Phase transitions in the pseudogap Anderson and Kondo models: Critical dimensions, renormalization group, and local-moment criticality

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by  Lars Fritz

Phase transitions in the pseudogap Anderson and Kondo models:
Critical dimensions, renormalization group, and local-moment criticality
Lars Fritz and Matthias Vojta
Institut fu¨r Theorie der Kondensierten Materie, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
(Dated: October15, 2004)
ThepseudogapKondoproblem,describingquantumimpuritiescoupledtofermionicquasiparticles
5 withapseudogapdensityofstates,ρ(ω)∝|ω|r,showsarichzero-temperaturephasediagram,with
0 differentscreenedandfreemomentphasesandassociatedtransitions. Weanalyzeboththeparticle-
0 hole symmetric and asymmetric cases using renormalization group techniques. In the vicinity of
2 r = 0, which plays the role of a lower-critical dimension, an expansion in the Kondo coupling
is appropriate. In contrast, r = 1 is the upper-critical dimension in the absence of particle-hole
n
symmetry, and here insight can be gained using an expansion in the hybridization strength of the
a
Anderson model. As a by-product,we show that the particle-hole symmetric strong-coupling fixed
J
pointforr<1isdescribedbyaresonantlevelmodel, andcorrespondstoanintermediate-coupling
3
fixedpointintherenormalizationgrouplanguage. Interestingly,thevaluer=1/2playstheroleofa

secondlower-criticaldimensionintheparticle-holesymmetriccase,andtherewecanmakeprogress
]
l by a novel expansion performed around a resonant level model. The different expansions allow a
e
completedescriptionofallcriticalfixedpointsofthemodelsandcanbeusedtocomputeavarietyof
-
r propertiesnearcriticality,describinguniversallocal-momentfluctuationsattheseimpurityquantum
t phasetransitions.
s
.
t
a
m I. INTRODUCTION coupling,J ,belowwhichtheimpurityspinisunscreened
c
evenatlowesttemperatures. Also,thebehaviordepends
-
d sensitivelyonthepresenceorabsenceofparticle-hole(p-
Non-trivial fixed points and associated phase transi-
n h) asymmetry, which can arise, e.g., from a band asym-
tions in quantum impurity problems have been subject
o
metry at high energies or a potential scattering term at
c ofconsiderableinterestinrecentyears,withapplications
the impurity site. A comprehensive discussion of possi-
[ for impurities in correlated bulk systems, in transport
ble fixed points and their thermodynamic properties has
through nanostructures, and for strongly correlated lat-
2 been given in Ref. 6 based on the NRG approach.
tice models in the framework of dynamical mean-field
v
3 theory. Many of those impurity phase transitions occur Until recently, analytical knowledge about the critical
4 in variations of the well-known Kondo model1 which de- properties of the pseudogap Kondo transition was lim-
5 scribes the screening of localized magnetic moments by ited. Previous works employed a weak-coupling renor-
8 metallic conduction electrons. A paradigmatic example malization group (RG) method, based on an expansion
0
of an intermediate-coupling impurity fixed point can be in the dimensionless Kondo coupling j=N J . It was
4 0 K
0 found in the two-channel Kondo effect. found that an unstable RG fixed point exists at j =r,
/ Non-metallic hosts, where the fermionic bath density corresponding to a continuous phase transition between
t
a ofstates(DOS)vanishesattheFermilevel,offeradiffer- the free and screened moment phases2. Thus, the per-
m ent route to unconventionalimpurity physics. Of partic- turbative computation of critical properties within this
- ular interest is the Kondo effect in so-called pseudogap approachis restrictedto smallr. Interestingly,the NRG
d systems2,3,4,5,6,7,8, where the fermionic bath density of studies6 showedthatthefixed-pointstructurechangesat
n statesfollowsapowerlawatlowenergies,ρ(ω) N0 ω r r = r∗ 0.375 and also at r=1, rendering the relevant
o ∝ | | ≈ 2
(r > 0). Such a behavior arises in semimetals, in cer- case of r=1 inaccessible from weak coupling. In the p-h
c
: tain zero-gap semiconductors, and in systems with long- symmetriccase,forr 1 thephasetransitionwasfound
v range order where the order parameter has nodes at the to disappear, and the≥im2purity is always unscreened in-
i
X Fermisurface,e.g.,p-andd-wavesuperconductors(r =2 dependent of the value of J . In contrast, in the asym-
K
r and 1). Indeed, in d-wave high-Tc superconductors non- metric case the phase transition is present for arbitrary
a trivialKondo-likebehaviorhasbeen observedassociated r >0. Numerical calculations6,7 indicated that the criti-
withthemagneticmomentsinducedbynon-magneticZn cal fluctuations in the p-h asymmetric case change their
impurities9,10. Note that the limit r corresponds character at r=1: whereas for r<1 the exponents take
→ ∞
to a system with a hard gap. non-trivialr-dependentvaluesandobeyhyperscaling,ex-
ThepseudogapKondoproblemhasattractedsubstan- ponents are trivial for r>1 and hyperscaling is violated.
tial attention during the last decade. A number of These findings suggest to identify r=1 as upper-critical
studies2,3,4 employed a slave-boson large-N technique; “dimension” of the problem, whereasr=0 plays the role
significant progress and insight came from numerical ofalower-critical“dimension”. As known,e.g.,fromthe
renormalization group (NRG) calculations5,6,7 and the critical theory of magnets11, the description of the tran-
local moment approach8. It was found that a zero- sitions using perturbative RGrequiresdifferent theoreti-
temperature phase transition occurs at a critical Kondo calformulationsneartheupper-criticalandlower-critical
2
dimensions, i.e., the φ4 theory and the non-linear sigma bleinthepresentcaseofapseudogapdensityofstates,as
model in the magnetic case. theproblemcannotbedescribedusinglinearlydispersing
In this paper, we provide a comprehensive analytical fermionsin onedimension. Furthermore,integratingout
account of the phase transitions in the pseudogap An- the fermions from the problem, in order to arrive at an
derson and Kondo models, including the proper theories effectivestatisticalmechanicsmodelcontainingimpurity
for the critical “dimensions”. This is made possible by degreesoffreedomonly,cannotbe performedeasily: the
working with the Anderson instead of the Kondo model fermionicdeterminantsarisingin(1+r)dimensionscan-
–the degreesoffreedomofthe Andersonmodelturnout notbesimplyevaluated. Thisimpliesthatthepseudogap
toprovideanaturaldescriptionofthelow-energyphysics Kondo model does not map onto a one-dimensional(e.g.
