Non-trivial fixed point in a twofold orbitally degenerate Anderson impurity model

0.18 MB·English

by  Michele Fabrizio

Non-trivial fixed point in a twofold orbitally degenerate Anderson impurity model
Michele Fabrizio,1,2 Andrew F. Ho,3 Lorenzo De Leo,1 and Giuseppe E. Santoro1,2,4
1International School for Advanced Studies (SISSA),
and Instituto Nazionale per la Fisica della Materia (INFM) UR-Trieste SISSA, Via Beirut 2-4, I-34014 Trieste, Italy
2The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O.Box 586, I-34014 Trieste, Italy
3School of Physics and Astronomy, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK.
4 4INFM Democritos National Simulation Center, Via Beirut 2-4, I-34014 Trieste, Italy
0 (Dated: February 2, 2008)
0
WestudyatwofoldorbitallydegenerateAndersonimpuritymodelwhichshowsanon-trivialfixed
2
pointsimilartothatofthetwo-impurityKondomodel,butremarkablymorerobust,asitcanonlybe

n destabilized byorbital- or gauge-symmetrybreaking. Theimpuritymodelis interestingperse,but
a hereour interest is rather in thepossibility that it might berepresentative of a strongly-correlated
J
lattice model close to a Mott transition. We argue that this lattice model should unavoidably

8 encounter the non-trivial fixed point just before the Mott transition and react to its instability by
2 spontaneous generation of an orbital, spin-orbital or superconductingorder parameter.

] PACSnumbers:
l
e
-
r When ametalis drivenby strongcorrelationstowards orbital a = 1,2 and spin σ, d† impurity ones and n =
t aσ d
s a Mottinsulator,anincoherentcomponentof the single- d† d . For Hˆ = 0, the AIM has a large SU(4)
. aσ aσ aσ K
at particle spectrum slowly moves away from the quasipar- Psymmetry which is lowered down to a spin SU(2) times
m ticle resonance and smoothly transforms into the Mott- an orbital O(2) by a Hund’s rule-like coupling:
Hubbardside-bands. Analogousbehaviorisdisplayedby
- 2 2
d an Anderson Impurity Model (AIM) from the mixed va- HˆK =K Tˆx + Tˆy , (2)
n lence to the Kondo regime. This is suggestive of similar (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
o physicalprocessesunderlyingthedynamicsofAIM’sand where Tˆi = 1 2 d† τi d , i = x,y,z,
c 2 σ a,b=1 aσ ab bσ
[ of strongly-correlated electron systems across the Mott are orbital pseudoP-spiPn operators with τi’s the Pauli

metal-to-insulator transition (MIT), even though a rig- matrices. For later convenience, we also intro-
2
v orous relationship holds only in infinite dimensions, as duce the impurity spin, Sˆi = 12 a α,βd†aασαiβdaβ,
8 shown by Dynamical Mean Field Theory (DMFT). Fur- and the impurity spin-orbital PopePrators Wˆij =
2 thermore, DMFT shows that the Mott-Hubbard bands 1 d† σi τj d .
3 splitofffromthequasiparticleresonancequitebeforethe 2 a,b α,β aα αβ ab bβ
5 MIT occurs[1], suggesting that it is rather the physics PIn thPe Kondo regime, U much largerthan the conduc-
0 tion bandwidth, two electrons get trapped by the im-
of the AIM in the Kondo regime to be significant near
3 purity in a configuration identified by total spin, S, to-
the MIT. In that limit, the AIM maps onto a Kondo
0 tal pseudo-spin, T, and their z-components,with energy
/ model with the number of conduction channels always
t E(S,Sz;T,Tz)=K T (T +1) (Tz)2 . By Pauli prin-
a such as to perfectly screen the impurity. For that reason h − i
m one would exclude that the appealing non-Fermi liquid ciple, two electron configurations have either S = 1 and
- physics of the overscreened multi-channel Kondo effect T =0, or S =0 and T =1. If K >0, the lowest energy
d may ever appear close to the MIT. configuration has S = 1 and T = 0, the conventional
n
That expectation is partly wrong. In this Letter we Hund’s rules. The impurity behaves effectively as a spin
o
c study the phase diagram of an AIM which does contain S=1 which may be Kondo-screened by the two conduc-
: a non-trivial fixed point. We show that the physical be- tion channels (δ = π/2 phase shift). On the contrary, if
v
i havior around this fixed point resembles that displayed K <0thenon-degenerateS =0T =1andTz =0state
X by the two-impurity Kondo model. We also argue that has lowest energy. Here we do not expect any Kondo
r the lattice model, where the physics of the above AIM effect, i.e. δ = 0. This situation is analogous to the two
a
shouldberelevant,necessarilyencountersthisfixedpoint S=1/2 impurity Kondo model (2IKM) in the presence
just before the MIT, and discuss possible consequences. of a direct exchange between the impurities. There, it
We consider the two-orbitalAIM Hamiltonian: is known[2, 3, 4] that under particular circumstances an
unstablefixedpoint(UFP)separatestheKondo-screened
Hˆ = ǫ c† c + V c† d +H.c. regime from the one in which the two-impurities couple
AIM k k,aσ k,aσ k k,aσ aσ
kX,a,σh (cid:16) (cid:17)i together into a singlet. In our model (1), that circum-
U stance is realized thanks to the O(2) orbital symmetry,
+ (n 2)2+Hˆ , (1)
2 d− K as shown later, hence an analogous UFP should exist.
We have studied the AIM (1) in the Kondo regime
where c† are conduction-band creation operators in by Wilson’s Numerical Renormalization Group (NRG),
k,aσ
2
2
<S2>
2 <T2>
1.5 <4T2>
z
1.5 1
1 1 <S2>
UFP
0.5
0.5 0
-0.1 0 0.5
0 10 20
K G/K(G=0)
* z c z
0
-1 -0.5 0 0.5 1
K
FIG. 2: Average impurity quantum numbers as function of
K. In the left inset the behavior around the UFP is shown
in more detail, while in the right inset hS2i along the path
FIG.1: NRGflowofthelowest energiesofthelevelslabeled towards the 2IKMis displayed (see text).
by(Q,Tz,S),whereQishalfoftheaddedcharge. Theright
and left flows correspond to a relative deviation δK∗/K∗ =
±4·10−3 from the fixed point value K∗, respectively. The δ =0, or been absorbed by the conduction sea, δ =π/2:
phase diagram is sketched in theinset.
U
Hˆ = t f† f +H.c. + ∗ (n 2)2
∗ −Xaσ ∗(cid:16) 0,aσ 1,aσ (cid:17) 2 0−
closely following the original work by Jones and Varma 2
for the 2IKM[2, 4]. We have also developed a comple- +JS∗S~0·S~0+JT∗(cid:16)Tˆ0z(cid:17) , (3)
mentary analysis based on abelian bosonization, which
provides a better characterizationof the UFP. where “0” labels the first available site of the Wilson
chain, i.e. the actual first site for K < K , δ = 0, and
∗
In the inset of Fig. 1 we sketch the phase diagram of
the second site for K > K , δ = π/2. Numerically we
∗
(1)intheKondoregime,asdeterminedbytheflowofthe
find J 2U 2J 32t upon approaching
T∗ ∗ S∗ ∗
low energy spectrum obtained by NRG. At fixed Kondo ∼ ∼ ∼ → ∞
the UFP, see Fig. 3. The diverging t implies a singu-
∗
exchange we find indeed an UFP upon varying K. For
lar( (K K )−2) impurity-contributionto the specific
∗
K > K∗, the model asymptotically flows to a Kondo- heat∝, just−like in the 2IKM. Additional information are
screened fixed point, see right panel in Fig. 1, while for
provided by the Wilson ratios R = (c δχ )/(χ δc ),
i v i i v
K <K <0itflowstowardsanon-Kondo-screenedfixed
∗ where i = S,Tz refer to spin and orbital susceptibili-
point, see left panel. The intermediate crossover region,
ties,i.e. to thoseresponsefunctions relatedto conserved
also shown in Fig. 1, identifies the UFP[5]. We notice
quantities,hence accessibleby Fermiliquid theory[6,7].
