Block diagonalization using similarity renormalization group flow equations

Abstract

By choosing appropriate generators for the Similarity Renormalization Group (SRG) flow equations, different patterns of decoupling in a Hamiltonian can be achieved. Sharp and smooth block-diagonal forms of phase-shift equivalent nucleon-nucleon potentials in momentum space are generated as examples and compared to analogous low-momentum interactions ("V low k "). Here s is a flow parameter and the flow operator G s specifies the type of SRG [4]. Decoupling between low-energy and highenergy matrix elements is naturally achieved in a momentum basis by choosing a momentum-diagonal flow operator such as the kinetic energy T rel or the diagonal of H s ; either drives the Hamiltonian toward band-diagonal form. This decoupling leads to dramatically improved variational convergence in fewbody nuclear systems compared to unevolved phenomenological or chiral effective field theory (EFT) potentials [5,6]. Renormalization Group (RG) methods that evolve NN interactions with a sharp or smooth cutoff in relative momentum, known generically as V low k , rely on the invariance of the two-nucleon T matrix [7,8]. These approaches achieve a block-diagonal form characterized by a cutoff (see left plots in Figs. 1 and 2). As implemented in Refs. [7,8], the high-momentum matrix elements are defined to be zero, but this is not required.Block-diagonal decoupling of the sharp V low k form can be generated using SRG flow equations by choosing a blockdiagonal flow operator (see Refs. [9,10] for details),with projection operators P and Q = 1 − P . In a partial-wave momentum representation, P and Q are step functions defined by a sharp cutoff on relative momenta. This choice for G s , which means that η s is nonzero only where G s is zero, suppresses off-diagonal matrix elements such that the Hamiltonian approaches a block-diagonal form as s increases. If one considers a measure of the off-diagonal coupling of the Hamiltonian,then its derivative is easily evaluated by applying the SRG equation, Eq. (1):Thus, the off-diagonal QH s P block will decrease in general as s increases [9,10]. The right plots in Figs. 1 and 2 result from evolving the N 3 LO potential from Ref.[11] using the block-diagonal G s of Eq. (2) with = 2 fm −1 until λ ≡ 1/s 1/4 = 0.5 fm −1 . The parameter λ is a quantitative measure of the progress toward block diagonalization made by the SRG evolution. The agreement between V low k and SRG potentials for momenta below is striking. A similar degree of universality is found in the other partial waves. Deriving an explicit connection between these approaches is the topic of an ongoing investigation.The evolution with λ of two representative partial waves ( 3 S 1 and 1 P 1 ) are shown in Figs. 3 and 4. The evolution of the "off-diagonal" matrix elements (meaning those outside the P H s P and QH s Q blocks) can be roughly understood from the dominance of the kinetic energy on the diagonal. Let the indices p and q run over indices of the momentum states in the P and Q spaces, respectively. To good approximation we can replace P H ...

Nuclear structure in the framework of the Unitary Correlation Operator Method

“…The particular physical system and application under consideration determines which basis and generator is used in the flow evolution. In this respect the SRG approach is very flexible and can be adapted to all kinds of bandor block-diagonalizations in any basis of choice [62,68]. This flexibility is an advantage of the SRG scheme as compared to the UCOM transformation, which is tailored for a very specific type of correlations.…”

“…In this way, the Hamiltonian is driven towards a diagonal form in the harmonic oscillator basis. Using various projection operators one can design generators that drive the Hamiltonian towards a block-diagonal structure in a given basis [68]. This flexibility of the SRG technique holds great potential for further refinements and applications of the approach.…”

Similarity renormalization group evolution of three-nucleon forces in a hyperspherical momentum representation

“…The T rel -SRG flow of two-nucleon forces (2NF) has been studied in detail using both momentum representation [1,[4][5][6][7] and in a discrete harmonic oscillator (HO) basis [8].…”

Alternative similarity renormalization group generators in nuclear structure calculations

“…2), the P gen H s Q gen and Q gen H s P gen blocks do get systematically eliminated with increasing s (decreasing λ) as anticipated based on general arguments presented in Ref. [37] (see Eq. (4) there).…”