at the quantum phase transitions near r = 1 as well as Ising) model with long-ranged interactions, in contrast
2
close to and above r = 1. We shall consider epsilon- toe.g. thespin-bosonmodel14. Indeed,thephasetransi-
type expansions in the hybridization, the on-site energy, tions in the pseudogap Kondo model and the sub-ohmic
and the interaction strength. Those expansions lead to spin-boson model are in different universality classes15.
different theories for the p-h symmetric and asymmet- ThereforewebelievethatourcombinedRGanalysispro-
ric cases. Interestingly, in the pseudogap Kondo model vides a unique tool for analyzing the pseudogap Kondo
the phase transitions near the lower-critical and upper- problem.
criticaldimensionarenotadiabaticallyconnected,asthe
fixed point structure changes both at r =r∗ and r = 1.
Thus the present quantum impurity problemhas a mor2e A. Models
complicated flow structure than the critical theory of
magnets, where the (2+ǫ) and (4 ǫ) expansions are The starting point of our discussion will be the single-
−
believed to describe the same critical fixed point. impurityAndersonmodelwithapseudogaphostdensity
In the p-h symmetric case of the pseudogap Kondo of states, = A+ b:
H H H
problem the line of non-trivial phase transitions termi- = ε f†f +U n n +V f†c (0)+h.c. ,(1)
nates at two lower-critical dimensions (!), r = 0 and HA 0 σ σ 0 f↑ f↓ 0 σ σ
Λ
rin=ter12ac.tNinegarrerso=na12ntwleevfienldmaondeelx,ptaongseitohnerarwouitnhdpaerntounr-- Hb = dk|k|rkc†kσckσ (cid:0) (cid:1)
Z−Λ
bative RG, to provide access to the critical fixed point,
where we have represented the bath, , by linearly
b
with the expansion being controlled in the small param- H
dispersing chiral fermions c , summation over repeated
eter(21−r),seeSec.IV. Interestingly,theweak-coupling spin indices σ is implied, ankσd cσ(0) = dk k rckσ is the
expansion for the Kondo model, presented in Sec. III, | |
conduction electron operator at the impurity site. The
provides a different means to access the same critical R
spectral density of the c (0) fermions follows the power
σ
fixed point, but with the small parameter being r; the law ω r below the ultra-violet (UV) cutoff Λ; details of
two expansions can be expected to match. | |
the density of states at high energies are irrelevant for
In the p-h asymmetric case an expansion can be done the discussion in this paper. The four possible impurity
inthehybridizationaroundthevalence-fluctuationpoint stateswill be labelled with , for the spin-carrying
oftheAndersonmodel. Bareperturbationtheoryissuffi- |↑i |↓i
states, e for the empty and d for the doubly occupied
cientforallr >1;forr<1aperturbativeRGprocedure state. P|riovidedthatthecond|ucitionbandisp-hsymmet-
is required to calculate critical properties, with the ex-
ric, the above model obeys p-h symmetry for U = 2ε
0 0
pansion being controlled in the small parameter (1 r). – this p-h symmetry can be considered as SU(2) p−seu-
−
Inparticular,thisidentifiesr =1astheupper-criticaldi-
dospin, i.e., the full symmetry of the model is SU(2)
spin
mensionofthe(asymmetric)pseudogapKondoproblem, SU(2) . Asymmetryofthehigh-energypartofthe
charge
andconsequentlyobservablesacquirelogarithmiccorrec- ×
conductionbandhasthesameneteffectasasymmetryof
tions for r = 1. A brief account on the p-h asymmetric the impurity states; we will always assume that the low-
caseandthe expansionaroundr =1 hasbeen givenin a
energypartofthebandisasymptoticallysymmetric,i.e.,
recentpaper12. We note that the flowof the asymmetric the prefactorof ω r in the DOSis equalfor positiveand
Andersonmodelinthemetalliccase,r =0,wasdiscussed | |
negative ω.
by Haldane13: here all initial parameter sets with finite The transformation
hybridizationsflow towardsthe strong-coupling(singlet)
f f†,
fixed point. σ → σ
For all cases listed above, we show that the critical c c† (2)
kσ → kσ
propertiesoftheAndersonandKondomodelsareidenti-
converts all particles into holes and vice versa, formally
cal,andwecalculatevariousobservablesinrenormalized
ε (ε +U ), V V . Physically, the roles of the
perturbation theory. To label the fixed points, we will 0 →− 0 0 0 →− 0
states and areinterchanged,aswellasthe states
follow the notation of Ref. 6. |↑i |↓i
e and d . It is useful to consider another transforma-
Before continuing, we emphasize that standard tools | i | i
tion,
for metallic Kondo models, such as bosonization, Bethe
ansatz,andconformalfieldtheory,arenoteasilyapplica- f f , f f†,
↑ → ↑ ↓ → ↓
3
ck↑ →ck↑, ck↓ →c†k↓, (3) a) r = 0 v2
SC
whichtransforms d , e . Here,the spinful
|↑i↔| i |↓i↔| i
doubletofimpuritystatesistransformedintothespinless
doublet and vice versa, i.e., the two SU(2) sectors are
interchanged. LM LM´
In the so-called Kondo limit of the Anderson model −∞ 0 ∞ ε = −u/2
charge fluctuations are frozen out, and the impurity site
is mainly singly occupied. Via Schrieffer-Wolff transfor- v2
b) 0 < r < 1/2
mation one obtains the standard Kondo model, =
+ , with H SSC
K b
H H
SCR SCR’
=J S s(0) (4)
K K
H ·
where the impurity spin S is coupled to the conduction LM LM´
electron spin at site 0, sα(0) = c†σ(0)σσασ′cσ′(0)/2, and −∞ 0 ∞ ε = −u/2
σα is the vector of Pauli matrices. The Kondo coupling
is related to the parameters of the Anderson model (1) c) 1/2 ≤ r < 1 v2
through:
SSC
1 1
J =2V2 + . (5)
K 0 ε U +ε
(cid:18)| 0| | 0 0|(cid:19) LM LM´
The Kondo limit is reached by taking U0 → ∞, ε0 → −∞ 0 ∞ ε = −u/2
, V , keeping J fixed. In the absence of p-h
0 K
s−y∞mmetry→th∞e Schrieffer-Wolff transformation also gen- d) r ≥ 1 v2
erates a potential scattering term in the effective Kondo
model1.