that (1) has a larger impurity Hilbert space then the
2IKM,whichcontains,besides theS =1andtheTz =0 RS and RTz are shown in Fig. 4 and appear to vanish
at the UFP. By analogy with the 2IKM, there are two
S = 0 configurations, also the Tz = 1 S = 0 dou-
± other susceptibilities which are instead expected to be
blet,absentinthe2IKM.Inspiteofthat,thelow-energy
singular: the susceptibility χSTz to a field which couples
spectra at the UFP’s are the same for both models. In to the relative spin operator Wˆiz, i = x,y,z, (the stag-
Fig. 2 we plotthe groundstate averagevalues of the im-
gered spin-susceptibility in the 2IKM), and the pairing
purityoperatorsS~2,T~2and(Tz)2. Forlargeandpositive
K, the impurity freezes into the S = 1 T = 0 configu-
ration while, for very negative values, into the Tz = 0
S = 0 one. At the UFP S2 = 1/2, T2 = 3/2 and 10000
(Tz)2 =1/4. ToprovethhatoiurUFP ishconinectedwith 5
h i 2 1000 0
the 2IKM one, we have added to (1) a term Gz Tˆz , -0.02 -0.01 0 0.01
with Gz > 0, which pushes upward the energy(cid:16)of t(cid:17)he 100 32 t
*
Tz = 1 S = 0 doublet absent in the 2IKM. In the in- 2 U*
set of±Fig. 2 we plot S2 as function of Gz at the UFP, 10 2J JS*
whosepositiondepenhds oinG too. We dofind that S2 T*
z h i 1
smoothly reaches the 2IKM unit value for large G . -0.1 -0.08 -0.06 -0.04 -0.02
z K
The approach to the two stable fixed points, K < K
∗
and K >K , can be described by the local perturbation FIG. 3: K-dependence of the effective couplings in Eq. (3).
∗
leftbehindbytheimpuritywhichhaseitherdisappeared, In theinset theregion around K =0 is shown.
3
3 electrons as chiral one-dimensional fermions c (x) =
3 a,σ
8/3 1/√2παF exp[ iφ (x)], where φ are chiral free
2.5 2 a,σ − a,σ a,σ
Bose fields, α a shortdistance cut-off, andthe Klein fac-
2 1 tors F are Grassman variables enforcing proper anti-
a,σ
0 0 0.5 1 R commutationrelations. Next,weintroducethecombina-
1.5 S
RTz tions: φc =(φ1↑+φ1↓+φ2↑+φ2↓)/2, φs =(φ1↑−φ1↓+
1 φ2↑ φ2↓)/2,φf =(φ1↑+φ1↓ φ2↑ φ2↓)/2,φsf =(φ1↑
K − − − −
* φ1↓ φ2↑+φ2↓)/2. Afterapplyingthecanonicaltransfor-
0.5 −
mationexp iSˆzφ (0) exp iTˆzφ (0) ,Eq.(4)canbere-
s f
0-0.1 0 0.1 fermionizedhviaΨb(x)i=1/h√2παFbeixp[ iφb(x)],where
K b = c,s,f,sf[9]. For a particular value o−f Jz = Jz, the
S T
endresultisaneffectivemodelwhereonlyΨ iscoupled
sf
FIG. 4: Spin (RS) and orbital (RTz) Wilson ratios as func- to the impurity, just like in the 2IKM[8].
