Applications of Renormalization Group Methods in Nuclear ... · “The method in its most general form can I think be understood as a way to arrange in various theories that the degrees

Applications of Renormalization GroupMethods in Nuclear Physics – 2

Dick FurnstahlDepartment of PhysicsOhio State University

HUGS 2014

Outline: Lecture 2

Lecture 2: SRG in practiceRecap from lecture 1: decouplingImplementing the similarity renormalization group (SRG)Block diagonal (“Vlow ,k ”) generatorComputational aspectsQuantitative measure of perturbativeness

Outline: Lecture 2


Why did our low-pass filter fail?

Basic problem: low k and high kare coupled (mismatched dof’s!)

E.g., perturbation theoryfor (tangent of) phase shift:

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+ · · ·

Solution: Unitary transformationof the H matrix =⇒ decouple!

En = 〈Ψn|H|Ψn〉 U†U = 1= (〈Ψn|U†)UHU†(U|Ψn〉)= 〈Ψn|H|Ψn〉

Here: Decouple using RG0 100 200 300

Elab

(MeV)

−20

0

20

40

60

phas

e sh

ift

(deg

rees

)

1S

0

AV18 phase shifts

after low-pass filter

k = 2 fm−1

Why did our low-pass filter fail?

Basic problem: low k and high kare coupled (mismatched dof’s!)

E.g., perturbation theoryfor (tangent of) phase shift:

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+ · · ·

Solution: Unitary transformationof the H matrix =⇒ decouple!

En = 〈Ψn|H|Ψn〉 U†U = 1= (〈Ψn|U†)UHU†(U|Ψn〉)= 〈Ψn|H|Ψn〉

Here: Decouple using RG0 100 200 300

Elab

(MeV)

−20

0

20

40

60

phas

e sh

ift

(deg

rees

)

1S

0

AV18 phase shifts

after low-pass filter

k = 2 fm−1

Aside: Unitary transformations of matricesRecall that a unitary transformation can be realized as unitarymatrices with U†αUα = I (where α is just a label)

Often used to simplify nuclear many-body problems, e.g., bymaking them more perturbative

If I have a Hamiltonian H with eigenstates |ψn〉 and an operator O,then the new Hamiltonian, operator, and eigenstates are

H = UHU† O = UOU† |ψn〉 = U|ψn〉

The energy is unchanged: 〈ψn|H|ψn〉 = 〈ψn|H|ψn〉 = En

Furthermore, matrix elements of O are unchanged:

Omn ≡ 〈ψm|O|ψn〉 =(〈ψm|U†

)UOU†

(U|ψn〉

)= 〈ψm|O|ψn〉 ≡ Omn

If asymptotic (long distance) properties are unchanged, H and Hare equally acceptable physically =⇒ not measurable!

Consistency: use O with H and |ψn〉’s but O with H and |ψn〉’sOne form may be better for intuition or for calculationsScheme-dependent observables (come back to this later)

Outline: Lecture 2


S. Weinberg on the Renormalization Group (RG)From “Why the RG is a good thing” [for Francis Low Festschrift]“The method in its most general form can I think be understood asa way to arrange in various theories that the degrees of freedomthat you’re talking about are the relevant degrees of freedom forthe problem at hand.”

Improving perturbation theory; e.g., in QCD calculations

Mismatch of energy scales can generate large logarithmsRG: shift between couplings and loop integrals to reduce logsNuclear: decouple high- and low-momentum modes

Identifying universality in critical phenomena

RG: filter out short-distance degrees of freedomNuclear: evolve toward universal interactions

Nuclear: simplifying calculations of structure/reactions

Make nuclear physics look more like quantum chemistry!RG gains can violate conservation of difficulty!Use RG scale (resolution) dependence as a probe or tool















Two ways to use RG equations to decouple Hamiltonians

“Vlow k ”

Λ0

Λ1

Λ2

k’

k

Lower a cutoff Λi in k , k ′,e.g., demanddT (k , k ′; k2)/dΛ = 0

Similarity RG

λ0

λ1

λ2

k’

k

Drive the Hamiltonian towarddiagonal with “flow equation”[Wegner; Glazek/Wilson (1990’s)]

=⇒ Both tend toward universal low-momentum interactions!

