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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  uij  S 
r  r  3  uij 

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 2t 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
functionalof 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   jl i l   nij 


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 eiE0 t   Yn eiEn 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)  
eiE n E 0 t  c.c
iE 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) 
eiE n E 0 t  c.c
iE 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 ))g1/ 4 (r1)... g1/ 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()  in0u()
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) it  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 2and
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 ' Sq  q'  Sq'

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
 dxK(x, x')u(x)  u(x')
a fourth-order integro-differential equation
1
n
0(x)

T0 (x) = xx(x, x ) x x' 
1 4 2 43
2
4
Oneparticle
density matrix

K(x, x') = 
x ) w''(x
1 4 2 4x')
3
142 (x,
2 43
Twoparticle 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)  Hn1 (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)  L2n2  
 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
ma2
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