The one-dimensional Kondo lattice model at partial band filling

Abstract

The Kondo lattice model introduced in 1977 describes a lattice of localized magnetic moments interacting with a sea of conduction electrons. It is one of the most important canonical models in the study of a class of rare earth compounds, called heavy fermion systems, and as such has been studied intensively by a wide variety of techniques for more than a quarter of a century. This review focuses on the one-dimensional case at partial band filling, in which the number of conduction electrons is less than the number of localized moments. The theoretical understanding, based on the bosonized solution, of the conventional Kondo lattice model is presented in great detail. This review divides naturally into two parts, the first relating to the description of the formalism, and the second to its application. After an all-inclusive description of the bosonization technique, the bosonized form of the Kondo lattice Hamiltonian is constructed in detail. Next the double-exchange ordering, Kondo singlet formation, the RKKY interaction and spin polaron formation are described comprehensively. An in-depth analysis of the phase diagram follows, with special emphasis on the destruction of the ferromagnetic phase by spin-flip disorder scattering, and of recent numerical results. The results are shown to hold for both antiferromagnetic and ferromagnetic Kondo lattice. The general exposition is pedagogic in tone.

Acknowledgements

I wish to thank my past and present collaborators, most notably Graeme Honner, Ian McCulloch and Raymond Chan, for the research undertaken to progress this field of study.

Notes

A brief overview of the strongly correlated electron systems is given in Appendix A.

While this notation is fairly standard for the Kondo lattice, note that the localized electrons in the manganese oxides are in the d band, cf. section 1.2.1 below.

A summary of the single-impurity Kondo model results can be found in Appendix B.

Adding a large antiferromagnetic Heisenberg interaction between the local moments produces a spin-gapped metal [206] with unconventional pairing fluctuations [42].

The effect of phonons on the 1D Kondo lattice model has been studied by Gulácsi et al. via bosonization [77, 78]. Details can be found in section 7.3.

More accurately, the single-impurity Kondo model turns out to be a local Fermi liquid. For details, see Andrei et al. [13]. The concept of a local Fermi liquid was introduced by Newns and Hewson [162] who used it to interpret experimental data on rare earth compounds. It corresponds to a non-interacting multilevel resonant model.

A similar enhancement is observed for the spin susceptibility.

The bands corresponding to the 4f orbitals are energetically narrow, or atomic-like, with a delta-function density of states ρ(ϵ) ≈ δ(ϵ − ϵ _f ), where ϵ _f is the f level. Note that these are not bands in the usual sense; band theory begins from a basis of non-interacting delocalized Bloch states (cf. Appendix D.1), and provides a poor description of the properties of partially filled 4f shells, in which the f electrons are localized and interact strongly with each other. For comparison with the energies of well-defined conduction bands, it is however convenient and conventional to consider a 4f ‘band’ for a given compound with a 4f ⁿ nominal occupation on constituent rare earth atoms. The f level ϵ _f is then the energy for the process 4fⁿ  → 4f^{n  − 1} of removing an f electron. For example, in SmS ϵ _f is the energy for the process 4f ⁶ → 4f ⁵ [89, 230].

According to standard band theory, V_k would be zero, since the states in different bands are orthogonal (cf. Appendix D.1). However, as noted above, the f orbitals are strongly correlated, and do not constitute a band in the rigorous sense.

The same notation is used as in equations (1.4) and (1.5) i.e., S _j represent the pseudo-spin operators for the f-electrons.

This in spite of the fact that double-exchange ferromagnetic ordering has been observed in rare earth compounds as far back as 1979. See Varma [231] for theory, and Batlogg et al. [19] for experiments on the thulium compound TmSe_0.83Te_0.17.

For a different derivation, see Van Vleck [229].

See also Yafet [252] for 1D.

Note that more complicated magnetic structures can arise with the stacking of 2D planes in a crystal [4].

Note, however, that the lattice problem in a certain sense precedes the work on the impurity Kondo Hamiltonian: Fröhlich and Nabarro [62] considered the lattice case in 1940 in their work on the magnetic ordering of nuclear spins.

For more details about the Kondo model, see Appendix B.

For a review, see Andrei et al. [13]. The exact solution of the single-impurity Anderson model is reviewed by Tsvelick and Wiegmann [227].

Following convention, N here denotes the degeneracy of the localized spin orbitals. Elsewhere N denotes the number of lattice sites.

Spin gaps are also observed in bosonization treatments of the 1D half-filled Kondo lattice. See Fujimoto and Kawakami [64] and Le Hur [133] for J>0 and Le Hur [132] for J<0. These treatments are not for the pure Kondo lattice of equation (1.12), but include also a direct interaction between the localized spins.

The fact that the hopping amplitude is renormalized by a factor 2 is the reason that the effective mass m* = 2m _e as J → ∞ for the spin polaron in the Kondo lattice with one conduction electron.

Some of the basic concepts of bosonization can be traced back further; an approximate 3D treatment was given by Bloch in 1934 [212].

The common examples of two-component systems are the two-band models [76], and the two-chain problem [56]. Both of these cases were solved using a field theoretical bosonization.

In the literature the Luttinger model is sometimes understood to imply also certain interactions, specifically forward scattering interactions as originally considered by Tomonaga [219] and by Luttinger [142]. This is not implied here; the Luttinger model simply refers to the introduction of the Dirac seas. Following modern convention [235], the Luttinger model with forward scattering interactions is called the Tomonaga–Luttinger model.