In the absence of an external magnetic field all above
models preserveSU(2) spin symmetry. Spin anisotropies
turn out to be irrelevant at the critical fixed points, see LM FImp LM´
AppendixD. Theeffectofamagneticfieldwillbebriefly
−∞ 0 ∞ ε = −u/2
discussed in Sec. VIII.
FIG. 1: Schematic RG flow diagrams for the particle-hole
B. Summary of results
symmetricsingle-impurityAndersonmodelwithapseudogap
DOS,ρ(ω)∝|ω|r. Thehorizontalaxisdenotestherenormal-
Our main results are summarized in the RG flow dia- ized on-site level energy ε (related to the on-site repulsion u
gramsin Figs.1 and2, for the p-hsymmetric andasym- by u=−2ε), the vertical axis is the renormalized hybridiza-
tion v. The thick lines correspond to continuous boundary
metric cases, respectively.
phasetransitions; thefull(open)circles arestable(unstable)
In the symmetric case, the ranges of exponent values
fixed points, for details see text. All fixed points at non-
r =0, 0<r< 1, 1 r <1, and r 1 lead to quite dif-
2 2 ≤ ≥ zeroεhaveamirrorimageat−ε,relatedbytheparticle-hole
ferent behavior, and are shown separately in Fig. 1. No
transformation (3). a) r=0, i.e., the familiar metallic case.
transition occurs for r = 0: for any non-zero hybridiza-
For any finite v the flow is towards the strong-coupling fixed
tiontheflowistowardsthemetallicKondo-screenedfixed point (SC), describing Kondo screening. b) 0<r< 1: The
2
point(SC).Thiswell-knownfixedpointcanbeidentified local-moment fixed point (LM) is stable, and the transition
asthestablefixedpointofaresonantlevelmodel;wear- tosymmetricstrongcoupling(SSC)iscontrolledbytheSCR
gue below that this is actually an intermediate-coupling fixedpoint. For r→0, SCRapproaches LM,and thecritical
fixed point. behaviorat SCR is accessible via an expansion in theKondo
For 0 <r < 1, small values of the hybridization leave coupling j. In contrast, for r → 21, SCR approaches SSC,
2 andthecriticalbehaviorcanbeaccessedbyexpandinginthe
the impurity spin unscreened provided that ε < 0, i.e.,
0 deviation from SCR, i.e., in ε = −u/2. c) 1 ≤r<1: v is
there is a stable local-moment fixed point (LM) corre- 2
still relevant at u = 0. However, SSC is now unstable w.r.t.
spondingtoε= ,v =0. Atransitionlineatnegative
−∞ finite u. At finite v, the transition between the two stable
ε,withanunstablefixedpoint(symmetriccritical,SCR)
fixedpoints LM and LM’ is controlled by SSC(which is now
at finite v, ε, separates the flow towards LM from the a critical fixed point!). d) r≥1: v is irrelevant, and the only
| |
flowtothe symmetricstrong-couplingfixedpoint(SSC). transition is a level crossing (with perturbative corrections)
Thestrong-couplingfixedpointdisplaysitsintermediate- occurring at v = u = 0, i.e., at the free-impurity fixed point
couplingpropertiesnowinafiniteresidualentropyanda (FImp).
finitemagneticmoment,seeSec.IVB. Asr 0theSCR
→
4
fixed point merges with LM, in a manner characteristic v2
foralower-criticaldimension,i.e.,withdivergingcorrela- a) r* < r < 1
tion lengthexponent. A secondcriticalfixedpoint SCR’
exists for ε > 0 which separates the symmetric strong-
couplingphase(SSC)fromonewithafreechargedoublet ACR
(LM’).
As r 1 the symmetric critical fixed points merge LM ASC
with the→str2ong-coupling one, again in a manner charac- −∞ 0 ∞ ε
teristicforalower-criticaldimension. Forr 1 thefixed
points SCR andSCR’ ceaseto exist; the str≥on2g-coupling b) r ≥ 1 v2
SSC fixed point becomes infrared unstable, and controls
the LM – LM’ transition. Finally, the structure of the
flow changes again at r = 1: for r 1 the unstable
→
strong-coupling fixed point (SSC) moves towards v = 0,
i.e., the free-impurity fixed point (FImp), and for r 1 LM VFl ASC
≥
no non-trivial fixed point remains. −∞ 0 ∞ ε
For maximal p-h asymmetry,realizedin the Anderson
model through U = , one has to distinguish exponent
0
ranges r = 0, 0 < r∞ r∗, r∗ < r < 1, and r 1. In FIG. 2: Schematic RG flow diagram for the maxi-
the metallic case r =≤0 any non-zero hybridizat≥ion gen- mally particle-hole asymmetric pseudogap Anderson impu-
erates flow to strong coupling with complete screening rity model. The horizontal axis denotes the on-site impurity
energy, ε, the vertical axis is the fermionic coupling v, the
– the strong-coupling fixed point is the same as in the
p-h symmetric situation, as p-h symmetry is marginally basarienoFni-gsi.te1.repau)lsri∗on<isrfi<xed1:atvui0s =rele∞va.nTt,haensdymthbeoltsraanre-
irrelevant at strong coupling. For all r >0 the situation
sition is controlled by an interacting fixed point (ACR). As
is drastically different: small V leaves the moment un- ∗
0 r → r ≈ 0.375, p-h symmetry at the critical fixed point is
screened,whereaslargeV0directstheflowtowardsanew, dynamically restored, and ACR merges into the SCR fixed
p-h asymmetric, strong-coupling fixed point (ASC). The point of Fig. 1 – this cannot be described using the RG of
character of the critical fixed point separating the two Sec.V. Inthemetallicr=0situation,studiedbyHaldane13,
phases depends6 on r: for 0 < r < r∗ p-h symmetry is the flow from any point with v 6= 0 is towards the screened
restored,andthe criticalfixedpointis theoneofthep-h singlet fixed point with ε=∞. b) r≥1: v is irrelevant, and
symmetric model. For r∗ < r < 1 there is a separate thetransitionisalevelcrossingwithperturbativecorrections,
occuringatv=ε=0,i.e.,thevalence-fluctuationfixedpoint
critical fixed point (ACR) which is p-h asymmetric, i.e.,
(VFl).
locatedatfinitevandε. Forr 1thecriticalfixedpoint
→
moves towardsv 0, and for r 1 the phase transition
→ ≥
becomes a levelcrossing (with perturbative corrections),
model with maximal p-h asymmetry.
controlled by the valence-fluctuation fixed point (VFl),
Taken together, the above observations show that r =
see Fig. 2.
