Applications of Renormalization Group Methods in Nuclear Physics 2 - - PowerPoint PPT Presentation

Applications of Renormalization Group Methods in Nuclear Physics – 2

Dick Furnstahl

Department of Physics Ohio State University

HUGS 2014

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

Why did our low-pass filter fail?

Basic problem: low k and high k are coupled (mismatched dof’s!) E.g., perturbation theory for (tangent of) phase shift:

k|V|k+

• k′

k|V|k′k′|V|k (k2 − k′2)/m +· · ·

Solution: Unitary transformation

• f the H matrix =

⇒ decouple! En = Ψn|H|Ψn U†U = 1 = (Ψn|U†)UHU†(U|Ψn) =

• Ψn|

H| Ψn Here: Decouple using RG

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees) 1S0 AV18 phase shifts after low-pass filter

k = 2 fm

−1

Why did our low-pass filter fail?

Basic problem: low k and high k are coupled (mismatched dof’s!) E.g., perturbation theory for (tangent of) phase shift:

k|V|k+

• k′

k|V|k′k′|V|k (k2 − k′2)/m +· · ·

Solution: Unitary transformation

• f the H matrix =

⇒ decouple! En = Ψn|H|Ψn U†U = 1 = (Ψn|U†)UHU†(U|Ψn) =

• Ψn|

H| Ψn Here: Decouple using RG

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees) 1S0 AV18 phase shifts after low-pass filter

k = 2 fm

−1

Aside: Unitary transformations of matrices

Recall that a unitary transformation can be realized as unitary matrices with U†

αUα = I

(where α is just a label) Often used to simplify nuclear many-body problems, e.g., by making 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†

U OU† U|ψn

• =

ψm| O| ψn ≡ Omn If asymptotic (long distance) properties are unchanged, H and H are equally acceptable physically = ⇒ not measurable! Consistency: use O with H and |ψn’s but O with H and | ψn’s One form may be better for intuition or for calculations Scheme-dependent observables (come back to this later)

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

• 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 as a way to arrange in various theories that the degrees of freedom that you’re talking about are the relevant degrees of freedom for the problem at hand.”

• 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 as a way to arrange in various theories that the degrees of freedom that you’re talking about are the relevant degrees of freedom for the problem at hand.” Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms RG: shift between couplings and loop integrals to reduce logs Nuclear: decouple high- and low-momentum modes Identifying universality in critical phenomena RG: filter out short-distance degrees of freedom Nuclear: evolve toward universal interactions

• 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 as a way to arrange in various theories that the degrees of freedom that you’re talking about are the relevant degrees of freedom for the problem at hand.” Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms RG: shift between couplings and loop integrals to reduce logs Nuclear: decouple high- and low-momentum modes Identifying universality in critical phenomena RG: filter out short-distance degrees of freedom Nuclear: 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., demand dT(k, k′; k2)/dΛ = 0 Similarity RG

λ0 λ1 λ2 k’ k

Drive the Hamiltonian toward diagonal 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 ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Flow equations in action: NN only

In each partial wave with ǫk = 2k2/M and λ2 = 1/√s

dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′) +

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′)

Decoupling and phase shifts: Low-pass filters work!

Unevolved AV18 phase shifts (black solid line) Cutoff AV18 potential at k = 2.2 fm−1 (dotted blue) = ⇒ fails for all but F wave Uncut evolved potential agrees perfectly for all energies Cutoff evolved potential agrees up 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

2 4 6

r [fm]

• 0.1

0.1 0.2 0.3 0.4

|ψ(r)|

2 [fm −3]

uncorrelated correlated 2 4 6

r [fm]

−100 100 200 300 400

V(r) [MeV]

2 4 6

r [fm]

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

Consequences of a repulsive core revisited

2 4 6

r [fm]

• 0.1

0.1 0.2 0.3 0.4

|ψ(r)|

2 [fm −3]

uncorrelated correlated 2 4 6

r [fm]

−100 100 200 300 400

V(r) [MeV]

2 4 6

r [fm]

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

3S1 deuteron probability density

softened

• riginal

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 approach Not a problem for many-body methods using HO matrix elements

Consequences of a repulsive core revisited

2 4 6

r [fm]

• 0.1

0.1 0.2 0.3 0.4

r

2|ψ(r)| 2 [fm −1]

uncorrelated correlated 2 4 6

r [fm]

−100 100 200 300 400

V(r) [MeV]

2 4 6

r [fm]

0.05 0.1 0.15 0.2 0.25 0.3 0.35

r

2|ψ(r)| 2 [fm −1]

Argonne v18 λ = 4.0 fm

• 1

λ = 3.0 fm

• 1

λ = 2.0 fm

• 1

3S1 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 approach Not 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 20

Matrix Size [Nmax]

−29 −28 −27 −26 −25 −24 −23 −22 −21 −20

Ground State Energy [MeV] Original

expt. VNN = N

3LO (500 MeV)

Helium-4

Softened with SRG

VNNN = N

2LO

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 4 8 12 16

Ground-State Energy [MeV]

Lithium-6

expt.