Flow equations in action: NN only

In each partial wave with εk = ~2k2/M and λ2 = 1/√

s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q

(εk + εk ′ − 2εq)Vλ(k ,q)Vλ(q, k ′)



s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q




s

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +∑

q


Decoupling and phase shifts: Low-pass filters work!

Unevolved AV18 phase shifts(black solid line)

Cutoff AV18 potential atk = 2.2 fm−1 (dotted blue)=⇒ fails for all but F wave

Uncut evolved potential agreesperfectly for all energies

Cutoff evolved potential agreesup to cutoff energy

F-wave is already soft (π’s)=⇒ already decoupled

Low-pass filters work! [Jurgenson et al. (2008)]NN phase shifts in different channels: no filter

Uncut evolved potential agrees perfectly for all energies

Low-pass filters work! [Jurgenson et al. (2008)]NN phase shifts in different channels: filter full potential

All fail except F-wave (D?) =⇒ already soft (π’s) =⇒ already decoupled

Low-pass filters work! [Jurgenson et al. (2008)]NN phase shifts in different channels: filtered SRG works!

Cutoff evolved potential agrees up to cutoff energy

Consequences of a repulsive core revisited

0 2 4 6r [fm]

-0.1

0

0.1

0.2

0.3

0.4

|ψ(r)

|2 [fm

−3]

uncorrelatedcorrelated

0 2 4 6r [fm]

−100

0

100

200

300

400

V(r)

[MeV

]

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

|ψ(r)

|2 [fm

−3]

Argonne v18

3S1 deuteron probability density

Probability at short separations suppressed =⇒ “correlations”

Short-distance structure⇔ high-momentum components

Greatly complicates expansion of many-body wave functions


0 2 4 6r [fm]

-0.1

0

0.1

0.2

0.3

0.4

|ψ(r)

|2 [fm

−3]


0 2 4 6r [fm]

−100

0

100

200

300

400

V(r)

[MeV

]

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

|ψ(r

)|2 [

fm−

3]

Argonne v18

λ = 4.0 fm-1

λ = 3.0 fm-1

λ = 2.0 fm-1

3S

1 deuteron probability density

softened

original

Transformed potential =⇒ no short-range correlations in wf!

Potential is now non-local: V (r)ψ(r) −→∫

d3r′ V (r, r′)ψ(r′)A problem for Green’s Function Monte Carlo approachNot a problem for many-body methods using HO matrix elements


0 2 4 6

r [fm]

-0.1

0

0.1

0.2

0.3

0.4

r2|ψ

(r)|

2 [

fm−

1]


0 2 4 6

r [fm]

−100

0

100

200

300

400

V(r

) [M

eV

]

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r2|ψ

(r)|

2 [

fm−

1]

Argonne v18

λ = 4.0 fm-1

λ = 3.0 fm-1

λ = 2.0 fm-1

3S

1 deuteron probability density

Transformed potential =⇒ no short-range correlations in wf!

Potential is now non-local: V (r)ψ(r) −→∫

d3r′ V (r, r′)ψ(r′)A problem for Green’s Function Monte Carlo approachNot a problem for many-body methods using HO matrix elements

HO matrix elements with SRG flow

We’ve seen that high and low momentum states decouple

Does this help for harmonic oscilator matrix elements?

Consider the SRG evolution from R. Roth et al.:

Yes! We have decoupling of high-energy from low-energy states

Revisit the convergence with matrix size (Nmax)

Harmonic oscillator basis with Nmax shells for excitations

2 4 6 8 10 12 14 16 18 20Matrix Size [N

max]

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

Gro

un

d S

tate

En

erg

y [

MeV

]

Original

expt.

VNN

= N3LO (500 MeV)

Helium-4

Softenedwith SRG

VNNN

= N2LO

ground-state energy

Jurgenson et al. (2009)

2 4 6 8 10 12 14 16 18

Matrix Size [Nmax

]

−36

−32

−28

−24

−20

−16

−12

−8

−4

0

4

8

12

16

Gro

un

d-S

tate

En

erg

y [

MeV

]

Lithium-6

expt.

