Applications of Renormalization Group Methods in Nuclear Physics 2

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 ( V low,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 (k 2 k 2 )/m + Solution: Unitary transformation of the H matrix = decouple! E n = Ψ n H Ψ n U U = 1 = ( Ψ n U )UHU (U Ψ n ) = Ψ n H Ψ n Here: Decouple using RG 20 0 100 200 300 E lab (MeV) phase shift (degrees) 60 40 20 0 1 S0 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 αu α = 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 ψ n and an operator O, then the new Hamiltonian, operator, and eigenstates are H = UHU Õ = UOU ψ n = U ψ n The energy is unchanged: ψ n H ψ n = ψ n H ψ n = E n Furthermore, matrix elements of O are unchanged: O mn ψ m Ô ψ n = ( ψ m U ) UÔU ( U ψ n ) = ψ m Õ ψ n Õmn If asymptotic (long distance) properties are unchanged, H and H are equally acceptable physically = not measurable! Consistency: use O with H and ψ n s but Õ with H and ψ n s One form may be better for intuition or for calculations Scheme-dependent observables (come back to this later)

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

Two ways to use RG equations to decouple Hamiltonians k V low k k Similarity RG k k Λ 2 Λ 1 Λ 0 Lower a cutoff Λ i in k, k, e.g., demand dt (k, k ; k 2 )/dλ = 0 λ 0 λ 1 λ 2 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 ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (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 0.4 0.3 uncorrelated correlated 400 300 0.25 0.2 3 S1 deuteron probability density ψ(r) 2 [fm 3 ] 0.2 0.1 V(r) [MeV] 200 100 ψ(r) 2 [fm 3 ] 0.15 0.1 Argonne v 18 0 0 0.05-0.1 0 2 4 6 r [fm] 100 0 0 2 4 6 r [fm] 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 0.4 0.3 uncorrelated correlated 400 300 0.25 0.2 3 S1 deuteron probability density softened ψ(r) 2 [fm 3 ] 0.2 0.1 V(r) [MeV] 200 100 ψ(r) 2 [fm 3 ] 0.15 0.1 Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm -1 0 0 0.05-0.1 0 2 4 6 r [fm] 100 original 0 0 2 4 6 r [fm] Transformed potential = no short-range correlations in wf! Potential is now non-local: V (r)ψ(r) d 3 r 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 r 2 ψ(r) 2 [fm 1 ] 0.4 0.3 0.2 0.1 0 uncorrelated correlated -0.1 0 2 4 6 r [fm] V(r) [MeV] 400 300 200 100 0 100 r 2 ψ(r) 2 [fm 1 ] 0.35 0.3 0.25 0.2 0.15 0.1 0.05 3 S1 deuteron probability density Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm -1 0 0 2 4 6 r [fm] Transformed potential = no short-range correlations in wf! Potential is now non-local: V (r)ψ(r) d 3 r 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 (N max ) Harmonic oscillator basis with N max shells for excitations Ground State Energy [MeV] 20 21 22 23 24 25 26 27 28 Softened with SRG Helium-4 ground-state energy Jurgenson et al. (2009) V NN = N 3 LO (500 MeV) V NNN = N 2 LO Original expt. 29 2 4 6 8 10 12 14 16 18 20 Matrix Size [N max ] Ground-State Energy [MeV] 16 12 Lithium-6 8 ground-state energy 4 V 0 NN = N 3 LO (500 MeV) V 4 NNN = N 2 LO 8 h- Ω = 20 MeV Original 12 16 20 Softened with SRG 24 λ = 2.0 fm 1 28 λ = 1.5 fm 1 32 expt. 36 2 4 6 8 10 12 14 16 18 Matrix Size [N max ] 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) = d 3 r V λ (r, r ) Roughly gives action of potential on long-wavelength nucleons Central part (S-wave) [Note: The V λ 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 U s : H s = U s HU s T + V s, where s is the flow parameter. Differentiating wrt s: dh s ds = [η s, H s ] with η s du s ds U s = η s. η s is specified by the commutator with Hermitian G s : η s = [G s, H s ], which yields the unitary flow equation (T held fixed), dh s ds = dv s ds = [[G s, H s ], H s ]. Very simple to implement as matrix equation (e.g., MATLAB) G s determines flow = many choices (T, H D, H BD,... )

