Similarity Renormalization Group for Nuclear Physics

Welcome to the SRG home page!

This website is maintained by Dick Furnstahl (furnstahl.1@osu.edu) and his collaborators (including Eric Anderson, Scott Bogner, Eric Jurgenson, Robert Perry, Lucas Platter, and Achim Schwenk). It will be updated at irregular intervals with new pictures, movies, and links. You are invited to send email with suggestions for new content or questions on what you see.

The research results presented here were supported in part by the National Science Foundation under Grant Nos. PHY-0354916 and PHY-0653312 and by the UNEDF SciDAC Collaboration under DOE grant DE-FC02-07ER41457.

Recent changes to this page:

• 25-Jul-2008 --- Updated links to papers on the SRG in nuclear physics.
• 08-Nov-2007 --- Added reference to recent book: The Flow Equation Approach to Many-Particle Systems by Stefan Kehrein.
• 19-Sep-2007 --- Added links to some unpublished explorations of the SRG.
• 18-Sep-2007 --- Added links to talks on the SRG for nuclear physics.

Contents

• Background and references
• Talks on the SRG and Nuclear Physics
• Unpublished Explorations of the SRG
• Movies of the SRG in Action
• Plots and Images
• Links to Other SRG-Related Sites

Overview

Background:

The similarity renormalization group (SRG) for low-energy nuclear physics is based on unitary transformations that suppress off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables, and provides a method to consistently evolve many-body potentials and other operators.

General references on the SRG and related Hamiltonian Flow Equations

The Flow Equation Approach to Many-Particle Systems, Stefan Kehrein (Springer, Berlin, 2006).

Renormalization of Hamiltonians, S.D. Glazek and K.G. Wilson, Phys. Rev. D 48, 5863 (1993).

Flow-Equations for Hamiltonians, F. Wegner, Annalen der Physik (Leipzig) 3, 77 (1994).

Perturbative renormalization group for Hamiltonians, S.D. Glazek and K.G. Wilson, Phys. Rev. D 49, 4214 (1994).

Asymptotic freedom and bound states in Hamiltonian dynamics, S.D. Glazek and K.G. Wilson, Phys. Rev. D 57, 3558 (1998), [hep-th/9707028].

Limit cycles in quantum theories, S.D. Glazek and K.G. Wilson, Phys. Rev. Lett. 89, 230401 (2002), [hep-th/0203088].

Flow Equations and Normal Ordering, E. Koerding and F. Wegner, cond-mat/0509801.

Flow Equations and Normal Ordering. A Survey, F. Wegner, cond-mat/0511660.

Limit cycles of effective theories, S.D. Glazek, Phys. Rev. D 75, 025005 (2007), [hep-th/0611015].

References on the SRG for Nuclear Physics:

Similarity renormalization group for nucleon-nucleon interactions, S.K. Bogner, R.J. Furnstahl, and R.J. Perry, Phys. Rev. C 75, 061001(R) (2007), [nucl-th/0611045]. This is probably the best place to start for learning about how the SRG can be applied to low-energy nuclear physics.

Are low-energy nuclear observables sensitive to high-energy phase shifts?, S.K. Bogner, R.J. Furnstahl, R.J. Perry, and A. Schwenk, Phys. Lett. B 649, 488 (2007), [nucl-th/0701013]. This paper shows how the SRG decouples low-energy from high-energy physics.

The Unitary Correlation Operator Method from a similarity renormalization group perspective, H. Hergert and R. Roth, Phys. Rev. C. 75, 051001(R) (2007), [nucl-th/0703006]. The UCOM method of unitary transformations is related to the SRG.

Three-body forces produced by a similarity renormalization group transformation in a simple model, S.K. Bogner, R.J. Furnstahl, and R.J. Perry, Ann. Phys. (NY) 323, 1478 (2008), [arXiv:0708.1602]. This is a pedagogical paper that shows in detail using a very simple model (only 2x2 matrices!) how the SRG works for two- and three-body interactions. A diagrammatic approach to the SRG equations is introduced and applied.

Convergence in the no-core shell model with low-momentum two-nucleon interactions, S.K. Bogner, R.J. Furnstahl, P. Maris, R.J. Perry, A. Schwenk, and J.P. Vary, Nucl. Phys. A801, 21 (2008) [arXiv:0708.3754]. Application of the SRG with only NN interactions (no three-body) to few-body systems up to Li-7 to show the improved convergence with SRG running.

