From weak to strong correlation:

A new 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

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:

free quasiparticle density of states specific heat coefficient

The low energy dynamic susceptibilities and collective excitations can be calculated by taking account of repeated quasiparticle scattering.

spin susceptibility charge susceptibility

Models of systems with strong electron correlations

Hubbard Model

Anderson Impurity model

Periodic Anderson model

A simplified of electrons in the 3d bands of transition metals

A model of localized states of an impurity in a metallic host, or more recently as a model of a quantum dot

Essentially a lattice version of the impurity model with d or f electrons hybridized with a conduction band – model for heavy fermions

Can we relate the parameters of Fermi liquid theory to renormalized parameters that define these models?

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

We note:

This enables us to interpret the Fermi liquid parameters in terms of renormalizations of the parameters that specify these models

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

The renormalised parameters (RP) describe the fully dressed quasiparticles of 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!

Summary of Renormalized Perturbation Theory (RPT) approach

Kondo Limit --- only one renormalised parameter

N-fold Degenerate Anderson Model

The n-channel Anderson Model with n=2S

(renormalised Hund’s rule term)

Relation to Fermi Liquid theory

This would be the RPA approximation for bare particles fin the case

The plot shows how the renormalized parameters vary as the impurity levelmoves from below the Fermi level to above the Fermi level for a fixed valueof U with

In the Kondo regime

energy scales merge --- strong correlation regime

Renormalized Parameters calculated from the NRG energy levels

Renormalized Parameters from the NRG energy levels in a magnetic field

mean field regimeKondo regime

RPA regime?

Strong coupling condition

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.

Stage 2 Stages 3, 4?

Stage 1

Weak field strong correlation regime

Strong correlation result satisfied

Further comparison of direct RPT with Bethe Ansatz and NRG results

Magnetization as a function of mmagnetic field compared to NRG results

T=0, H=0, susceptibility compared with Bethe anasatz results as a function of U

Comparison of RPT and NRG results in the low field regime

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

Quantum Critical Points in Heavy Fermion Compounds

From a recent review by Si and Steglich -Science 329, 1161 (2010)

Candidate for the local “Kondo collapse” scenario

QCP

NFL

Quantum critical transitions in the symmetric model

weak J predominantly Kondo screening (U12=0)

strong J predominantly local screening

locally charged ordered state (U/D=0.05)

Exact RPT results for the low energy behaviour

Predictions based on continuity of these susceptibilities at the QCP:

Calculation of renormalized parameters by the NRG

(U12=0)

implies and Kondo resonance at Fermi level disappears

Results for large U

Universal curves Agreement with predictions

Convergence of energy scales for small U

Confirm predictions as J Jc (U12=0)

Calculations for J>Jc

At J=Jc z--> 0 so we lose the Kondo resonance at the Fermi levelbecause 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 analyticat the Fermi-level

We retain the equations as a description of a local Fermi liquid but in the NRGwe 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 pointin a similar way as for J<Jc but we must allow for the fact that we have modifiedthe conduction chain.

Fermi liquid 1 Fermi liquid 2

NFL

Results through the transition (U12=0)

SU(4) point

Local charge order transition

Predictions again confirmed

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:

Susceptibility through the transition (U12=0)

Dynamic Susceptibilities

Temperature dependence in the non-Fermi liquid regime?

We explore the idea of using temperature dependentrenormalized parameters from the NRG

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 periodicAnderson 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.