Renormalised Perturbation Theory ● Motivation ● Illustration with the Anderson impurity model ● Ways of calculating the renormalised parameters ● Range of Applications ● Future Developments Work in collaboration with Johannes Bauer, Winfried Koller, Dietrich Meyer and Akira Oguri

Renormalisation in Field Theory Aim to eliminate divergences Certain quantities are taken into account at the beginning so one works with (i) the final mass --- absorb all mass renormalisations (ii) the final interaction or charge---absorb all charge renormalisations (iii) the final field---absorb all field renormalisations Parameters characterising the renormalised perturbation expansion ; (i) renormalised mass m(ii) renormalised interaction g (iii) renormalised field ~ The expansion is carried out in powers of g and the counter terms cancel renormalisations which have already been taken into account ~ ~       

Form of Perturbation Expansion for  heory Renormalisation conditions: and separated out 

wide band limit

Apply the same procedure to the Anderson model definition of renormalised parameters renormalised interaction

Finite Order Calculations in Powers of Two methods of calculation: Method 1: With counter terms: Method 2: Without counter terms Step 2: Calculate the renormalised parameters in perturbation theory in powers of U using Step 3: Invert to the required order to find the bare parameters in terms of the renormalised ones Step 4: Express the quantity calculated in terms of the renormalised parameters Step 1: Calculate the quantity using perturbation theory in the bare interaction U The three counters are determined by the renormalisation conditions

Example of Method 2: Susceptibility calculation to order Step 1: Step 2: Step 3: Step 4: same result as calculated using counter terms

Low Order Results Zero Order Friedel Sum Rule Define free quasiparticle DOS Specific heat coefficient First Order Spin susceptibilities and charge Second Order Impurity conductivity symmetric model All these results are exact (Ward identities, Yamada)

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

Calculation of and using the NRG NRG chain Given  d and V the excitations  n of the non- interacting system are solution of the equation: Non-interacting Green’s function Interacting Case We require the lowest single particle E p (N) and hole E h (N) excitations to satisfy this equation for a chain of length N This gives us N-dependent parameters

 Kondo regime

Quasiparticle Interactions We look at the difference between the lowest two-particle excitations E pp (N) and two single particle excitations 2 E p (N). This interaction U pp (N) will depend on the excitations and chain length N. We can define a similar interaction U hh (N) between holes U ph (N) and between a particle and hole If they are all have the same value for large N, independent of N then we can identify this value with U In the Kondo limit we should find ~ ~ ~~

Overview of renormalised parameters in terms of ‘bare’ values Full orbital >>>> mixed valence >>>> Kondo regime >>>>> mixed valence >>>>> empty orbital

Note accurate values for large values of discretisation parameter Full orbital >>>> mixed valence >>>> Kondo regime

Overview for U>0 as a function of the occupation value n d Strongest renormalisations in the case of half-filling

Overview for U<0 as a function of the occupation n d Features can be interpreted in terms of a magnetic field using a charge to spin mapping

Applications using this approach Systems in a magnetic field H We develop the idea of field dependent parameters—like running coupling constants----appropriate to the value of the magnetic field for symmetric model with and Dynamic spin susceptibilities in a magnetic field --- impurity and Hubbard models Quantum dot in a magnetic field field and finite bias voltage Antiferromagnetic states of Hubbard model

Renormalised parameters a a function of the magnetic field value Parameters are not all independent: Mean field regime U= 

Without particle-hole symmetry

Induced Magnetisation Comparison with Bethe ansatz for localised model U=3  BA AM Charge fluctuations playing a role

Low Temperature behaviour in a magnetic field All second order coefficients have a change of sign at a critical field hc where 0<h c <T*

Susceptibility Impurity contribution to conductivity

  (h) changes sign at h=h c in the Kondo regime Impurity contribution to conductivity Conductance of quantum dot G 2 (h) changes sign in this range

We look at the repeated scattering of a quasiparticle with spin up and a quasihole with spin down Spin and Charge Dynamics new vertex condition determines vertex in this channel Vertex in terms of U ~

Spin and charge irreducible Verticies charge spin

Imaginary part of dynamic spin susceptibility Note the different energy scales in the two cases NRG results using complete Anders-Schiller basis _______ RPT

Real part of dynamic spin susceptibility Imaginary parts of spin and charge dynamic susceptibilities spin charge

RPA Imaginary part of dynamic spin susceptibilities

Spin and charge dynamics in a magnetic field Irreducible vertices charge _|_ spin || spin U

Non-interacting Case U=0 _|_ || _|_ NRG compared with exact results

NRG compared with RPT in the interacting case _|_ || _|_

Comparison of NRG and RPT results in strong field limit _|_ Imaginary part of transverse susceptibility

Without Particle-Hole symmetry _|_ ||

Infinite Dimensional Hubbard model in magnetic field H Definition of renormalised parameters Free quasiparticle density of states Quasiparticle number for each spin type gives density Induced Magnetisation

Fully aligned state (U=6, h=0.26) at 5% doping. Comparison of quasiparticle band with interacting DOS Narrow spin down quasiparticle band predicted by Hertz and Edwards

U=6, h=0.05 5% doping Note the difference in vertical scales Real and imaginary parts of dynamic spin susceptibilities transverse susceptibility longitudinal susceptibility

Conductance through a quantum dot in a magnetic field

Outline of Calculation Leading non-linear corrections in the bias voltage V ds (Oguri) for H=0, Generalise to include a magnetic field H We calculate the self-energy in the Keldysh formalism to second order in the renormalised interaction which is known to be exact to second order in V ds for H=0. See poster J. Bauer with splitting also for finite voltage V ds with h=0 There is a critical value h=h c at which A 2 (h) changes sign signally the development of a two peak structure

Conductance versus bias voltage V ds in a magnetic field Results asymptotically valid for small V ds.

Renormalised paramameters for antiferromagnetic states of Hubbard model Calculation of renormalised parameters for antiferromagnetic states of the infinite dimensional Hubbard model for n=0.9 U=3U=6

Can we use temperature dependent running coupling constants ? The relation relating temperature and N dependence used in the NRG can be used to convert the N-dependence of the renormalised parameters into a T-dependence Using this for the susceptibility where is evaluated with the temperature dependent parameters. Note using the mean field result in this expression gives the mean field susceptibility

Temperature dependence of susceptibility compared to Bethe ansatz results U/  =5

Summary and Outlook  We can do a perturbation theory in terms of renormalised parameter for a variety of impurity models, which is asymptotically exact at low energies (including 2CKM).  We can calculate the renormalised parameters from NRG calculations very accurately.  We can generalise the approach to lattice models and calculate the renormalised parameters within DMFT, including an arbitrary magnetic field, and for broken symmetry states.  We can use the Keldysh formalism to look at steady state non-equilibrium for small finite bias voltages.  Can we extend the non-equilibrium calculations accurately into the larger bias voltage regime?  Can we extend the results for the self energy and response functions to higher temperatures?  Other methods of deducing the renormalised parameters independent of NRG? For references for our work on this topic see: