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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

1 Department of Physics, University of California, Davis, CA 95616, USA Fachbereich Physik, Philipps-Universit¨t Marburg, D-35032 Marburg, Germany a 3 Institut f¨r Physik, Universit¨t Mainz, 55099 Mainz, Germany u a 4 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 (Dated: February 2, 2008) 2

arXiv:cond-mat/0501430v1 [cond-mat.str-el] 18 Jan 2005

Employing the density matrix renormalization group method and strong-coupling perturbation theory, we study the phase diagram of the SU(2)×SU(2) Kondo lattice model in one dimension. We show that, at quarter ?lling, the system can exist in two phases depending on the coupling strength. The weak-coupling phase is dominated by RKKY exchange correlations while the strong-coupling phase is characterized by strong antiferromagnetic correlations of the channel degree of freedom. These two phases are separated by a quantum critical point. For conduction-band ?llings of less than one quarter, we ?nd a paramagnetic metallic phase at weak coupling and a ferromagnetic phase at moderate to strong coupling.

PACS numbers: 71.27.+a, 75.20.Hr, 75.30.Mb, 75.40.Mg

? fiα σ αβ fiβ , where

Although Landau’s theory of Fermi liquids is one of the cornerstones of modern condensed matter physics,1 many materials show metallic properties that do not ?t into this framework. For instance, it is well known that one-dimensional systems usually behave as Luttinger liquids rather than Fermi liquids.2 In higher dimensions, the proximity of a quantum critical point (QCP) is considered to be responsible for the non-Fermi liquid properties of systems like CeCu6?x Aux .3 The quadrupolar Kondo e?ect has been proposed as an alternative source of nonFermi liquid behavior. This e?ect is described by the twochannel Kondo model4,5 . This model has a non-Fermi liquid ground-state because of frustration in the screening of a localized impurity by two degenerate conduction electron channels if the degeneracy of the channels M is greater than twice the impurity spin S.6 The singleimpurity two-channel Kondo model can provide an adequate description of dilute systems like Th1?x Ux Ru2 Si2 7 or Y1?x Ux Pd3 ,8 but it does not incorporate the lattice e?ects that become relevant in fully concentrated compounds like UBe13 5 . These materials are described by the two-channel Kondo lattice model (KLM). In this paper, we address the question of what happens when two of these fundamental sources of non-Fermi liquid behavior coincide. For this purpose we study the SU(2)×SU(2) Kondo lattice model in one spatial dimension. The Hamiltonian reads H = ?t c? ci+1,mσ + H.c. imσ

imσ

(Si =

1 2)

is de?ned by Si =

1 2

αβ

1 + J 2

imαβ

Si · c? σ αβ cimβ imα

,

where t > 0 is the conduction electron hopping amplitude, taken to be the same in both bands, and c? (cimσ ) imσ creates (annihilates) an electron on lattice site i, 1 ≤ i ≤ L (L being the number of lattice sites), with channel ?avor m = + or ? and spin projection σ =↑ or ↓. The Heisenberg spin operator for the localized f -electrons

? ? the f -operators satisfy the constraint fi↑ fi↑ + fi↓ fi↓ = 1 and σ is a vector of Pauli spin matrices. The conduction band ?lling is de?ned as nc = nc+ +nc? = (Nc+ +Nc? )/L with Nc± the number of conduction electrons in channel m = ±, respectively; nc = 1 corresponds to the quarter ?lled system. The KLM may be derived from the more fundamental periodic Anderson model in the limit of strong Coulomb repulsion where the Kondo coupling is usually antiferromagnetic (AF), J > 0,9 or alternatively in the “extended Kondo limit”.10 In the following, we measure all energies in units of t. Not much is known about the ground-state phase diagram of the two-channel KLM. Tsvelik and Ventura11 investigated this model using a mean-?eld analysis and found that at half ?lling the system exists in two phases. One is dominated by RKKY exchange interaction effects, and the other by Kondo screening. A QCP separates these two regimes. A generalized one-dimensional two-channel KLM with an additional Heisenberg interaction, JH , between the f -spins was studied by Andrei and Orignac.12 In the limit of strong JH , they ?nd that the system is in a superconducting phase with odd-frequency singlet pairing of the electrons. In in?nite spatial dimensions, the ground state is characterized by superconducting or magnetic phases, which may coexist or compete.13 The focus on the two-channel system below or at quarter ?lling is also motivated by the relation to the singlechannel system at half ?lling or less. In particular, the quarter-?lled case for the two-channel model is analogous to the half-?lled case for the single-channel model in that there is one conduction electron per impurity spin, which leads to complete screening at strong coupling. In the following, we will show that the situation in the two-channel model below quarter ?lling is qualitatively similar to the single-channel case below half ?lling, while exactly at quarter ?lling it is quite di?erent. The phase diagram of the single-channel model is well understood, at least qualitatively.14,15 In the low carrier limit, this system

