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Physical review. B, Condensed matter

Finite-size-scaling study of the gap formation in the ‘‘Kondo-necklace’’ model

Paolo Maria Santini, Jenő Sólyom

Published Sep 15, 1992• 24 citations
Publisher

Abstract

The ``Kondo-necklace'' model is studied by exact numerical diagonalization on finite chains up to N=12 sites. The gap extrapolated to infinite systems can be well fitted by a Kosterlitz-Thouless-like form with a critical coupling (J/W${)}_{\mathit{c}}$=0.24. Although this value is appreciably smaller than the best estimate from real-space renormalization, our results indicate strongly a finite critical coupling in contrast to the one-dimensional Kondo lattice.

DOI
10.1103/physrevb.46.7422

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arXiv:cond-mat/0605396•Physical Review B•Nov 9, 2006•9 citations
The ``Kondo-necklace'' model is studied by exact numerical diagonalization on finite chains up to N=12 sites. The gap extrapolated to infinite systems can be well fitted by a Kosterlitz-Thouless-like form with a critical coupling (J/W)c{)}_{\mathit{c}})c​=0.24. Although this value is appreciably smaller than the best estimate from real-space renormalization, our results indicate strongly a finite critical coupling in contrast to the one-dimensional Kondo lattice.
We investigate a gap-formation mechanism of the one-dimensional Kondo-necklace model at zero temperature. An analytic treatment using the bosonization method shows that the model undergoes the Kosterlitz-Thouless transition at the critical coupling Jc=0{J}_{c}=0Jc​=0, which is described by the quantum sine-Gordon model. This result is contrary to the previous investigations on the ground-state phase transition, where some finite critical coupling values have been predicted. To clarify the validity of our analytic calculation result, we estimate the singlet-triplet gap by employing White's density matrix renormalization group method; the numerical calculation is carried out for the systems up to 200 sites with open boundary condition. As a result, we find that the excitation gap observed in the large JJJ region becomes smaller quickly with the decrease of JJJ. However, the data show the existence of a small gap for J> 0.25J>~0.25J> 0.25, which seems to support Jc=0{J}_{c}=0Jc​=0.
We analyze the one-dimensional Kondo necklace model, at zero temperature, with an anisotropy parameter $\ensuremath{\eta}$ in the interaction of the conduction chain, by means of the density matrix renormalization group. We calculate the energy gap and estimate the quantum critical points that separate a Kondo singlet state from an antiferromagnetic state, assuming a Kosterlitz-Thouless tendency. We also observe the correlation functions and the structure factors that support our critical points. The resulting phase diagram is presented and compared to that reported previously using Lanczos calculations. It is shown that the quantum critical points vary very slowly with $\ensuremath{\eta}$, but when $\ensuremath{\eta}$ approaches zero, they drop abruptly.
We performed numerical diagonalization studies of the periodic Anderson Model on finite chains up to 6 sites, in the integral valence limit for T=0. We find that in the weak coupling (Kondo) region the hybridization between f and conduction electrons induces pronounced correlations between the f spins. These correlations are ferromagnetic for low filling and become antiferromagnetic close to half-filling, with a smooth transition between the limiting cases. In no limit can the physics be completely described by a Ruderman–Kittel–Kasuya–Yosida effective spin Hamiltonian, as f and conduction band electrons are always correlated. To study the onset of long range order we considered also a simpler model for the half-filled case, called ‘‘Kondo necklace.’’ We studied the model at T=0 by numerical diagonalization and finite-size scaling, on chains up to 12 sites. We find that below a finite critical value of the Kondo coupling there exists a gapless phase with quasi long range antiferromagnetic correlations.
Abstract Using the density matrix renormalization group method, we study the one‐dimensional Kondo necklace model including two anisotropies: η in the XY interaction of the conduction chain, and Δ in the local Kondo coupling between localized and conduction spins. For different values of the anisotropies, we calculate the energy gap and estimate the quantum critical points in which a phase transition from antiferromagnetic to spin liquid states takes place, assuming a Kosterlitz–Thouless‐like behavior. For the case η=0\eta = 0η=0 , the system remains in a spin liquid state for the different Δ considered, but when η≠0\eta \ne 0η=0 , Δ markedly affects the location of the critical points. We also calculate correlation functions for a specific case, in order to support our gap results.
We study single-hole dynamics in the bilayer Heisenberg and Kondo necklace models. Those models exhibit a magnetic order-disorder quantum phase transition as a function of the interlayer coupling ${J}_{\ensuremath{\perp}}$. At strong coupling in the disordered phase, both models have a single-hole dispersion relation with band maximum at $\mathbit{p}=(\ensuremath{\pi},\ensuremath{\pi})$ and an effective mass at this $\mathbit{p}$ point which scales as the hopping matrix element ttt. In the Kondo necklace model, we show that the effective mass at $\mathbit{p}=(\ensuremath{\pi},\ensuremath{\pi})$ remains finite for all considered values of ${J}_{\ensuremath{\perp}}$ such that the strong-coupling features of the dispersion relation are apparent down to weak coupling. In contrast, in the bilayer Heisenberg model, the effective mass diverges at a finite value of ${J}_{\ensuremath{\perp}}$. This divergence of the effective mass is unrelated to the magnetic quantum phase transition, and at weak coupling the dispersion relation maps onto that of a single hole doped in a planar antiferromagnet with band maximum at $\mathbit{p}=(\ensuremath{\pi}∕2,\ensuremath{\pi}∕2)$. We equally study the behavior of the quasiparticle residue in the vicinity of the magnetic quantum phase transition both for a mobile and static hole. In contrast to analytical approaches, our numerical results do not unambiguously support the fact that the quasiparticle residue of the static hole vanishes in the vicinity of the critical point. The above results are obtained with a generalized version of the loop algorithm to include single-hole dynamics on lattice sizes up to $20\ifmmode\times\else\texttimes\fi{}20$.