Table Of ContentNatural Orbitals Renormalization Group Approach to the Two-Impurity Kondo
Critical Point
Rong-Qiang He,1,2 Jianhui Dai,3 and Zhong-Yi Lu1
1Department of Physics, Renmin University of China, Beijing 100872, China
2Institute for Advanced Study, Tsinghua University, Beijing 100084, China
3Department of Physics, Hangzhou Normal University, Hangzhou 310036, China
(Dated: January 9, 2015)
The problem of two magnetic impurities in a normal metal exposes the two opposite tendencies
in the formation of a singlet ground state, driven respectively by the single-ion Kondo effect with
5
conduction electrons to screen impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction
1
betweenthetwoimpuritiestodirectlyformimpurityspinsinglet. However,whetherthecompetition
0
between these two tendencies can lead to a quantum critical point has been debated over more
2
than two decades. Here, we study this problem by applying the newly proposed natural orbitals
n renormalization group method to a lattice version of the two-impurity Kondo model with a direct
a exchangeK betweenthetwoimpurityspins. Themethodallows forunbiasedaccessing theground
J state wave functions and low-lying excitations for sufficiently large system sizes. We demonstrate
8 the existence of a quantum critical point, characterized by the power-law divergence of impurity
staggeredsusceptibilitywithcriticalexponentγ =0.60(1),ontheantiferromagneticsideofK when
] theinterimpuritydistanceRisevenlatticespacing,whileacrossoverbehaviorisrecoveredwhenR
l
e isoddlatticespacing. Theseresultshaveultimatelyresolvedthelong-standingdiscrepancybetween
- the numerical renormalization group and quantum Monte Carlo studies, confirming a link of this
r
t two-impurityKondo critical point toa hidden particle-hole symmetry predicted bythelocal Fermi
s liquid theory.
.
t
a
PACSnumbers: 75.10.Hf,71.15.Dx,75.20.Hr,75.40.Cx
m
-
d
I. INTRODUCTION staggered susceptibility and coefficient of specific heat
n
at a finite ratio of the RKKY interaction to the Kondo
o
temperature(≈2.2). Ontheotherhand, the subsequent
c It is well-recognized that the competition between
[ well-controlledquantumMonteCarlo(QMC)studiesdid
the single-ion Kondo effect and the Ruderman-Kittel-
not find such a divergence, instead, a crossoverbehavior
1 Kasuya-Yosida (RKKY) interaction, the two inevitable
inthecorrespondingquantitiesatverylowbutstillfinite
v forces in any Kondo systems with more than one local
temperatureswasobserved[14,15]. Sincethen,although
4 impurity magnetic moment, plays a crucial role in cor-
3 the effective field theory analysis suggested the occur-
related systems ranging from dilute magnetic alloys to
8 rence of critical points in several variants of the TIKM
heavy fermion compounds [1, 2]. The indirect RKKY
1 [16–21], the strong debate in the TIKM studies, espe-
interaction, namely the interimpurity interaction medi-
0 ciallyinnumericalstudies,hasstillremaineduntiltoday
. ated by conduction electrons via a short-range (on-site)
1 [22–25].
Kondo coupling, oscillates and decays with the inter-
0
impurity distance R and Fermi momentum 2k [3–5].
5 F Almost all the previous investigations, except for the
1 When the RKKY interaction grows toward the strong QMCstudies[14,15],relyonadecompositionofthecon-
: antiferromagneticlimit,thequantummany-bodyground ductionelectronsintoodd/evenchannelswithrespectto
v
state will evolve from a collective Kondo singlet state[6]
i the impurity center. Under such a decomposition, the
X into the interimpurity singlet state lockedby the RKKY
originaltwo-impurityproblemismappedeffectivelyonto
r interaction. However, whether a distinct separation or two-channel/two-impurity problem defined on the mo-
a a quantum critical point exists between the two singlet
mentum or energy space, resulting in various impurity
ground states has remained elusive. This theoretical is-
couplings which are energy-dependent in general. The
suehasafundamentalimportanceasitcloselycorrelates
obtainedtwo-channel/two-impuritymodel,capturingthe
withthe criticaldivergenceorscalingbehaviorofseveral
low-energypropertiesofthe originalproblem,providesa
physicalquantities,oremergentenergyscalesinrealistic
base for the NRG and effective field theory studies. In
materials[7–9].
particular, the early NRG calculations [11, 12] assume
It is remarkable that even for the simplest case with “energy-independent” coupling constants in odd/even
only two local impurities, i.e., the two-impurity Kondo channels. The QMC simulations [14, 15], on the other
model (TIKM)[10], the evidences for a quantum critical hand, suffer from finite temperatures. Other NRG[22–
point separating the two distinct singlet states do not 24] or DMRG[25] studies, where no evidence was found
converge. On the one hand, the early numerical renor- foraquantumcriticalpointaftertakingintoaccountthe
malization group (NRG) studies[11–13] revealed an un- “energy-dependent” coupling strengths or starting from
stable fixed point characterized by diverging impurity a real-spacetwo-impuritylattice model, indeedshow the
2
importance of lattice geometry details.
