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Renormalised Perturbation Theory
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Motivation
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Illustration with the Anderson impurity model
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Ways of calculating the renormalised parameters
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Range of Applications
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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 f
The expansion is carried out in powers of g~ and the counter terms l1 l2 l3
renormalisations which have already been taken into account
cancel
Form of Perturbation Expansion for f4 Theory
and
Renormalisation conditions:
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:
The three counters are determined by the renormalisation conditions
Method 2: Without counter terms
Step 1: Calculate the quantity using perturbation theory in the bare interaction U
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
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
Non-interacting Green’s function
Given ed and V the excitations w=en of the noninteracting system are solution of the equation:
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
~o
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
U=3pD
Mean field regime
Parameters are not all independent:
Without particle-hole symmetry
Induced Magnetisation
AM
BA
U=3pD
Charge fluctuations playing a
role
Comparison with Bethe ansatz for localised model
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*
Impurity contribution to
conductivity
Susceptibility
Impurity contribution to
conductivity
s2(h) changes sign at h=hc in
the Kondo regime
Conductance of quantum
dot
G2(h) changes sign in this range
Spin and Charge Dynamics
We look at the repeated scattering of a quasiparticle with spin up and a
quasihole with spin down
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
------- NRG results using complete Anders-Schiller basis
_______ RPT
Note the different energy scales in the two cases
Real part of dynamic spin susceptibility
spin
charge
Imaginary parts of spin and charge
dynamic susceptibilities
Imaginary part of dynamic spin susceptibilities
RPA
Spin and charge dynamics in a magnetic field
Irreducible vertices
charge
U
_|_ spin
|| spin
Non-interacting Case U=0
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_|_
||
NRG compared with exact results
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NRG compared with RPT in the
interacting case
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Imaginary part of transverse susceptibility
_|_
_|_
Comparison of NRG and RPT results in strong field limit
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
Real and imaginary parts of dynamic spin susceptibilities
transverse susceptibility
U=6,
longitudinal susceptibility
h=0.05
5% doping
Note the difference in vertical scales
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
U=3
U=6
Calculation of renormalised parameters for antiferromagnetic states of the infinite
dimensional Hubbard model for n=0.9
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/pD=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/