Originalh- Ω = 20 MeV

Softened with SRG

ground-state energy

VNN

= N3LO (500 MeV)

VNNN

= N2LO

λ = 1.5 fm−1

λ = 2.0 fm−1

Graphs show that convergence for soft chiral EFT potentialis accelerated for evolved SRG potentialsRapid growth of basis still a problem; what else can we do?

importance sampling of matrix elementse.g., use symmetry: work in a symplectic basis

Visualizing the softening of NN interactionsProject non-local NN potential: Vλ(r) =

∫d3r ′ Vλ(r , r ′)

Roughly gives action of potential on long-wavelength nucleons

Central part (S-wave) [Note: The Vλ’s are all phase equivalent!]

Tensor part (S-D mixing) [graphs from K. Wendt et al., PRC (2012)]

=⇒ Flow to universal potentials!

Basics: SRG flow equations [e.g., see arXiv:1203.1779]

Transform an initial hamiltonian, H = T + V , with Us:

Hs = UsHU†s ≡ T + Vs ,

where s is the flow parameter. Differentiating wrt s:

dHs

ds= [ηs,Hs] with ηs ≡

dUs

dsU†s = −η†s .

ηs is specified by the commutator with Hermitian Gs:

ηs = [Gs,Hs] ,

which yields the unitary flow equation (T held fixed),

dHs

ds=

dVs

ds= [[Gs,Hs],Hs] .

Very simple to implement as matrix equation (e.g., MATLAB)

Gs determines flow =⇒ many choices (T , HD, HBD, . . . )

SRG flow of H = T + V in momentum basis

Takes H −→ Hs = UsHU†s in small steps labeled by s or λ

dHs

ds=

dVs

ds= [[Trel,Vs],Hs] with Trel|k〉 = εk |k〉 and λ2 = 1/

√s

For NN, project on relative momentum states |k〉, but generic

dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′)+∑

q


Vλ=3.0(k , k ′) 1st term 2nd term Vλ=2.5(k , k ′)

First term drives 1S0 Vλ toward diagonal:

Vλ(k , k ′) = Vλ=∞(k , k ′) e−[(εk − εk ′)/λ2]2 + · · ·



dHs

ds=

dVs


√s


dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′)+∑

q




Vλ(k , k ′) = Vλ=∞(k , k ′) e−[(εk − εk ′)/λ2]2 + · · ·



dHs

ds=

dVs


√s


dVλdλ

(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′)+∑

q




Vλ(k , k ′) = Vλ=∞(k , k ′) e−[(εk − εk ′)/λ2]2 + · · ·

Outline: Lecture 2



General form of the flow equation: dHsds = [[Gs,Hs],Hs]

“Vlow k ” SRG (“T” generator)

General rule: Choose Gs to match the desired final pattern


General form of the flow equation: dHsds = [[Gs,Hs],Hs]

SRG (“BD” generator) SRG (“T” generator)

General rule: Choose Gs to match the desired final pattern

Block diagonalization via SRG [Gs = HBD]

Can we get a Λ = 2 fm−1 Vlow k -like potential with SRG?

Yes! Use dHsds = [[Gs,Hs],Hs] with Gs =

(PHsP 0

0 QHsQ

)

What are the best generators for nuclear applications?




(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)





(PHsP 0

0 QHsQ

)


Custom potentials via the SRG

Can we tailor the potential to other shapes with the SRG?

Consider dHsds = [[Gs,Hs],Hs] in the 1P1 partial wave

with a strange choice for Gs









































Outline: Lecture 2


S-wave NN potential as momentum-space matrix

〈k |VL=0|k ′〉 =

∫d3r j0(kr) V (r) j0(k ′r) =⇒ Vkk ′ matrix

Momentum units (~ = c = 1): typical relative momentumin large nucleus ≈ 1 fm−1 ≈ 200 MeV

What would the kinetic energy look like on right?

Comments on computational aspects

Although momentum is continuous in principle, in practicerepresented as discrete (gaussian quadrature) grid:

=⇒

Calculations become just matrix multiplications! E.g.,

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+· · · =⇒ Vii +∑

j

VijVji1

(k2i − k2

j )/m+· · ·

100× 100 resolution is sufficient for two-body potential

Discretization of integrals =⇒ matrices!Momentum-space flow equations have integrals like:

I(p,q) ≡∫

dk k2 V (p, k)V (k ,q)

Introduce gaussian nodes and weights kn,wn (n = 1,N)

=⇒∫

dk f (k) ≈∑

n

wn f (kn)

Then I(p,q)→ Iij , where p = ki and q = kj , and

Iij =∑

n

k2n wn VinVnj →

∑n

VinVnj where Vij =√

wiki Vij kj√

wj

Lets us solve SRG equations, integral equation for phaseshift, Schrodinger equation in momentum representation,. . .