K
J
More seriously, these numerical results seem to t
stronglycontradictageneralphaseshift argumentbased R
onthelocalFermiliquidtheory[26,27],whichstatesthat
aphasetransitionmustexistbetweenthetwostablefixed
FIG. 1: (Color online) Schematic view of a standard two-
points if a TIKMpreservesparticle-hole(PH) symmetry impurity Kondomodel with impurity separation R=3.
[28, 29]. As a matter of fact, there are two types of
PH symmetries associated with a standard TIKM, cor-
responding to the cases with the interimpurity distance argument and other relevant numerical results are given
R being even or odd, or in the lattice case, the impurity in the appendixes.
mirrorcenterbeingonsiteorbond,respectively. Itisthe
first type of PH symmetry, namely the distance R being
evenorthe mirrorcenter being on site, thatcanguaran- II. MODEL AND RESULTS
tee a phase transition [29, 30]. Although the phase shift
argument comes from the odd/even decomposition with
Our studied TIKM is a standard one-dimensional lat-
a number of simplifications including the sphericalplane
tice model[32], as schematically shown in Fig. 1, which
wave and linear dispersion approximations, its validity
is described by the following Hamiltonian,
shouldnotdependonignoringenergydependenceofcou-
pling constants and other lattice details. Otherwise, if H =H +H +H ;
0 Kondo RKKY
there is no any phase transition in this case, it wouldin- H =−t [c† c +h.c.],
0 iσ i+1σ
dicate a breakdown of the local Fermi liquid picture of Piσ (1)
the single-impurity Kondo problem. Here we would like HKondo =J Sj ·s(Rj),
toemphasizethatsuchaphasetransitioncouldbeeither j=P1,2
H =KS ·S ,
afirst-ordertransitionoraquantumcriticalpoint. Inthe RKKY 1 2
case of only the second type of PH symmetry preserved,
where H describes the non-interactingconductionband
0
there is no such a guarantee, in other words, either a
with c being the annihilation operator of a conduction
iσ
phase transition or a crossover takes place between the
electron located at the i-th site with spin component σ
two stable fixed points.
and t being the nearest-neighbor hopping integral, J is
In our point of view, the most straightforward ap-
the short-range (on-site) Kondo coupling between each
proach to challenge or confirm this argument and fur- impurity spin S and the spin s(R ) of a conduction
j j
ther clarify the discrepancies among various numerical
electron passing by the j-th impurity site, and K is
studies is to directly solve the ground state of a stan- the direct exchange interaction between the two impu-
dard TKIM for sufficiently large systems without using rity spins. Such a TIKM can be realized in nanoscale
decomposition. Infact,Afflecketal.[29]alreadyoutlined devices where the observed Kondo signature varies with
three conditions for such a decisive numerical study: (1) tunable RKKY interaction[33–35]. Notice that in real-
the studied model preservesthe first type ofPH symme- istic materials the direct interaction K, which is finite
try; (2) a model parameter is varied to pass the interim- even at vanishing Kondo coupling, can be mediated as a
purity singlet to the Kondo singlet; (3) sufficiently low superexchangeviaotherfilledorbitalsasinsomelayered
temperatures are accessible and other model parameters f-electron compounds[36].
are fine-tuned.
