ARTICLE IN PRESS

0304-8853/$

doi:10.1016

�CorrespLondon SW

E-mail a

Journal of Magnetism and Magnetic Materials 310 (2007) 1133–1135

www.elsevier.com/locate/jmmm

Equilibrium and non-equilibrium dynamics of quasiparticles for locallycorrelated systems in a magnetic field

J. Bauera,b,�, A.C. Hewsona, A. Ogurib

aDepartment of Mathematics, Imperial College, London SW7 2AZ, UKbDepartment of Material Science, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan

Available online 15 November 2006

Abstract

We present renormalised perturbation theory calculations for the description of the dynamics of quasiparticles in the Anderson

impurity model in magnetic field. Our calculations are compared with NRG spectra and good agreement is found for magnetic fields of

the order of the Kondo temperature, when an appropriate perturbation series is chosen. We point out how this approach can be

generalised to the non-equilibrium situation, where it is relevant for the description of quantum dots.

r 2006 Elsevier B.V. All rights reserved.

PACS: 72.10.F; 73.61; 71.10.A

Keywords: Strongly correlated electrons; Renormalised perturbation theory; Kondo effect in magnetic field; Quantum dots

Strongly correlated local systems are characterised by alow energy scale T�. Quantities, like the static and dynamicresponse functions, depend crucially on this scale which forlocal systems can be identified with the Kondo temperatureTK. Such systems, subject to a magnetic field h ¼ gmBB=2,show an enhanced susceptibility as compared to the weakinteraction case [1]. This can be understood in terms of thescale T�, as for h�T�, and for strongly correlated systemT� is small, so the field plays an important role for thebehaviour of the system. It is, therefore, of great interest todescribe such systems accurately whilst subjected to amagnetic field.

As a standard model for local strong correlation physicswe consider the Anderson impurity model (AIM), which inthe wide conduction band limit is usually described interms of three bare parameters, localised level�d;s ¼ �d þ sh, hybridisation with the band D and theinteraction strength U. For simplicity we assume particle-hole symmetry in the following. The low energy behaviouris completely characterised by three Fermi liquid para-meters [2], the renormalised level and width ~�d;sðhÞ, ~DðhÞ,

- see front matter r 2006 Elsevier B.V. All rights reserved.

/j.jmmm.2006.10.729

onding author. Department of Mathematics, Imperial College,

7 2AZ, UK. Tel.: +44 20759 41474.

ddress: j.bauer@imperial.ac.uk (J. Bauer).

and the quasiparticle interaction ~UðhÞ. It turns out to bevery useful to include their dependence on the field h [1].These parameters can be calculated by different methods,and we choose to deduce them from the low energyexcitations in numerical renormalisation group (NRG)computations [3]. A detailed description of the low energyphysics of the AIM in magnetic field in terms of theseparameters has been given in Ref. [1].The purpose of this paper is to demonstrate how this

approach can be extended to the single particle dynamicsup to energies of the order of T� in a renormalisedperturbation theory (RPT) [2]. The quality of ourapproximations can be tested against spectra obtainedfrom NRG calculations.Such an RPT approach is best described in the

functional integral formulation, where the action of theAIM is given by

SAIM ¼Xs

Z b

0

dtdt0dsðtÞG0;sðt; t0Þ�1dsðt0Þ þU

Z b

0

dt nd;"ðtÞnd ;#ðtÞ

with

G0;sðtÞ ¼1

b

Xn

e�iton1

ion � �d;s þ iDsgnðonÞ. (1)

ARTICLE IN PRESS

-2 0 2 4 6 8 10 120

1

2

3

4

5

ω/TK

ρ d,σ

(ω)

RP

SORPT

phRPT

NRG

Fig. 2. For h=TK ¼ 2:65 we plot rdsðoÞ in terms of only renormalised

parameters (RP) in comparison with a second order (SO) RPT and the

repeated scattering RPT (ph). As a benchmark for comparison serve NRG

spectra [1].

J. Bauer et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1133–11351134

The RPT is defined by

SAIMð�d ;s;D;UÞ ¼ SrAIMð~�d;s; ~D; ~UÞ þ Sr

ctðl1; l2; l3Þ. (2)

This means that perturbation theory can be carried out interms of the renormalised action Sr

AIMð~�d ;s; ~D; ~UÞ, but inorder to avoid overcounting of renormalisation effects, thecounter term action Sr

ctðl1; l2; l3Þ has to be included. Theparameters li are determined by the renormalisationconditions [2]

~Ssð0; hÞ ¼ 0;q ~Ssð0; hÞ

qo¼ 0; ~UðhÞ ¼ ~G";#ð0; hÞ, (3)

where ~G";#ð0; hÞ ¼ zðhÞ2G";#ð0; hÞ is the antisymmetrised fullvertex at zero frequency.

