Hybridization oscillation in the one-dimensional Kondo-Heisenberg model with Kondo holes

We use the density matrix renormalization group method to study the properties of the one-dimensional Kondo-Heisenberg model doped with Kondo holes. We find that the perturbation of the Kondo holes to the local hybridization exhibits spatial oscillation pattern and its amplitude decays exponentially with distance away from the Kondo hole sites. The hybridization oscillation is correlated with both the charge density oscillation of the conduction electrons and the oscillation in the correlation function of the Heisenberg spins. In particular, we find that the oscillation wavelength for intermediate Kondo couplings is given by the Fermi wavevector of the large Fermi surface even before it is formed. This suggests that heavy electrons responsible for the oscillation are already present in this regime and start to accumulate around the to-be-formed large Fermi surface in the Brillouin zone.


Results
We start with the following Hamiltonian for the 1D Kondo-Heisenberg model with Kondo holes, The subscript "0" indicates the corresponding results for the clean Kondo lattice as a background for comparison.
We first discuss the results for n c < 1, namely away from the half filling. Figure 2 presents some typical results with n c = 0.2 and 0.8 for different Kondo couplings, J K /t. As expected, the resulting δ < n 0 i c at the Kondo hole sites is distinctly different from that induced by an attractive nonmagnetic impurity. We observe spatial oscillations in almost all the cases and the amplitude of the oscillation decays gradually with distance away from the hole sites. A straightforward comparison of the results at different J K /t indicates that the wavelength of the oscillation changes dramatically from the weak coupling regime to the strong coupling regime. Moreover, there seems to be a close correlation of the oscillation wavelength in all three quantities in the strong coupling regime. In the weak coupling regime, on the other hand, the oscillation is hardly seen in  δ i , whereas the charge density δn i c and the spin correlation, δχ i , also exhibit different oscillation patterns, suggesting that the conduction electrons are not strongly coupled with the local spins.
To see these results more clearly, we present in Fig. 3 the Fourier transforms of  δ i , δn i c , and δχ i in the momentum space for n c = 0.2, where the features are more pronounced. We see that all spectra exhibit peak structures. For J K /t = 0.8 in the weak coupling regime, δ k ( )  is negligible small with only a tiny peak at k = π. In contrast, δχ(k) shows a sharp peak at k = π due to the strong inter-site antiferromagnetic correlations between neighboring spins, and δn k ( ) c has two peaks at π = = k n k 2 c F c and k = π where the former corresponds to the small Fermi surface of the conduction electrons and the latter comes from the influence of the spin chain as is the case for the small peak at k = π in  δ i . The weak oscillations in both  δ i and δχ i indicate that the conduction electrons and the local spins are effectively decoupled (or only very weakly coupled). For J K /t = 1.6 in the intermediate coupling regime, a sharp peak appears at in all three quantities. Here + k F c f is the Fermi wavevector of the large Fermi surface, indicating that the conduction electrons are now coupled to the local spins and the large Fermi surface starts to take effect. For J K /t = 2.4, a new peak emerges at k = 2πn c , which corresponds to either k . Further increasing J K /t seems to suppress the peak at + k 2 F c f but keep the peak at + k 4 F c f almost unchanged. We note that the + k 4 F c f or k 4 F c oscillation may be understood as a special one-dimensional feature that originates from the spinless hole Fermi surface due to complete spin-charge separation following the formation of the Kondo singlets between the conduction electrons and the local spins 26 .
by number, we attempt to ascribe this peak to π + k 4 (mod 2 ) F c f associated with the large Fermi surface because of its coexistence with the + k 2 F c f peak, whereas the k 2 F c peak is absent in these regimes. We note that the critical Kondo coupling is about J K /t ≈ 3.0 for n c = 0.2 and J H /t = 0.5 in the clean Kondo lattice 24 , as determined from the maximum change in the distribution function of conduction electrons in the momentum space. It is quite unexpected that the hybridization oscillation is determined by + k F c f at J K /t = 1.6 and 2.4 before the large Fermi surface is even formed. This implies that heavy electrons are already present in this intermediate coupling regime and start to accumulate around + k F c f in the momentum space. Our results for n c < 1 suggest three regimes of the Kondo lattice physics depending on the magnitude of J K /t: (1) the weak coupling regime where the conduction electrons and the local spins are effectively decoupled (or only very weakly coupled); (2) the strong coupling regime where the two components are coupled to give rise to a well-defined large Fermi surface and complete spin-charge separation in 1D (corresponding to the + k 4 F c f peak); (3) the intermediate regime where the large Fermi surface is not yet formed, but preformed heavy electrons are already present and all three quantities,  δ i , δn i c and δχ i , exhibit similar oscillation pattern (corresponding to the + k 2 F c f peak). These preformed heavy electrons are primarily responsible for the charge density and hybridization oscillations induced by the Kondo holes, whereas local spin singlets are not fully established so that the hybridization must be collective in nature and the global spin singlet state must involve highly nonlocal entanglement between neighboring spins and conduction electrons.
