Conserving many body approach to the fully screened, infinite U Anderson model

0.17 MB·English

by  Eran Lebanon

Conserving many body approach to the fully screened, infinite U Anderson model.
Eran Lebanon1, Jerome Rech1,2, P. Coleman1 and Olivier Parcollet2
1 Center for Materials Theory, Serin Physics Laboratory,
Rutgers University, Piscataway, New Jersey 08854-8019, USA.
2 Service de Physique Th´eorique, CEA/DSM/SPhT - CNRS/SPM/URA
2306 CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, FRANCE
6 Using a Luttinger Ward scheme for interacting gauge particles, we present a conserving many
0 body treatment of a family of fully screened infinite U Anderson models that has a smooth cross-
0 over into the Fermi liquid state, with a finite scattering phase shift at zero temperature and a
2 Wilson ratio greater than one. We illustrate our method, computing the temperature dependence
ofthethermodynamics,resistivityandelectron dephasingrateanddiscussitsfutureapplication to
n
non-equilibrium quantumdots and quantumcritical mixed valent systems.
a
J
PACSnumbers: 73.23.-b,72.15.Qm,72.10.Fk
2

]
l The infinite U Anderson model is a centralelement in OurstartingpointisaSchwingerbosonrepresentation
e
the theory of magnetic moments, in their diverse man- for the SU(N) infinite U Anderson model. Schwinger
-
r ifestations within antiferromagnets, heavy electron sys- bosonscanbeusedtodescribemagneticallyorderedlocal
t
s tems and quantum dots. While the underlying physics moments, and in the limit of large N, this description
.
t ofthe Andersonimpurity modeliswellunderstood,with becomesexact[8]. Recentprogress[9,10]hasshownhow
a
a wide variety of theoretical techniques available for its this scheme can be expanded to encompass the Fermi
m
description[1],therearemanynewrealmsofphysicsthat liquid physics of the Kondo effect. through the study
-
d relateto it,suchasquantumcriticalmixedvalent[2]and of a family of “multichannel” Kondo model, in which
n heavy fermion materials[3] or voltage biased quantum the number of channels K commensurate with n = 2S,
b
o dots[4]whereourtoolsandunderstandingofequilibrium (K =2S), where S is the spin of the local moment. The
c
impurity physics are inadequate. corresponding infinite U Anderson model is
[

These considerations motivate us to seek new theo-
1 retical tools with the flexibility to explore the physics = ǫ c† c + V c† χ†b +H.c
5v of the Anderson model in a lattice and non-equilibrium H ~Xkνα ~k ~kνα ~kνα √N ~Xkναh ~kνα ν α i
1 environment. One of the well-established tools of many + ǫ b†b λ(Q 2S). (1)
0 bodyphysicsistheLuttingerWard(LW)scheme[5]. This 0 σ σ− −
1 scheme permits a systematic construction of many-body Xσ
0
6 approximations which preserve important interrelation- Herec† createsa conductionelectronwithmomentum
ships between physical variables, such as the Luttinger ~kνα
0 ~k, channel index ν [1,K], and spin index α [1,N].
/ sum rule, or the Korringa relationship between spin re- ∈ ∈
t The bilinear product b†χ between a Schwinger boson
a laxationand spinsusceptibility, thatare implied by con- α ν
m servationlaws. RecentworkontheapplicationoftheLW b†α, and a slave fermion (or holon) operator χν creates a
localizedelectronintheconstrainedHilbertspace,which
- scheme to gauge theories[6, 7] provides new insight into
d hybridizes with the conduction electrons. The √N de-
how such conserving approximations can be extended to
n nominator in the hybridization ensures a well-defined
encompass Fermi liquids in which ultra-strong interac-
o large N limit. The energy of a singly occupied impu-
c tions are replaced by constraints on the Hilbert space.
rity ǫ is taken to be negative. The conserved operator
: 0
v Inthispaperweshowhowtheseinsightscanbeapplied Q = b†b + χ†χ replaces the no-double occu-
i to the infinite U Andersonmodel to obtaina description α α α ν ν
X pancyPconstraintoPftheinfiniteU Andersonmodel,where
ofthesmoothcross-overfromlocalmomenttoFermiliq-
Q=2S =K is required for perfect screening.
