Phase diagram of the one-dimensional, two-channel Kondo lattice model

0.16 MB·English

by  T. Schauerte

Phase diagram of the one-dimensional, two-channel Kondo lattice model
T. Schauerte,1 D.L. Cox,1 R.M. Noack,2 P.G.J. van Dongen,3 and C. D. Batista4
1Department of Physics, University of California, Davis, CA 95616, USA
2Fachbereich Physik, Philipps-Universit¨at Marburg, D-35032 Marburg, Germany
3Institut fu¨r Physik, Universit¨at Mainz, 55099 Mainz, Germany
4Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
(Dated: February 2, 2008)
5
0
Employing the density matrix renormalization group method and strong-coupling perturbation
0
theory,westudythephasediagramoftheSU(2)×SU(2)Kondolatticemodelinonedimension. We
2
showthat,atquarterfilling,thesystemcanexistintwophasesdependingonthecouplingstrength.

n The weak-coupling phase is dominated by RKKY exchange correlations while the strong-coupling
a phase is characterized by strong antiferromagnetic correlations of the channel degree of freedom.
J These two phases are separated by a quantum critical point. For conduction-band fillings of less

8 thanonequarter,wefindaparamagneticmetallicphaseatweakcouplingandaferromagneticphase
1 at moderate to strong coupling.

] PACSnumbers: 71.27.+a,75.20.Hr,75.30.Mb,75.40.Mg
l
e
-
r Although Landau’s theory of Fermi liquids is one of (S = 1) is defined by S = 1 f† σ f , where
at.st tmhaencyormnaetresrtioanlsesshoofwmomdeetranlliccopnrdoepnesretdiems tahttaetrdpohnysoitcsfi,1t tahneidfσ-oips2earavteocrtsorsaotfisPfyautlhiesipcionnsmt2raPatirniαctβefs.i†i↑αTfih↑αe+βcofiβni†↓dfui↓ct=ion1
m into this framework. For instance, it is well known that bandfillingisdefinedasn =n +n =(N +N )/L
c c+ c− c+ c−
one-dimensionalsystemsusuallybehaveasLuttingerliq-
- with Nc± the number of conduction electrons in channel
d uidsratherthanFermiliquids.2Inhigherdimensions,the m = , respectively; n = 1 corresponds to the quar-
c
n proximity of a quantum critical point (QCP) is consid- ±
ter filled system. The KLM may be derived from the
o eredtoberesponsibleforthenon-Fermiliquidproperties
more fundamental periodic Anderson model in the limit
c of systems like CeCu Au .3 The quadrupolar Kondo
[ 6−x x of strong Coulomb repulsion where the Kondo coupling
effect has been proposedas analternative sourceof non- is usually antiferromagnetic (AF), J > 0,9 or alterna-
1 Fermiliquidbehavior. Thiseffectisdescribedbythetwo- tively in the “extendedKondo limit”.10 In the following,
v channel Kondo model4,5. This model has a non-Fermi
we measure all energies in units of t.
0 liquid ground-state because of frustration in the screen-
3 Not much is known about the ground-state phase di-
4 ingofalocalizedimpurityby twodegenerateconduction agram of the two-channel KLM. Tsvelik and Ventura11
electron channels if the degeneracy of the channels M
1 investigated this model using a mean-field analysis and
0 is greater than twice the impurity spin S.6 The single- foundthatathalffilling the systemexists intwophases.