We would like to point out that previous mean-field calculations predicted different periodicity for the three quantities, with k 2 F c for δn i c and  δ i and + k 2 F c f for δχ i in all parameter regimes 13 . This is in contradiction with our results where all three quantities have the same periodicity in the intermediate and strong coupling regimes. At the moment it is not clear what causes this discrepancy. It could be due to the one dimensional nature (with strong quantum fluctuations) of our DMRG calculations. However, our calculations on a ladder (L = 50) yield similar oscillation patterns. Also, it is natural to imagine that all three quantities should be strongly entangled and have the same periodicity at least in the strong coupling regime. Moreover, the same + k 2 F c f oscillation for the spin correlation δχ i obtained in our approach and the mean-field approach seems to support the idea that our results may still be valid in higher dimensions. If this is the case, one may attempt to think that the mean-field calculations yield wrong predictions for treating incorrectly (as simple hybridization bands) the correlated states of collectively entangled conduction electrons and local spins. A thorough investigation of this discrepancy may lead to a deeper understanding of the various approaches to the Kondo lattice problem. Careful analysis and numerical calculations in higher dimensions will be crucial in order to clarify the role of quantum fluctuations in determining the oscillation pattern. As a consequence, the observation of the hybridization oscillation at the small Fermi wavevector in the hidden order phase of Th-doped URu 2 Si 2 may probably need to be reinterpreted or reexamined in the normal state 16 .
To obtain a quantitative understanding of the spatial decay of the oscillation, we consider two examples where the spectra are governed by one dominant sharp peak in the momentum space. The correlation functions are then fitted using the following form in real space, where the oscillation wavelength λ is the same for the two quantities as discussed above and x i = i − L/2 − 1 is the distance of the i-th lattice site from the Kondo holes.  θ n / is the phase shift and  ξ n / denotes the characteristic decay length of the oscillation. Figure 4 shows two examples at n c = 0.2 and 0.8. We find an excellent agreement for both δ i  and δn i c . Both quantities exhibit similar decay lengths,  ξ ξ ξ = = n . On the other hand, the oscillation in δχ i in the strong coupling regime has a smaller decay length, despite that it exhibits the same periodicity. This suggests that the oscillation in  δ i is most affected by the charge density oscillation for n c < 1. We note that the above data do not follow a power law decay that is typically expected in 1D systems. This may originate from the finite Kondo energy and indicate that the hybridization perturbation is different from the usual Friedel oscillation.
Next we study briefly the half-filling case at n c = 1, where the density oscillation is suppressed due to the particle-hole symmetry and the finite charge gap. We therefore only need to consider the local hybridization δ i  and the spin correlation function δχ i . Figure 5 compares the two quantities on a logarithmic-linear scale as a function of the distance from the Kondo holes. We see that  δ i and |δχ i | exhibit similar slope (  ξ ∝ −1 ) with distance and similar oscillation pattern originating from the antiferromagnetic correlations in the spin chain (see the inset in Fig. 5). The good linearity confirms clearly the exponential decay of the oscillation amplitude and provides a further support for our analysis for n c < 1. We obtain ξ = 3  for J K /t = 0.8 and ξ = 1  for J K /t = 1.6. No oscillation is observed for larger J K /t. This indicates that J K /t = 1.6 locates near the boundary of weak and intermediate couplings for n c = 1. The very rapid suppression of the oscillation at n c = 1 suggests that the charge oscillation (for n c < 1) plays the role of an enhancer for the hybridization oscillation. The absence of spatial oscillation for larger J K /t is consistent with our observed π periodicity for n c < 1, but contradicts the mean-field calculations which predicted a wavevector π = = k k n 2 F c c in the hybridization oscillation 13 . This difference may provide a possible clue for understanding the discrepancy. In our calculations, the absence of the spatial oscillation for larger J K /t may be attributed to the effective localization of the redundant conduction electron at the impurity site caused by the large hybridization gap created by the Kondo coupling at its undoped nearest-neighbor sites. The crossover from weak to intermediate couplings reflects the competition between the kinetic energy and the hybridization energy. It would be interesting to check if this charge localization is present in the mean-field calculations.