r
a uidbehavior,inwhichtheimportantconservingrelation- The characteristic “Kondo scale” of our model is
shipsbetweenthephysicalpropertiesoftheground-state,
such as the Friedel sum rule and the Yamada-Yosida Γ K/N π ǫ
0
T =D exp | | , (2)
relationships between susceptibilities and specific heat, K
(cid:18)πD(cid:19) (cid:26)− Γ (cid:27)
are preserved without approximation. Our method is
scalable to a lattice, and can also be extended to non- where D is the conduction electron half bandwidth, Γ=
equilibrium settings. To demonstrate the method we πρV2 is the hybridization width and ρ is the conduction
compute the temperature dependent thermodynamics, electron density of states. This relationship follows from
resistivity and electron dephasing rate in a single impu- general leading logarithmic scaling of the model. The
ritymodel,anddiscusshowthemethodcanbeextended non-trivial prefactor in the Kondo temperature depends
to encompass a non-equilibrium environment. on the third order terms in the beta function.
2
The next step is to constructthe LW functional Y[G]. O(N) O(1) O(1/N)
The variation of Y[G] with respect to the full Green’s Y[G] = + + +... + +... +...
functions G of the conduction, Schwinger boson and
ζ (a)
slave fermion fields, Σ = δY/δG , self-consistently de-
ζ ζ 0.4
termines the self-energiesΣ ofthese fields[6, 7]. Y[G] is
eFqeuyanlmtaonthdeiasgurmamofs,agllrtowuop-epdζairntipcloewirerresdoufc1ib/Nlef(rFeeige.1n(ear)g)y. δπ/c00..23 m{G}|b
In our procedure we neglect all but the leading order 0.1 |I
0
O(N) term in the LW functional, and then impose the 0.1 0.2 0.3 −1 0 1
full self-consistency generated by this functional. (b) 1/N (c) ε/TK
In the formal large N limit, the conduction electron
self-energy is of order O(1/N), and at first sight, should FIG. 1: (a)LW functional grouped in powers of 1/N. Solid
line - Schwinger bosons, dashed lines - holons, double lines -
be neglected. However, the effects of these terms on the
conductionelectrons. LoopscontributeO(N),vertices1/√N.
free energy are enhanced by the number of channels K
(b) Circles: phase shifts extracted from the calculated t-
and spin components N, to give a leading order O(N)
matrices, for K = 1, nχ 0.04. Solid line stands for
contributiontothefreeenergy: itispreciselytheseterms (K nχ)/(NK), dashed line≈shows the phase shift without
−
that produce the Fermi liquid behavior[10]. The com- imposingselfconsistencyontheconductionelectronpropaga-
plex issue of exactly when the conduction electron self- tors. (c)Thebosonspectrumisgappedatlowtemperatures.
energiesshouldbeincludedisnaturallyresolvedbyusing
the leading order LW functional, subsequently treating
modelto nowbe extendedtoinfiniteU. Moreover,these
1/N as a finite parameter,using the conductionelectron
relationships are all satisfied by the use of the leading
self-energiesinsidetheself-consistencyrelations. Theex-
LW functional.
plicit expressions for the self-energies are then
There are many important conserving relationships
Σ (τ) = V2G (τ)G ( τ) that are preserved, provided the spinons and holons de-
χ b c
−
Σ (τ) = (K/N)V2G (τ)G (τ) velopagap. The firstofthese is the Friedelsumrule (or
b χ c
− inthelattice,theLuttingersumrule),accordingtowhich
Σ (τ) = (1/N) V2G (τ)G ( τ) (3)
c b χ − the sum of the conduction electron phase shifts must be
equal to the total charge K n on the impurity, whith
where Gb, Gχ and Gc are the fully dressed imaginary n as the ground-state holon−ocχcupancy, so
time Green’s functions of the boson, holon and conduc- χ
tion electrons respectively. Equations (3) are solved self
π K n T
χ K
consistently,adjustingthechemicalpotentialλtosatisfy δc = − +O , (4)
N K (cid:18) D (cid:19)
the averagedconstraint Q =K.
h i
A key step in the derivation of the Ward identities is Fig. 1. contrasts the dependence of phase shift obtained
the presence of a scale in the excitationspectrum, which from the t-matrix in our conserving scheme, with that
manifests itself mathematically in our ability to replace obtainedusingΣ fromtheleadingorder1/N expansion.
c
Matsubarasummationsbyintegralsatlowtemperatures: The relationship (4) is obeyed at each value of N in our
approximation.
