Quantum critical properties of the Bose-Fermi Kondo Model in a large-N limit

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by  Lijun Zhu

Quantum critical properties of the Bose-Fermi Kondo model in a large-N limit
Lijun Zhua, Stefan Kirchnera, Qimiao Sia, and Antoine Georgesb
aDepartment of Physics & Astronomy, Rice University, Houston, TX 77005-1892, USA
bCentre de Physique Th´eorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Studies of non-Fermi liquid properties in heavy fermions have led to the current interest in the
Bose-FermiKondomodel. Hereweuseadynamicallarge-N approachtoanalyzeanSU(N)×SU(κN)
5 generalization ofthemodel. Weestablishtheexistenceinthislimitofanunstablefixedpointwhen
0 the bosonic bath has a sub-ohmic spectrum (|ω|1−ǫsgnω, with 0<ǫ<1). At the quantum critical
0 point, theKondo scale vanishes and thelocal spin susceptibility (which is finite on theKondo side
2 for κ<1) diverges. We also find an ω/T scaling for an extended range (15 decades) of ω/T. This
scaling violates (for ǫ ≥ 1/2) the expectation of a naive mapping to certain classical models in an
n
extradimension; it reflects the inherent quantumnatureof thecritical point.
a
J
PACSnumbers: 75.20.Hr,71.10.Hf,71.27.+a,71.28.+d
6

]
l Heavy fermion systems show unusual quantum criti- is the condensed slave-bosonmean field theory[15]; here,
e
cal properties, which have motivated the interest in how the QCP is trivial: the effect of the Kondo interaction
-
r the Kondo effect is destroyed near a magnetic quantum disappears completely as soon as the static slave-boson
t
s phase transition[1, 2, 3, 4]. One theoretical approach amplitudevanishes. Adynamicalmethodisnecessaryto
.
t studies Kondo lattice systems through a self-consistent capture the quantum critical properties.
a
m Bose-Fermi Kondo model[1]. Here, a local magnetic mo- In this paper, we study the model using a dynami-
ment not only interacts antiferromagnetically with the cal large-N method[16, 17, 18]. The spin symmetry is
-
d spins of a conduction electron bath, it also is coupled to taken to be SU(N) and the number of conduction elec-
n adissipativebosonicbath. (Forworkinarelatedcontext, tron channels M κN. The large-N limit is expected
o seeRef.5.) Theconductionelectronsalonewouldleadto to capture the qua≡ntum-critical behavior of the physical
c
a Kondo singlet for the ground state and spin-1 charge- cases(seebelow). Thislimit,withpurelyKondocoupling
[ 2
e Kondo resonances in the excitation spectrum[6]. The to fermions was extensively studied in Refs. [16, 17] and

3
bosonic bath – characterizing the fluctuating magnetic withpurelybosoniccouplinginRef.[18](seealso[9,10]).
v
field provided by spin fluctuations – competes with the Our primary results are three-fold. First, we establish
3
Kondo effect; a sufficiently strong bosonic coupling de- the existence of a non-trivial QCP even in the large-N
9
2 stroystheKondoeffect. Aquantumcriticalpoint(QCP) limit. Second, we show that the QCP is non-Gaussian
6 arises, where the electronic excitations are of non-Fermi not only for small ǫ but also for ǫ 1/2. This demon-
0 liquid nature. When such a Kondo-destroyingcriticality stratesthe inherentlyquantumnatu≥reofthe QCP:stan-
4
is embedded into the criticality of a magnetic ordering, dard mapping of a quantum critical point to a classical
0
the critical theory becomes distinct from the standard critical point at extra dimensions[23] would imply that
/
at paramagnon theory[1]. This picture has been called lo- ǫ=1/2 is the upper critical dimension (c.f. the classical
m cally quantum critical. Direct evidence for the destruc- spin chains with 1/r2−ǫ interactions[19, 20]); the differ-
- tionofthe KondoeffecthasrecentlyemergedintheHall ence arises from the Berry phase of the quantum spin.
d measurements of a heavy fermion metal[7]. Third, we determine the critical exponents and ampli-
n
tudes and the universal scaling functions near the QCP,
o Previous studies of the Bose-Fermi Kondo model are
as well as the properties on the approach towards the
c based on an ǫ-expansion renormalization group (RG)
: bosonicfixedpoint. Wewillrestricttoκ M/N <1for
v analysis[1, 8, 9, 10, 11, 12] [the definition of ǫ appears in ≡
whichthe spinsusceptibility of the multi-channelKondo
i Eq. (2) below; ǫ > 0 corresponds to sub-ohmic bosonic
X fixed point is finite[16, 17, 21].