0 plays the roleof a lower-criticaldimension: as r 0+,
We finally discuss the general case of finite p-h asym- →
thecorrelationlengthexponentdiverges,andthesecond-
metry,moredetailswillbegiveninSec.VI. Powercount-
order transition turns into a Kosterlitz-Thouless transi-
ing showsthatLM (SSC) arealways(un)stable w.r.t. p-
tion at r = 0. Interestingly, in the symmetric case the
h asymmetry. The symmetric critical SCR fixed point is
correlation length exponent also diverges as r 1−,
stable w.r.t. p-h asymmetry for small r. In contrast, for → 2
r < 1 SCR is unstable towards p-h asymmetry, as it is and the transition between LM and SSC disappears for
2 r 1: r = 1 is a second lower-criticaldimension for the
clo∼setoSSCinthisregime. Thisrequirestheexistenceof ≥ 2 2
p-h symmetric problem. In the asymmetric case, there
a specific r value where this change in character occurs:
is a transition between LM and ASC for all r > 0, and
the is precisely r = r∗ 0.375 where p-h asymmetry
≈ r =1isequivalenttotheupper-criticaldimension,above
at SCR is marginal6. Upon increasing r beyond r∗ the
whichthe criticalfixedpointis non-interacting(actually
p-hasymmetriccriticalfixedpoint(ACR)splits offfrom
a level crossing).
SCR. In other words, upon approaching r∗ from large r
the ACR fixed point moves towards small effective p-h
asymmetry, and at r = r∗ ACR merges into SCR, im-
plying p-h symmetry is dynamically restored. As stated C. Outline
above,thedescriptionofACRusinganexpansionaround
VFlconsequentlybreaksdownasr r∗+. Neitherfrom The rest of this paper is organized as follows: Sec. II
numerics6 nor from the presentRG→arethere indications introduces the observables to be evaluated in the course
for the existence of a second asymmetric critical fixed of the paper, together with their expected scaling be-
pointbesides ACR;thus, the criticalpropertiesfor finite havior near criticality. In Sec. III we briefly review the
p-h asymmetry are always equivalent to the ones of a standard weak-coupling perturbative RG for the Kondo
5
model, which is suitable to describe the quantum phase Theimpuritycontributiontothetotalsusceptibilityis
transition for small r. Sec. IV discusses the particle- defined as
hole symmetric Andersonmodel. Starting from the non-
interacting case, ε0 = U0 = 0, we first discuss the χimp(T)=χimp,imp+2χu,imp+(χu,u−χbu,uulk), (8)
physics of the resulting non-interacting resonant level
model – interestingly this can be identified with a sta- whereχbulk isthesusceptibilityofthebulksysteminab-
u,u
ble intermediate-coupling fixed point. We then use a sence of the impurity. For an unscreened impurity spin
perturbative expansion in U0 to access the critical fixed of size S = 21 we expect χimp(T → 0) = 1/(4T) in the
points for r < 1. In Sec. V we turn to the situation low-temperature limit, and this is precisely the result in
2
with maxima∼l p-h asymmetry, i.e., U = , and show the whole LM phase. A fully screened moment will be
0
∞
that an expansion in the hybridization provides access characterized by Tχimp = 0; note that the SSC fixed
to the critical properties for r > 1 as well as for r < 1. point displays a finite value of Tχimp for r >0. At crit-
In Sec. VI we consider the case of general p-h asym∼me- icality χimp does not acquire an anomalous dimension18
try. Sec. VIII briefly describes the effect of a magnetic (incontrasttoχloc below), becauseitis aresponsefunc-
field: the pseudogap model is shown to permit a sharp tion associated to the conserved quantity Stot. Thus we
transition as function of a field applied to the impurity expect a Curie law
for couplings larger than the zero-field critical coupling.
C
In Sec. VII we compare the physics of the Anderson and lim χ (T)= imp , (9)
imp
Kondomodels,arguingthatthetransitionsinbothmod- T→0 T
els fall in the same universality classes. A brief discus-
where the prefactor C is in general a non-trivial uni-
imp
sion of applications concludes the paper. All renormal-
versal constant different from the free-impurity value
ization group calculations will employ the field-theoretic
S(S +1)/3. Apparently, Eq. (9) can be interpreted as
RG scheme16 together with dimensional regularization the Curie response of a fractional effective spin19.
and minimal subtraction of poles, with details given in
The local impurity susceptibility is given by
the appendices; one-loop RG results can equivalently be
obtained using the familiar momentum-shell method. χ (T)=χ , (10)
loc imp,imp
which is equivalent to the zero-frequency impurity spin
II. OBSERVABLES AND SCALING autocorrelation function. In the unscreened phase we
haveχ 1/T asT 0;wecanconsiderthisasarising
loc
∝ →
Toestablishnotationsandtopavethe wayforthe RG from the overlap of the local impurity moment with the
analysis below, we introduce a few observables together total,freelyfluctuating,momentofS =1/2,andsowrite
with their expected scaling properties.
m2
lim χ (T)= loc . (11)
loc
T→0 4T
A. Susceptibilities
The quantity m turns out to be a suitable order
loc
parameter6,7 for the phase transitions between an un-
Magnetic susceptibilities are obtained by coupling an
screenedandascreenedspin: itvanishescontinuouslyas
external magnetic field to the bulk electronic degrees of
t 0−, where t is the dimensionless measure of the dis-
freedom in Hb, ta→ncetocriticality;intheKondomodelt=(JK Jc)/Jc,
−
−Huα(x)(c†σσσασ′cσ′)(x) (6) wThχereaissinnotthpeinAnneddertsootnhmeovadleuleto=f 1(V/04−foVr0ct)</V00c(.inTchouns-,
loc
and to the impurity part A, K, trast to Tχimp). Remarkably, mloc =0 at the SSC fixed
H H point for r<1, although Tχ =r/8 there.
imp
−Himp,α(fσ†σσασ′fσ′) , −Himp,αSˆα (7) The phase transitions occurring for 0 < r < 1 are de-
scribedby interactingfixed points, andthus obey strong
fortheAnderson(1)andKondo(4)models,respectively. hyperscalingproperties,includingω/T scalingindynam-
The bulk field Hu varies slowly as function of the space icalquantities20. Forinstance,thelocaldynamicsuscep-
coordinate,andHimpisthemagneticfieldatthelocation tibility will follow a scaling form
of the impurity.