Original

h

• Ω = 20 MeV

Softened with SRG ground-state energy

VNN = N

3LO (500 MeV)

VNNN = N

2LO

λ = 1.5 fm

−1

λ = 2.0 fm

−1

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

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

Visualizing the softening of NN interactions

Project non-local NN potential: V λ(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 ds U†

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 = ǫk|k and λ2 = 1/ √ s For NN, project on relative momentum states |k, but generic dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′)+

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′) 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−[(ǫk − ǫk′)/λ2]2 + · · ·

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 = ǫk|k and λ2 = 1/ √ s For NN, project on relative momentum states |k, but generic dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′)+

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′) Vλ=2.5(k, k′) 1st term 2nd term Vλ=2.0(k, k′) First term drives 1S0 Vλ toward diagonal: Vλ(k, k′) = Vλ=∞(k, k′) e−[(ǫk − ǫk′)/λ2]2 + · · ·

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 = ǫk|k and λ2 = 1/ √ s For NN, project on relative momentum states |k, but generic dVλ dλ (k, k′) ∝ −(ǫk − ǫk′)2Vλ(k, k′)+

• q

(ǫk + ǫk′ − 2ǫq)Vλ(k, q)Vλ(q, k′) Vλ=2.0(k, k′) 1st term 2nd term Vλ=1.5(k, k′) First term drives 1S0 Vλ toward diagonal: Vλ(k, k′) = Vλ=∞(k, k′) e−[(ǫk − ǫk′)/λ2]2 + · · ·

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

Two ways to use RG equations to decouple Hamiltonians

General form of the flow equation: dHs

ds = [[Gs, Hs], Hs]

“Vlow k” SRG (“T” generator) General rule: Choose Gs to match the desired final pattern

Two ways to use RG equations to decouple Hamiltonians

General form of the flow equation: dHs

ds = [[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 dHs

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Block diagonalization via SRG

[Gs = HBD]

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

ds = [[Gs, Hs], Hs] with Gs =

PHsP QHsQ

• What are the best generators for nuclear applications?

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Custom potentials via the SRG

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

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

with a strange choice for Gs

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

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 momentum in 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 practice represented 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

VijVji 1 (k2

i − 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

k2

n wn VinVnj →

• n
• Vin

Vnj where

• Vij = √wiki Vij kj

wj

Lets us solve SRG equations, integral equation for phase shift, Schr¨

• dinger equation in momentum representation,

. . . In practice, N=100 gauss points more than enough for accurate nucleon-nucleon partial waves

MATLAB Code for SRG is a direct translation!

The flow equation d

dsVs = [[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 can be fed to a differential equation solver, with the right side:

% V_s is a vector of the current potential; convert to square matrix V_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 equation if (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 returned dVds = reshape(rhs_matrix, tot_pts*tot_pts, 1);

Pseudocode for SRG evolution

1

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

2

Calculate (or input) the initial Hamiltonian and Gs matrix elements (including any weight factors)

3

Reshape the right side [[Gs, Hs], Hs] to a vector and pass it to 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-90

Instructive computational project for undergraduates!

Once there are discretized matrices, the solver is the same with any size basis in any number of dimensions! Still the same solution code for a many-particle basis Any basis can be used

For 3NF, harmonic oscillators, discretized partial-wave momentum, and hyperspherical harmonics are available An accurate 3NF evolution in HO basis takes ∼ 20 million matrix elements = ⇒ that many differential equations

Outline: Lecture 2

Lecture 2: SRG in practice

Recap from lecture 1: decoupling Implementing the similarity renormalization group (SRG) Block diagonal (“Vlow ,k”) generator Computational aspects Quantitative measure of perturbativeness

• 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 as a way to arrange in various theories that the degrees of freedom that you’re talking about are the relevant degrees of freedom for the problem at hand.” Improving perturbation theory; e.g., in QCD calculations Mismatch of energy scales can generate large logarithms RG: shift between couplings and loop integrals to reduce logs Nuclear: decouple high- and low-momentum modes Identifying universality in critical phenomena RG: filter out short-distance degrees of freedom Nuclear: 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