tions of K. Notice that the K = 0, SU(4)-point, as well as
To locate the UFP, we follow the same strategy of
largeK,S =1impurity(shownintheinset),valuescoincide
with known results. Ref.[8]: weassumeK largecomparedtotheconduction-
bandwidth and searchfor an accidentalgroundstate de-
generacyinthatpartoftheeffectiveHamiltonianinvolv-
susceptibility χSC inthe Cooperchannelc†1↑c†2↓+c†2↑c†1↓. ing just the impurity and the Fb’s:
Those are not accessible by Fermi liquid theory. Yet one
can get a rough estimate of them by the corresponding Hˆ = Hˆ +λ Sˆz 2+λ Tˆz 2
imp K S T
scattering amplitudes at zero external frequencies. They
(cid:16) (cid:17) (cid:16) (cid:17)
are given respectively by ΓSTz =−2U∗+JS∗−JT∗ and + JW⊥ F†F† Wˆ−−+F F† Wˆ−++H.c. .(5)
ΓSC =2U∗−3JS∗−JT∗,hencearenegative(correspond- 2παh s f f s i
ing to anenhancementof the response)and divergesim-
λ and λ are cut-off dependent functions of Jz, Jz.
ilarly approaching the UFP. The physics underneath is S T S T
For a specific K < 0, we find that the impurity state
the same of the 2IKM, and has been exhaustively dis- ∗
0 S =0,Sz =0;T =1,Tz =0 is degenerate with:
cussed in Ref. [4]. The UFP has a residual entropy
| i≡| i
ln√2. Away from the UFP, this entropy is quenched
cosθ
sbpeelocwificahteeamtpceoreaffitucrieenstcγale T1∗/T∼.|KTh−e Kre∗st|2o,fimthpelyiminpgua- |1i≡ √2 (cid:16)Ff |0,0;1,+1i+Ff† |0,0;1,−1i(cid:17)
∗
∼ sinθ
rity entropy is quenched at higher temperatures of order + F 1,+1;0,0 F† 1, 1;0,0 , (6)
TK ∼|K|. Atthe UFP,γ is finite, whilstboth χSTz and √2 (cid:0) s | i− s | − i(cid:1)
χ display a lnT singularity.
SC
| | where θ depends on the Hamiltonian parameters. For
The stability of the UFP is more easily accessed by
our model (1), θ should be equal to π/4 to reproduce
abelian bosonization, following Ref. [8] on the 2IKM. In
the observed UFP average values of S2, T2, and (Tz)2.
the large U limit, (1) maps onto the Kondo model
2
If we added the term G Tˆz , θ should increase with
z
3 (cid:16) (cid:17)
Hˆ = ǫ c† c +Hˆ + Jij Wˆijωˆij Gz, reaching the 2IKM value of θ = π/2 for large Gz.
s−d k k,aσ k,aσ K W The Klein factors in (6) show that 0 and 1 differs
kX,a,σ iX,j=1 | i | i
by one fermion, justifying the introduction of a fictitious
3
+ JSi Sˆiσˆi+JTi Tˆiτˆi , (4) ferTmhioenlocwo-nenneecrtgiyngHtahmatiltdoonuiabnletc:lofse† |to0ith=e|U1Fi.P, Hˆ ,
Xi=1(cid:16) (cid:17) UFP
is then obtained by projection onto the above doublet-
where σˆi, τˆi and ωˆij are, respectively, the conduction- subspace. Including up to dimension 3/2 operators,
electron spin, orbital and spin-orbital densities at the
impurity site. As usual in abelian bosonization we al- Hˆ =H +λ Ψ† (0) Ψ (0) f†+f (7)
low for anisotropy: Jx = Jy = Jz, and similarly for Ji UFP 0 0h sf − sf i(cid:0) (cid:1)
aannidsoJtWirjo.piWcKe founrdtohemroaSdsseulm(4eS)Jh6 Waijs=aScJoW⊥nt,infouroui,sjO6=(2z). ThTe +λ1∂xhΨ†sf(0)−Ψsf(0)i(cid:0)f†−f(cid:1)+δK∗f†f ,
spin
×
O(2)orbital U(1)charge symmetry, which is useful to de- with H0 the free Hamiltonian for the Ψb(x)’s, and δK∗
×