Withthesedefinitions,aspatiallyuniformfieldapplied ω T1/ν
χ′′ (ω,T)= B1 Φ , (12)
to the whole system corresponds to Hu = Himp = H. loc ω1−ηχ 1 T t
Response functions can be defined from second deriva- (cid:18) (cid:19)
tives of the thermodynamic potential, Ω = TlnZ, in which describes critical local-moment fluctuations, and
the standard way17: χ measures the bulk −response to the local static susceptibility obeys
u,u
a field applied to the bulk, χ is the impurity re-
imp,imp
sponse to a field applied to the impurity, and χu,imp is χ (T)= B2 Φ T1/ν . (13)
the cross-response of the bulk to an impurity field. loc T1−ηχ 2 t
(cid:18) (cid:19)
6
Here, η is a universal anomalous exponent, which con- under RG flow (which is equivalent to decreasing T).
χ
trolstheanomalousdecayofthetwo-pointcorrelationsof Theso-calledg-theorem21providesaproofofthisconjec-
the impurity spin, and Φ are universal crossoverfunc- turefor systemswithshort-rangedinteractions;formost
1,2
tions(for the specificcriticalfixedpointandforfixedr), quantum impurity problems this appears to apply. In-
whereas are non-universal prefactors. Furthermore, terestingly, the pseudogap Kondo problem provides an
1,2
B
ν is the correlation length exponent, describing the flow explicit counter-example, as the two critical fixed points
away from criticality: when the system is tuned through obey S < S , with the RG flow being from SCR
SCR ACR
the transition, the characteristic energy scale T∗, above to ACR (!), see Sec. V for details. (For another counter-
which critical behavior is observed, vanishes as20 example see Ref. 22.)
T∗ tν; (14)
∝| |
the dynamical critical exponent z can be set to unity in C. T matrix
the present (0+1)-dimensional problem. Note that at
criticality, t = 0, the relation (13) reduces to χ (T)
loc ∝ AnimportantquantityinanAndersonorKondomodel
T−1+ηχ.
is the conduction electronT matrix, describing the scat-
Hyperscaling can be used to derive relations between
teringofthecelectronsofftheimpurity. ForanAnderson
critical exponents. The susceptibility exponent η and
χ model, the T matrix is just given by T(ω) = V2G (ω)
the correlation length exponent ν of a specific transi- 0 f
whereG isthefullimpurityf electronGreen’sfunction.
tion are sufficient to determine all critical exponents as- f
For a Kondo model, it is useful to define a propagator
sociated with a local magnetic field7. In particular, the
T 0 local susceptibility away from criticality obeys GT of the composite operator Tσ = fσ†fσ′cσ′, such that
→ the T matrix is given by T(ω)=JK2GT(ω). As with the
χ (t>0) t−γ, γ =ν(1 η ), localsusceptibility,weexpectascalingformoftheT ma-
loc χ
∝ − trixspectraldensityneartheintermediate-couplingfixed
Tχloc(t<0) ( t)γ′, γ′ =νηχ, (15) points similar to Eq. (12). In particular, at criticality a
∝ −
power law occurs:
which can be derived from a scaling ansatz for the im-
purity part of the free energy7. The last relation implies
1
the order parameter vanishing as T(ω) . (17)
∝ ω1−ηT
m ( t)νηχ/2. (16)
loc
∝ − Remarkably, we will find the exact result η = 1 r for
T
−
Note that hyperscaling holds for all critical fixed points r < 1, i.e., for all interacting critical fixed points con-
of the pseudogap Kondo problem with 0<r <1. sidered in this paper the T matrix follows T(ω) ω−r.
∝
NRGcalculationshavefoundpreciselythiscriticaldiver-
gence, for both the symmetric and asymmetric critical
B. Impurity entropy points5,23. At the trivialfixedpoints (LM, ASC) the be-
havior of the T matrix follows from perturbation theory
in the hybridization to be T(ω) ωr.
In general, zero-temperature critical points in quan-
∝
tum impurity models can show a finite residual entropy Notably, the T matrix can be directly observed
[in contrast to bulk quantum critical points where the in experiments, due to recent advances in low-
entropy usually vanishes with a power law, S(T) Ty]. temperature scanning tunneling microscopy, as has
Forthemodelsathand,theimpuritycontribution∝tothe been demonstrated, e.g., with high-temperature
low-temperature entropy is obtained by a perturbative superconductors23,24,25.
evaluation of the thermodynamic potential and taking
the temperature derivative. This will yield epsilon-type
expansions for the ground state entropy S (T = 0),
imp
D. Phase shifts
with explicit results given below.
Notethattheimpuritypartofthethermodynamicpo-
tential will usually diverge with the cutoff, i.e., we have Fixed points which can be described in terms of free
Ω = E TS , where E is the non-universal fermions can be characterized by the s-wave conduc-
imp imp imp imp
−
(cutoff-dependent) impurity contribution to the ground- tion electron phase shift, δ (ω), which can be related
0
stateenergy. However,theimpurityentropyS isfully to the conduction electron T matrix through δ (ω) =
imp 0
universal,and the UV cutoff canbe sent to infinity after argT(ω). A decoupled impurity simply has a phase shift
taking the temperature derivative of Ω . δ = 0, whereas a p-h symmetric Kondo-screened impu-
imp 0
Thermodynamic stability requires that the total en- rity in a metallic host has a low-energy phase shift of
tropy of a system decreases upon decreasing tempera- δ (ω) = πsgn( ω). A detailed discussion for the pseu-
0 2 −
ture, ∂ S(T) > 0. This raises the question of whether dogapmodelhasbeengiveninRef. 6,inthebodyofthe
T
the impurity part of the entropy, S has to decrease paper we will simply quote the results.
