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Renormalised Perturbation Theory

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.

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Renormalised Perturbation Theory

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  1. 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

  2. Renormalisation in Field Theory Aim to eliminate divergences Certain quantities are taken into account at the beginning so one works with (i) thefinal 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; ~ ~ ~ f (i) renormalised massm (ii) renormalised interactiong (iii) renormalised field ~ l1 l2 l3 The expansion is carried out in powers of g and the counter terms cancel renormalisations which have already been taken into account

  3. Form of Perturbation Expansion for f Theory 4 separated out and Renormalisation conditions:

  4. wide band limit

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

  6. 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

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

  8. Low Order Results Zero Order Friedel Sum Rule Specific heatcoefficient Define free quasiparticle DOS First Order Spin susceptibilitiesandcharge Second Order Impurity conductivity symmetric model All these results are exact (Ward identities, Yamada)

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

  10. Calculation of and using the NRG NRG chain Non-interacting Green’s function Given ed and V the excitations w=en of thenon-interacting 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

  11. ~o Kondo regime

  12. 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

  13. Overview ofrenormalisedparameters in terms of ‘bare’ values Full orbital >>>> mixed valence >>>> Kondo regime >>>>> mixed valence >>>>> empty orbital

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

  15. Overview for U>0 as a function of the occupation value nd Strongest renormalisations in the case of half-filling

  16. 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

  17. 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

  18. Renormalised parameters a a function of the magnetic field value U=3pD Mean field regime Parameters are not all independent:

  19. Without particle-hole symmetry

  20. Induced Magnetisation AM BA U=3pD Charge fluctuations playing a role Comparison with Bethe ansatz for localised model

  21. 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*

  22. Impurity contribution to conductivity Susceptibility

  23. Impurity contribution to conductivity Conductance of quantum dot s2(h) changes sign at h=hc in the Kondo regime G2(h) changes sign in this range

  24. 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

  25. Spin and charge irreducible Verticies charge spin

  26. Imaginary part of dynamic spin susceptibility ------- NRG results using complete Anders-Schiller basis_______RPT Note the different energy scales in the two cases

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

  28. Imaginary part of dynamic spin susceptibilities RPA

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

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

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

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

  33. Without Particle-Hole symmetry _|_ ||

  34. 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

  35. 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

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

  37. Conductance through a quantum dot in a magnetic field

  38. 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 whichA2(h) changes sign signally the development of a two peak structure

  39. Conductance versus bias voltage Vds in a magnetic field Results asymptotically valid for small Vds.

  40. 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

  41. 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

  42. Temperature dependence of susceptibility compared to Bethe ansatz results U/pD=5

  43. 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 anarbitrary magnetic field, and forbroken 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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