SRG flow of H = T + V in momentum basis Takes H H s = U s HU s in small steps labeled by s or λ dh s ds = dv s ds = [[T rel, V s ], H s ] with T rel k = ɛ k k and λ 2 = 1/ s For NN, project on relative momentum states k, but generic dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k )+ q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k ) V λ=3.0 (k, k ) 1st term 2nd term V λ=2.5 (k, k ) First term drives 1 S 0 V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

SRG flow of H = T + V in momentum basis Takes H H s = U s HU s in small steps labeled by s or λ dh s ds = dv s ds = [[T rel, V s ], H s ] with T rel k = ɛ k k and λ 2 = 1/ s For NN, project on relative momentum states k, but generic dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k )+ q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k ) V λ=2.5 (k, k ) 1st term 2nd term V λ=2.0 (k, k ) First term drives 1 S 0 V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

SRG flow of H = T + V in momentum basis Takes H H s = U s HU s in small steps labeled by s or λ dh s ds = dv s ds = [[T rel, V s ], H s ] with T rel k = ɛ k k and λ 2 = 1/ s For NN, project on relative momentum states k, but generic dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k )+ q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k ) V λ=2.0 (k, k ) 1st term 2nd term V λ=1.5 (k, k ) First term drives 1 S 0 V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

Two ways to use RG equations to decouple Hamiltonians General form of the flow equation: dhs ds = [[G s, H s ], H s ] V low k SRG ( T generator) General rule: Choose G s to match the desired final pattern

Two ways to use RG equations to decouple Hamiltonians General form of the flow equation: dhs ds = [[G s, H s ], H s ] SRG ( BD generator) SRG ( T generator) General rule: Choose G s to match the desired final pattern

Block diagonalization via SRG [G s = H BD ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q ) 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 = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s

S-wave NN potential as momentum-space matrix k V L=0 k = d 3 r j 0 (kr) V (r) j 0 (k r) = V kk 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 (k 2 k 2 )/m + = V ii+ j V ij V ji 1 (k 2 i k 2 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 k 2 V (p, k)v (k, q) Introduce gaussian nodes and weights {k n, w n } (n = 1, N) = dk f (k) w n f (k n ) n Then I(p, q) I ij, where p = k i and q = k j, and I ij = n k 2 n w n V in V nj n Ṽ in Ṽ nj where Ṽ ij = w i k i V ij k j wj Lets us solve SRG equations, integral equation for phase shift, Schrödinger 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 ds V s = [[T, H s ], H s ] is solved by discretizing, so it is just matrix multiplication. If the matrix V s 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 N max ) 2 Calculate (or input) the initial Hamiltonian and G s matrix elements (including any weight factors) 3 Reshape the right side [[G s, H s ], H s ] to a vector and pass it to a coupled differential equation solver 4 Integrate V s to desired s (or λ = s 1/4 ) 5 Diagonalize H s 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

Flow of different N 3 LO chiral EFT potentials 1 S 0 from N 3 LO (500 MeV) of Entem/Machleidt 1 S 0 from N 3 LO (550/600 MeV) of Epelbaum et al. Decoupling = perturbation theory is more effective k V k + k k V k k V k (k 2 k 2 )/m + = V ii+ j V ij V ji 1 (k 2 i k 2 j )/m +

Flow of different N 3 LO chiral EFT potentials 3 S 1 from N 3 LO (500 MeV) of Entem/Machleidt 3 S 1 from N 3 LO (550/600 MeV) of Epelbaum et al. Decoupling = perturbation theory is more effective k V k + k k V k k V k (k 2 k 2 )/m + = V ii+ j V ij V ji 1 (k 2 i k 2 j )/m +

Convergence of the Born series for scattering Consider whether the Born series converges for given z T (z) = V + V 1 z H 0 V + V 1 z H 0 V 1 z H 0 V + If bound state b, series must diverge at z = E b, where (H 0 + V ) b = E b b = V b = (E b H 0 ) b For fixed E, generalize to find eigenvalue η ν [Weinberg] 1 E b H 0 V b = b = 1 E H 0 V Γ ν = η ν Γ ν From T applied to eigenstate, divergence for η ν (E) 1: T (E) Γ ν = V Γ ν (1 + η ν + η 2 ν + ) = T (E) diverges if bound state at E for V /η ν with η ν 1

