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.

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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