Similarity Renormalization Group for Few-Body Systems

Internucleon interactions evolved via flow equations yield soft potentials that lead to rapid variational convergence in few-body systems. The Similarity Renormalization Group (SRG) [1, 2] provides a compelling method for evolving internucleon forces to softer forms by decoupling lowfrom high-momentum matrix elements [3, 4]. A series of unitary transformations parameterized by s (or λ ≡ s−1/4) is implemented through a flow equation: Hs = UsHU † s ≡ Trel + Vs =⇒ dHs ds = [[Gs,Hs],Hs] , (1) where Trel is the relative kinetic energy. Applications to nuclear physics to date in a partial-wave momentum basis have used Gs = Trel [3], so the flow equation for each matrix element is (with ǫk ≡ 〈k|Trel|k〉 = ~ k/m) d ds 〈k|Vs|k 〉 = −(ǫk − ǫk′) 2〈k|Vs|k 〉+ ∑ q (ǫk + ǫk′ − 2ǫq)〈k|Vs|q〉〈q|Vs|k 〉 . (2) The flow of off-diagonal matrix elements is dominated by the first term, which drives them rapidly to zero. This partially diagonalizes the momentum-space potential, leading to decoupling [4]. Pictures showing different initial NN potentials evolving to band-diagonal form can be viewed at the SRG website [5]. In the left panel of Fig. 1, the 1S0 phase shift for the Argonne v18 NN potential is shown up to 800MeV lab energy. The phase shifts for the SRG potential Vs are indistinguishable at any λ because the evolution is exactly unitary at the two-body level. To test decoupling, the original and evolved SRG potential (to λ = 2 fm) are smoothly set to zero for momenta above kmax = 2.2 fm . The SRG phases are unchanged up to the corresponding Elab, so high momenta are not needed. The AV18 phases are completely changed because even low-energy observables have contributions from high momentum, which has led to the misconception that high-energy phase shifts are important for nuclear structure [4]. E-mail address: [email protected] 2 SRG for Few-Body Systems 0 100 200 300 400 500 600 700 800 E lab [MeV] −100 −75 −50 −25 0 25 50 ph as e sh if t [ de g. ] AV18 or any V s AV18 [k max = 2.2 fm] V s [k max = 2.2 fm] 1 S 0 0 1 2 3 4 5 6 k max [fm −1 ] −9 −8 −7 −6 −5 −4 −3 −2 −1 0 E tr ito n [M eV ] λ = ∞ λ = 3 fm λ = 2 fm N 3 LO (500 MeV) n = 8 Figure 1. Left: Decoupling in the S0 phase shift for the Argonne v18 NN potential [4]. Right: Decoupling in the triton with the NLO chiral EFT potential of Entem and Machleidt [6]. A similar story for the triton ground-state energy with a chiral EFT NLO potential is seen in the right panel, which shows the energy as a function of kmax. Because of decoupling, the full answer for smaller λ is reached for lower kmax. A consequence is faster convergence in variational and ab initio few-body calculations, as shown in the left panel of Fig. 2 for two NLO potentials, where the energy is plotted against the size of a harmonic oscillator basis. A complete study with NN potentials in the no-core shell model is given in ref. [6]. 10 20 30 N max 10 20 30 N max −8.5 −8.0 −7.5 −7.0 −6.5 −6.0 −5.5 −5.0 E t [ M eV ] initial λ = 4 fm λ = 3 fm λ = 2 fm λ = 1 fm 550/600 MeV 600 MeV 7.6 7.8 8 8.2 8.4 8.6 8.8 E b ( 3 H) [MeV] 24 25 26 27 28 29 30 31 E b ( 4 H e) [ M eV ] NN potentials SRG N 3 LO (500 MeV) N 3 LO λ=1.0 λ=3.0 λ=1.25 λ=2.5 λ=2.25 λ=1.5 λ=2.0 λ=1.75 Expt. A=3,4 binding energies SRG NN only, λ in fm Figure 2. Left: Convergence in the triton [4]. Right: Tjon line traced out by SRG-evolved NN potentials, labeled by λ [6]. The commutators in Eq. (1) imply that the evolving Hamiltonian will have many-body interactions to all orders (i.e., insert second-quantized operators). Thus there will always be a truncation and the evolution will only be approximately unitary. The present calculations evolve only the NN part, which explains why different converged triton energies are seen in Fig. 2. This is a controlled R.J. Furnstahl 3 approximation in the range of λ for which the variation is comparable to the truncation error inherent in the initial EFT Hamiltonian. The variation is seen to be natural in Fig. 2 and the left panel of Fig. 3, which also shows the improved convergence (decreasing error bars) for smaller λ [6]. 0.5 1 1.5 2 2.5 3 3.5 λ [fm] −34 −33 −32 −31 −30 E gs [ M eV ] SRG NCSM Li