As shown in Fig. 1, the two impurities are located
In this paper, we reexamine whether or not there is a R=|R −R |latticespacingsapart. Themodelexhibits
2 1
quantum critical point in a standard TIKM by applying thelatticeinversionsymmetrywithrespecttothemirror
anewlydevelopednumericalmethod,i.e.,thenaturalor- center of the two impurities. As illustrated by Affleck et
bitalsrenormalizationgroup(NORG)[31]. Differentfrom al.[29], at half filling of the conduction band the model
theconventionalNRG,theNORGkeepsfaithfullyallthe exhibits the first(or second)type ofPHsymmetry when
lattice geometry details and does not need a decomposi- R is even (or odd). Throughout the present study we
tion into odd/even channels and mapping onto the mo- fix t = 1/2 and take half filling of the conduction band,
mentum or energyspace. Inparticular,the groundstate whiletheperiodicboundaryconditionisimposedandthe
wave function as well as the relevant low energy excita- length of the chain is denoted by L. When K = 0, the
tions in electronic bath with several impurities can be modelbecomestheonestudiedbyFyeetal. usingQMC
calculated accurately in a well-controllable manner for method [14, 15].
a very large system. Therefore, the NORG is particu- Motivated by the suggestion of Affleck et al.[29], we
larly pertinent for studying the zero temperature quan- mainly consider the two representative cases, i.e., the
tum phase transition in the two-impurity Kondo prob- interimpurity distance being odd (R = 1,3) or even
lem. (R = 2,4), respectively. We note that the case with
In the following we will present the numerical results even R was not thoroughly considered in the previous
anddiscussthemainobservations. Abriefdescriptionof QMC calculations[14, 15]. However, the required first
the NORG method and an illustration of the phase shift type of PH symmetry is realized only in this situation.
3
We also notice that actually in most of the previous nu-
40
merical studies [11, 12, 14, 15, 25] only indirect RKKY 0.25
L=4n
interaction is considered, corresponding to K =0. 0.00 L=16
Inthepresentwork,theeigenvaluesandwavefunctions 32 SS12-0.25 LL==3624
of the ground state and low-lying states of Hamiltonian
L=128
(1) were first directly solved by the NORG method for -0.50
24 Kc
sufficiently large systems. The relevant physical quan- -0.75
tities were then calculated. Here the proper physical s 0.0 0.1 K 0.2 0.3
quantities to characterize the ground state structure of
16
the systemanddescribe its responseto externalfieldare
respectivelythe interimpurityspin-spincorrelationfunc- L=4n+2
L=14
tion hS1 · S2i (h···i denotes ground state expectation 8 L=30
value) and impurity staggered susceptibility χ that is L=62
s
defined as χs = 0∞dτh[S1z(τ)−S2z(τ)][S1z−S2z]i with 0 L=126 Kc
Sz(τ)=eτHSze−RτH. To monitor the change of physical -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
j j K
property of the system upon varying couplings K and
J, we thus calculated hS1·S2i and χs as functions of K FIG. 3: (Color online) Impurity staggered susceptibility χs
and interimpurity spin-spin correlation hS ·S i (inset) as
andJ. Thecorrespondingquantumphaseswereachieved 1 2
by extrapolating hS ·S i and χ to the thermodynamic functions of K for J =1 and R=2 with different lengths L.
1 2 s Itis similar to thecase of J =1and R=4(not shown). For
limit, namely 1/L → 0, as described in Appendix C for
L=4n+2representedbysolidlines[37],χ showsadivergent
details. peak at K = Kodd(L) which moves to thse right and finally
c
As a benchmark test, we carried out the calculations converges to K ≡K (L=∞)=0.1555 as L→∞[30]. And
c c
for a set of parameters with K = 0, R = 1, J = 0.728 forL=4nrepresentedbydottedlines,χ showsapeak(with
s
and 0.8, and with varying lattice sizes L, respectively. discontinuity) at K = Keven(L) which moves to the left and
c
Figure 2 shows the calculated impurity staggered sus- finallyconvergestothesameKc(determinedintheL=4n+2
ceptibilities along with finite size extrapolations to the case)asL→∞andmeanwhilethepeakheightincreasesand
thermodynamic limit. As we see, the results in the ther- finally diverges. hS1·S2i jumps where χs peaks. The dis-
continuity of hS ·S i indicates a quantum phase transition,
modynamic limit fall within the error bars of those cal- 1 2
which is furthercharacterized as aquantum critical point by
culated by the early QMC [15]. This also confirms the
thedivergence of χ .
early QMC results. We remind that the case of K 6= 0 s
wasnotstudiedbyQMCcalculationssincethiswouldin-
troduce technical difficulties in QMC.By contrast,there
is no extra difficulty by using the NORG approach, as We first study the cases of interimpurity distance
shown below. R = 2 and 4 with Kondo coupling J = 1 respectively,
inwhichthereisthefirsttypeofPHsymmetry. Figure3
showsthecalculatedimpuritystaggeredsusceptibilityχs
and interimpurity spin-spin correlation hS ·S i at zero
1 2
14
temperature as functions of K for various lattice sizes.
