Random Phase Approximation and Extensions

Random Phase Approximation and Extensions

Peter Schuck

IPN Orsay – LPMMC Grenoble

Collaboration: D.S. Delion, M. Jemai, J. Dukelsky, M. Tohyama Jemai, Delion, P.Sch, PRC88(2013)044004

Outline

1. Standard RPA and the quasi boson approximation

2. Improved ground state with Coupled Cluster Theory (CCT) 3. Extension to Self-Consistent RPA (SCRPA);renormalised RPA 4. Some Results

5. Discussion

6. Higher RPA’s; odd-particle number RPA;Second RPA 7. Conclusions

Random Phase Approximation from the Nuclear Physics Point of View

The nucleus is a SELFBOUND system of FOUR different fermions:

neutrons, spin up/down—-protons, spin up/down

Ground state: HARTREE-FOCK Mean-Field

relativistic and non-relativistic

Excited states: QUADRUPOLE DEFORMATIONS, BREATHING (COMPRESSION) MODE, etc. →

TIME DEPENDENT HF

id

dtρ = [hˆ ^HF, ˆρ]

Small amplitude...Linear response → ˆ

ρ = ˆρ0+ δρ This leads tostandardRPA eqs:

[Ων− (εk− εk^′)]δρk,k^′= (n⁰_k′−n⁰_k)X

l,l^′

v_k,l^′;k^′lδρl,l^′

δρph≡X_ph δρhp≡Y_ph -

h^HF from ENERGY DENSITY FUNCTIONAL (EFFECTIVE FORCES) (about 12 adjustable parameters). Microscopic nucleon- nucleon force unknown!!

Energy Density Functional:

ε(ρ, τ, ∇ρ, τρ, ...)

ρ(r) =X

i

φ(r)φ^∗(r) τ (r) =X

i

∇φ(r)∇φ^∗(r)

minimisation with respect toφ’s→ HF eqs

h^HF[φk]φi(r) = εiφi(r)

Vibrations around HF minimum (RPA):

A B B^∗ A^∗

 X^ν Y^ν



=Eν

 X^ν

−Y^ν



withAph;p^′h^′ = δ²ε δρphδρp^′h^′

andBph;p^′h^′ = δ²ε δρphδρh^′p^′

GROUND STATE ENERGY:

E0=E^HF +X

ν

X

ph

Eν|Y_ph^ν|² - (Applications by other speakers)

Some appreciated properties of RPA HF: always some symmetries are broken!

-

1)Translational Invariance 2)Rotational Invariance 3)Particle Number (BCS) etc.

-

HF-RPA: Goldstone mode atEν=0(Spurious mode) Translation: _2Am^P2 =E_kin^total

-

Conservation laws, Ward Identitiesfullfilled!!

Sum rule, etc.

Very well established scheme in nuclear physics!

EXTENSIONS OF RPA THEORY.

-

SELFCONSISTENT-RPA -

also

renormalised RPA -

standard RPA→ Quasi boson approximation → -

First:ideal bosons→

Reminder: HFB theory for fermions

BCS ground state

|BCSi = Πk(uk +v_ka⁺_ka⁺_¯

k)|vaci ∝e

P

kvk uka⁺_ka⁺_¯

k|vaci quasi-particles:

α⁺_k =u_ka⁺_k −v_ka¯k

Then, we have the ’killing’ property

αk|BCSi =0.

Theu,v coefficients can be determined from minimisation of a sum-rule

E_k =h{αk, [H, α⁺_k]}i

h{αk, α⁺_k}i ; {.., ..} = anticommutator The minimisation leads to standard BCS eqs

 h ∆

∆⁺ −h^∗

 u v



=E

u v

 ,

Hartree-Fock Bogoliubov theory for bosons

The Bogoliubovunitarytransformation for bosons is q^†_ν=X

α

Uναb^†_α−Vναbα

 ↔ [b^†_α=X

ν

[Uανq^†_ν+Vανqν] .