In practice, N=100 gauss points more than enough foraccurate nucleon-nucleon partial waves

MATLAB Code for SRG is a direct translation!

The flow equation dds Vs = [[T ,Hs],Hs] is solved by

discretizing, so it is just matrix multiplication.

If the matrix Vs is converted to a vector by “reshaping”, it canbe fed to a differential equation solver, with the right side:

% V_s is a vector of the current potential; convert to square matrixV_s_matrix = reshape(V_s, tot_pts, tot_pts);H_s_matrix = T_matrix + V_s_matrix; % form the Hamiltontian

% Matrix for the right side of the SRG differential equationif (strcmp(evolution,’T’))

rhs_matrix = my_commutator( my_commutator(T_matrix, H_s_matrix), ...H_s_matrix );

elseif (strcmp(evolution,’Wegner’))rhs_matrix = my_commutator( my_commutator(diag(diag(H_s_matrix)), ...

H_s_matrix), H_s_matrix );

[etc.]

% convert the right side matrix to a vector to be returneddVds = reshape(rhs_matrix, tot_pts*tot_pts, 1);

Pseudocode for SRG evolution

1 Set up basis (e.g., momentum grid with gaussianquadrature or HO wave functions with Nmax)

2 Calculate (or input) the initial Hamiltonian and Gs matrixelements (including any weight factors)

3 Reshape the right side [[Gs,Hs],Hs] to a vector and pass itto a coupled differential equation solver

4 Integrate Vs to desired s (or λ = s−1/4)5 Diagonalize Hs with standard symmetric eigensolver

=⇒ energies and eigenvectors

6 Form U =∑

i |ψ(i)s 〉〈ψ

(i)s=0| from the eigenvectors

7 Output or plot or calculate observables

Many versions of SRG codes are in use

Mathematica, MATLAB, Python, C++, Fortran-90Instructive computational project for undergraduates!

Once there are discretized matrices, the solver is the samewith any size basis in any number of dimensions!

Still the same solution code for a many-particle basisAny basis can be used

For 3NF, harmonic oscillators, discretized partial-wavemomentum, and hyperspherical harmonics are availableAn accurate 3NF evolution in HO basis takes ∼ 20 millionmatrix elements =⇒ that many differential equations

Outline: Lecture 2









Flow of different N3LO chiral EFT potentials1S0 from N3LO (500 MeV) of Entem/Machleidt

1S0 from N3LO (550/600 MeV) of Epelbaum et al.

Decoupling =⇒ perturbation theory is more effective

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+· · · =⇒ Vii +∑

j

VijVji1

(k2i − k2

j )/m+· · ·

Flow of different N3LO chiral EFT potentials3S1 from N3LO (500 MeV) of Entem/Machleidt

3S1 from N3LO (550/600 MeV) of Epelbaum et al.

Decoupling =⇒ perturbation theory is more effective

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+· · · =⇒ Vii +∑

j

VijVji1

(k2i − k2

j )/m+· · ·

Convergence of the Born series for scattering

Consider whether the Born series converges for given z

T (z) = V + V1

z − H0V + V

1z − H0

V1

z − H0V + · · ·

If bound state |b〉, series must diverge at z = Eb, where

(H0 + V )|b〉 = Eb|b〉 =⇒ V |b〉 = (Eb − H0)|b〉

For fixed E , generalize to find eigenvalue ην [Weinberg]

1Eb − H0

V |b〉 = |b〉 =⇒ 1E − H0

V |Γν〉 = ην |Γν〉

From T applied to eigenstate, divergence for |ην(E)| ≥ 1:

T (E)|Γν〉 = V |Γν〉(1 + ην + η2ν + · · · )

=⇒ T (E) diverges if bound state at E for V/ην with |ην | ≥ 1

Convergence of the Born series for scattering

Consider whether the Born series converges for given z

T (z) = V + V1

z − H0V + V

1z − H0

V1

z − H0V + · · ·

If bound state |b〉, series must diverge at z = Eb, where

(H0 + V )|b〉 = Eb|b〉 =⇒ V |b〉 = (Eb − H0)|b〉

For fixed E , generalize to find eigenvalue ην [Weinberg]

