunctions:
€
unr(r) = Ln−22 2r
n
⎛
⎝ ⎜
⎞
⎠ ⎟
C. Two interacting particles in a 1D harmonic C. Two interacting particles in a 1D harmonic potential – Spin singletpotential – Spin singlet
Parabolic trap
€
H =P 2
4+
ω02
2 X 2
Center of Mass6 7 4 8 4
+ p2 +ω0
2
8x 2 +
1
x 2 + a2
Soft Coulomb repulsion
1 2 4 3 4
Relative Motion6 7 4 4 4 8 4 4 4
€
Enm = ω0 n + m 3( )
WEAK CORRELATION >>1
€
Ψnm (X, x) = φn (X)ψ m (x)
€
Enm = ω0 n + 2m( )
STRONG CORRELATION <<1
n0(x) n0(x)
n,m non-negative integers
Evolution of exact excitation energiesEvolution of exact excitation energies
€
2 ω0
E/
0
€
(1,0)
€
(0,1)
€
(2,0)
€
(1,1)
€
(3,0)€
(4,0)€
(5,0)
€
(3,1)
€
(1,2)
€
(2,1)
€
(0,2)
WEAK CORRELATION
STRONG CORRELATION
Kohn’s mode
Breathing mode
Exact excitation energies (lines) vs Exact excitation energies (lines) vs QCM energies (dots)QCM energies (dots)E
/0
€
2 ω0
WEAK CORRELATION
STRONG CORRELATION
(1,0)
(0,1)(2,0)
(1,1)(3,0)(0,2)(2,1)(4,0)(1,2)(3,1)(5,0)
3.94
2.63
Strong Correlation LimitStrong 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
€
nm = ω0 2 + 3 3k + 6k(k −1) 2 − 3( ) − (−1)m 2 − 3( )k
€
k = n + m −1( )
(3,0)
(1,2)
3
3.944.46 3.46
(2,0)
(0,2)
22.63
even (1,0)
(1,1)
(3,0),(1,2)
odd
(2,1)(0,3)
Other planned applicationsOther planned applicationsPeriodic system: Replace bands of
single –particle excitations by bands of collective modes
Luttinger liquid in a harmonic trap
€
g1D =2h2a3D
ma⊥2
3D scattering length
radius of tube
€
=a⊥
Conclusions and speculations IConclusions 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
- Possible nonlocal refinement of the plasmon pole approximation in GW calculations
- The theory of dispersive Van derWaals forces, especially in complex geometries
- 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 IIConclusions and speculations II
-This kernel should help us to study an importance of the space and time nonlocalities in the KS formulation of time-dependent CDFT.
2. 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