where the coefficientsUandV are determined by minimisation of

eν= h0|[qν, [H,q_ν^†]]|0i

h0|[qν,qν^†]|0i ; H=X

tb^†b+X

vb^†b^†bb

|0i ≡ |HFBi qν|HFBi =0, |HFBi =e^PVUb†b†|vaci The minimisation leads to the following set of equations

 h[U,V] ∆[U,V]

∆^∗[U,V] h^∗[U,V]

 U V



=E

 U

−V

 ,

with

h[U,V] = h0|[b, [H,b^†]]|0i ; ∆[U,V] =gh0|bb|0i =gUV . (1)

= h0|[b, [H,b]]|0i .

SCRPA for particle-hole excitations

RPA excitation operator in the particle-hole channel is Q^†_ν=X

ph

hX_ph^νA^†_ph−Y_ph^νA_phi ,

A^†_ph=X

ν

[X_ph^νQ^†_ν+Y_ph^νQ_ν]

A^†_ph=a^†_pah ∼B^†_ph Q⁺_B =XB⁺−YB where a^†,a are fermion creation/destruction operators.

-

It is like Bogoliubov unitary transformation for ph pairs!

-

The operator should have the properties

Q_ν^†|RPAi = |νi , Qν|RPAi =0. |RPAiB ∼e

Pz_p1p2h1h2B⁺

p1h1B⁺

p2h2|HF) QB|RPAi =0

Evidence of quasi-boson approximation from response fct

(ω −ek+eq)Rkq,k^′q^′ = n⁰_q−n⁰_k ω −e_k+eq

[δkk^′δqq^′+X

k1q1

v¯kq1qk1Rk1q1k^′q^′]

with

R^t−t_kq,k^′′q^′ = −ihT(a⁺_kaq)t(a⁺_q′ak^′)t^′i

= −iΘ(t−t^′)h(a⁺_kaq)t(a⁺_q′ak^′)t^′i −iΘ(t^′−t)h(a⁺_q′ak^′)t^′(a⁺_kaq)ti Equation of motion

(i ∂

∂t −e_k+e_q)R_kq,k^t−t′′q^′ = δ(t−t^′)h[(a⁺_ka_q),a⁺_q′a_k^′]i + ...

Then

h[(a⁺_kaq),a⁺_q′ak^′]i = (nq−nk)δkk^′δqq^′

with nk = ha⁺_kaki ∼n_k⁰= 1 or 0. This is quasi-boson approximation!

Existence of RPA Vacuum

|0i ≡ |Z⁽²⁾i =e^Z(2)|HFi; Z⁽²⁾= 1 4

Xz_p⁽²⁾_1p2h1h2A^†_p1h1A^†_p2h2; Qν|Z⁽²⁾i =0 ??

Q_ν =X

ph

hX_ph^νa^†_ha_p−Y_ph^νa^†_pa_hi

+1 2

X

php1p2

η^ν_p1pp2ha⁺_p2a_p1a⁺_pa_h−1 2

X

phh1h2

η^ν_ph1hh2a^†_h

1a_h2a^†_pa_h

with

Y_ph^ν =X

p1h1

X_p^ν_1h1z_p⁽²⁾

1ph1h, z_pp^′_hh^′ =X

ν

(X⁻¹)^ν_phY_p^ν′h^′ (2)

and

η_p^ν_1pp2h=X

h1

X_p^ν_1h1z_pp⁽²⁾

2hh1, η_ph^ν_1hh2 =X

p1

X_p^ν_1h1z_pp⁽²⁾

1hh2. (3)

Approximation:a⁺_p2ap1a⁺_pah→ ha⁺_p2ap1ia⁺_pah, etc.

In order to determine the amplitudes X,Y of (11) we define a generalised sum rule

Ων =1 2

h0|[Qν, [H,Q_ν^†]]|0i h0|[Qν,Qν^†]|0i .

= 1

h0|[Qν,Qν^†]|0i X

µ

(Eµ−E0)|h0|Qν|µi|²

which we minimise with respect to X,Y .