imp
7
III. WEAK-COUPLING RG FOR THE KONDO The anomalous exponent of the local susceptibility eval-
MODEL uates to
Inthissectionwebrieflysummarizetheweak-coupling ηχ =r2+ (r3). (23)
O
RGforthepseudogapKondomodel(4),asfirstdiscussed
by Withoff and Fradkin2. Perturbative RG is performed A comparison of the above results with numerical data
is given in Figs. 3, 5, 6, and 7 below.
around J =0, i.e., the local-moment fixed point (LM):
K
thiswillallowtoaccessthe(p-hsymmetric)criticalfixed Mostimportantly,thecontinuoustransitioncontrolled
point SCR, which is located close to LM for small DOS by the fixed point (19), which exists only for r > 0,
exponents r. evolves smoothly into the Kosterlitz-Thouless transition
at r = 0, j = 0, which separates the antiferromagnetic
andtheferromagneticmetallicKondomodel. Thisisalso
A. Lower critical dimension: Expansion around the indicated by the divergence of the correlation length ex-
local-moment fixed point ponent(20)asr 0+. Thus,r =0canbeidentifiedasa
→
lower-critical“dimension”ofthepseudogapKondoprob-
lem. It is interesting to compare the present expansion
TheRGflowequationfortherenormalizedKondocou-
pling j, to two-loop order, reads26 withthe(2+ǫ)expansionforthenon-linearsigmamodel,
appropriateformagnetsclosetothelower-criticaldimen-
j3 sion. The expansion is done about the ordered magnet,
β(j)=rj j2+ . (18) thus the LM phase with ln 2 residual entropy takes the
− 2
role of the ordered state in the pseudogap Kondo prob-
This yields an infrared unstable fixed point at lem.
r2
j∗ =r+ + (r3) (19)
2 O IV. PARTICLE-HOLE SYMMETRIC
ANDERSON MODEL
which controls the transition between the decoupled LM
andtheKondo-screenedSSCphases. Thesmall-j expan-
In the following sections of the paper we shift our at-
sion(18)–whichisnothingbutthegeneralizationofAn-
tentionfromthe Kondomodelto theimpurity Anderson
dersonpoorman’sscaling27 tothepseudogapcase–can-
model with a pseudogapdensity of states. This formula-
notgiveinformationaboutthestrong-couplingbehavior,
tion will provide new insights into the RG flow and the
and it can only describe critical properties for small r.
critical behavior of both the Anderson and Kondo mod-
(Inthep-hsymmetriccase,thefixedpointstructuredoes
not change within the exponent range 0 < r < 1, thus els.
2 The coupling between impurity and bath is now the
the present expansion is in principle valid up to r = 1.)
2 Anderson hybridization term, V , which turns out to be
AddingapotentialscatteringtermV givesafinitep-h 0
0 marginal in a RG expansion around V =0 for the DOS
asymmetry. Under RG, we find that V renormalizes to 0
exponent r = 1 (in contrast to the Kondo coupling J
zero for r > 0, β(V) = rV. Thus, within the range of K
which is marginal for r =0). As we will show in Sec. V,
applicability ofthe weak-couplingRG,p-hasymmetry is
theAndersonmodelprovidestherelevantlow-energyde-
irrelevant. (Strictly, this applies for r <r∗, see Ref. 6.)
greesoffreedomforthep-hasymmetricpseudogaptran-
sition near its upper-critical dimension.
Interestingly, also the p-h symmetric version of the
B. Observables near criticality
Anderson model allows to uncover highly non-trivial
physics,in particular the special roleplayedby the DOS
We quote a few properties of the criticalregime which exponent r = 1, where the transition disappears in the
have been determined in Ref. 26. Expanding the beta 2
presence of p-h symmetry. Thus we start our analysis
function(18)aroundthefixedpointvalue(19)yieldsthe
with the particle-hole symmetric Anderson model (1),
correlation length exponent ν:
i.e., we keep U = 2ε and discuss the physics as func-
0 0
−
tion of V and ε .
1 r2 0 0
=r + (r3). (20)
ν − 2 O
Thelow-temperatureimpuritysusceptibilityandentropy A. Trivial fixed points
at criticality are given by
For vanishing hybridization, V = 0, the symmetric
0
1
Tχ = (1 r)+ (r2), (21) Anderson model (1) features three trivial fixed points:
imp
4 − O for ε < 0 the ground state is a spinful doublet – this
0
3π2 representsthelocal-momentfixedpoint(LM).Forε >0
S = ln2 1+ r3 + (r5). (22) 0
imp 8 O wefindadoubletofstates(emptyanddoublyoccupied),
(cid:18) (cid:19)
8
denoted as LM’ and related to LM by the p-h transfor- i.e., the hybridization is relevant only for r <1.
mation(3). BothLMandLM’havearesidualentropyof ToperformRGwithinthefield-theoreticscheme16,we
S = ln2. At ε = 0 a level crossing between the two introduce a renormalized hybridisation v according to
imp 0
doublets occurs, i.e., all four impurity states are degen- V = (Z µr¯/√Z)v, where µ is a renormalization energy
0 v
erate – this is the free-impurity fixed point (FImp), with scale, and Z and Z are the interaction and field renor-
v
residual entropy ln4. The impurity spin susceptibilities malization factors. The RG flow equation for v is found
are to be
1/4 LM β(v)= r¯v+v3. (30)
−
Tχ = 1/8 FImp , (24)
imp 0 LM’ Remarkably, this result is exact to all orders in per-

turbation theory: the cubic term arises from the only
the conduction electron phase shift is zero at all these self-energy diagram of the f fermions, and no vertex
fixed points. The hybridization term, V0, is irrelevant at renormalizations occur (Zv = 1). This implies that the
boththe LMandLM’fixedpoints for r >0,whereasfor low-energy physics of the non-interacting resonant level
r = 0 it is marginally relevant, as shown by the RG in model is controlled by the stable intermediate-coupling
Sec. IIIA. fixed point located at
1 r
v∗2 =r¯= − (31)
B. Resonant level model: Intermediate-coupling 2
fixed point
for 0 r <1, which also applies to the familiar metallic
≤
case r=0.
