RG Methods for Nuclei and Neutron Stars Achim Schwenk University of - PowerPoint PPT Presentation

RG Methods for Nuclei and Neutron Stars Achim Schwenk University of Washington / TRIUMF (2006-) supported by US DOE and GSI

Outline 1) Motivation and introduction 2) Renormalization group approach to nucleonic matter 3) Applications to electronic systems 4) Some low-momentum results 5) Summary

1) Motivation and introduction Crab pulsar Many-body physics of nucleonic matter Supernova July 4, 1054 Density Functional Theory A>100 Supernova SN1987a Coupled Cluster, Shell Model A<100 Exact methods A ≤ 12 Nuclei with extreme Low-mom. GFMC, NCSM interactions compositions of n,p Lattice QCD “Femtoscience” structural dep. on N/Z Chiral EFT interactions (low-energy theory of QCD) Connections to QCD quark mass dep. of shell QCD Lagrangian structure Nuclear superfluidity adapted from A. Richter @ INPC2004

Cold atoms in the vicinity of Feshbach resonances Fermi gases with resonant interactions exhibit universal properties for large scattering lengths, no scales associated with interaction at low energy can constrain properties of extreme Feshbach resonances in 6 Li low-density neutron matter in lab experiments with resonantly-tuned atomic 6 Li or 40 K universal energy ξ =0.51±0.04 Duke 2005, (2002) ξ =0.27±0.12 Innsbruck 2004 Thermodynamics continuous across resonance! Bourdel et al., PRL 91 (2003) 020402.

Nuclear matter under extreme conditions in supernovae desire systematic approach that includes sun in neutrinos with SuperK bound states: d, α ,… and resonances: nn,pp( 1 S 0 ), n α (P 3/2 ), αα (0 + ,2 + ),… on equal footing Supernova SN1987a Virial Equation of State of low-density nuclear matter composed of n,p, α with second virial coefficients directly from NN, N α , αα phase shifts Horowitz, AS, nucl-th/0507033, nucl-th/0507064. gives model-independent results for neutrinosphere in supernovae, n ~ 10 11 -10 12 g/cm 3 , T ~ 4 MeV resulting α particle mass fraction disagrees with all EoS models used in supernova simulations

Extreme conditions in neutron stars Neutron star cooling is dominated by ν emission from interior nucleons beta decay in 1N or 2N reactions, ν bremsstrahlung,… ν emission suppressed due to superfluidity (costs pairing energy to break nn, pp pair) except for enhancement of ν emission below superfluid T c ∆ ~ 50 keV ∆ ~ 120 keV ∆ ~ 100 keV Yakovlev, Pethick, ARAA 42 (2004) 169. Blaschke et al., A&A 424 (2004) 979. found too rapid cooling for P-wave superfluid gaps ∆ P > 30 keV Need reliable, microscopic predictions for nucleonic pairing gaps!

Pairing in halo nuclei and ν emission neutron stars: ν emission suppressed due to superfluidity, costs pairing energy to break nn, pp pair from P. Garrett Sarazin et al., PR C70 (2004) 031302(R). 3/2 - 8.81 beta decay of nn halo suppressed due to pairing, as for ν emission in neutron star cooling

Many-body calculations are traditionally based on different nucleon-nucleon (NN) interactions fit to NN scattering below ( for low momenta k < 2.1 fm -1 ) Argonne v 18 k k Details not constrained for momenta k > 2.1 fm -1 or distances r < 0.5 fm This is where NN models are strong and thus difficult to handle.

Unifying nuclear forces with the Renormalization Group k Scattering amplitude (T matrix) in mom. space in matrix form V NN ( k ’ ,k ) Integrating out high-momentum modes leads k ’ to effective interaction V low k (which reproduces the low-momentum T matrix) k Λ V low k Λ Changes of effective interaction with cutoff Λ k ’ given by RG equation

Low-momentum interactions for low-energy phenomena, desire low-momentum interactions, no need for model-dependent high-momentum modes k k Λ in matrix form integrate out high-mom. modes V low k with k > Λ using the V NN ( k ’ , k ) Renormalization Group (RG) Λ Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1. k ’ k ’ leads to universal low-momentum interaction V low k for all V low k

Benchmark BCS gaps in neutron matter uses free-space NN interaction for pairing and free spectrum low-density ( crust ): 1 S 0 pairing higher-density ( core ): 3 P 2 pairing Baldo et al., PR C58 (1998) 1921. np gaps nn gaps phase-shift equivalent NN interactions give identical BCS gaps resolves charge-dependence of NN force:

Beyond BCS theory: induced interactions Simple case: dilute gases (perturbative ) Gorkov et al., JETP 13 (1961) 1018, Heiselberg et al., PRL 85 (2000) 2418. screening/vertex corr. benchmark + induced interactions Induced interactions lead to a significant reduction of the gap [universal reduction for 2 spin states] RG approach to include particle-hole intermediate states away from Fermi surface

