NTNU Faculty of Natural Sciences Department of Physics

NTNU Faculty of Natural Sciences Department of Physics

Exam TFY 4210 Quantum theory of many-particle systems, spring 2017

Lecturer: Assistant Professor Pietro Ballone Department of Physics

Phone: 73593645 Examination support:

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Rottmann: Matematisk Formelsamling Rottmann: Matematische Formelsammlung Barnett & Cronin: Mathematical Formulae

The exam has 5 problems, with subproblems (i), (ii), ...

All subproblems have the same weight.

The sum the weights is 125% of the full mark .

There are 7 pages in total. Some useful formulas are given on the last page

Thursday, 1 June, 2017

09.00-13.00h

(i) Compute the matrix element:

x0|aˆ_αaˆ_βˆa^:_αˆa^:_β |0y (1) for Fermions and for Bosons.

Distinguish the case α‰β and α“β.

(ii) Consider a many-electron system.

The number of particles is given by the operator:

Nˆ “ ÿ

α

ˆ

a^:_αaˆ_α (2)

where ˆa^:_α, ˆa_α are creation and annihilation operators for the stateα.

Show that:

rN ,ˆ aˆ_αs “ ´ˆa_α (3)

rN ,ˆ ˆa^:_αs “ˆa^:_α (4) (iii) Let us consider the Boson operators a^:_λ and a_λ, and let fpa^:_λq or fpa_λq be polynomial functions of their argument.

For instance:

fpa_λq “ c₀`c<sub>1</sub>a<sub>λ</sub>`c₂a²_λ...`c_naⁿ_λ (5) Show that:

ra_λ, fpa^:_λqs “ Bfpa^:_λq

Ba^:_λ (6)

and:

ra^:_λ, fpa_λqs “ ´Bfpa_λq

Ba_λ (7)

The time ordered correlation function of two operators Aˆand Bˆ is defined as:

χ^T_ABptq ” ´ixΨ₀ |TrAptqˆ Bp0qs |ˆ Ψ₀y (8) where|Ψ₀yis the ground state, the time dependence in the Heisenberg representation is:

Aptq “ˆ e^iHtˆ Aeˆ ^´iHtˆ (9) and the time ordering operator is by:

TrAptˆ 1qBˆpt2qs “

$

’’

&

’’

%

Aptˆ ₁qBˆpt₂q t₁ ąt₂

Bptˆ ₂qAptˆ ₁q t₂ ąt₁ (10)

(Notice: there is no (-1) factor associated to the interchange of Fermion operators).

(i) Compute the Fourier transform:

χ^T_ABpωq “ lim

ηÑ0^`

ż`8

´8

χ^T_ABptqe^iωt´η|t|dt (11) and show that it is given by:

χ^T_ABpωq “ ´iÿ

n

ˆ A_0nB_n0

ω´ωn0 `iη ´ B<sub>0n</sub>A<sub>n0</sub> ω`ωn0´iη

˙

(12) where A_0n “ xΨ₀ | Aˆ | Ψ_ny, tΨ₀,Ψ₁, ...u are eigenstates of the Hamiltonian, and

¯

hω_n0 “E_n´E₀ ą0.

The causal version of the same correlation function is given by:

χABptq ” ´iθptqxΨ0 | rAptq,ˆ Bp0qs |ˆ Ψ0y (13) where r.., ..s is the commutator.

(ii) Compute the Fourier transform of χ_ABpωq and compare it to that ofχ^T_AB. (iii) Comment on the position of the poles in the complex ωplane forχ^T_ABpωqand χ_ABpωq.

(i) The exchange-correlation energy functional of a many-electron system in 1D is given by:

E_XCrρs “ ż

αrρpxqs^4{3dx` 1 2

ż Kpρq

„dρpxq dx

2

dx (14)

where α is a positive numerical coefficient.

Compute the exchange-correlation potential:

µ_XCpxq “ δE_XC

δρpxq (15)

(ii) According to Hartree-Fock, the total energy epr_sq per particle of the spin unpolarised homogeneous electron liquid is:

epr_sq “e_kpr_sq `e_xpr_sq “ 2.21

r_s² ´0.916

r_s (16)

wherer_sis the Wigner-Seitz radius (r_s“ r3{p4πρqs^p1{3q,ρbeing the electron density), e_kpr_sqis the kinetic energy per particle ande_xpr_sqis the exchange energy per particle.