It proves useful to discuss the ε0 = U0 = 0 case, i.e., The intermediate-coupling nature of the stable fixed
the physics on the vertical axis of the flow diagrams in point, with associated universal properties, is consistent
Fig.1. Thisnon-interactingsystemisknownasresonant with the results known from the exact solution of the
level model, as the two spin species are decoupled. The problem,e.g.,auniversalconductionelectronphaseshift,
problemcanbesolvedexactly: thef electronself-energy a universal crossover in the temperature-dependent sus-
is ceptibility etc.
We proceed with calculating a number of observables
Σf =V02Gc0 (25) for the pseudogap resonant level model. Interestingly,
this can be done in two ways: either (i) via the ex-
where G is the bare conduction electron Green’s func-
c0 act solution of the problem, i.e., by integrating out the
tion at the impurity location R = 0. In the low-energy
c fermions exactly [leading to the propagator (26)], or
limit the f electron propagator is then given by
equivalently(ii)byevaluatingperturbativecorrectionsto
the FImp fixed point using the RG result (31), utilizing
G (iω )−1 =iω iA sgn(ω ) ω r (26)
f n n− 0 n | n| standard renormalized perturbation theory, and noting
that all corrections beyond second order in v vanish ex-
where the ω r self-energy term dominates for r < 1,
| n| actly within this scheme. Details of the calculation are
and the prefactor A is
0 in Appendix A.
πV2 We start with evaluating spin susceptibilities – note
A0 = cos0πr . (27) that we have kept two spin species in the model. The
2 zero-temperature dynamic local susceptibility is propor-
tionaltothebubbleformedwithtwof propagators(26),
Before stating results for observables it is interesting
to tackle the problem using RG techniques, with an ex-
ω1−2r (0 r <1)
pansioninthehybridizationstrengthV0 aroundthefree- χ′lo′c(ω)∝ δ(ω)ω (r ≤1) , (32)
impurityfixedpoint(FImp,V0 =0). Westudytheaction (cid:26) T ≥
where the case of r = 1 receives logarithmic corrections,
β 2
= dτ f¯ ∂ f +V (f¯c (0)+c.c.) see below. The low-temperature limit of the impurity
σ τ σ 0 σ σ
S
Z0 (cid:20) susceptibility is found to be
(cid:2) Λ (cid:3)
+ dk k rc¯ (∂ k)c (28) r
| | kσ τ − kσ Tχimp(T)= , (33)
Z−Λ (cid:21) 8
where c (0) is the bath fermionfield at the impurity po- the impurity entropy is
σ
sition as above. Power counting w.r.t. the V = 0 fixed
0
point, using dim[f]=0, dim[c(0)]=(1+r)/2, yields Simp =2rln2, (34)
1 r where the two last equations are valid for 0 r <1; for
dim[V0]= − r¯, (29) r 1 the resonantlevelmodel flows to the fr≤ee-impurity
2 ≡ ≥
9
fixed point (FImp) with properties listed in Sec. IVA. (27), and we have assumed 0 < r < 1. The interac-
TheconductionelectronphaseshiftneartheFermilevel, tion term has been written in a p-h symmetric form. A
determined in Ref. 6, is note is in order regarding the cutoff: The originalmodel
hadaUVcutoffΛ,andthissetstheupperboundforthe
(1 r)π (0 r <1)
δ0(ω) = −π 2 (r =≤1) . (35) e|ωnner|rgybechuatvoifforfoorftthheefsppercotpraalgadteonrs(it2y6)o,fi.teh.,eΛfisfenromwiotnhse.
sgn( ω)  2ln|Λ/ω|
−  (ωr−1) (r >1) The RG to be performed below can be understood as
O progressivereductionof this cutoff (althoughwe will use
Interestingly, the resonant level model describes a the field-theoretic scheme where the cutoff is implicitely
screenedimpurityonlyinthemetalliccase. Forthepseu- sent to infinity at an early stage).
dogapcase,r >0,Eqs.(33)and(34)showthattheimpu- Dimensionalanalysisw.r.t. theU =0situation,using
0
rityisonlypartiallyscreened: inamodeloffreefermions dim[f]=(1 r)/2, results in
we have a residual entropy! We will see below that the −
resonant level model fixed point (31) can be identified
dim[U ]=2r 1 ǫ, (37)
0
with the symmetric strong-coupling fixed point (SSC) of − ≡−
Gonzalez-BuxtonandIngersent6 ,introducedforthep-h
hence, for r > 1/2 the interaction term is relevant and
symmetric Kondo and Anderson models.
the SSC fixed point is unstable.
We continuewith the RGanalysisof(36). To perform
a perturbative expansion in U using the field-theoretic
C. Expansion around the resonant level fixed point 0
scheme16,weintroducearenormalizedfieldandadimen-
sionless coupling according to
After having analyzed the behavior of the Anderson
model in the non-interacting case, we proceed to study
f = √Zf , (38)
the stability of the resonant level fixed point w.r.t. a fi- σ Rσ
nite interaction strength U , keeping p-h symmetry. Im- µ−ǫA2Z
0 U = 0 4u (39)
portantly, this fixed point, characterized by a finite hy- 0 Z2
bridizationstrengthbetweenimpurityandbath,isstable
forsmallr[seeEq.(37)];weconcludethatitcanbeiden- where µ is a renormalization energy scale as usual, and
tified with the SSC fixed point of Ref. 6. ( ǫ) is the bare scaling dimension of U0; we have ab-
−
Numerical results5,6 indicate that the quantum phase sorbed the non-universal number A0 appearing in the
transition between LM and SSC disappears as r is in- dynamictermoftheaction(36)inordertoobtainauni-
creasedto 1,wherethep-hsymmetriccriticalfixedpoint versal fixed point value of u.
2
(SCR) merges with the SSC fixed point. We shall show The complete RG analysis, needed to determine the
that an expansion around the SSC fixed point captures flow of u, is presented in Appendix B, here we restrict
the physics of the SCR fixed point for r < 1. Thus, this ourselves to the final results. P-h symmetry prohibits
2
expansion describes the same critical fix∼ed point as the the occurence of even powers of u in the beta function
weak-coupling expansion of Sec. III, but approaching it of u, therefore the lowest contributions arise at two-loop
from r = 1 instead of r = 0. (The r values 0 and 1 order. Remarkably, no singular propagator renormaliza-
2 2
are two lower-critical dimensions for the p-h symmetric tions occur, thus
pseudogap Kondo problem.)
