From weak to strong correlation : A new renormalization group

1. From weak to strong correlation: Anew renormalization group approach to strongly correlated Fermi liquids Alex Hewson , Khan Edwards, Daniel Crow, Imperial College London, U.K Yunori Nishikawa, Osaka City University, Japan; Johannes Bauer, MPI, Stuttgart

2. Fermi Liquid Theory The low energy single particle excitations of the system are quasiparticles with energies in 1-1 correspondence with those of the non-interacting system ie. interaction between quasiparticles We have a number of exact results at T=0: • specific heat coefficient free quasiparticle density of states charge susceptibility spin susceptibility The low energy dynamic susceptibilities and collective excitations can be calculated by taking account of repeated quasiparticle scattering.

3. Models of systems with strong electron correlations Hubbard Model A simplified of electrons in the 3d bands of transition metals Anderson Impurity model A model of localized states of an impurity in a metallic host, or more recently as a model of a quantum dot Periodic Anderson model Essentially a lattice version of the impurity model with d or f electrons hybridized with a conduction band – model for heavy fermions

4. Can we relate the parameters of Fermi liquid theory to renormalized parameters that define these models? We note: 1. Quasiparticles should correspond to the low energy poles in the single-electron Green’s function 2. The quasiparticle interactions should correspond to the low energy vertices in a many-body perturbation theory This enables us to interpret the Fermi liquid parameters in terms of renormalizations of the parameters that specify these models

5. Renormalised Parameters: Anderson Model Four parameters define the model Local Green's function Use substitution New form of Green’s function renormalized parameters quasiparticle Green’s function Interaction interaction between quasiparticles

6. Summary of Renormalized Perturbation Theory (RPT) approach • The renormalised parameters (RP) describe the fully dressed quasiparticlesof Fermi liquid theory. • They provide an alternative specification of the model • We can develop a renormalised perturbation theory (RPT) to calculate the behaviour of the model under equilibrium and steady state conditions using the free quasiparticle propagator, • In powers of with counter terms to prevent overcounting, determined by the conditions: • Exact low temperature results for the Fermi liquid regime are obtained by working only to second order only!

8. Kondo Limit --- only one renormalised parameter N-fold Degenerate Anderson Model The n-channel Anderson Model with n=2S (renormalised Hund’s rule term)

9. Relation to Fermi Liquid theory This would be the RPA approximation for bare particles fin the case

10. Renormalized Parameters calculated from the NRG energy levels energy scales merge --- strong correlation regime The plot shows how the renormalized parameters vary as the impurity level moves from below the Fermi level to above the Fermi level for a fixed value of U with In the Kondo regime

11. Renormalized Parameters from the NRG energy levels in a magnetic field RPA regime? mean field regime Kondo regime Strong coupling condition

12. Can we derive these results from perturbation theory? We calculate the parameters directly from the definitions in four stages: 1. We use mean field theory to calculate the renormalised parameters in extremely large field h1 (>>U) 2. Extend the calculation to include RPA diagrams in the self-energy 3. We use the renormalized parameters in field h1 to calculate the renormalized self-energy in a reduced field h2, and calculate the renormalized parameters in the reduced field. 4. We set up a scaling equation for the renormalized parameters to reduce the field to zero.

13. Stage 1 Stage 2 Stages 3, 4 ?

14. Weak field strong correlation regime Strong correlation result satisfied

15. Further comparison of direct RPT with Bethe Ansatz and NRG results Magnetization as a function of m magneticfield compared to NRG results T=0, H=0, susceptibility compared with Bethe anasatz results as a function of U

16. Comparison of RPT and NRG results in the low field regime

17. Quantum critical points of a two impurity Anderson model This model has two types of quantum critical points Local singlet transition - Local charge order transition

18. Quantum Critical Points in Heavy Fermion Compounds NFL QCP Candidate for the local “Kondo collapse” scenario From a recent review by Si and Steglich -Science 329, 1161 (2010)

19. Quantum critical transitions in the symmetric model strong J predominantly local screening locally charged ordered state (U/D=0.05) weak J predominantly Kondo screening (U12=0)

20. Exact RPT results for the low energy behaviour Predictions based on continuity of these susceptibilities at the QCP:

21. Calculation of renormalized parameters by the NRG (U12=0) implies and Kondo resonance at Fermi level disappears

22. Results for large U Universal curves Agreement with predictions

23. Convergence of energy scales for small U Confirm predictions as J Jc (U12=0)

24. Calculations for J>Jc At J=Jc z--> 0 so we lose the Kondo resonance at the Fermi level because the self-energy develops a singularity At J=Jc there is a sudden change in the NRG fixed point from one for an even (odd) chain to that for an odd (even) one We can no longer use the RPT as we assumed the self-energy to be analytic at the Fermi-level We retain the equations as a description of a local Fermi liquid but in the NRG we treat the first conduction site as an effective impurity because now the impurities are decoupled as a local singlet. We then calculate the renormalized parameters for J>Jc from the NRG fixed point in a similar way as for J<Jc but we must allow for the fact that we have modified the conduction chain.

25. Results through the transition (U12=0) NFL Fermi liquid 1 Fermi liquid 2

26. Local charge order transition SU(4) point Predictions again confirmed

27. Leading low temperature correction terms in Fermi liquid regime 1 These corrections to the self-energy can be calculated exactly using the second order diagrams in the RPT:

28. Susceptibility through the transition (U12=0)

29. Dynamic Susceptibilities

30. Temperature dependence in the non-Fermi liquid regime? We explore the idea of using temperature dependent renormalized parameters from the NRG

31. Conclusions We have demonstrated that it is possible to obtain accurate results for the low energy behaviour of the Anderson model in the strong correlation regime by introducing a magnetic field to suppress the low energy spin fluctuations which lead to the large mass renormalisations, and then slowly reduce the field to zero, renormalizing the parameters at each stage. This approach should be applicable to a wide range of strong correlation models such as the Hubbard and periodic Anderson model. The results for the two impurity model support the Kondo collapse conjecture that at the quantum critical point in some heavy fermions systems the f-states at the Fermi level disappear and no longer contribute to a large Fermi surface (eg. Yb2Rh2Si2). The convergence to a single energy scale T* which goes to zero at the QCP suggest how w,T scaling can arise.