Physicallya verylargenegative K,namely a largeferro-
12 magnetic coupling, tends to lock the two impurity spins
J=0.728, L=4n+2
intoatriplet,characterizedbyhS ·S ibeingpositiveand
J=0.728, L=4n 1 2
J=0.728, QMC close to the triplet value 1. This has been shown in the
4
1s0 inset of Fig. 3. The resulting triplet is then screened by
J=0.8, L=4n+2 theconductionelectronsthroughthetwo-stageKondoef-
J=0.8, L=4n fect process[10]. In particular, we see that this tendency
8
J=0.8, QMC extends to K = 0. This means that the indirect RKKY
interaction, which is due to the second or higher order
6 corrections to the expectation value of hS ·S i between
1 2
two impurity spins (when K = 0), is ferromagnetic at
half filling. Thus the ground state for K = 0 is always
0.00 0.01 0.02 0.03 0.04
inthe Kondoregime. Inorderto reachthe borderofthe
1/L
interimpuritysingletstateregime,wetunedthestrength
FIG. 2: (Color online) Finite size extrapolation of impurity
of K in a wider range from ferromagnetic to antiferro-
staggered susceptibility χ at K = 0 and at J = 0.728 and
s
magnetic.
0.8respectivelyforR=1. HereL=22, 24, 30, ..., 510,and
512, respectively. The endpoint at 1/L = 0 of each curve is For a large positive K, namely a large antiferromag-
determinedbyaquadraticpolynomialfitoftheleftmost four netic coupling, the ground state shall escape from the
points of that curve calculated by NORG. Here the NORG Kondo screening phase, entering the interimpurity spin
results are consistent well with those of Fye’s QMC[15].
singlet phase. This is clearly seen from the interimpu-
4
rity spin-spin correlation hS ·S i (the inset of Fig. 3),
1 2
which is negative and approaching the singlet value −3 0.8
4
on the large positive K side. In this case, there is the
development of interimpurity singlet driven by the an-
0.6
tiferromagnetic correlation. In the intermediate regime
/
of K ∼ 0.15, however, we find a rather sharp peak in )0
L
the impurity staggered susceptibility. With increasing L=62
L 0.4
lattice size, the peak goes diverging and the position of 1 ( L=94
s
the peak convergesto K , indicating that the divergence L=126
c
of impurity staggered susceptibility is an intrinsic prop- L=190
0.2
ertyfreeofthe finite sizeeffect. Furthermore,whenK is L=254
aroundthelocationoftheimpuritystaggeredsusceptibil- L=382
ity peak, the corresponding interimpurity spin-spin cor-
0.0
relationhS1·S2i exhibits a concomitantjump acrossthe -2 -1 0 1 2 3 4
onsetofantiferromagneticcorrelationsignedby hS1·S2i (K Kc)(L L0)1/
beginning to be negative. This feature indicates a quan-
tum phase transition taking place at K = K from the FIG.5: (Coloronline)Finite-sizedatacollapseforthecaseof
c
J =1 and R=2 with different lengths L. This gives rise to
Kondosingletregimetotheinterimpuritysingletregime,
χ ∼|K−K |−γ with γ =0.60±0.01. L =−9.2 and ν =1
whichisfurthercharacterizedasaquantumcriticalpoint s c 0
are determined by fitting data with formula |K (L)−K |∼
by the divergence of the impurity staggered susceptibil- (L−L )−1/ν [30]. c c
ity. 0
The location of the critical point K can be precisely
c
determined by fitting the location of the divergent peak
culated the impurity staggeredsusceptibility χ (K) as a
s
inthe largeL limit, asillustratedinAppendix C.Figure
function of K in a range centered at the critical point
4showsthecriticalpointK asafunctionofKondocou-
c K with different system sizes, reported in Fig. 5. Us-
c
pling J, which defines a phase boundary separating the
ing a standard numerical technique[38], we can derive
Kondo singlet phase (K < K ) from the interimpurity
c that χ (K) diverges towards K in power law rather
s c
singlet phase (K > K ). As we see, the phase diagrams
c thanlogarithmicallyinthethermodynamiclimit,namely
for interimpuritydistance R=2andR=4are basically χ (K)∼|K−K |−γ withγ =0.60±0.01. Hencethecrit-
s c
the same, while the function K (J) shows a quadratic
c icaldivergenceisfasterthanthelogarithmiconebutstill
behavior at the small J limit and linear behavior at the slowerthan|K−K |−1,indicative ofacontinuousphase
c
large J limit.
transitionincomputationalstudies[38]. Thisexponentγ
Toquantifythequantumcriticality,weelaboratelycal- has not been reported in literatures to our knowledge.