2) RG approach to interacting Fermi systems Shankar, RMP 66 (1994) 129. Cutoff Λ around the Fermi surface defines an effective theory for low-mom. particles and holes start from full space with vacuum V low k integrate out mom. shells successively effective interaction, dressed quasiparticles Simpler problem in truncated space, RG focuses on low-lying modes

One-loop particle-hole RG Change of effective 4-point vertex: [ ] ZS ’ ZS Intermediate states: thin from momentum shells, thick from fast particles/holes + RG treats momentum dependences on equal footing (in contrast to one-channel resummations, no momentum extrapolations needed) + RG maintains all symmetries - Currently: treat pairing correlations explicitly after ph RG Future: include BCS channel with RG equation for pairing gap

At one-loop, renormalization due to change in ph phase space ZS ZS ’ On the Fermi surface keep only quasi-local a(q 2 ,q ’ 2 ) in RGE to extrapolate for off the Fermi surface, on V low k level dep. small Patch k Fermi surface [similar to N-patch scheme for 2d Hubbard: wave vect. k const. 4-point vertex for momenta in patch]

Efficacy of the RG method V low k Start from free-space two-body interaction After two decimations: builds up many-body correlations (~ parquet)

cutoffs fast ph Phase space argument in RG equation: ZS ZS ’ In long-wavelength limit can calculate RGE analytically q=0 eliminates ZS flow, define qp interaction + forward amplitude in RGE

Illustration for a schematic model Spinless fermions with simple interaction in units of bare dos, coupling can be related to P-wave scattering volume AS, Friman et al., nucl-th/0207004. angles ~ momentum transfers Flow of A( q ,q ’ fixed) small q - interplay of ZS and ZS ’ channels at one-loop large q - large q amplitude runs first - ZS and ZS ’ channels counteract flow

RG flow of qp interaction and Fermi liquid parameters [m * /m = 6/5] small q ’ large q ’ Flow according to l>1 Landau parameters generated in RG, no truncations or model for mom. dependences needed

Renormalization of effective mass and qp strength Should be solving RGE for 2-pt function Use approximate method for quadratic dispersion - Fermi liquid relation for running of eff. mass - static/adaptive method for qp strength 1. static density-independent mean value in RGE 2. start with adapt self-consistently approximates k-mass by V low k contribution resulting factor agrees well with self-consistent in-medium ladders for static z factor

Results for neutron matter Find increase of effective mass at low squares: static z factor triangles: adaptive z factor densities see Wambach et al., NP A555 (1993) 128. ph RG V low k Fermi liquid parameters decrease of F 0 small effects on G 0

RG also yields non-forward scattering amplitude for transport, pairing,… S-wave pairing in neutron matter With induced interactions: max. S-wave gap ∆ ≈ 0.8 MeV BCS gaps: max. ∆ ≈ 3 MeV AS, Friman, Brown, NPA 713 (2003) 191. spin fluctuations suppress S-wave pairing in neutron matter

Polarization effects on P-wave pairing For P-waves: spin-orbit, tensor forces crucial Screening effects? Spin fluctuations attractive in S=1 Pethick, Ravenhall, Ann. NY Acad. Sci. 647 (1991) 503; Jackson et al., NPA 386 (1982) 125. For P-wave pairing: first perturbative assessment AS, Friman, PRL 92 (2004) 082501. (Note: 2 nd order / V low k < 50%) Induced spin-orbit forces deplete P-wave pairing P-wave gaps < 10 keV possible Confirmed in above neutron star cooling simulations

Extensions Quasiparticle interations, n/p pairing, phase diagram of asymmetric matter AS, Friman, Furnstahl, in prep. Beyond one-loop RG: higher loops suppressed by 1/N ~ Λ /k F for regular Fermi surfaces Shankar Renormalization of currents/operators see Halboth, Metzner, PR B61 (2000) 7364. Effective interactions among valence nucleons in finite nuclei see el. excitations in atoms Murthy, Kais, Chem. Phys. Lett. 290 (1998) 199.

3) Applications to electronic systems Zanchi, Schulz, PR B61 (2000) 13609, Halboth, Metzner, PRL 85 (2000) 5162, Salmhofer, Honerkamp, PTP 105 (2001) 1, Binz et al., Ann. Phys. 12 (2003) 704. 2d Hubbard model: U t ‘ t Patch k Fermi surface k 1 k 2 N-patch scheme: 4-point V(k i ) constant for k i in same patch wave vect. k k 3 k 4 RG approach studies interference of infrared instabilities via flow towards the Fermi surface

Eliminate momentum shells according to flow V=U typical flow of couplings due to build-up and interference of instabilities identify leading instabilities: spin-density wave, d-wave SC unbiased RG phase diagram