The RG expansion below will be performed around Z =1 (40)
an intermediate-coupling fixed point, in contrastto most
analytical RG calculations which expand around trivial to all orders in perturbation theory. The RG flow equa-
(i.e. weakorstrong-coupling)fixedpoints. Strategically, tion for the renormalizedinteraction u, arising now only
one could think about a double expansion in V0 and U0. from two-loop vertex renormalizations,is found to be
However,this is not feasible, as the marginaldimensions
forbothcouplingsaredifferent,r =1andr = 1,respec-
tively. Therefore we choose to first integrate 2out the c β(u)=ǫu 3(π−2ln4)u3. (41)
− π2
fermionsexactly,andthenusestandardRGtoolsforthe
expansion in U0. Next-to-leadingordercontributionswouldrequireafour-
Consequently, the starting point is the action loop calculationwhich we do not attempt here. For pos-
= f¯σ(ωn)[iA0sgn(ωn)ωn r]fσ(ωn) (36) itive ǫ, i.e., r < 12, Eq. (41) yields a pair of unstable
S | | fixed points at finite u (in addition to the stable one at
Xωn u=0);the correlatio|n|lengthexponentofthe transition,
+ βdτU f¯f 1 f¯f 1 Eq. (43), diverges as r 1−. Thus, the behavior below
Z0 0(cid:18) ↑ ↑− 2(cid:19)(cid:18) ↓ ↓− 2(cid:19) r = 1 is similar to the s→tan2dard behavior above a lower-
2
where the f fermions are now “dressed” by the conduc- criticaldimension(e.g. inthenon-linearsigmamodelfor
tion lines. A is the non-universal number given in Eq. bulk magnet case).
0
10
D. r=0
1
Clearly,the unstable finite-u fixed points predicted by 0.8
theperturbativeRGequation(41)donotnecessarilyex-
ist for the metallic case r = 0, as ǫ = 1 is possibly out- 0.6
ν
sidetheconvergenceradiusoftheexpansion. Indeed,the 1 /
KondoRGofSec.IIIAshowsthat,atr=0,theLMand 0.4
LM’ fixed points are unstable w.r.t. finite impurity cou-
pling. As the resonant-level fixed point at u = 0 is sta- 0.2
ble, weconclude thatthe flowis directlyfromLM(LM’)
to SC, and SC represents the familiar strong-coupling 0
0 0.2 0.4 0.6 0.8 1
Kondo fixed point, with complete screening of the spin. r
The RG flow is in Fig. 1a.
FIG. 3: Inverse correlation length exponent 1/ν obtained
from NRG,atboth thesymmetric(squares) andasymmetric
E. 0<r< 1 (triangles) critical points, together with the analytical RG
2
results from the expansions in r [Sec. III, Eq. (20), solid], in
For r values smaller than 1, both the v = v∗,u = 0 (12 −r) [Sec. IV, Eq. (43), dashed], and in (1−r) [Sec. V,
2 Eq. (72), dash-dot]. The numerical data have been partially
fixed point (SSC) and the v = 0, ε = fixed points
| | ∞ extracted from Ref. 7 using hyperscaling relations; for the
(LM, LM’) are stable, and should be separated by criti-
symmetric model data are from Ref. 5.
cal fixed points. The RG equation (41) yields a pair of
infrared unstable fixed points at
between the two stable fixed points LM and LM’. The
π2
u∗2 = ǫ+ (ǫ2) (42) resulting flow diagram is in Fig. 1d.
3(π 2ln4) O
−
with ǫ = 1 2r. These two fixed points represent SCR
− H. Observables near criticality
andSCR’,seetheflowdiagraminFig.1b. Notethatp-h
symmetry also dictates that the flow trajectories out of
the SSC fixed point are horizontal in the u-v2 diagram. Here we discuss critical properties of the SCR fixed
Therefore,closeto r = 1 theSCRandSCR’fixedpoints point, the properties of SCR’ are identical when trans-
2 lated from spin to chargedegrees of freedom. The corre-
arecompletely describedby the fixedpointcoupling val-
ues v∗ (31) and u∗ (42). lation length exponent follows from expanding the beta
function (41) around its fixed-point value:
1
F. 1 ≤r<1 =2 4r+ (ǫ2) (43)
2 ν − O
For r > 1 (r = 1) the self-interaction u is a A comparisonwith results from NRG is shownin Fig. 3.
2 2
(marginally) relevant perturbation at the SSC fixed Close to r = 1, the analytical expression nicely matches
2
point,and(41)doesnotyieldadditionalnon-trivialfixed the numerical results, however, higher-order corrections
points. AsLMandLM’arestable,wecanconcludethat in the expansion quickly become important.
the flow is fromSSC directly towardsLM (LM’) for pos- We continue with the quantities introduced in Sec. II.
itive (negative) U . Hence, SSC has become a critical As usual for an expansion where the non-linear coupling
0
fixed point, controlling the transition between LM and has an infrared unstable fixed point (as occurs above the
LM’, which occurs at U = 0 for any finite V . We shall lower-critical dimension in standard situations), the UV
0 0
not consider this transition in greater detail, apart from cutoffneedstobekeptexplicitly,andintermediatequan-
statingitscorrelationlengthexponent,1/ν = ǫ. Fig.1c titieswilldivergewiththeUVcutoff(seee.g. thecalcula-
−
displaystheflowdiagramarisingfromthisdiscussion,be- tionoftheimpurityentropy). However,thesedivergences
ing consistent with the numerical results of Ref. 6. will cancel in the final expressions for universal observ-
ables, and this is an important check for the consistency
of our calculations.
G. r≥1
The physics of the symmetric Anderson model for 1. Local susceptibility
r 1 is easily discussed: the hybridization term V is
0
≥
irrelevant for all U , Eq. (29). The free-impurity fixed The local susceptibility at the SSC fixed point, i.e., at
0
point(FImp)istheonlyremainingfixedpointatU =0. tree level, follows the power law χ ω1−2r = ωǫ. To
0 loc
∝
It is unstable w.r.t. finite U , andcontrolsthe transition obtaincorrectionsto the tree-levelresult,one introduces
0