As a comparison, we likewise study the cases of in-
terimpurity distance R = 1 and 3 with Kondo coupling
J = 1 respectively. There is now the second type of PH
101 12 symmetry, which however does not guarantee any phase
0 8 transition. In Fig. 6 we report the calculated impurity
10
staggered susceptibility χ and interimpurity spin-spin
s
10-1 4 correlation hS1 ·S2i as functions of the direct exchange
0 R=2 interaction K. Similar to the cases of R = 2 and 4, for
10-2 0 4 8 12 16 20 R=4 a large negative (positive) K, the hS1 ·S2i approaches
Kc the triplet (singlet) value, namely 1 (−3) (the inset of
10-3 0.15 Fig.6), corresponding to the Kondo4(inte4rimpurity) sin-
0.10 glet regime. However, there is no any jump or disconti-
10-4 nuityforhS1·S2iuponK varying. Actually,asshownin
0.05 theinsetofFig. 6,thechangeofhS ·S ifromtheKondo
1 2
-5
10 0.00 singlet regime to the interimpurity singletregime is very
0.0 0.2 0.4 0.6 0.8 1.0 smooth across the middle value −1 between the singlet
-6 4
10 and triplet values, indicating co-existence of the Kondo
0.01 0.1 1 10
J andinterimpuritysingletsinthisintermediateregime,in
other words, a crossover taking place. This can be fur-
FIG.4: (Coloronline)Phasediagramofdirectexchangeinter-
action K versus Kondo coupling J: calculated critical point ther confirmed by examining the behavior of impurity
Kc as a function of J. This defines a phase boundary sepa- staggeredsusceptibility χs as a function of K as follows.
rating the Kondo singlet phase (small K) from the interim- Inaddition,thehS ·S i,beingnegativeatK =0,shows
1 2
purity singlet phase (large K) at zero temperature. Kc(J) is that the indirect RKKY interaction is antiferromagnetic
quadratic at the small J limit (lower inset) and linear at the at K =0, contrary to the cases of R=2 and 4.
large J limit (upperinset).
From Fig. 6 we see that the χ (K) curves for L= 4n
s
5
Using the NORG calculations we have demonstrated
8
0.25 thatthereisindeedaquantumcriticalpointforaTIKM
0.00 with the first type of PH symmetry. The criticality is
6 SS12-0.25 crihtayrsatcatgegriezreeddbsyustcheeptpiboiwlietyr-lχaw, cdoivnecrogmenitcaentolfytahcecoimmppua--
s
-0.50 niedbyajump ofthe interimpurityspin-spincorrelation
hS ·S i, and thus belongs to a genuine impurity quan-
-0.75 1 2
-2 -1 0 1
s4 K tum phase transitionwhere the impurity contributionto
thegroundstateenergybecomessingular[39–41]. More-
over,as presented in above sections, we have also shown
L=4n L=4n+2 that to resolve this longstanding issue depends on not
2 L=32 L=30 only the required PH symmetry but also the tunable di-
L=64 L=62
rect RKKY interaction. Meanwhile, the position of crit-
L=128 L=126
L=256 L=254 ical point is closely related to the lattice details. The
0 newly developed NORG method, which keeps lattice de-
-2 -1 0 1 2
K tailsandisfreeofdecompositionandfinitetemperatures,
makes examination on all these aspects possible.