bath]. This approach has been successful, partly due to
r The Hamiltonian for the model is
a thefactthatthelinearinǫcontributiontotheanomalous
dimensionofthelocalspinoperatorturnsouttobeexact = (J /N) S s + E c† c
toallordersinǫ[10,11,12]. Inspiteofthis,theapproach HMBFK K Xα · α pX,α,σ p pασ pασ
is after all perturbative in ǫ; moreover, it is capable of
+ (g/√N)S Φ+ w Φ† Φ . (1)
calculating only a limited number of physical quantities. · p p · p
Xp
(Themodellacksconformalinvariance,makingboundary
conformal field theory inapplicable.) For the case with Here, a local moment S interacts with fermions c
pασ
an Ising spin-anisotropy, a self-consistent version of the and bosons Φ . The spin and channel indices are
p
model has been studied[13] by a Quantum Monte-Carlo σ = 1,...,N and α = 1,...,M, respectively, and
method[14]. With spin rotational invariance, the only Φ (Φ +Φ† ) contains N2 1 components. We
≡ p p −p −
approach that has been used, beyond the ǫ-expansion, will fiPrst consider a flat conduction electron density of
2
statesandasubohmic bosonicspectrum. Denoting 0 = G F GB
G
T c (τ)c† (0) and = T Φ(τ)Φ†(0) for each
c−ohmτpoσnαent, σwαe haiv0e N0(ωGΦ) ≡ −h π1τImG0(ω +ii00+) = N0 S f= Gf + G0
(ω < D = 1/2N ) and A (ω) 1Im (ω +i0+) =
| | 0 Φ ≡ π GΦ
[δ(ω w ) δ(ω+w )] being
p − p − p
P
S =
A (ω)=[K2/Γ(2 ǫ)]ω 1−ǫsgn(ω), (2) B
Φ 0 − | |
for ω < Λ 1/τ (Γ is the gamma function). We con- FIG.1: Thelarge-N Feynmandiagramsfor theself-energies.
0
| | ≡
sider the conduction electrons in the fundamental rep-
resentation of the SU(N) SU(M) group, and the local
× two types of interactions. To proceed, we seek for scale-
momentinanantisymmetricrepresentationwhoseYoung
invariant solutions of the following form at T =0,
tableaux is a single column of N/2 boxes. We can then
δwσr,iσt′e/2S, winithtertmhes coofnpstsreauidnot-ferNσm=i1onfσs†,fσSσ=σ′ N=/2f,σ†wfσh′ic−h Gf(τ) = A1/|τ|α1sgnτ, GB(τ)=B1/|τ|β1, (7)
isenforcedbyaLagrangemulPtiplier iµ. The Kondocou- for the asymptotic regime τ τ0. Using these, the
pling, (JK/N) σσ′ fσ†fσ′ −δσ,σ′/2 c†ασ′cασ, for each saddle-point equations become≫
channelα,willbPedec(cid:0)oupledintoaB†(cid:1) c† f /√N in-
α σ ασ σ ω1−α1
teraction using a dynamical Hubbard-PStratonovich field G−f1(ω) = A C =ω+κN0B1Cβ1ωβ1
Bα(τ). Itisconvenienttoconsiderg andJK asindepen- 1 α1−1
dentlyvarying,sincetheuniversalpropertiesonlydepend −(K0g)2A1C1−ǫ+α1ω1−ǫ+α1 (8)
on the ratio g/TK0. G−1(ω) = ω1−β1
Renormalizationgroupanalysis: Wewillfirstestablish B B β C
1 1 β1
that a non-trivial QCP survives the large-N limit. We
1
havegeneralizedtheRGequationsofRef.11toarbitrary = J −ΣB(0)+N0A1(α1+1)Cα1+1ωα1(9)
N and M. In the large-N limit, the RG beta functions, K
perturbative only in ǫ, become where C π exp[iπ(2−x)/2]. We have used λ +
x−1 ≡ Γ(x) sin[π(2−x)/2]
Σ (0) = 0, which follows from Eq. (5). We have also
β(g¯)= 2g¯ ǫ/2 g¯+g¯2 κ¯j2 , f
− − − assumed 0 < α1,β1 < 1, which turns out to be valid in
β(¯j)= ¯j ¯j(cid:0) κ¯j2 g¯+g¯2 , (cid:1) (3) most cases; exceptions will be specified below.