This leads to the RPA-type of equations of the form Ak1k2k₁^′k₂^′ Bk1k2k₁^′k₂^′

Bk^∗1k2k₁^′k₂^′ A^∗k1k2k₁^′k₂^′

! X_k^ν′ 1k₂^′

Y_k^ν′ 1k₂^′

!

= Ων

X_k^ν

1k2

−Y_k^ν

1k2

 ,

where

Ak1k2k₁^′k₂^′ = h0|h δQk1k2

hH, δQ_k^†′ 1k₂^′

ii|0i , and

Bk1k2k₁^′k₂^′ = −h0|h δQ^†_k1k2h

H, δQ^†_k′ 1k₂^′

ii|0i .

where

δQ_k^†

1k2= Ak1k2

√n_k2−n_k1 , Ak1k2 =a^†_k

1ak2 , are the normalised pair creation operators and

n_k = h0|a^†_ka_k|0i , are the single particle occupation numbers

The Bogoliubov orthonormality relations allow us to invert the operator (14) A^†_k

1k2

√n_k2−n_k1 =X

ν

X_k^ν∗_1k2Q^†_ν+Y_k^ν∗_1k2Qν
.

The double commutators inA, B contain the occupation numbersnk and hA^†Ai or hAAi. The latter can be expressed byX,Y amplitudes via killing relationQ|RPAi =0.

hA^†Ai =F[X,Y] hAAi =G[X,Y]

With inversion and killing condition, we get with CC wave fct n_k =n_k[X,Y]

and thus

 A[X,Y] B[X,Y] B^∗[X,Y] A^∗[X,Y]

 X Y



=E

 X

−Y



Leads to a fullySelf-consistent scheme, very similar to HFB eqs for bosons.

Here, HFB for fermion pairs.

Linearising withX →1,Y →0in matrix→standard RPA.

Determination ofoptimal single particle basis

Minimisation of ground state energy with respect to s.p. basis→ h[H,Q^†]i = h[H,a^†_kak^′]i = Ψ[X,Y; φ] =0

Very natural result, since just another Equation of Motion! Again onlynk

andhAAi enter and, thus, s.p. basis gets coupled toX,Y amplitudes, selfconsistently.

Application to Lipkin model

For simplicity, we consider two levelLipkin model ———- 1 -

———- 0 H= εJ0−V

2(J+J++J−J−) with[J−,J+] = −2J₀, [J0,J±] = ±J±and

J₀=1 2

X

m

(c_1m^† c_1m−c_0m^† c_0m) J+=X

m

c_1m^† c_0m J−= (J+)^†

We try exponential with two body operator:

|zi =e^zJ+J+|HFi Using following operator withz = _N^1Y_X andη =_N^2Y_X

Q^†=XJ+−YJ−+ ηJ−J₀ we have→ Q|RPAi =0!!

Again: J−J₀→J−hJ₀i

Scheme can be and has been worked out for general many body problem.Two particle case exact withoutηterm !.

0 0.2 0.4 0.6 0.8 1 1.2

χ 0

0.2 0.4 0.6 0.8 1

Ω/ε

s-RPA SCRPA [Eq.(36)]

SCRPA Exact

N=4

(a)

χ =V(N−1)/ε

0 0.2 0.4 0.6 0.8 1 1.2 χ

0 0.2 0.4 0.6 0.8 1

Ω/ε s-RPA

SCRPA [Eq.(36)]

SCRPA Exact

N=14 (b)

0 1 2 3 4 5 6 7 8 9 10

0 0.5 1 1.5 2 2.5

%Error = 100.(E₀^Exact - E₀^SCRPA)/E₀^Exact N=14

The HUBBARD Model

We treat a ring with 6 sites and half filling, i.e. 6 electrons.

H= −tX

hijiσ

c_i^†_σc_jσ+UX

i

nˆ_i+ˆni− , ˆn_iσ =c_iσ^†c_iσ (4)

Two site problem again exact in SCRPA!