Decoupling in the similarity renormalization group for nucleon-nucleon forces, E.D. Jurgenson, S.K. Bogner, R.J. Furnstahl, and R.J. Perry, Phys. Rev. C 78, 014003 (2008) [arXiv:0711.4252]. This paper explores in detail how decoupling plays out in the SRG.

Block diagonalization using srg flow equations, E. Anderson, S.K. Bogner, R.J. Furnstahl, E.D. Jurgenson, R.J. Perry, and A. Schwenk, Phys. Rev. C 77, 037001 (2008) [arXiv:0801.1098]. This paper shows how one can choose different generators for the SRG to get different patterns of decoupling in a Hamiltonian. In particular, block diagonalization.

The impact of bound states on similarity renormalization group transformations, S.D. Glazek and R.J. Perry, Phys. Rev. D (in press) [arXiv:0803.2911].

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Talks on the SRG for Nuclear Physics

• Decoupling with the Similarity Renormalization Group (SRG), 10-minute talk by Eric Jurgenson at the APS Division of Nuclear Physics meeting, October, 2007.
• Operator Evolution Via the Similarity Renormalization Group, 10-minute talk by Eric Anderson at the APS Division of Nuclear Physics meeting, October, 2007.
• Similiary Renormalization Group For Few-Body Systems, talk by Dick Furnstahl at the EFB20, Pisa, Italy, September, 2007.
• Low-Momentum Interactions for Few- and Many-Body Systems, talk by Scott Bogner at the EFB20, Pisa, Italy, September, 2007.
• The Similarity Renormalization Group --- In Pictures, talk by Eric Anderson at the National Nuclear Physics Summer School, Tallahassee, FL, July, 2007.
• Decoupling with the SRG, talk by Eric Jurgenson at the National Nuclear Physics summer School, Tallahassee, FL, July, 2007.
• Three Nucleon Forces and the Similarity Renormalization Group, talk by Robert Perry at TRIUMF Three-Body workshop, Vancouver, BC, March, 2007.

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Unpublished Explorations of the SRG

• The SRG Differential Equation in Pictures. We use color contour plots to examine the terms in the SRG differential equation that uses the relative kinetic energy to evolve.
• N3LOW Pictures. Pictures of the low-momentum chiral EFT potential N3LOW by Entem and Machleidt.

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Movies of the SRG and V_{low k} in Action

The Similarity Renormalization Group (SRG) and V_{low k} potentials evolve as a parameter is lowered. For the SRG, it is lambda, which is a measure of the spread of the momentum-space potential about the diagonal. For V_{low k}, it is the momentum cutoff Lambda. There is a rough correspondence between the potentials at numerically similar values of lambda and Lambda. The units of the potentials are fm (hbar²/M = 1).

The movies here depict the evolution of the potentials (and related quantities) with lambda and Lambda.

SRG Evolved V_srg

• V_srg evolved from the Argonne V18 potential
• V_srg evolved from the N3LO Entem/Machleidt potential with 500 MeV and 600 MeV cutoffs
• V_srg evolved from the N3LO Epelbaum et al. potential with 600/550 MeV cutoff
• V_srg evolved from the N3LOW Entem/Machleidt potential with 400 MeV cutoff

SRG Evolved Operators (other than the hamiltonian)

• Operators (or their matrix elements in the deuteron) evolved from the Argonne V18 potential

Cutoff Evolved V_{low k}

• V_{low k} evolved from the Argonne V18 potential
• V_{low k} evolved from the N3LO Entem/Machleidt potential with 500 MeV and 600 MeV cutoffs
• V_{low k} evolved from the N3LO Epelbaum et al. potential with 600/550 MeV cutoff

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Pictures

Included here are both color contour plots and ordinary figures.

SRG Evolved V_srg

• V_srg evolved from the Argonne V18 potential
• V_srg evolved from the N3LO Entem/Machleidt potential with 500 MeV and 600 MeV cutoffs
• V_srg evolved from the N3LO Epelbaum et al. potential with 600/550 MeV cutoff

Deuteron Properties for V_srg and V_{low k}

Shown are the binding energy and asymptotic D/S ratio, which are observables, and the D-state probability, which is not, as a function of lambda or Lambda.

• SRG potentials
• V_{low k} potentials

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Links to other SRG-Related Sites

• Stan Glazek is a leading expert on the SRG (and one of its inventors). Here we link to some videos created by Stan and his students.
  • Three-dimensional limit cycle videos.
  • Asymptotic freedom videos.
• Flow Equations for Hamiltonians page from the University of Heidelberg. Many references are available here.

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