− − −
(cid:0) (cid:1) Threesolutionsarisedependingonthecompetitionbe-
where, g¯ (K g)2, and¯j N J . There is an unstable
0 0 K tween the Kondo and bosonic coupling terms. A dom-
fixed poin≡t at (g¯∗,¯j∗) = (≡ǫ/2+(1 κ)ǫ2/4,ǫ/2). Here,
inating Kondo or bosonic term leads to the Kondo or
−
χ(τ) 1/τ η and the anomalous dimension η =ǫ.
bosonic phase, respectively. When the two terms are of
∼ | |
Saddle-point equations: Our primary objective is to
the same order, the critical fixed point emerges.
study the saddle-point equations of the large-N limit, Multichannel Kondo and bosonic fixed points: When
which have the following form (cf. Fig. 1): β <1 ǫ+α , the Kondo coupling dominates over the
1 1
−
bosonic coupling on the RHS of Eq. (8). The leading
G−1(iω ) = 1/J Σ (iω ); Σ (τ)= (τ)G ( τ)
B n K − B n B −G0 f − ordersolutionis thenidentical to thatof the model with
G−f1(iωn) = iωn−λ−Σf(iωn); only a Kondo coupling[16]. The leading exponents are
Σ (τ) = κ (τ)G (τ)+g2G (τ) (τ), (4) α1 =1/(1+κ) and β1 =κ/(1+κ).
f 0 B f Φ
G G Inthe opposite case,with β >1 ǫ+α , the bosonic
1 1
−
together with a constraint equation, term dominates over the Kondo term. Matching the dy-
namical parts of Eq. (8) leads to
G (τ =0−)=(1/β) G (iω )eiωn0+ =1/2. (5)
f f n 2 ǫ πǫ
Xiωn α1 = ǫ/2; A21 = 4π(K−g)2 tan 4 . (10)
0
The dynamical spin susceptibility is
Together with Eq. (6), they combine to yield the follow-
ing result for the dynamical spin susceptibility
χ(τ)= G (τ)G ( τ). (6)
f f
− −
2 ǫ πǫ 1 1
Here, λ is the saddle point value of iµ, G (τ) = χ(τ)= − tan . (11)
B (cid:18) 4π 4 (cid:19)(K g)2 τ ǫ
T B (τ)B†(0) and G (τ)= T f (τ)f†(0) . 0 | |
h τ α α i f −h τ σ σ i
The first and second terms on the RHS of the last TheexponentagreeswiththatofRefs.8,9,10,11,12,18.
line of Eq. (4) reflect the Kondo and bosonic couplings, IfJ isstrictly0,G (ω)vanishes. Forsmallbutfinite
K B
respectively. They capture the competition between the J , Eq. (9), which (more precisely, the term in between
K
3
0(cid:31)(!=0;T)(cid:1)T1K01000100 (cid:15)=0g:g1;==(cid:20)240=0TT1K0K=02 00(cid:0)700(cid:31)(!;T=10T)(cid:1)TKK 1010 ggg===1958080TTTKK00K0 0(cid:31)(!=0;T)(cid:1)TK00:0:111 (cid:15)=0:gg9;==(cid:20)04=TK01=2 000(cid:0)70)(cid:1)T(cid:31)(!;T=10TKK100:(cid:0)0141 ggg===414:008TTTK0K0K0
(a) 0:01(b) (a) (b)
0:01 10(cid:0)6 0:00T01=TK0 0:01 1 10(cid:0)6 0:0001!=T0:K001 1 100 0:001 10(cid:0)6 0:0T00=1TK0 0:01 1 10(cid:0)6 10(cid:0)4 !0=:0T1K0 1 100
FIG.2: Staticanddynamicallocalsusceptibilitiesforǫ=0.1. FIG. 3: Static and dynamical susceptibilities for ǫ=0.9.
The red solid line corresponds to the quantum critical point,
g=g .
c
1 for ω T∗ to A (ω) 1 for ω T∗,
ωκ/(1+κ) | | ≪ f ∼ ω1−ǫ/2 | | ≫
and inserting the ansatz into 1/J = Σ (ω = 0) =
K B
the two equalities) comes with the assumption β1 < 1, βdτ (τ)G (τ), we find T∗ (g g)2/ǫ. Correspond-
cannot be satisfied. A solution does however exist for 0 G0 f ∼ c−
Ringly, the susceptibility diverges as g approaches gc:
β > 1, in which case G (ω) no longer diverges. It now
1 B
follows from G−B1(ω)∼const+ωβ1−1 that χ(T,ω =0,g .gc)∼(gc−g)−γ, γ =2(1−ǫ)/ǫ.(16)
β =1+ǫ/2. (12) Ourresultssofarrepresentanessentiallycompletean-
1
alytical solution at zero temperature. The solution at fi-
The non-divergence of G reflects the irrelevant nature nitetemperaturesisconsiderablymoredifficulttoobtain.
