renormalised perturbation theory ● motivation ● illustration with the anderson impurity model...
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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/
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