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arXiv:cond-mat/0405379 · Physical review. B, Condensed matter

Spin and charge gaps of the half-filledN-leg Kondo ladders

J. C. Xavier

Published Oct 14, 2003• 11 citations
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Abstract

In this work, we study N-leg Kondo ladders at half-filling through the density matrix renormalization group. We found nonzero spin and charge gaps for any finite number of legs and Kondo coupling $J>0.$ We also show evidence of the existence of a quantum critical point in the two-dimensional Kondo lattice model, in agreement with previous works. Based on the binding energy of two holes, we did not find evidence of superconductivity in the two-dimensional Kondo lattice model close to half filling.

DOI
10.1103/physrevb.68.134422

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In this work, we study N-leg Kondo ladders at half-filling through the density matrix renormalization group. We found nonzero spin and charge gaps for any finite number of legs and Kondo coupling J>0.J>0.J>0. We also show evidence of the existence of a quantum critical point in the two-dimensional Kondo lattice model, in agreement with previous works. Based on the binding energy of two holes, we did not find evidence of superconductivity in the two-dimensional Kondo lattice model close to half filling.
The interplay between the Kondo effect, indirect magnetic interaction, and geometrical frustration is studied in the Kondo lattice on the one-dimensional zigzag ladder. Using the density-matrix renormalization group, the ground-state and various short- and long-range spin- and density-correlation functions are calculated for the model at half filling as a function of the antiferromagnetic Kondo interaction down to J=0.3tJ=0.3tJ=0.3t, where ttt is the nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown to lead to at least two critical points: Starting from the strong-JJJ limit, where almost local Kondo screening dominates and where the system is a nonmagnetic Kondo insulator, antiferromagnetic correlations between nearest-neighbor and next-nearest-neighbor local spins become stronger and stronger, until at ${J}_{\mathrm{c}}^{\mathrm{dim}}\ensuremath{\approx}0.89t$ frustration is alleviated by a spontaneous breaking of translational symmetry and a corresponding transition to a dimerized state. This is characterized by antiferromagnetic correlations along the legs and by alternating antiferro- and ferromagnetic correlations on the rungs of the ladder. A mechanism of partial Kondo screening that has been suggested for the Kondo lattice on the two-dimensional triangular lattice is not realized in the one-dimensional case. Furthermore, within the symmetry-broken dimerized state, there is a magnetic transition to a ${90}^{\ensuremath{\circ}}$ quantum spin spiral with quasi-long-range order at ${J}_{\mathrm{c}}^{\mathrm{mag}}\ensuremath{\approx}0.84t$. The quantum-critical point is characterized by a closure of the spin gap (with decreasing JJJ) and a divergence of the spin-correlation length and of the spin-structure factor S(q)S(q)S(q) at wave vector $q=\ensuremath{\pi}/2$. This is opposed to the model on the one-dimensional bipartite chain, which is known to have a finite spin gap for all J>0J>0J>0 at half filling.
The relationship between charged and spin-excitation gaps of the half-filled Hubbard model, the symmetric periodic Anderson model, and the Kondo lattice model is considered for a general ddd-dimensional bipartite lattice. In a previous paper [G. S. Tian, Phys. Rev. B 58, 7612 (1998)], it was shown that the quasiparticle gaps of these models at half filling are always larger than their spin excitation gaps. In the present paper, we establish a theorem, which states that the charged gap is also greater than the corresponding spin gap in these models. This conclusion, which has been reached previously for one-dimensional systems based on variational and the density-matrix renormalization-group numerical calculations, is thus put on a rigorous and more general footing.
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec,{E}_{c},Ec​, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec{E}_{c}Ec​ of the same type as in N/2N/2N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
The phase diagram of the two-leg $t\text{\ensuremath{-}}{J}_{z}$ ladder is explored, using the density-matrix renormalization group method. Results are obtained for energy gaps, electron density profiles, and correlation functions for the half filled and quarter filled cases. The effective Lagrangian velocity parameter ${v}_{\ensuremath{\rho}}$ is shown to vanish at half filling. The behavior of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full $t\text{\ensuremath{-}}J$ ladder, but scaled up by a factor of about 2 in coupling. Near half filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter filling the system appears to be in a Tomonaga-Luttinger phase.
Using the density-matrix renormalization-group (DMRG) method we study a two-channel Kondo lattice model on a half-filled ladder. Our model involves an on-site s-wave and a nearest neighbor d-wave coupling between the local moments and the conduction electrons on the ladder. By changing the relative strength of the two Kondo interactions we examine the evolution of the system from a conventional Kondo insulator with a singlet at each site to a special kind of semimetallic state formed by overlapping of Zhang-Rice-like singlets. The DMRG is used to study how the spin and charge correlation functions evolve between these two regimes.