5 impurity two-channel Kondo model can provide an ade-
One is dominated by RKKY exchange interaction ef-
0 quatedescriptionofdilutesystemslikeTh U Ru Si 7
1−x x 2 2 fects, and the other by Kondo screening. A QCP sep-
/ or Y U Pd ,8 but it does not incorporate the lattice
t 1−x x 3 arates these two regimes. A generalized one-dimensional
a effects that become relevant in fully concentrated com-
m two-channelKLMwithanadditionalHeisenberginterac-
pounds like UBe 5. These materials are described by
13 tion,J ,betweenthef-spinswasstudiedbyAndreiand
H
d- the two-channel Kondo lattice model (KLM). Orignac.12 In the limit of strong JH, they find that the
Inthispaper,weaddressthequestionofwhathappens
n systemisinasuperconductingphasewithodd-frequency
when two ofthese fundamental sourcesof non-Fermiliq-
o singletpairingofthe electrons. Ininfinite spatialdimen-
c uid behavior coincide. For this purpose we study the sions,thegroundstateischaracterizedbysuperconduct-
v: SU(2)×SU(2) Kondo lattice model in one spatialdimen- ingormagneticphases,whichmaycoexistorcompete.13
sion. The Hamiltonian reads
i Thefocusonthetwo-channelsystembeloworatquar-
X
ter filling is also motivated by the relation to the single-
ar H = −tX(cid:16)c†imσci+1,mσ+H.c.(cid:17) channel system at half filling or less. In particular, the
imσ
quarter-filledcaseforthetwo-channelmodelisanalogous
1
+ J S c† σ c , tothehalf-filledcaseforthesingle-channelmodelinthat
2 X i·(cid:16) imα αβ imβ(cid:17)
thereisoneconductionelectronperimpurityspin,which
imαβ
leadstocompletescreeningatstrongcoupling. Inthefol-
where t > 0 is the conduction electron hopping ampli- lowing,wewillshowthatthesituationinthetwo-channel
tude,takentobethesameinbothbands,andc† (c ) model below quarter filling is qualitatively similar to the
imσ imσ
creates(annihilates)anelectrononlattice site i, 1 i single-channel case below half filling, while exactly at
≤ ≤
L (L being the number oflattice sites),with channelfla- quarter filling it is quite different. The phase diagram
vor m = + or and spin projection σ = or . The of the single-channel model is well understood, at least
Heisenberg spin−operator for the localized↑f-el↓ectrons qualitatively.14,15 In the low carrier limit, this system
2
displays ferromagnetic order with complete polarization, duction electron. For J <J , there is a region of incom-
c
S = (L N )/2, where n 1.16,17 In the strong- pleteferromagnetism,similartoonethathasbeenfound
tot c c
− ≪
coupling limit, the ground state is ferromagnetic for all in the periodic Anderson model.21 While we cannot rule
n .18 Exactly at half filling, the single-channel model is out that this region is due to a continuous transition to
c
known to be a Kondo insulator.15,19 This is a quantum the complete ferromagnetic phase, the local spin profiles
disordered phase in which the conduction electrons are in the incomplete ferromagnetic regime show small fer-
bound into local singlet states with the impurity spins, romagnetic domains (corresponding to polarizations be-
andboththespinandchargecorrelationfunctionsdecay tween 25% and 90% of the complete value), suggesting
exponentially in space and time. that phase separation may occur here. For n 1, we
c
≥
In order to calculate the ground-state properties of find a singlet ground state for all couplings J.
theone-dimensionaltwo-channelKLM,weusethefinite- Exactlyatquarterfilling,wefindtwophasesasafunc-
system algorithm20 of the DMRG to calculate gaps, tion of J. At weak coupling (J < 2.0), the electrons
equal-time correlation functions, and the total spin of of different flavors generate indepen∼dent RKKY interac-
the ground state. We keep up to 1000 states per block tions betweenthe localizedmoments. We observestrong
on lattices of up to L =50 sites and obtain a maximum correlationsofthef-spinsatawavevectorq =π/2,asin
discarded weight of 10−5. Fig. 1 shows the total spin the single-channelmodelatquarter filling.22 Atstronger
S per site extrapolated to the thermodynamic limit coupling (J >2.0), the systemis in a channelAF phase,
tot
for various couplings and conduction electron densities. where the co∼rrelations of the channel degree of freedom
Here S is calculated directly by taking the expecta- decay as 1/r. The two phases are separated by a QCP.
tot
tion value of S2 in the ground state and also by ex-
tot
amining the degeneracy of excited states in various S Distance r
z 0 10 20 30 40
sectors. A finite-size extrapolation is then used to de-
(a)
termine whether S =(L N )/2 (complete ferromag- 0.1
tot c
netism), S < (L N )/2−but finite (incomplete ferro- 0.05 M=2, J=0.5
magnetismt)o,torStot−=0c(paramagnetism)inthethermo- S(r)f 0 MM==22,, JJ==14..00
dynamic limit. At quarter filling (n =1), the nature of -0.05
c
the ordering is also indicated for S =0 phases. -0.1
tot
0
(b)
1 -1
complete ferromagnet
0.9 ipnacroammapglenteet ferromagnet (r)|f -2 MM==22,, JJ==01..50
aannttiiffeerrrroommaaggnneett ((csphiann)nel) g|S -3 M=1, J=0.5
0.8 l -4
+t) 0.7 -50 5 10 15 20 25 30
J/(J 0.6 Distance r
0.5 FIG. 2: Magnetic correlation function Sf(r)=hSz(0)Sz(r)i
of thef-spins for the two-channel KLM at nc =1 in (a) and
0.4 forthesingle-andtwo-channelmodels,bothatquarter-filling,
in (b).