compose into U(1)spin U(1)orbital U(1)charge plus two the deviation from the fixed-point value K∗. λ0 and λ1
× ×
discrete symmetries: (i) a Πspin rotation: c c are model dependent parameters. As expected, Eq. (7)
x a,σ ↔ a,−σ
and d d ; and (ii) a Πorb rotation: c has the same form as in the 2IKM[8]. The UFP Hamil-
a,σ ↔ a,−σ x 1σ ↔
c and d d . By abelian bosonization[9], we tonian [first line of Eq. (7)] is a resonant level model in-
2σ 1σ 2σ
write the s-wa↔ve scattering components of conduction volvingoneMajoranafermionΨ† Ψ hybridisingwith
sf− sf
4
f† +f. The combination f† f is free and is respon- withinDMFT,weexpectthattheself-consistencycondi-
−
sible for the ln√2 UFP residual entropy. The relevant tionwhichrelatestheimpurityGreen’sfunctionwiththe
term (dimension 1/2) proportional to δK describes the localGreen’s function of the bath enlargesthe UFP into
∗
deviationfromthe UFP,whiletheλ -termistheleading awholeregionwherethemodelundergoesaspontaneous
1
irrelevantoperator(dimension3/2). Otherpossiblyrele- symmetry breaking. This would open up new screening
vant operators are instead not allowed by the symmetry channels for those degrees of freedom which survive be-
properties of (4), which have to be preserved by Hˆ low T down to T T +K 2/T (K < 0) and are
UFP K ∗ K K
∼ | |
too. For instance, among the particle-hole symmetry responsibleofthefinite entropyattheUFP.Ifthe band-
breakingtermsallowedinthe2IKM[4],onlythemarginal structure lacks nesting or Van Hove singularities,orbital
one, which does not spoil the UFP properties, may ap- or spin-orbital instabilities are not competitive with the
pear inour model, since the relevantoperator,bosoniza- Cooper instability[11]. This suggests a superconducting
tion of which is given by Ψ†(0)+Ψ (0) f f† [8], is region just before the MIT, which would be remarkable
here forbidden by U(1) h f. In facft, whii(cid:0)le −f is (cid:1)invari- since the bare scattering amplitude in the Cooper chan-
orbital
nel is U +K, hence repulsive for U U K . We
antunder a U(1)orbital rotationparametrizedby a phase believe that this phase is analogous to∼thecs≫tron|gl|y cor-
α, due to the Klein factors in (6), Ψ transforms into
f relatedsuperconductivityrecentlyidentifiedbyDMFTin
e2iαΨ . Indeed all relevantperturbations which destabi-
f
a model for tetravalent alkali doped fullerenes[12]. The
lizetheUFPcorrespondtophysicalinstabilitiesofmodel
lattermodelmapsbyDMFTontoathreefolddegenerate
(1), unlike what happens in the 2IKM. For instance,
AIM with inverted Hund’s rules, mimicking a t H dy-
the relevant terms Ψc(0)±Ψ†c(0) f −f† , of dimen- namical Jahn-Teller effect. Although different f⊗rom our
sion 1/2, break U(1(cid:2))charge. Theref(cid:3)o(cid:0)re, gaug(cid:1)e symmetry
model (1)-(2), it contains the essential physics we have
breakingdestabilizes the UFP, whichexplains the singu-
described in this work; namely the competition between
lar behavior of χSC. Analogously, χSTz is the response
the Kondo- and an intra-impurity-screening mechanism.
to a field which breaks SU(2) Πorb and allows the
spin × x We acknowledge helpful discussions with E. Tosatti.