Numerical coefficients are in Rydberg energy units.

Compute the pressure P as a function of the density, with pressure defined as:

P “ ´ ˆBE

BV

˙

N

(17) where E is the system ground state energy, V is the volume, and the derivative is computed at constant number of particles.

Is there an optimal density for the homogeneous electron liquid, and, in such a case, could you estimate this optimal density?

The order n term in the perturbative expansion of the time ordered correlation function χ^T_ABptq is:

1 n!

ˆ

´i

¯ h

˙nż8

´8

dt₁...

ż8

´8

dt_nxΦ₀ |TrAˆ_IptqBˆ_IHˆ₁pt₁qHˆ₁pt₂q...Hˆ₁pt_nqs |Φ₀y (18) For the sake of definiteness, assume that Aˆ and Bˆ are single particle operators:

Aˆ“ ÿ

αβ

Aαβˆa^:_αˆaβ (19)

Bˆ “ ÿ

γδ

B_γδˆa^:_γˆa_δ (20)

and the perturbation Hamiltonian contains a pair interaction term:

Hˆ_1I “ 1 2

ÿ

abcd

v_abcdˆa^:_aˆa^:_bˆa_cˆa_d (21)

(i) List all the pairing schemes of creation and annihilation operator for the order n“0 term of Eq. 18.

(ii) Count all the pairing schemes for the n “ 1 term (you don’t need to write them down) and verify that they are 4!“24

Argue that in general the number of all pairing schemes is p2n`2q! for the order n term of Eq. 18.

(iii) Write down the integral corresponding to the zero order diagram:

Figure 1: Zero order diagram

Please use the reciprocal space notation (consistent with the labels on the figure).

Consider a system of Fermions interacting through the pair potential:

vprq “e²e^´λr

r (22)

whose Fourier transform is:

v_q “ 4πe²

q²`λ² (23)

To first order in the interaction strength, the energy of the state that arises from the non-interacting state with momentum occupation numbers N_kσ is given by:

ErN_kσs “ ÿ

kσ

¯ h²k²

2m N_kσ` 1 2V

ÿ

kσk¹σ¹

rv₀´v_k´k¹δ_σσ¹sN_kσN_k¹_σ¹ (24)

(a) Substitute N_kσ “ N_kσ^p0q ` δN_kσ (where N_kσ^p0q “ Θpk_F ´kq are the ground state occupation numbers) to obtain the Landau energy functional. Give explicit expressions for the quasi-particle energy and for the Landau interaction function.

(b) Calculate the Landau parameter F₁^s and the effective mass of the quasi- particle.

What happens for λÑ0?

Commutation relations for Bosons:

rˆa_α,aˆ_βs “ rˆa^:_α,ˆa^:_βs “0 (25) rˆa_α,ˆa^:_βs “δ_αβ (26) Anti-commutation relations for Fermions:

tˆa_α,ˆa_βu “ tˆa^:_α,ˆa^:_βu “ 0 (27) tˆa_α,ˆa^:_βu “δ_αβ (28) Fourier transform:

fpωq “ ż8

´8

fpτqe^iωτdτ (29)

fpτq “ ż8

´8

fpωqe^´iωτdω

2π (30)

Special relation:

1

x˘iη “P ˆ1

x

˙

¯iπδpxq (31)

Chain-rule for thermodynamic derivatives:

V B

BV “ ´ρ B Bρ “ r_s

3 d

dr_s (32)

In this equationrsis the Wigner-Seitz radiusrs “ r3{p4πρqs^p1{3q,ρ being the electron density.

Landau energy functional for the normal electron liquid:

ErN_k,σs “E₀` ÿ

k,σ

E_k,σδN_k,σ`1 2

ÿ

k,σ,k¹,σ¹

f_k,σ,k¹_,σ¹δN_k,σδN_k¹_,σ¹ (33)

• E_k,σ is the isolated quasi-particle energy;

• fk,σ,k¹,σ¹ is the Landau interaction function;

• δN_k,σ is the deviation of the quasi-particle distribution from the ground state one (T “0 K).