Perturbation theory can be carried out as usual, onlythat the parameters in the unperturbed propagators (1) arereplaced by the renormalised quantities. RPT is asympto-tically exact and thus, we expect that a good approxima-tion for the low energy description can be obtained alreadyin low order perturbation theory.

The shift of the Kondo peak from the Fermi level of onespin component of the spectral density in a magnetic field,or equivalently the magnitude of the Kondo splitting forthe full spectral density, has been subject of considerablestudy in recent years. The free quasiparticle Green’sfunction which solely depends on the field dependentrenormalised parameters (RP) can describe this Kondosplitting. The comparison with asymptotic exact resultsand with NRG calculation shows that the magnitude islargely overestimated and corrections from a renormalisedself-energy are necessary (see Figure 2). In constructing anappropriate perturbation expansion the simplest dynamiccorrection comes from a second order (SO) diagram. Theresulting correction is, however, found still not to besufficient. Therefore, we consider an approximation, whichfocuses on the most important process in the Kondoregime, which consists of spin fluctuations. Mathemati-cally, this is represented by repeated scattering ofquasiparticle and quasihole with opposite spin and atypical diagram is depicted in Fig. 1. For dynamic chargeand spin susceptibilities, it was shown that consideringsuch a repeated scattering of renormalised quasiparticlesyields accurate results [4]. Note that it is important toinclude counter terms to enforce the renormalisationconditions (3) in such a calculation. This is easily done

>

>

> >

<

σ σ σ

−σ

−σ

~U1~

U1

Fig. 1. Contribution to the renormalised self-energy from repeated

particle-hole scattering (opposite spin) with effective interaction~U1 ¼ ~U � l3.

for the self-energy and also possible for the effectiveinteraction ~U1 ¼ ~U � l3 by considering exact identities forthe full vertex [4].We have carried out such a calculation and determined

the renormalised self-energy and spectral density in theapproximations described. We find that for a range ofmagnetic fields from 0 up to a few TK that, whilst only thecorrection from the SO renormalised self-energy gives anoverestimated magnitude of the Kondo splitting, therepeated scattering process yields good results, whencompared with NRG spectra. For an illustration, wedisplay the result for one component of the spectral densityrdsðoÞ ¼ �ImGdsðoÞ=p for a finite magnetic field h ’

2:65TK in Fig. 2. Bare parameters for the symmetricAnderson model are chosen as U=pD ¼ 4. One can see thatthe SO perturbation theory gives a dynamic correction inthe right direction, albeit too small, whereas the repeatedprocess renders a dynamic correction of the rightmagnitude. Differences in the peak form of RPT andNRG can be attributed to broadening effects. We can seethat, whilst for lower energies the agreement is very good,the RPT results for large energies become inaccurate. This,however, is expected, since for higher energies otherprocesses, such as charge fluctuations, will start to playan important role and need to be included in therenormalised self-energy. Our conclusion is, that for thelow energies and not too large magnetic fields ht3TK, themost important contributions to the renormalised self-energy are included in the processes shown in Fig. 1.It is straightforward to generalise such calculations to

the non-equilibrium situation [5,6], where it is relevant forthe description of quantum dot experiments [7]. We arecarrying out such a calculation at present, and details andresults will be presented elsewhere.

ACH wishes to thank the EPSRC for support throughthe Grant GR/S18571/01. JB thanks the Gottlieb Daimler-

ARTICLE IN PRESSJ. Bauer et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1133–1135 1135

and Karl Benz-Foundation, the DAAD, JSPS and EPSRCfor financial support, and for the hospitality at Osaka CityUniversity. AO acknowledges the support by the Grant-in-Aid for Scientific Research for JSPS.

References

[1] A.C. Hewson, et al., Phys. Rev. B 73 (2006) 045117.

[2] A.C. Hewson, Phys. Rev. Lett. 70 (1993) 4007.

[3] A.C. Hewson, A. Oguri, D. Meyer, Eur. Phys. J. B 40 (2004) 177.

[4] A.C. Hewson, J. Phys. Condens. Matter 18 (2006) 1815.

[5] A. Oguri, J. Phys. Soc. Japan 74 (2005) 110.

[6] A.C. Hewson, J. Bauer, A. Oguri, J. Phys. Condens. Matter 17 (2005)

5413.

[7] A. Kogan, et al., Phys. Rev. Lett. 93 (2004) 166602.