relevant terms[10] Ψ† (0)+Ψ (0) f f† [4, 8] and
Ψ†(0) Ψ (0) fh fsf† . Besidsefsthios(cid:0)et−wosu(cid:1)sceptibili- TChOiFsINw2o0r0k1 ahnadsFbIReeBn200p2a,rtalnyd EsuPpSpRoCrte(dUKb)y. MIUR
s ± s −
(cid:2)ties,alsoχTa an(cid:3)d(cid:0)χSTa,w(cid:1)itha=x,y,arelogarithmically
diverging, being related to fields breaking U(1) .
orbital
Wenowturntoouroriginalmotivationanddiscussthe
possible relevance of the above results to the physics of [1] A. Georges et.al., Rev.Mod. Phys. 68, 13 (1996).
theMotttransition. Takealatticemodelwithanon-site [2] B. Jones and C.M. Varma, Phys. Rev. Lett. 58, 843
interaction of the same form as in (1)-(2), with inverted (1987), B. Jones at.al., ibid, 61, 125 (1988), B.A. Jones
Hund’s coupling K <0. This may occur if the electrons and C.M. Varma, Phys.Rev. B 40, 324 (1989).
areJahn-Tellercoupledtotwodegenerateweaklydisper- [3] I.AffleckandA.W.W.Ludwig,Phys.Rev.Lett.68,1046
sive optical phonons by g R q1RTˆRx+q2RTˆRy , where [4] I(.19A9ffl2)e.ck, A.W.W. Ludwig, B.A. Jones, Phys. Rev. B
qiR are the phonon coordPinat(cid:16)es on site R. T(cid:17)his cou- 52, 9528 (1995).
plinggivesrisetoaretardedelectron-electroninteraction [5] All the results presented throughout refer to a fixed
which reduces to (2) with K g2/ω when the typical Kondoexchangeequalto0.05inunitsofhalf-bandwidth,
0
≃− which is our unit of measure for energy.
phonon frequencyω is muchlarger than the quasiparti-
0 [6] P. Nozi`eres, J. Low Temp. Phys. 17, 31 (1974).
cle bandwidth. Alternatively, two single-band Hubbard
[7] L. Mih´aly and A. Zawadowskii, J. Phys. Lett. (France)
planes/chains coupled by J RS~1R ·S~2R, where 1 and 39, L-483 (1978).
2 refer to the two planes/chPains and J > 0, would also [8] J. Gan, Phys. Rev. B 51, 8287 (1995).
display a similar behavior. [9] G. Zar´and and J. von Delft, Phys. Rev. B 61, 6918
(2000).
When K =0 the lattice model should undergo a MIT
[10] Under Πorb, f ↔ −f, Ψ ↔ −Ψ†, Ψ ↔ Ψ† and
at some finite U in the absence of nesting. If K U , x f f sf sf
c | |≪ c Ψc ↔−Ψc.
the physics of the metallic phase near the MIT should
[11] TheBethe-Salpeterequationsforbulksusceptibilitiesde-
resemblethat ofthe AIM,Eq. (1), inthe Kondoregime. pend not only upon the irreducible vertices, which are
Since the width of the quasiparticle resonance, i.e. the local in DMFT, but also on products of lattice Green’s
effective Kondo temperature T , vanishes at the MIT, functions in the appropriate particle-particle (p-p) or
K
T U U[1], the system is forced to enter the crit- particle-hole(p-h)channels.Whilethisproductinthep-
K c
∼ − pchannelisalwayssingularbytheCooperphenomenon,
ical region around the unstable fixed point, K T ,
K
| | ∼ in the p-h channels singularities may arise only due to
before the MIT occurs. However the instability of the
band structure effects. Therefore, even though local sus-
AIMaroundthe UFPtowardstheorbitalO(2)orcharge
ceptibilities hence irreducible vertices are singular in p-
U(1) symmetry breakings should transform in the lat- p and p-h channels, the bulk p-p susceptibility is more
tice model into a true bulk instability. Namely, at least likely thedominant one.
5
[12] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti,
Science 296, 2364 (2002).