0 0.5 1 1.5 2 2.5 3 3.5 U

0 0.5 1 1.5 2 2.5 3 3.5

ε

|q|=π

ph −RPA standard Exact ph −SCRPA

sp

ch

sp

0 1 2 3 4 5

U 0

1 2 3 4 5 6

ε

|q|=2π/3

ph −RPA standard Exact ph −SCRPA

ch

sp

sp

0 1 2 3 4 5 6 U

0 1 2 3 4 5

ε

|q|=π/3

ph −RPA standard Exact ph −SCRPA

ch

sp

SCRPA in the particle-particle (hole-hole) channel. The pairing or Picket Fence Model.

H=

Ω

X

i=1

(εi− λ)Ni−G

Ω

X

i,j=1

P_i^†Pj

where

Pi =c¯ici , P_i^†= (Pi)^† , Ni =c_i^†ci+c_¯^†

ic¯i

The pp-RPA operator is A^†µ=X

p

X_p^µP_p^†−X

h

Y_h^µP_h^†

Self-Consistent mashinerie gives following results. 2 particle case again exact standard RPA:E ∝√

1−G; SCRPA:E ∝√ 1+G (exact(!)). Screening has changed sign from attraction to repulsion !!

The Picket Fence Model

Estimate of violation of Pauli principle X

pp^′

hM_pM_p^′i =X

ph

hM_hM_pi M_p=c_p⁺c_p+c_¯p⁺c¯p;etc.

Above sumrule is violated by about 4 percent.

DISCUSSION

-

Killing condition in SCRPA nearly fullfilled. Then theory nearly Raleigh-Ritz variational.In all models two particle case exact.

Pleasent properties of RPA remain fullfilled: Goldstone mode appears→ Delion

This is a very strong property!! Difficult to obtain with other approaches.

Sum rules satisfied!

Can be formulated with Green’s functions and at finite temperature.

An approximation to SCRPA→ renormalised RPA:

In standard RPA one only replaces

n⁰_k →n_k =n_k[X,Y]

Yields often appreciable improvement over standard RPA; much easier than SCRPA. (F. Catara et al., PLB 306(1993)197; PRB 51(1995)4569)

So far some problems in ’deformed’ region: transition from ’spherical’ to deformed region is discontinuous like a first order phase transition what should not be. Further work is in progress on this problem. It seems that same problem appears with CCT (Dukelsky)

CONCLUSIONS

SCRPA has quite satisfying properties. Realistic applications to nuclear shell model and comparison with large scale shell model calculations are in progress.

RPA for odd particle number systems

It is interesting that our vacuum state|Ziis also the vacuum to an odd number RPA operator (for calculating N±1 systems)

qa,µ|Zi =qr,µ|Zi =0 with

q^†_a,µ=X

p

y_p^µa^†_p−1 2

X

hh^′p

Y_hh^µ′:pa^†_ha^†_h′ap q_r^†_,µ=X

h

y_h^µa_h−1 2

X

pp^′:h

Y_pp^µ′:ha^†_hapap^′

under the condition X

p

y_p^µz_pp^′_hh^′ =Y_hh^µ′:p^′ , X

h

y_h^µz_pp^′_hh^′ =Y_pp^µ′:h^′

We minimise again energy weighted sum rule for an average single particle energy and obtain

hZ|{δq, [H,q^†]}i =ehZ|{δq,q^†}|Zi Double commutator leads to three particle correlation fct.

Factorisation into products of s.p. and two particle correlation fcts preserves the property that two particle system stays exact. This may be a good approximation in general. Two body correlation fct can be evaluated by SCRPA as before. Should give quite reasonable results.

Conclusions

Instead of evaluating RPA eqs with HF we evaluate it with|Zi(CCT).

We obtain naturally a fully selfconsistent RPA scheme in X, Y amplitudes.

Results in non trivial model cases very encouraging!

Extension to odd particle numbers.

Collaborators: D. Delion, J. Dukelsky, M. Jemai, M. Tohyama.