FIG. 6: (Color online) Impurity staggered susceptibility χ
s
and interimpurity spin-spin correlation hS ·S i (inset) as
1 2
functions of K for R = 1. J = 1 is fixed with different
HerewewouldliketoremindthatinthepreviousNRG
lengths L. For L=4n+2, χ looks diverging at some value
s
ofK =K . ButthedivergingtendencyissuppressedandK studiesonTIKM(orthecorrespondingtwo-impurityAn-
c c
movesto−∞asL→∞[30,37]. Therefore,theχ curvesfor derson model) the occurrence of a critical point requires
s
L=4n+2andL=4nbothconvergetothesameonein the vanishing even/odd channel asymmetry[24] or energy-
thermodynamic limit n → ∞. The case of R = 3 is similar independence of coupling constants [22, 23]. However,
(not shown). the even/odd channel asymmetry or energy-dependence
of coupling constants is an inevitable consequence of de-
compositionwhenmappingalatticeTIKMontoacontin-
converge to a smooth and non-divergent one as L → ∞ uousmodelinmomentumorenergyspace. Onthe other
[37]. In contrast, each χ (K) curve for finite L=4n+2 hand,thephaseshiftargument[16,21,29]foraquantum
s
seemingly has a divergent peak. However, further cal- phase transition in a TIKM does not reply on even/odd
culations [30] show that the value of K at which χ (K) channel asymmetry or energy-independence of coupling
s
diverges for L=4n+2 moves to the negative infinity as constants. Therefore, our NORG study on a standard
L→∞. Itturnsoutthateventuallytheχ (K)curvesfor latticeTIKMdirectlyconfirmsthelong-cherishedpredic-
s
bothL=4n+2andL=4nconvergetothesamesmooth tion of the link between the first type of PH symmetry
andnon-divergentoneinthethermodynamiclimit. This and the two-impurity Kondo critical point [29].
indicates the absence of a quantum critical point. This
conclusion is in agreement with the previous QMC cal-
culations for the case of R=1 (or R=3). The present study clearly shows that the previous nu-
merical studies actually do not contradict each other
for their own sake because the discrepancies are mainly
due to the fact that they adopted different parameter
III. DISCUSSION AND OUTLOOK
regions or approximations. In particular, the energy-
dependence of the coupling constants appeared in these
The studied TIKM offers a new ingredient, namely studiesshouldnotbeanobstacleforstudyingthecritical
the interimpurity exchange K, which competes with the point provided that the first type of PH symmetry and
single-ion Kondo effect of individual impurities. At zero antiferromagnetic interimpurity interaction are realized.
temperature, two different regimes of the ground state, Astherequiredsymmetryoccursathalffillinginalattice
i.e.,theKondosingletandthe inter-impuritysinglet,are modelwithreflectionsymmetryaboutasite (orinterim-
respectivelyrealizedwhenK isvariedfromthelargefer- purity distance is even),the two-impurityKondocritical
romagnetictothelargeantiferromagnetic. Crossoverbe- point should be observed experimentally in realistic ma-
tween these two regimes is expected to be a generic fea- terials with tunable interimpurity exchange interaction.
ture since the relevant degrees of freedom due to quan-
tum impurities are finite. Thus it comes as a surprise
whenanunstablenon-Fermiliquidfixedpointoraquan- Finally,theNORGmethoddevelopedherecanbealso
tum critical point separating the two regimes was firstly used to solve the multi-impurity models in two or three
evidenced by the NRG studies[11, 12]. The existence dimensions[32], or used as impurity solvers in the dy-
of such a critical point in the TIKM has been then de- namicalmean-field theory. Therefore we expect that the
bated by various approaches even in the presence of PH NORG method can be used to study other challenging
symmetry[14, 15, 22–25]. problems in correlated electron systems.
6
Acknowledgments tonian, H =H0+HK, takes a general form as follows,
This work is supported by National Natural Sci- H0 = Z dEEnψe†,Eψe,E +ψo†,Eψo,Eo, (B1)
ence Foundation of China (Grant Nos. 91121008 and
1M1O19S0T02o4f)CahnidnaN(aGtiroannatlNPoro.g2r0a1m1CfoBrAB0a0s1i1c2R).esJe.aDr.chwaosf HK = Z dEdE′ng1ψe†,E~σψe,E′ ·(S1+S2)o (B2)
aZlhseojisaunpgpPorrtoevdinicnepaatrHtabnygzthheouQNiaonrjmiaanlgUSncihvoelrasritsyhi(puno-f + Z dEdE′ng2ψo†,E~σψo,E′ ·(S1+S2)o