B
of the Kondo interaction at the bosonic fixed point. In the following, we resort to numerical studies.
Critical fixedpoint: TheKondoandbosonictermsare Numerical results: For low-energy analysis, we find
of the same order when β = 1 ǫ+α . In this case, it important to solve the integral eqs. (8,9) in real fre-
1 1
−
matching the dynamical parts leads to quencies and using a logarithmic discretization [supple-
mented by linearly-spaced points; typically 250 points
α =ǫ/2; β =1 ǫ/2; (13)
1 1 in total for the frequency range ( 10,10)]. The con-
−
−
2 (1+κ)ǫ πǫ ǫ πǫ vergence is determined in terms of A (ω) and A (ω):
A2 = − tan ; B = tan .(14) B f
1 4π(K g )2 4 1 −4πN A 4 for each, the sum over the entire frequency range of
0 c 0 1
therelativedifferencebetweentwoconsecutiveiterations
The critical bosonic coupling g needs to be numerically
c is less than 10−5. We choose κ = 1/2 and N (ω) =
0
determinedbymatchingthestaticpart: 1/J Σ (0)=
K− B (1/π)exp( ω2/π). The nominal bare Kondo scale is
0. The resulting dynamical spin susceptibility is: −
T0 (1/N )exp( 1/N J ) 0.06, for fixed J = 0.8.
K ≡ 0 − 0 K ≈ K
2 (1+κ)ǫ πǫ 1 1 Thebosonicbathspectralfunction[cf. Eq.(2)]iscutoff
χ(τ)= − tan . (15) smoothly at Λ 0.05. K2 is fixed at Γ(2 ǫ).
(cid:18) 4π 4 (cid:19)(K0gc)2 τ ǫ ≈ 0 −
| | In Fig. 2a), we show the static local spin susceptibil-
We make two observations at this point. First, for ity χ(T) as a function of temperature for ǫ = 0.1. For
infinitesimal ǫ, our result agrees not only with the re- vanishing and relatively small bosonic coupling g, χ(T)
sult of the ǫ-expansion for the large-N model [c.f. the saturates to a finite value in the zero-temperature limit
subsection containing Eq. (3)] but also that of the ǫ- – which characterizes the Kondo phase. At a critical
expansion for the the physically relevant SU(2) single- coupling[22] of g/T0 56, χ(T) diverges in a power-law
K ≈
channelmodel[11,12]. Thisimpliesthatthelarge-Nlimit fashion. Beyond this threshold coupling, χ(T) continues
captures the quantum critical behavior of the physical to have a power-law divergence. Its amplitude, on the
systems, even though it yields a multichannel behavior other hand, decreases with increasing g. These features
on the Kondo side. This is reminiscent of the effects of are not restricted to small ǫ; similar results are given in
spin anisotropies in the single-channel models: though Fig.3a),forǫ=0.9. Theyareconsistentwiththeanalyt-
the bosonic fixed point with Ising anisotropy (whose lo- icalresults: Eq.(16)statesthatχ(T =0)shoulddiverge
cal susceptibility contains a finite Curie constant[11]) is as g approaches g from below; for g > g , on the other
c c
verydifferentfromitscounterpartwithSU(2)symmetry, hand, Eq. (11) implies that the susceptibility amplitude
η is the same for the QCPs of the two cases[11]. Second, should indeed decrease as g increases.