dω
T ThequasiparticlespectrumintheFermiliquidground-
→Z 2π
Xiωn state is intimately related to the conduction electron
phase shifts via Nozi´eres Fermi liquid theory[17]. The
This transformation is the key to derivations of the changeinthe scatteringphaseshiftinresponsetoanap-
Friedel sum rule, the Yamada-Yosida and Shiba rela- plied field, or a change in chemical potential, is directly
tionships between susceptibilities, specific heat and spin related to the spin, charge and channel susceptibilities,
relaxation rates[12, 13, 14]. There is long history of which in turn, leads to a generalized “Yamada-Yosida
attempts to develop conserving approximations to the identity”,distributingtheNK degreesoffreedomamong
infinite U Anderson model [15, 16], but each has been the spin, flavor and charge sectors:
thwartedbythedevelopmentofscale-invariantX-raysin-
gularitiesinthe gaugeorslaveparticlespectra forwhich γ N2 1χ K2 1χ χ
s f c
NK =K − +N − + . (5)
the above replacement is illegal. In the multi-channel γ0 N +K χ0 K +N χ0 χ0
s f c
approach,the gauge particles - the formation of a stable
Kondo singlet when 2S =K leads to the formation of a Here γ stands for the specific heat coefficient γ = C/T,
gap in the spinon and holon spectrum [see Fig. 1(c)], and χ , χ and χ stand for the spin, flavor and charge
s f c
which permits a Fermi liquid spectrum to develop at susceptibilities. χ0 denotes the corresponding suscep-
s,f,c
lower energies. This feature permits all of the manip- tibilities in the absence of the impurity. Our identity (5)
ulations previously carried out on the finite U Anderson reduces to the known results for the Wilson ratio in the
3
0.15
0.4
2 1
mp 0.2 )}
Si 1 η 0.5
+i 0.1
0 ε
0 0 0.05 0.1 m{t( 0
ρ I −0.005 0 0.005
π 0.05
χ −
K 1
T
0
0 −0.4 −0.2 0
0.1 1 10 100 1000 ε/D
T/T
K
FIG. 3: t-matrix for ǫ0/D = 0.2783, Γ/D = 0.16, N = 4
FIG. 2: Upper pannel: The impurity contribution to the −
and K = 1 (TK/D = 0.002). Main frame: T/D=0.1, 0.08,
entropy vs. temperature. Inset: at low temperatures the en- 0.06, 0.04, 0.02, and 0.01. Inset: T/10−4D=10, 8, 6, 4, 2, 1,
tropybecomeslinearandγsaturates. Lowerpannel: impurity and 0.5. The dot marks sin2 π(K nχ)/NK
susceptibility vs. temperature. For parameters see Fig.3. { − }
self-consistency in a small magnetic field[11]. Figure (2)
onechannelmodel[13]andtheKondolimit[18]. Sinceour
shows the impurity susceptibility χ(T). The susceptibil-
scheme preserves the Friedel sum rule for each channel,
ity is peaked slightly below the Kondo temperature and
it automatically satisfies this relationship.
saturates to a constant at a lower temperature.
Various thermodynamic quantities can be extracted
One of the important conservation laws associated
from the numerical solution of the self-consistency equa-
tions. The development of a fully quenched Fermi liquid with the conservation of spin, is the Korringa Shiba re-
lation between the dynamical and static spin suscepti-
state is succinctly demonstrated by calculating the en-
bility. On the assumption that the gauge particles are
tropy, using a formula of Coleman, Paul and Rech[7]
gapped, the LW derivation carried out by Shiba on the
S = Tr dǫdnζ Imln G−1 +G′Σ′′ , (6) finite U Anderson model, some thirty years ago, can
− ζZ dT h (cid:0)− ζ (cid:1) ζ ζi be simply generalized to the infinite U model, to ob-
tain χ′′(ω)/ω =(Nπ/2K)(χ/N)2. This relationship
where the trace denotes a sum over the spin and chan- |ω=0
guarantees that the power-spectrum of the magnetiza-
nelindices ofthe variousfields andn is the distribution
ζ tion is linear at low frequencies, ensuring that the spin
function (Fermi for ζ =c,χ andBose for ζ =b). G and
ζ response function decays as 1/t2 in time. Remarkably, a
Σ respectively denote the retardedGreenfunctions and
ζ Fermi liquid, with slow gapless spin excitations is sand-
selfenergies. Thesingleanddoubleprimesmarkthereal
wiched beneath the spinon-holon continuum.