2 displays ferromagnetic order with complete polarization, Stot = (L ? Nc )/2, where nc ? 1.16,17 In the strongcoupling limit, the ground state is ferromagnetic for all nc .18 Exactly at half ?lling, the single-channel model is known to be a Kondo insulator.15,19 This is a quantum disordered phase in which the conduction electrons are bound into local singlet states with the impurity spins, and both the spin and charge correlation functions decay exponentially in space and time. In order to calculate the ground-state properties of the one-dimensional two-channel KLM, we use the ?nitesystem algorithm20 of the DMRG to calculate gaps, equal-time correlation functions, and the total spin of the ground state. We keep up to 1000 states per block on lattices of up to L = 50 sites and obtain a maximum discarded weight of 10?5 . Fig. 1 shows the total spin Stot per site extrapolated to the thermodynamic limit for various couplings and conduction electron densities. Here Stot is calculated directly by taking the expectation value of S2 in the ground state and also by extot amining the degeneracy of excited states in various Sz sectors. A ?nite-size extrapolation is then used to determine whether Stot = (L ? Nc )/2 (complete ferromagnetism), Stot < (L ? Nc )/2 but ?nite (incomplete ferromagnetism), or Stot = 0 (paramagnetism) in the thermodynamic limit. At quarter ?lling (nc = 1), the nature of the ordering is also indicated for Stot = 0 phases.

1 0.9 0.8 J/(J+t) 0.7 0.6 0.5 0.4 0.3 0 0.5 1 nc 1.5 2

complete ferromagnet incomplete ferromagnet paramagnet antiferromagnet (channel) antiferromagnet (spin)

duction electron. For J < Jc , there is a region of incomplete ferromagnetism, similar to one that has been found in the periodic Anderson model.21 While we cannot rule out that this region is due to a continuous transition to the complete ferromagnetic phase, the local spin pro?les in the incomplete ferromagnetic regime show small ferromagnetic domains (corresponding to polarizations between 25% and 90% of the complete value), suggesting that phase separation may occur here. For nc ≥ 1, we ?nd a singlet ground state for all couplings J. Exactly at quarter ?lling, we ?nd two phases as a function of J. At weak coupling (J < 2.0), the electrons ? of di?erent ?avors generate independent RKKY interactions between the localized moments. We observe strong correlations of the f -spins at a wave vector q = π/2, as in the single-channel model at quarter ?lling.22 At stronger coupling (J > 2.0), the system is in a channel AF phase, ? where the correlations of the channel degree of freedom decay as 1/r. The two phases are separated by a QCP.

0 0.1 0.05 0 -0.05 -0.1 0 -1 lg|S f(r)| -2 -3 -4 -5 0 5 10 15 20 Distance r 25 30 10 Distance r 20 30 (a)

M=2, J=0.5 M=2, J=1.0 M=2, J=4.0

40

Sf(r)

(b)

M=2, J=0.5 M=2, J=1.0 M=1, J=0.5

FIG. 2: Magnetic correlation function Sf (r) = Sz (0)Sz (r) of the f -spins for the two-channel KLM at nc = 1 in (a) and for the single- and two-channel models, both at quarter-?lling, in (b).

FIG. 1: Ground-state phase diagram of the two-channel KLM as a function of conduction band ?lling nc . The AF channeland spin-ordered phases at quarter ?lling are associated with a singlet ground state. The crosses indicate polarizations that extrapolate in the thermodynamic limit to values between 25% and 90% of complete polarization.