der Grant No. 2012QDL037). Computational resources
wereprovidedbythePhysicalLaboratoryofHighPerfor- + Z dEdE′ng3ψe†,E~σψo,E′ ·(S1−S2)o
mance Computing in RUC and National Supercomputer
Center in Guangzhou with Tianhe-2 Supercomputer. + Z dEdE′ng4ψo†,E~σψe,E′ ·(S1−S2)o,
where g (i = 1,2,3,4) are energy-dependent coupling
i
constants reflecting the lattice details. The first type of
Appendix A: Method PH transformation of the fields implies
ψ →ψ†
For a quantum impurity model, by generalizing quan- e,E e,−E
tumrenormalizationgroupapproachintonaturalorbitals ψ →ψ† , (B3)
o,E o,−E
spacethroughiterativeorbitalrotations,wecanrealizea
numerical many-body approach, namely the NORG[31], while the second type of PH symmetry implies
with polynomial, no longer exponential, computational
complexity (O(N4 )) in the number of electron bath ψ →ψ†
bath e,E o,−E
sites(N ). Itturnsoutthatdozens,orevenhundreds,
bath ψ →ψ† . (B4)
of bath sites can be dealt with in practice. In addition, o,E e,−E
the NORG works in a Hilbert space constructed from a
Inthe Fermiliquidpicture,thezerotemperaturefixed
setofnaturalorbitals,thusitcanworkonaquantumim-
point can be characterized by the phase shifts (δ ,δ )
purity model with any lattice topological structure. In e o
for both components at the Fermi energy E = 0. This
the present work, we have improved the efficiency of the
amountsto the following relationsbetweenthe incoming
NORG method. The computational complexity is fur-
and outgoing operators
ther reduced from O(N4 ) to O(N3 ), in which the
bath bath
impurity orbitals are no longer involved into orbital ro-
ψout =e2iδeψin
tations,andhencethetransformedHamiltoniansbecome e,E e,E
simpler. Thisenablesus tocalculatethe eigenvaluesand ψout =e2iδoψin . (B5)
o,E o,E
wavefunctionsofgroundstatesandlow-lyingexcitations
for larger system sizes. The largest system size we have Hence δe = δo in the first type of PH symmetry, and
reached is L = 1022 (see Appendix C). Errors of the δe+δo = 0 in the second type of PH symmetry. In the
NORG are controllable. In this paper the relative errors limitofK →−∞,bothchannelsareintheKondoscreen-
of all data calculated by the NORG are less than 10−3. ing phase, indicating δe,o =±π/2. While, in the limit of
K → ∞, the Kondo effect is completely suppressed by
interimpurity correlation,leading to δ =δ =0. There-
e o
fore, by varying −∞<K <∞, a phase transition must
take place at certain value of K when the first type of
c
Appendix B: Phase shift argument and particle-hole
PHis maintained, as sketchedin Fig. 7(b). By contrast,
symmetry
no phase transition is guaranteed for the second type of
PH symmetry (Fig. 7(a))
We reiterate the phase shift argument for a phase Here we would like to remind that for a lattice model
transition between the collective Kondo screening sin- with inversion symmetry and specified Kondo coupling
glet state and the RKKY interaction-locked interimpu- at half filling, the first or second type of PH symmetry
rity singlet state in the presence of the first type of will be preserved when the interimpurity separation R
particle-hole (PH) symmetry[17, 29]. In the field the- is even or odd, respectively. In general, a long-range hy-
orytreatmentofthetwo-impurityKondomodel(TIKM), bridizationorKondocouplingcanbeintroducedinorder
a decomposition of electron field into even/odd compo- tomaintainthe requiredPHsymmetry. The observation
nents ( ψ ,ψ ) is introduced(the spindegree offree- described in the main text indicates that a TIKM with
e,E o,E
dom is implied), both components are dependent on the the simplest on-site Kondo coupling which preserves the
quasi-particleenergyE measuredfromthe Fermi energy requiredPHsymmetryisenoughtosupportanemergent
E whichissettobezero. TheobtainedeffectiveHamil- quantum critical point.
F
7
(a) δ odd R (δ =−δ)
δ e o 5
π2 e
0
δ R=2
o 4 R=4
−π2
−∞ +∞ K K
T
/c
K
(b) δ δ(δ) even R (δ =δ) 3
e o e o
π2
0
2
−∞ K +∞ K
c
Kondo singlet interimpurity singlet
0.0 0.2 0.4 0.6 0.8 1.0 1.2
FIG.7: (Coloronline)Schematicviewofthephaseshiftδ 1/J
e(o)
intheeven(odd)channeloftheconductionbandattheFermi FIG. 9: (Color online) K /T versus 1/J. In the NRG
c K
energy. studies[12,13],K /T waspredictedtobeaconstant,about
c K
2.2.
-0.1
of the NORG method.