and imaginarypartrespectively. The impurity contribu-
We now turn to a discussion of the electron scatter-
tiontotheentropyS =S S ,whereS istheentropy
imp 0 0
− ing off the impurity, which is determined by the conduc-
of the bare conduction band, is shown in the upper pan-
tion electron t-matrix, t(z) = Σc , where G0(z) =
nel of Fig. 2. An inspection of low temperatures shows 1−G(c0)Σc c
a linear behavior of the impurity contribution to the en- ~k(z−ǫ~k)−1 is the bare localconductionelectronprop-
tropy. The latter marks the saturation of the specific Pagator. Various important physical properties, such as
heat coefficient γ = ∂S /∂T at low temperatures the impurity resistivity and the electron dephasing rate
imp imp
and the formation of a local Fermi liquid. inthe dilute limitcanbe relatedto thisquantity. Figure
To calculate the spin susceptibility, we couple a small 3 shows the imaginary part of t(ǫ+iη): as temperature
magnetic field to the magnetization, defined as is lowered towards the Kondo temperature, a resonance
peak starts to develop around the Fermi energy. The
N +1
M = sgn(α ) b†b + c† c . peak continues to evolve with temperature and finally
− 2 (cid:18) α α ~kνα ~kνα(cid:19) saturates a decade below the Kondo temperature. The
Xα X~k,ν
fullydevelopedresonanceispinnedtotheLangrethsum-
The impurity magnetization is obtained by subtracting rule[12]valueofsin2δ /πρ,attheFermienergypresented
c
the magnetization of the free conduction sea from M. by the dot in the inset of Fig.3.
When we apply a field, we must keep track of the ver- The impurity contribution to the resistivity R is re-
i
tex corrections that arise from the field-dependence of lated to a thermal average of the t matrix,
−
the self-energies. These vertex functions are essential to
maintain the conservation laws. We compute the mag- 3m2n ∂f −1
i −1
R = dǫ Imt(ǫ+iη) . (7)
netic field vertex functions by iterating the solutions to i 2e2ρk2 (cid:26)Z (cid:18)−∂ǫ(cid:19)| | (cid:27)
F
4
0.3 corporation of spin relaxation effects on the boson lines,
suggestthatthiswillbearobustmethodtoexaminehow
22k / 3mnFi0.2 0.2ρ(0) / ni sqthpueiannvatounltmdagedleeocdttser.poTennhddeeespnehcmaese[it2nh4go]edoffsfecctahtnseeadvliossoltvrbeibewuuittsihoendvotfuoltnascgttueidoinny
2ρe(0) 0.1 0.1−1τπ φ anOdnneoiosef itnhemsatgrnikeitnicgalfleyatduorpeesdomf etshoescSocphiwciwngireers[b25o]s.on
2
R approach to the Anderson and Kondo models, is the co-
existence of a gapless Fermi liquid, sandwiched at low
0 0
10−2 10−1 100 101 102 103 energies between a gapped spinon and holon fluid. The
T/T gapinthespin-chargedecoupledexcitationsappearsinti-
K
matelylinkedtothedevelopmentoftheFermiliqidWard
FIG. 4: The impurity contribution to the resistivity Ri and identities. The methods presented in this paper can be
dephasing rateτ−1 as afunction of temperature. Solid lines:
φ scaleduptodescribethedenseAndersonandKondolat-
theparameters used in Fig. 3, dashed lines: Γ=0.1D.
tice models, where the Friedel sum rule is replaced by
the Luttinger sum rule[7]. In the lattice, the U(1) lo-
cal symmetry of the impurity model will in general be
where k is the Fermi wavelength and f is the Fermi
F replaced by a global U(1) symmetry associated with the
distribution function. Figure 4 shows R (T). The resis-
i pair-condensedSchwingerbosons[26]. Inthissetting,the
tivity increases as the temperature decreases, saturating
gapped spinons and holons are propagating excitations,
at a value determined by the scattering phase shift.
whose gap is fundamental to the large Fermi surface. It
As a final applicationof the t-matrix, we compute the
is this very gap that we expect to collapse at a quan-
electrondephasingrateτ−1 thatcontrolsthefielddepen-
φ tum criticalpoint. The currentwork,whichincludes the
dence of weak electron localization. In the cross-over to
effectsofvalencefluctuations,providesapowerfulframe-
the Fermi liquid state, the formation of the Kondo reso-
work for examining this new physics.
nance gives rise to a peak in the inelastic scattering[19]
and the electron dephasing rate [20]. In the dilute limit, This work was supported by DOE grant number DE-
the impurity contribution to the dephasing rate is given FE02-00ER45790 and an ACI grant of the French Min-
by τ−1 = 2n [ Imt πρt2] [19, 21, 22]. Fig. 4 shows istry of Research. We are grateful to Natan Andrei and
the dφephasingi −rate o−n the| |Fermi surface computed from Gergely Zarand for discussions related to this work.
t,showinghowthescatteringscaleswiththeKondotem-
perature T in the Kondo regime of the model.