0.3
0 0.5 1 1.5 2
n Fig.2(a)showsthatinthetwo-channelmodelthespin-
c
spincorrelationfunctiondecaysslowlyforweakcoupling
FIG.1: Ground-statephasediagramofthetwo-channelKLM (J = 0.5 or 1.0) and is short-ranged for strong coupling
as afunction of conduction bandfillingnc. TheAFchannel- (J = 4.0), i.e., numerically zero for more than two lat-
andspin-orderedphasesatquarterfillingareassociated with tice spacings. The transition between the two behaviors
asingletgroundstate. Thecrossesindicatepolarizationsthat occurs at J 2.0 (not shown in Fig. 2), where the cor-
extrapolate in the thermodynamic limit to values between relations ext≈end over roughly two lattice sites. Fig. 2(b)
25% and 90% of complete polarization.
shows a comparison between the spin-spin correlations
functions S (r) of the single-channel and two-channel
f
Fig.1showsalargeregionofcompleteferromagnetism KLMonalogarithmicplot. ThegroundstateforM =1
for n < 1 above a certain critical value J which de- in the weak-coupling limit shows RKKY liquid behavior
c c
creases with decreasing n and tends to zero as n 0. with spatial oscillation characterized by a wave vector
c c
→
In the completely polarized phase, eachconduction elec- q = π/2. In the M = 2 case the correlations at J = 0.5
tronformsanitinerantsingletwiththef-spins. Delocal- decay so slowly that the asymptotic behavior cannot be
ization of these singlets leads to ferromagnetic ordering determined for the system sizes considered, as can be
of the remaining unscreened f-spins. The complete po- seen in Fig. 2(b). At larger J-values [e.g., J = 1.0 in
larization for all J at low conduction electron density is Fig. 2(b)], the correlation functions appear to decay ex-
in agreement with an exact argument16 for a single con- ponentially, consistent with the opening of a gap in the
3
spin excitation spectrum due to the transition into the which represent local Kondo singlets. In one spatial di-
channel-orderedphase. Fig. 3 shows the magnetic struc- mension, model (1) has been solved exactly using the
ture factor S (q) for the two-channel system. The am- Betheansatz.25Thegroundstateischaracterizedbycrit-
f
plitude of the peak at q =π/2 (the wavevectorexpected icalAFcorrelationsthatdecayas1/r,inagreementwith
for a RKKY liquid) becomes smaller with increasing J, our numerical findings at quarter filling [see Fig. 4(a)].
indicating that the magnetic correlationsbetween the f- This critical behavior is replaced by long-range AF or-
spins become weaker and finally vanish at J 2.0.23 dering for D > 1. Due to the SU(2) pseudo-spin in-
≈
It is known that the RKKY correlations of the variance of H, the AF ordering is present for all three
single-channel KLM at quarter filling become unstable τ-components. In particular, the order along the z axis
with increasing coupling toward a ferromagnetic ground corresponds to staggered orbital ordering while the x,y-
state.15,24 However, the two-channel model cannot be orderinggivesrisetoaBose-Einsteincondensationofex-
ferromagnetic at quarter filling because each f-spin is citons(particle-holesingletboundstatesbetweenthetwo
screenedby anelectron, so that some other kindof sym- channels).
metry breaking must lift the degeneracy of the ground For n < 1, the low-energy subspace in the strong-
c
state. Fig. 4 shows the staggered magnetization D(r) of coupling limit contains states with zero and one conduc-
the channel degree of freedom, with tion electron per site. While the local state with one
conduction electron can be described by the hard-core
D(i−j)= X mm′ hnimσ njm′σ′i. bosons of Eq. (2), the empty state acquires spin–1/2
σσ′mm′ character from the f spins. It can therefore be rep-
dwehcearyesnwimitσh=1/cr†imasσcfiumnσc.tioInnFofigt.h4e(ad)isotannecseeers=thia−t Dj (for)r rHweiastemhnitnletbiodn=iaasPnahmacbso†imnthsbteirmafo.inrAmedt sfterromnigoncoϕu†ipσlin=g, (t1he−effnebic)tfii†vσe,
J =4.0. Fig.4(b)showsthattheAFchannelcorrelations
become stronger with increasing coupling and the quasi-
t
long-range behavior develops for J >2.0. H0 =−2 X(cid:16)b†imϕiσϕ†i+1,σbi+1,m+H.c.(cid:17) .