1Eb − H0

V |b〉 = |b〉 =⇒ 1E − H0

V |Γν〉 = ην |Γν〉

From T applied to eigenstate, divergence for |ην(E)| ≥ 1:

T (E)|Γν〉 = V |Γν〉(1 + ην + η2ν + · · · )

=⇒ T (E) diverges if bound state at E for V/ην with |ην | ≥ 1

Weinberg eigenvalues as function of cutoff Λ/λ

Consider ην(E = −2.22 MeV)

Deuteron =⇒ attractiveeigenvalue ην = 1

Λ ↓ =⇒ unchanged

But ην can be negative, soV/ην =⇒ flip potential

Hard core =⇒ repulsiveeigenvalue ην

Λ ↓ =⇒ reduced

In medium: both reducedην 1 for Λ ≈ 2 fm−1

=⇒ perturbative (at least forparticle-particle channel)

2 3 4 5 6 7

Λ [fm-1

]

0

0.5

1

1.5

2

2.5

| ην |

free space, η > 0

3S

1 (E

cm = -2.223 MeV)







Λ ↓ =⇒ reduced









Λ ↓ =⇒ reduced









Λ ↓ =⇒ reduced



2 3 4 5 6 7

Λ [fm-1

]

0

0.5

1

1.5

2

2.5

| ην |

free space, η > 0

free space, η < 0

3S

1 (E

cm = -2.223 MeV)







Λ ↓ =⇒ reduced



2 3 4 5 6 7Λ [fm-1]

0

0.5

1

1.5

2

2.5

| ην |

free space, η > 0free space, η < 0

kf = 1.35 fm-1, η > 0

kf = 1.35 fm-1, η < 0

3S1 (Ecm = -2.223 MeV)

Weinberg eigenvalue analysis of convergence

Born Series: T (E) = V + V1

E − H0V + V

1E − H0

V1

E − H0V + · · ·

For fixed E , find (complex) eigenvalues ην(E) [Weinberg]

1E − H0

V |Γν〉 = ην |Γν〉 =⇒ T (E)|Γν〉 = V |Γν〉(1+ην+η2ν+· · · )

=⇒ T diverges if any |ην(E)| ≥ 1 [nucl-th/0602060]

AV18CD-BonnN3LO (500 MeV)

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η1S0

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

AV18CD-BonnN3LO

3S1−3D1

Lowering the cutoff increases “perturbativeness”

Weinberg eigenvalue analysis(repulsive) [nucl-th/0602060]

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

Λ = 10 fm-1

Λ = 7 fm-1

Λ = 5 fm-1

Λ = 4 fm-1

Λ = 3 fm-1

Λ = 2 fm-1

N3LO

1S0

1.5 2 2.5 3 3.5 4Λ (fm-1)

−1

−0.8

−0.6

−0.4

−0.2

0

η ν(E=0

)

Argonne v18

N2LO-550/600 [19]

N3LO-550/600 [14]

N3LO [13]

1S0

Pauli blocking in nuclear matter increases it even more!at Fermi surface, pairing revealed by |ην | > 1


Weinberg eigenvalue analysis(repulsive) [nucl-th/0602060]

−3 −2 −1 1

Re η

−3

−2

−1

1

Im η

Λ = 10 fm-1

Λ = 7 fm-1

Λ = 5 fm-1

Λ = 4 fm-1

Λ = 3 fm-1

Λ = 2 fm-1

N3LO

3S1−3D1

1.5 2 2.5 3 3.5 4Λ (fm-1)

−2

−1.5

−1

−0.5

0

η ν(E=0

)

Argonne v18

N2LO-550/600

N3LO-550/600

N3LO [Entem]

3S1−3D1



Weinberg eigenvalue analysis(ην at −2.22 MeV vs. density)

0 0.2 0.4 0.6 0.8 1 1.2 1.4kF [fm-1]

-1

-0.5

0

0.5

1η ν(B

d)

Λ = 4.0 fm-1

Λ = 3.0 fm-1

Λ = 2.0 fm-1

3S1 with Pauli blocking


Documents

Applications of Renormalization Group Methods in Nuclear ... · “The method in its most general form can I think be understood as a way to arrange in various theories that the degrees