Unitary correlation operator method from a similarity renormalization group perspective

We investigate how the unitary correlation operator method (UCOM), developed to explicitly describe the strong short-range central and tensor correlations present in the nuclear many-body system, relates to the similarity renormalization group (SRG), a method to band-diagonalize Hamiltonians by continuous unitary transformations. We demonstrate that the structure of the UCOM transformation, originally motivated from the physically intuitive picture of correlations in coordinate space, arises naturally from the SRG flow equation. Apart from formal considerations we show that the momentum space matrix elements of the effective interactions obtained in both schemes agree extremely well.

Figure 1

Three-dimensional plots of the momentum space matrix elements $v(q,q^{\prime })$ in the ${}^3{S}_1$ partial wave for the AV18 potential (a), the $V_{\mathrm{UCOM}}$ interaction with $I_{\vartheta }=0.09{\mathrm{fm}}^3$ (b), and the SRG interaction $V_{\alpha }$ for $\overline{\alpha }=0.0215{\mathrm{fm}}^4$ (c). The dots mark the plane of vanishing matrix elements.

Figure 2

Off-diagonal matrix elements $v(0,q)$ in the ${}^3{S}_1$ and ${}^3{S}_1-{}^3{D}_1$ partial waves. Panels (a) and (b) show the $V_{\mathrm{UCOM}}$ matrix elements for $I_{\vartheta }=$ $0.03{\mathrm{fm}}^3$ (——), $0.06{\mathrm{fm}}^3$ (–––), and $0.09{\mathrm{fm}}^3$ (·–·–·) in comparison to those of AV18 (-----). Panels (c) and (d) show the SRG-evolved matrix elements of $V_{\alpha }$ for flow parameter values $\overline{\alpha }=0.0055{\mathrm{fm}}^4$ (——), $0.0125{\mathrm{fm}}^4$ (–––), and $0.0215{\mathrm{fm}}^4$ (-–-–-), which have been chosen to reproduce the value of the corresponding $V_{\mathrm{UCOM}}$ matrix element at $q=q^{\prime }=0{\mathrm{fm}}^{-1}$.