∼ iσ,m
J=0.5 Inanalogytotheinfinite-U Hubbardmodel,the ground-
0.9 J=1.0 state wavefunction can be written as a direct product of
J=1.5
a charge, spin, and orbital component
J=2.5
0.6 J=4.0
q) |ψn{τ;σ}i = |nci⊗|τ1··· τL−Ni⊗|σ1··· σNi
(
Sf = Γ(n) ϕ† ϕ† b† b† 0
0.3 X i1i2···iN i1σ1··· iNσN j1τ1··· iL−NτN| i
i1<i2<...<iN
where τ;σ = (τ ,...,τ ;σ ,...,σ ) and 0 denotes
0 1 L−N 1 N
{ } | i
the vacuum of ϕ and b particles. The complete spin de-
generacy of H in the strong coupling limit is lifted in
0 0.2 0.4 0.6 0.8 1 0
q/π O(t2/J). The effective Hamiltonian in this order con-
tains only one term that lifts this degeneracy:
FIG. 3: Magnetic structure factor of the two-channel KLM
t2
modelasafunctionofwavevectorqforvariousJ atnc =1.0. H1 = 2J X(cid:16)ϕ†i+1,σbi+1,mnϕib†i−1,mϕi−1,σ +H.c.(cid:17) ,
iσ,m
Inordertounderstandournumericalresultsforstrong
coupling, we use the methods of Ref. 18 to derive an where nϕ = ϕ† ϕ . Here H exchanges two spins
effective Hamiltonian valid for t/J 1. At n = 1, i Pσ iσ iσ 1
c by hopping of a fermion over another to an empty next-
≪
the channel flavor is the only degree of freedom and the
nearest-neighbor site. In the same way as for the single-
low-energy spectrum can be described by a pseudospin–
channel KLM, it can be shown that the off-diagonal ele-
1/2 model. Accordingly, the effective Hamiltonian is a
mentsofH areallnon-positiveandtheHamiltonianma-
1
Heisenberg model for the channel spin,
trix in real-space is completely connected. The Perron-
16t2 Frobeniustheorem(see,forinstance,Ref.26)thenstates
H = 3J Xτi·τi+1 , (1) that the ground state is unique and that the coefficients
i ofthewave-functioncanbechosentobestrictlypositive.
The only spin state that has strictly positive coefficients
where τ = 21Pmm′b†imσmm′bim′ is defined in terms of in each of the subspaces of H1 is the one with maximum
hard-core bosons
totalspin. Therefore,thegroundstateofthetwo-channel
1 KLMbelowquarter-fillingisferromagneticinthestrong-
b† = (c† f† c† f†), (2)
im √2 im↑ i↓− im↓ i↑ couplinglimit,inaccordancewithournumericalfindings.
4
Distance r
qualitativelysimilar forlow bandfillings (n <1 in both
c
0 10 20 30 40
models) but quite different at n = 1. For n < 1, the
c c
(a) additionofaseconddegeneratebandofconductionelec-
0.1/r
trons does not alter the physical picture of the single-
)
(r 0 channel model if n is low or J is large. The similarity
D c
between the two phase diagrams for n < 1 further sug-
c
J=4.0
gests that the metallic ground state of the two-channel
J=3.0
J=2.0 KLMatweakcouplingisdeterminedbytwoindependent
0.025 J=1.0 Kondo effects in both channels. In contrast, at quarter-
(r) 0 filling(nc =1)thetwo-channelKLMexhibitsaquantum
D phase transition between this metallic phase and an in-
-0.025 sulator characterized by strong AF correlations of the
(b)
channel degrees of freedom. Our perturbative analysis
-0.05
0 5 10 15 shows that this insulating phase is present for any spa-
Distance r
tial dimension D andhas long-rangeorderingfor D >1.
Therefore, we also expect a QCP separating the metal-
FIG.4: StaggeredmagnetizationD(r)forthechanneldegree
lic and the insulating phase for D > 1. The AF chan-
of freedom for various J at quarter filling (nc = 1.0). The
nel (or excitonic) fluctuations diverge at the QCP and
longer distance behavior for J =4.0 (where there is a slower
falloff, approximately ∝ 1/r) is shown in (a) and a shorter can produce a deviation from the normal Fermi-liquid
range in r is shown for all four J-valuesin (b). behavior of the metallic state. In this way we see that
the deviations from the Fermi-liquid behavior which are
obtained in the dilute (impurity model) and the concen-
In this paper, we have mapped out the zero-tempera- trated(latticemodel)limitshaveacommonorigininthe
ture phase diagram of the two-channel KLM as a func- fluctuations of the channel degree of freedom.
tion of conduction band filling n and Kondo coupling
c
strength J. Our main results are that the phase dia- T.S., D.L.C., and C.D.B. acknowledge support from
grams of the two-channel and single-channel KLMs are U.S. Department of Energy.
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