Fig. 1 shows a large region of complete ferromagnetism for nc < 1 above a certain critical value Jc which decreases with decreasing nc and tends to zero as nc → 0. In the completely polarized phase, each conduction electron forms an itinerant singlet with the f -spins. Delocalization of these singlets leads to ferromagnetic ordering of the remaining unscreened f -spins. The complete polarization for all J at low conduction electron density is in agreement with an exact argument16 for a single con-

Fig. 2(a) shows that in the two-channel model the spinspin correlation function decays slowly for weak coupling (J = 0.5 or 1.0) and is short-ranged for strong coupling (J = 4.0), i.e., numerically zero for more than two lattice spacings. The transition between the two behaviors occurs at J ≈ 2.0 (not shown in Fig. 2), where the correlations extend over roughly two lattice sites. Fig. 2(b) shows a comparison between the spin-spin correlations functions Sf (r) of the single-channel and two-channel KLM on a logarithmic plot. The ground state for M = 1 in the weak-coupling limit shows RKKY liquid behavior with spatial oscillation characterized by a wave vector q = π/2. In the M = 2 case the correlations at J = 0.5 decay so slowly that the asymptotic behavior cannot be determined for the system sizes considered, as can be seen in Fig. 2(b). At larger J-values [e.g., J = 1.0 in Fig. 2(b)], the correlation functions appear to decay exponentially, consistent with the opening of a gap in the

3 spin excitation spectrum due to the transition into the channel-ordered phase. Fig. 3 shows the magnetic structure factor Sf (q) for the two-channel system. The amplitude of the peak at q = π/2 (the wavevector expected for a RKKY liquid) becomes smaller with increasing J, indicating that the magnetic correlations between the f spins become weaker and ?nally vanish at J ≈ 2.0.23 It is known that the RKKY correlations of the single-channel KLM at quarter ?lling become unstable with increasing coupling toward a ferromagnetic ground state.15,24 However, the two-channel model cannot be ferromagnetic at quarter ?lling because each f -spin is screened by an electron, so that some other kind of symmetry breaking must lift the degeneracy of the ground state. Fig. 4 shows the staggered magnetization D(r) of the channel degree of freedom, with D(i ? j) = m m′ nimσ njm′ σ′ .

σσ′ mm′

where nimσ = c? cimσ . In Fig. 4(a) one sees that D(r) imσ decays with 1/r as function of the distance r = i ? j for J = 4.0. Fig. 4(b) shows that the AF channel correlations become stronger with increasing coupling and the quasilong-range behavior develops for J > 2.0. ?

0.9 0.6 Sf(q) 0.3 0 0 0.2 0.4 q/π

FIG. 3: Magnetic structure factor of the two-channel KLM model as a function of wavevector q for various J at nc = 1.0.

which represent local Kondo singlets. In one spatial dimension, model (1) has been solved exactly using the Bethe ansatz.25 The ground state is characterized by critical AF correlations that decay as 1/r, in agreement with our numerical ?ndings at quarter ?lling [see Fig. 4(a)]. This critical behavior is replaced by long-range AF ordering for D > 1. Due to the SU(2) pseudo-spin invariance of H, the AF ordering is present for all three τ -components. In particular, the order along the z axis corresponds to staggered orbital ordering while the x, yordering gives rise to a Bose-Einstein condensation of excitons (particle-hole singlet bound states between the two channels). For nc < 1, the low-energy subspace in the strongcoupling limit contains states with zero and one conduction electron per site. While the local state with one conduction electron can be described by the hard-core bosons of Eq. (2), the empty state acquires spin–1/2 character from the f spins. It can therefore be rep? resented as a constrained fermion ?? = (1 ? nb )fiσ , i iσ ? b with ni = m bim bim . At strong coupling, the e?ective Hamiltonian has the form H0 = ? t b? ? ?? b + H.c. 2 iσ,m im iσ i+1,σ i+1,m .