K
Our calculation of the dephasing rate shows that our
method captures the cross-over into the coherent Fermi
[1] For a review see A.C. Hewson The Kondo problem to
liquid. However,italsohighlightsashortcomingthatwe Heavy Fermions (Cambridge Press, Cambridge 1993).
hopetoaddressinfuturework. Atlowtemperatures,the [2] A.T. Holmes , D. Jaccard and K. Miyake, Phys. Rev. B
electron dephasing rate has a quadratic dependence on 69, 024508 (2004).
temperature. The electron-electron scattering diagrams [3] P.Coleman andA.J.Schofield,Nature433,226(2005).
[4] L. Kouwenhovenand L.Glazman, Physics World, 1433
responsiblefortheseprocessesinvolveaninternalloopof
(2001).
gaugeparticles,whichonlyentersinthesubleadingO(1)
[5] J.M. Luttinger, and J.C. Ward, Phys. Rev. 118, 1417
terms of the LW functional shown in Fig. 1(a), which
(1960);J.MLuttinger,Phys.Rev.119,1153(1960);J.M.
are absent from the current work. On the other hand, Luttinger, Phys. Rev.121, 1251 (1961).
the low temperature, on-shell value of the conduction [6] J.-P. Blaizot, E. Iancu, & A. Rebhan Phys. Rev. D 63,
electronvertexisdirectlyrelatedtoγ byWardidentities 065003 (2001).
[21],providingapossiblefuturesimplificationfortreating [7] P. Coleman, I. Paul, and J. Rech, Phys. Rev. B 72,
094430 (2005).
these processes.
[8] D.P. Arovas, and A. Auerbach, Phys. Rev. B 38, 316
There are many directions for future development.
(1988).
We are particularly interested in the extension to non- [9] O. Parcollet, and A. Georges, Phys. Rev. Lett. 79, 4665
equilibrium quantum dots[4]. The theoretical under- (1997).
standing of the Kondo physics of quantum dots at finite [10] J. Rech, P. Coleman, G. Zar´and, and O. Parcollet,
voltagebiasisstillevolving. Anextensionofourmethod cond-mat/0507001.
[11] E. Lebanon, A.Schiller, and V.Zevin,Phys.Rev.B 64,
toa voltagebiasedinfinite Andersonmodelcanbe made
245338 (2001).
using a Keldysh generalization of the self-consistency
[12] D.C. Langreth, Phys.Rev. 150, 516 (1966).
equations[23]. Many features of our approach, its con- [13] K. Yamada, Prog. Theor. Phys. 53, 970 (1975); K.
serving properties, its inclusion of scaling properties up Yoshida and K. Yamada, ibid. 53, 1286 (1975).
tothirdorderinthebetafunction,anditssystematicin- [14] H. Shiba, Prog. Theo. Phys54, 967 (1975).
5
[15] Y.Kuramoto, Z.Phys. B 53, 37 (1983). [22] C. Bau¨erle et. al. Phys. Rev.Lett. 95, 266805 (2005).
[16] P.Coleman, Phys.Rev.B 28, 5255 (1983). [23] A. Posazhennikova, and P. Coleman, Phys. Rev. Lett.
[17] P.Nozi`eres & A. Blandin, J. Physique,41, 193, 1980. 94, 036802 (2005).
[18] A. Jerez, N. Andrei, and G. Zar´and, Phys. Rev. B 58, [24] E.Lebanon,andP.Coleman,Phys.Rev.Lett.95,046803
3814, (1998). (2005).
[19] G. Zar´and, L. Borda, J.v. Delft and N. Andrei, Phys. [25] H. Pothier et al., Phys. Rev. Lett. 79, 3490 (1997); A.
Rev.Lett. 93, 107204 (2004). Kaminski and L. I. Glazman, Phys. Rev. Lett. 86, 2400
[20] C. Van Haesendonck et al., Phys. Rev. Lett. 58, 1968 (2001); J. Kroha and A. Zawadowski, Phys. Rev. Lett.
(1987); R.P. Peters et al., Phys. Rev. Lett. 58, 1964 88, 176803 (2002); G. Gppert and H. Grabert, Phys.
(1987); F.J. Ohkawa, H. Fukuyama, and K. Yoshida, J. Rev. B 64, 033301 (2001).
Phys.Soc. Jpn. 52, 1701 (1983). [26] M. Hermele et al, Phys. Rev.B 70, 214437 (2004).
[21] T. Micklitz, A. Altland, T.A. Costi, and A. Rosch,
cond-mat/0509583 (2005).