K /T when R=2 and 4. To comparewith the early
c K
NRG result, we calculate the ratio K /T , where T is
c K K
-0.2 the single-ion Kondo temperature defined as[15]
2
S L=4n+2 T =D(ρJ)1/2exp(−1/ρJ), (C1)
1 K
S L=4n
-0.3 QMC whereD(=2here)istheconductionelectronbandwidth
and ρ (= 1/π here) the conduction electron density of
statesattheFermilevel. K /T isthenplottedinFig. 9.
c K
-0.4 While the generic behavior is similar for R=2 and R=
4, quantitative difference between them appears when
J <5. ItisseenthatinawiderangeofJ,K /T isclose
c K
0.00 0.01 0.02 0.03 0.04
to the value predicted by the NRG studies, K /T ≈
1/L c K
2.2.[12, 13]
FIG. 8: (Color online) hS1·S2i for R = 1, J = 0.728, and Determination of K when R is even. For even R the
c
K =0atzerotemperature. L=22,24,30,...,1022. Theend-
TIKMhasaquantumcriticalpointatK =K wherethe
c
point at 1/L=0 of each curve is determined by a quadratic
impurity staggered susceptibility χ diverges. For finite
polynomialfitoftheleftmost fourpointsofthatcurvecalcu- s
system size L = 4n+2, χ diverges at K = K (L). We
latedbytheNORG.OurNORGresultisconsistentwellwith s c
extracttheseK (L)fromχ curvescalculatedbyNORG
that of Fye’s QMC[15]. c s
for different values of L and extrapolate it to L=∞ by
fitting. To presentthe fitting details we showthe typical
case of J = 1 in Fig. 11. The perfect fitting shows that
Appendix C: Complementary numerical results |K (L)−K |∼(L−L )−1/ν with ν =1. For a series of
c c 0
valuesofJ,weshowtheresultinFig. 12. WhileJ spans
Impurity spin-spin correlation. Another benchmark acrossthree ordersofmagnitude, K acrosssevenorders
c
testforourNORGmethodistheimpurityspin-spincor- of magnitude. This shows the robustness of the NORG
relation, hS ·S i, in the ground state for R = 1, where method.
1 2
the QMC result is available. We choose J = 0.728 and Suppression of susceptibility peak when R is odd. The
K = 0, the same parameters as given in Fye’s QMC impurity staggered susceptibility χ shows a sharp peak
s
study[15]. The calculated result is shown in Fig. 8. inthecaseofR=3(orR=1)whenL=4n+2. Thisfea-
The system size varies as L = 22,24,30,...,1022. The tureseemsdifferenttowhathappenswhenL=4nwhere
endpoint at 1/L = 0 of each curve is determined by a nosharppeakisdetected. Inordertoclarifywhetherthis
quadratic polynomial fit of the leftmost four points of is an intrinsic property of the ground state in the ther-
that curve calculated by the NORG. Our NORG result modynamic limit L → ∞, we plot the peak position K
c
is consistent well with that of Fye’s QMC[15]. The rela- byvaryingthelatticesizeLfordifferentvaluesofJ (Fig.
tive differencebetweenthe twoendpointvaluesis justas 10). The result suggests that in the limit L → ∞, the
small as 1.4×10−4, which highlights the high accuracy peak position K of χ for L=4n+2 movesto negative
c s
8
0
J=1.0 0.15
-2
J=0.9 L)
(c0.14
K
-4
0.13
J=0.8
c
K 0.000 0.005 0.010 0.015
1/ 1/L
-6
FIG.11: (Coloronline)ThevalueofK =K (L)atwhichχ
c s
diverges for R = 2, J = 1, and L = 4n+2. The dots are
calculated by NORG. The line is determined by fitting the
dots with formula K (L) = K −a(L−L )−1, where K , a,
c c 0 c
-8 andL0 arefittingparameters. Thisperfectfittingshowsthat
|K (L)−K |∼(L−L )−1/ν with ν =1.
c c 0
-10 J=0.7
0.00 0.02 0.04 0.06
1/L
FIG. 10: (Color online) The value of K = K at which χ
c s
diverges for R = 3 and L = 4n+2. Dots denote the results
calculated bytheNORG.Linesaredeterminedbyfittingthe
dotswiththeformulaKc =−a(L−c)b,wherea,b,andcare infinity and χs’s for both L = 4n+2 and L = 4n con-
fittingparameters. TheK behaviorforR=1issimilar(not verge to the same one with a non-divergent behavior in
c
shown). This figure clearly shows that Kc decreases to −∞ thethermodynamiclimit. Therefore,nophasetransition
as L→∞. takes place in the case of R=3 (or R=1).
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1
12 0.01 0.8 10
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0.1 1.6 100
10 0.2 2
0.4 4
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