Weinberg eigenvalues as function of cutoff Λ/λ Consider η ν (E = 2.22 MeV) Deuteron = attractive eigenvalue η ν = 1 Λ = unchanged η ν 2.5 2 1.5 1 3 S1 (E cm = -2.223 MeV) free space, η > 0 0.5 0 2 3 4 5 6 7 Λ [fm -1 ]

Weinberg eigenvalues as function of cutoff Λ/λ Consider η ν (E = 2.22 MeV) Deuteron = attractive eigenvalue η ν = 1 Λ = unchanged But η ν can be negative, so V /η ν = flip potential

Weinberg eigenvalues as function of cutoff Λ/λ Consider η ν (E = 2.22 MeV) Deuteron = attractive eigenvalue η ν = 1 Λ = unchanged But η ν can be negative, so V /η ν = flip potential Hard core = repulsive eigenvalue η ν Λ = reduced η ν 2.5 2 1.5 1 0.5 3 S1 (E cm = -2.223 MeV) free space, η > 0 free space, η < 0 0 2 3 4 5 6 7 Λ [fm -1 ]

Weinberg eigenvalues as function of cutoff Λ/λ Consider η ν (E = 2.22 MeV) Deuteron = attractive eigenvalue η ν = 1 Λ = unchanged But η ν can be negative, so V /η ν = flip potential Hard core = repulsive eigenvalue η ν Λ = reduced In medium: both reduced η ν 1 for Λ 2 fm 1 = perturbative (at least for particle-particle channel) η ν 2.5 2 1.5 1 0.5 0 3 S1 (E cm = -2.223 MeV) free space, η > 0 free space, η < 0 k f = 1.35 fm -1, η > 0 k f = 1.35 fm -1, η < 0 2 3 4 5 6 7 Λ [fm -1 ]

Weinberg eigenvalue analysis of convergence 1 1 1 Born Series: T (E) = V + V V + V V V + E H 0 E H 0 E H 0 For fixed E, find (complex) eigenvalues η ν (E) [Weinberg] 1 E H 0 V Γ ν = η ν Γ ν = T (E) Γ ν = V Γ ν (1+η ν +η 2 ν + ) = T diverges if any η ν (E) 1 [nucl-th/0602060] Im η Im η 1 S0 3 S1 3 D 1 1 1 3 2 1 1 Re η 3 2 1 1 Re η AV18 CD-Bonn N 3 LO (500 MeV) 1 2 AV18 CD-Bonn N 3 LO 1 2 3 3

Lowering the cutoff increases perturbativeness Weinberg eigenvalue analysis (repulsive) [nucl-th/0602060] Im η 1 S0 1 0 1 S0 3 2 1 1 1 2 3 Re 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 3 LO η ν (E=0) 0.2 0.4 0.6 0.8 1 Argonne v 18 N 2 LO-550/600 [19] N 3 LO-550/600 [14] N 3 LO [13] 1.5 2 2.5 3 3.5 4 Λ (fm -1 )

Lowering the cutoff increases perturbativeness Weinberg eigenvalue analysis (repulsive) [nucl-th/0602060] Im η 3 S1 3 D 1 0 1 3 S1 3 D 1 3 2 1 1 1 2 3 Re 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 3 LO η ν (E=0) 0.5 1 1.5 2 Argonne v 18 N 2 LO-550/600 N 3 LO-550/600 N 3 LO [Entem] 1.5 2 2.5 3 3.5 4 Λ (fm -1 )

Lowering the cutoff increases perturbativeness Weinberg eigenvalue analysis (η ν at 2.22 MeV vs. density) 1 3 S1 with Pauli blocking η ν (B d ) 0.5 0 Λ = 4.0 fm -1 Λ = 3.0 fm -1 Λ = 2.0 fm -1-0.5-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 k F [fm -1 ] Pauli blocking in nuclear matter increases it even more! at Fermi surface, pairing revealed by η ν > 1