Flow of different N3LO chiral EFT potentials

1S0 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

VijVji 1 (k2

i − k2 j )/m+· · ·

Flow of different N3LO chiral EFT potentials

3S1 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

VijVji 1 (k2

i − k2 j )/m+· · ·

Convergence of the Born series for scattering

Consider whether the Born series converges for given z

T(z) = V + V 1 z − H0 V + V 1 z − H0 V 1 z − H0 V + · · ·

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

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

Convergence of the Born series for scattering

Consider whether the Born series converges for given z

T(z) = V + V 1 z − H0 V + V 1 z − H0 V 1 z − H0 V + · · ·

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]

1 Eb − H0 V|b = |b = ⇒ 1 E − H0 V|Γν = ην|Γν

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

T(E)|Γν = V|Γν(1 + ην + η2

ν + · · · )

= ⇒ 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 = ⇒ attractive eigenvalue ην = 1

Λ ↓ = ⇒ unchanged

2 3 4 5 6 7

Λ [fm

• 1]

0.5 1 1.5 2 2.5

| ην |

free space, η > 0

3S1 (Ecm = -2.223 MeV)

Weinberg eigenvalues as function of cutoff Λ/λ

Consider ην(E = −2.22 MeV) Deuteron = ⇒ attractive eigenvalue ην = 1

Λ ↓ = ⇒ unchanged

But ην can be negative, so V/ην = ⇒ flip potential

Weinberg eigenvalues as function of cutoff Λ/λ

Consider ην(E = −2.22 MeV) Deuteron = ⇒ attractive eigenvalue ην = 1

Λ ↓ = ⇒ unchanged

But ην can be negative, so V/ην = ⇒ flip potential

Weinberg eigenvalues as function of cutoff Λ/λ

Consider ην(E = −2.22 MeV) Deuteron = ⇒ attractive eigenvalue ην = 1

Λ ↓ = ⇒ unchanged

But ην can be negative, so V/ην = ⇒ flip potential Hard core = ⇒ repulsive eigenvalue ην

Λ ↓ = ⇒ reduced

2 3 4 5 6 7

Λ [fm

• 1]

0.5 1 1.5 2 2.5

| ην |

free space, η > 0 free space, η < 0

3S1 (Ecm = -2.223 MeV)

Weinberg eigenvalues as function of cutoff Λ/λ

Consider ην(E = −2.22 MeV) Deuteron = ⇒ attractive eigenvalue ην = 1

Λ ↓ = ⇒ unchanged

But ην can be negative, so V/ην = ⇒ flip potential Hard core = ⇒ repulsive eigenvalue ην

Λ ↓ = ⇒ reduced

In medium: both reduced

ην ≪ 1 for Λ ≈ 2 fm−1

= ⇒ perturbative (at least for particle-particle channel)

2 3 4 5 6 7

Λ [fm

• 1]

0.5 1 1.5 2 2.5

| ην |

free space, η > 0 free 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 + V 1 E − H0 V + V 1 E − H0 V 1 E − H0 V + · · ·

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

1 E − H0 V|Γν = ην|Γν = ⇒ T(E)|Γν = V|Γν(1+ην+η2

ν+· · · )

= ⇒ T diverges if any |ην(E)| ≥ 1

[nucl-th/0602060]

AV18 CD-Bonn N

3LO (500 MeV)

−3 −2 −1 1

Re η

−3 −2 −1 1

Im η 1S0

−3 −2 −1 1

Re η

−3 −2 −1 1

Im η AV18 CD-Bonn N

3LO

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

N

3LO

1S0 1.5 2 2.5 3 3.5 4

Λ (fm

• 1)

−1 −0.8 −0.6 −0.4 −0.2

ην(E=0)

Argonne v18 N

2LO-550/600 [19]

N

3LO-550/600 [14]

N

3LO [13]

1S0

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

N

3LO

3S1− 3D1 1.5 2 2.5 3 3.5 4

Λ (fm

• 1)

−2 −1.5 −1 −0.5

ην(E=0)

Argonne v18 N

2LO-550/600

N

3LO-550/600

N

3LO [Entem]

3S1− 3D1

Lowering the cutoff increases “perturbativeness”

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

0.2 0.4 0.6 0.8 1 1.2 1.4

kF [fm

• 1]
• 1
• 0.5

0.5 1

ην(Bd)

Λ = 4.0 fm

• 1

Λ = 3.0 fm

• 1

Λ = 2.0 fm

• 1

3S1 with Pauli blocking

Pauli blocking in nuclear matter increases it even more!

at Fermi surface, pairing revealed by |ην| > 1