J=0.5 J=1.0 J=1.5 J=2.5 J=4.0

In analogy to the in?nite-U Hubbard model, the groundstate wavefunction can be written as a direct product of a charge, spin, and orbital component |ψn {τ ; σ} = |nc ? |τ1 · · · τL?N ? |σ1 · · · σN

(n) i1 <i2 <...<iN

=

Γi1 i2 ··· iN ??1 σ1 · · · ??N σN b?1 τ1 · · · b?L?N τN |0 i i j i

0.6

0.8

1

where {τ ; σ} = (τ1 , ..., τL?N ; σ1 , ..., σN ) and |0 denotes the vacuum of ? and b particles. The complete spin degeneracy of H0 in the strong coupling limit is lifted in O(t2 /J). The e?ective Hamiltonian in this order contains only one term that lifts this degeneracy: H1 = t2 2J

? ? ?? i+1,σ bi+1,m ni bi?1,m ?i?1,σ + H.c. iσ,m

,

In order to understand our numerical results for strong coupling, we use the methods of Ref. 18 to derive an e?ective Hamiltonian valid for t/J ? 1. At nc = 1, the channel ?avor is the only degree of freedom and the low-energy spectrum can be described by a pseudospin– 1/2 model. Accordingly, the e?ective Hamiltonian is a Heisenberg model for the channel spin, H= 16t2 3J τ i · τ i+1 , (1)

i

1 where τ = 2 mm′ b? σ mm′ bim′ is de?ned in terms of im hard-core bosons

1 ? ? b? = √ (c? fi↓ ? c? fi↑ ) , im im↑ im↓ 2

(2)

where n? = σ ?? ?iσ . Here H1 exchanges two spins iσ i by hopping of a fermion over another to an empty nextnearest-neighbor site. In the same way as for the singlechannel KLM, it can be shown that the o?-diagonal elements of H1 are all non-positive and the Hamiltonian matrix in real-space is completely connected. The PerronFrobenius theorem (see, for instance, Ref. 26) then states that the ground state is unique and that the coe?cients of the wave-function can be chosen to be strictly positive. The only spin state that has strictly positive coe?cients in each of the subspaces of H1 is the one with maximum total spin. Therefore, the ground state of the two-channel KLM below quarter-?lling is ferromagnetic in the strongcoupling limit, in accordance with our numerical ?ndings.

4

0 10 Distance r 20 30 (a) 0.1/r D(r) 0 40

0.025 D(r) 0

J=4.0 J=3.0 J=2.0 J=1.0

-0.025 -0.05 0 10 5 Distance r

(b) 15

FIG. 4: Staggered magnetization D(r) for the channel degree of freedom for various J at quarter ?lling (nc = 1.0). The longer distance behavior for J = 4.0 (where there is a slower fallo?, approximately ∝ 1/r) is shown in (a) and a shorter range in r is shown for all four J-values in (b).

In this paper, we have mapped out the zero-temperature phase diagram of the two-channel KLM as a function of conduction band ?lling nc and Kondo coupling strength J. Our main results are that the phase diagrams of the two-channel and single-channel KLMs are

qualitatively similar for low band ?llings (nc < 1 in both models) but quite di?erent at nc = 1. For nc < 1, the addition of a second degenerate band of conduction electrons does not alter the physical picture of the singlechannel model if nc is low or J is large. The similarity between the two phase diagrams for nc < 1 further suggests that the metallic ground state of the two-channel KLM at weak coupling is determined by two independent Kondo e?ects in both channels. In contrast, at quarter?lling (nc = 1) the two-channel KLM exhibits a quantum phase transition between this metallic phase and an insulator characterized by strong AF correlations of the channel degrees of freedom. Our perturbative analysis shows that this insulating phase is present for any spatial dimension D and has long-range ordering for D > 1. Therefore, we also expect a QCP separating the metallic and the insulating phase for D > 1. The AF channel (or excitonic) ?uctuations diverge at the QCP and can produce a deviation from the normal Fermi-liquid 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 concentrated (lattice model) limits have a common origin in the ?uctuations of the channel degree of freedom. T.S., D.L.C., and C.D.B. acknowledge support from U.S. Department of Energy.

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