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 calculatedusing counter terms

Low Order Results

Zero OrderFriedel 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 Ep(N) and hole Eh(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 Epp(N) and two single particle excitations 2 Ep(N) . This interaction Upp(N) will depend on the excitations and chain length N.

We can define a similar interaction Uhh(N) between holes Uph(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 nd

Strongest renormalisations in the

case of half-filling

Overview for U<0 as a function of the occupation nd

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<hc<T*

Susceptibility

Impurity contribution to conductivity

(h) changes sign at h=hc in the Kondo regime

Impurity contribution to conductivity Conductance of quantum

dot

G2(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 verticescharge

_|_ 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 Vds (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 Vds for H=0. See poster J. Bauer with splitting also for finite voltage Vds with h=0

There is a critical value h=hc at which A2(h) changes sign signally the development of a two peak structure

Conductance versus bias voltage Vds in a magnetic field

Results asymptotically valid

for small Vds.

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=3 U=6

Can we use temperature dependent running coupling constants ?

The relation relating temperature and N dependence used in the NRG canbe used to convert the N-dependence of the renormalised parameters intoa 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: http://www.ma.ic.ac.uk/~ahewson/