Eq. (15) goes beyond the ǫ-expansion result described The quantum phase transition can also be seen in the
earlierinthe sensethatitis validbeyondinfinitesimal ǫ. dynamics. InFig.2b),weshowtheimaginarypartofthe
We have also determined the exponents of the sub- dynamical local spin susceptibility, χ′′(ω), at the lowest
leadingtermsforG andG ,findingα =ǫandβ =1. studied temperature (T = 10−7T0) and for ǫ = 0.1. On
f B 2 2 K
We next consider approaching the QCP from the the Kondo side, it vanishes in the zero-frequency limit
Kondo side. Defining T∗ to be the crossover scale with an exponent close to 0.33, consistent with the ana-
wherethespectralfunctionA (ω)changesfromA (ω) lyticalresult2α 1=1/3. Atg ,itdisplaysapower-law
f f 1 c
∼ −
4
with Ising anisotropy[13]. Our results are also relevant
1 in two other contexts. First, some generalized version of
9 the Bose-FermiKondomodelmaycapturethe physicsof
0:
D 0:01 impurities in high Tc superconductors[24, 25]. Results
(cid:1) similar to what we report here should shed light on the
)
T
000:1 (cid:1)(cid:31)(!; 1100(cid:0)(cid:0)64 TTTTT=====TTTTTKKKKK00000 =====1111100000(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)12345 dbbinyeeetntwnhaeemseycnicsoatnKeltmopenxraodttpioocefaarldntliyidessoaprodadfdresarruemscinshaegnsdnye.osatnSrelemyccoosmun,padwlg,ihnntigehctsheicahcloamsmvoeetpanapelotpsit[et2iya6oer]nts.
T T=TK0 =10(cid:0)6 We acknowledge the support of NSF Grant DMR-
10(cid:0)8 T=TK0 =10(cid:0)7 0424125 and the Robert A. Welch Foundation (LZ, SK,
10(cid:0)8 10(cid:0)6 10(cid:0)4 0:01 1 100 104 106 108 1010 and QS), DFG (SK), CNRS and Ecole Polytechnique
!=T (AG) and NSF Grant PHY99-07949 (LZ, QS, AG), as
well as the Lorentz Center, Leiden and KITP, UCSB.
FIG. 4: ω/T scaling of the dynamical spin susceptibility at
g=g , for ǫ=0.9 and κ=1/2. Thescaling exponentis 0.1
c
divergence,withanexponentcloseto0.9. Onthebosonic [1] Q. Si et al.,Nature(London) 413, 804 (2001).
side,theexponentremainsthesamewhiletheamplitude [2] P.Colemanetal.,J.Phys.Cond.Matt.13,R723(2001).
decreases as g increases. Both exponents agree with the [3] S. Burdin et al.,Phys.Rev. B 66, 045111 (2002).
analyticalresult of1 2α =1 ǫ=0.9. Similar results [4] T. Senthilet al.,Phys. Rev.Lett. 90, 216403 (2003).
1
for ǫ=0.9 are shown−in Fig. 3b−). [5] A.M. Sengupta and A. Georges, Phys. Rev. B52, 10295
(1995).
We now address the finite frequency and temperature
[6] A. C. Hewson, The Kondo Problem to Heavy Fermions
behavior at the QCP in some detail. Fig. 4 plots the
(Cambridge University Press, Cambridge, 1997).
dynamical spin susceptibility for ǫ = 0.9, clearly mani- [7] S. Paschen et al.,Nature (London) 432, 881 (2004).
festing an ω/T scaling. Remarkably, the scaling covers [8] J. L. Smith and Q. Si, Europhys. Lett. 45, 228 (1999);
an overall 15 decades of ω/T (from 10−8 to about 107)! Q.SiandJ.L.Smith,Phys.Rev.Lett. 77,3391 (1996).
Foreachtemperature,theresultfallsonthescalingcurve [9] A. M. Sengupta, Phys.Rev.B61, 4041 (2000).
until an ω of the order T0. Similar behavior is observed [10] M. Vojta et al.,Phys. Rev.B 61, 15152 (2000).
K [11] L. Zhu and Q. Si, Phys. Rev.B 66, 024426 (2002).
for other values of ǫ in the range 0 < ǫ < 1. The ω/T
[12] G.Zar´and and E.Demler, Phys.Rev.B66, 024427 (2002).
scalingreflectsthe interactingnatureofthe criticalfixed [13] D.Grempel and Q.Si, Phys.Rev.Lett.91,026401 (2003).
point[23]: there is no energy scale other than T; the re- [14] D. Grempel and M. Rozenberg, Phys. Rev. B60, 4702
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[18] S. Sachdev and J. Ye, Phys.Rev.Lett. 70, 3339 (1993).
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f
viate from ω/T scaling at a smaller frequency, reflecting [22] The dimensionless g Λ1−ǫ/2/T0 is of orderunity.
c K
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Our work serves as a basis for studies of the spin-
[25] J. Bobroff et al., Phys.Rev.Lett. 86, 4116 (1991).
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firstnon-perturbativeapproachto the Kondo-destroying
Y.L.Loh,V.TripathiandM.Turlakov,cond-mat/0405618;
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