Discussion and Conclusion
We note that there has been debate on the exact value of the critical Kondo coupling in the 1D Kondo-Heisenberg model. Different from the DMRG results 23,24,28 , some theories argue that the critical Kondo coupling should be at J K = 0 29 . In this case, there would be no preformed heavy electrons before the formation of the large Fermi  surface, but this would seem to be in contradiction with our observed existence of the weak coupling regime. We will not try to solve this difficult issue here due to the limitation of our numerical accuracy in distinguishing the real singularity (rather than a "rapid" change) of the distribution function in the momentum space. In this sense, our observed change at the "critical" Kondo coupling is actually a crossover. Whether or not it will lead to a true singular point in the thermodynamic limit requires more numerical or theoretical scrutiny. For the same reason, we also cannot exclude other possible values of the critical Kondo coupling, although it would then be inconsistent with both numerical tendency 24 and theoretical argument 29 . In either case, our numerical calculations yield different predictions from previous mean-field approximation on the oscillation pattern induced by the Kondo holes.
We would like to remark that there may be different understandings of the concept of "heavy electrons". A "standard" definition may depend on the existence of the Fermi liquid state with a large Fermi surface. In this case, heavy electrons are Landau quasiparticles with a large effective mass. However, this definition is limited to a small parameter range in the generic phase diagram. At high temperatures well above the Fermi liquid temperature, or in the antiferromagnetic phase with a small Fermi surface, one also discusses the emergence or existence of heavy electrons, as manifested in a logarithmically divergent specific heat coefficient 30 or a large cyclotron mass as measured in the de Haas-van Alphen (dHvA) experiment 31 . Especially in 1D, the ground state of the strong coupling limit may not be a Fermi liquid, even if it has a large Fermi surface. Therefore, we adopt in this work a more general "definition" of the concept of heavy electrons to reflect its usual meaning in the literature, namely, a composite state of conduction electrons and localized Heisenberg spins. Unambiguous experimental identification of the composite nature of heavy electrons is difficult due to its small energy scales. Some indirect signatures include the Fano interference in STM or the point-contact spectroscopy as first proposed by one of the authors 32 , and the hybridization gap in the optical conductivity 33 . Recent angle-resolved photoemission spectroscopy (ARPES) measurement 34 and quasiparticle interference experiment 35 also reveal some convincing evidences. Our work proposes another property of the composite state, namely a hybridization oscillation with a large Fermi wavevector even outside the Fermi liquid regime.
One may wonder if the obtained heavy electron behavior originates from the disturbance of the Kondo holes to the electronic structures of the neighboring sites rather than the intrinsic properties of the pure Kondo lattice. Due to the limitation of the DMRG method, we cannot calculate the spectral properties of the system in this work. However, we have previously calculated the local density of states of a doped periodic Anderson model on a square lattice using the determinant quantum Monte Carlo (DQMC) method 36 and found that, instead of producing heavy electrons, Kondo holes actually destroy heavy electron state on their neighboring sites. As a result, increasing the number of Kondo holes will lead to a percolation-type transition from the Kondo lattice physics to the single-ion Kondo physics.
To summarize, we use the density matrix renormalization group method to study the perturbation in the local hybridization, the charge density of the conduction electrons and the correlation function of the local spins induced by Kondo holes in the 1D Kondo-Heisenberg model. We find that all three quantities exhibit spatial oscillations whose amplitude decay exponentially with distance away from the Kondo holes. This suggests that the hybridization perturbation due to Kondo holes is different from the usual Friedel oscillation with a power law decay. Our results indicate that the hybridization oscillation is closely associated with the oscillations in the conduction electron charge density and the antiferromagnetic spin correlations. At half filling, where the charge density oscillation is suppressed, the antiferromagnetic spin correlations play the major role in determining the hybridization oscillation. Away from half filling, the charge density oscillation plays the dominant role and determines the wavelength and the decay length of the hybridization oscillation. We find three different regimes in the oscillation pattern. In the intermediate and strong coupling regimes, the wavelength is given by the Fermi wavevector of the large Fermi surface, implying the presence of heavy electrons around the large Fermi surface even before it is well formed. Moreover, the derived large decay length indicates that the Kondo lattice physics is highly nonlocal. This excludes any attempt based on local approximations and demands a proper treatment of the nonlocal and collective nature of the lattice hybridization in pursuit of a satisfactory solution to the Kondo lattice problem.