This is the physical content of Emery's identification [51] of α with the lattice spacing a. Emery's interpretation, which has wide currency, is discussed further in section 4.1.3.

An alternative decomposition, designed to investigate short-wavelength fluctuations with wavevectors |k| ≈ 2k _F, is used in section 3.2.2, cf. equations (3.13).

The standard Hubbard model, as considered here, does not contain impurities or other lattice imperfections that may act to localize the electrons below this scale.

A brief overview of the strongly correlated electron systems is given in Appendix A.

For more details on the canonical transformations in general, see section 1.2.3, including Appendix C, and section 5.1.2.

Within the framework of exact solutions, an extension of the Bethe Ansatz to the quantum inverse scattering method [123] allows correlation functions to be determined.

In more complicated cases it is conceivable that double-exchange may be mediated by two or several conduction electrons. Double-exchange is not, however, a collective effect involving a large number of conduction electrons.

The similarities between the transition in the Kondo lattice model, and that in the transverse-field Ising chain, have been investigated by Juozapavicius et al. [113] on the basis of this mapping.

For the step function cut-off Λ_α(k) = θ(α⁻¹ − |k|), equation (5.11) gives a ferromagnetic interaction at short-range, similar to the interaction determined by the continuous cut-off functions. However, due to interference set up by the discontinuity, the step-function cut-off determines a weak oscillating interaction at longer range (cf. figure 7). This is not shared by the smoother cut-off functions. As discussed in detail in sections 3.2.2 and 4.1.2, a discontinuously sharp cut-off on bosonic density fluctuations is unlikely in a real system, and the long-range oscillations are artefacts resulting from a singular cut-off function. Because of this, the step-function cut-off is not considered in this section, and attention is restricted to continuous cut-off functions.

This property is specific to the Kondo lattice [152]; for the single-impurity Kondo Hamiltonian at zero temperature, the spin coupling renormalizes to infinity for all J > 0, as discussed in section 2.2.

There are minor changes in the derivation for ferromagnetic couplings J < 0. For clearness in the exposition, it is however convenient to fix the sign of J as positive for the remainder of this section. It is straightforward to verify that all the results go through for J < 0 with trivial modifications. The phase diagram and ground-state properties for J < 0 are discussed in section 6.3.

A failure to realize this point may be the reason for the erroneous evaluation of the canonical transformation by Zachar et al. [262], see section 5.1.2.

For certain quasi-commensurate fillings, for example n = 1/2, the spontaneous magnetization still grows continuously, but the critical exponent may be reduced to its value 1/8 as in the constant transverse-field Ising chain, see also Pfeuty [177].

It is interesting to note that some of the features of the transition in the random transverse-field Ising chain are loosely similar to those of first-order transitions in random classical systems. See Fisher [58] for more details.

At partial conduction band filling this is assured by bosonization. In section 3.2.2 it was shown that α is limited by the interparticle spacing of the conduction electrons. The interparticle spacing is greater than a lattice spacing for a partially filled conduction band.

The spin polaron picture for the ferromagnetic ordering induced on the localized spins was used by Sigrist et al. [203]. It is equivalent to the double-exchange picture, and is perhaps more illustrative at low conduction band filling.

For more details see also section 3 and section 4.2.2.

A brief overview of the strongly correlated electron systems is given in Appendix A.

This limitation is still valid: even recent results [249, 250] do not cover the region of doping close to half-filling.

As mentioned in section 2.4.2, earlier results based on simple mean-field, slave-boson or Gutzwiller approximations [225] could not predict ferromagnetism close to half filling.

This is the reason why the second ferromagnetic region above n = 0.9 is plotted as a striped area in figure 18, where the accuracy was insufficient to determine the phase boundaries.

Here a full non-Abelian bosonization approach is used to study the fixed points of the 1D Kondo lattice, and it is shown that the fixed points differ from the ones of the single-impurity Kondo model.

Recall from section 2.1 that the RKKY interaction strictly diverges in 1D, and there is no lower bound on the ground-state energy for the RKKY Hamiltonian, even for arbitrarily small J [204]. The divergence is typical of perturbation expansions in 1D, and does not occur in higher dimensions.

In equation (7.3) the notation of Novais et al. [165, 166] was used, instead of equation (5.2).

As in the previous section, the calculated value, see section 5, of the constant A has been used here.

α/a measures the effective range of the double-exchange, see figure 13, which was shown to be equivalent to the width of the spin polarons, for details see sections 5.1.3 and 6.1.

A brief overview of the strongly correlated electron systems is given in Appendix A.

The rule of thumb is that if the Luttinger model representation (cf. Appendix G) involves the normal-ordering convention non-trivially, then the corresponding Bose representation in the condensed matter system is exact. If the Luttinger model representation does not require a non-trivial application of the normal-ordering convention, then the corresponding Bose representation in the condensed matter system is valid only over asymptotically large separations.

Bethe's highly detailed original paper is available in a new English translation by Mahan [144].

Only in the so-called Toulouse limit [221], the Anderson and Yuval [6, 7, 10, 11] solution is exact. The Toulouse limit corresponds to a certain value of J_z  = J _T, where J _T ≈ 0.97 [246]. In this limit, however, the Kondo model reduces to a simple quadratic non-interacting model.

For simplicity we use a convention, where k _B is unity.

nth order canonical transformation corresponds to 2nth order perturbation theory in H_V [257].

The formalism is thus written for 1D systems. The generalization to higher dimensions is straightforward, and usually involves only a replacement of scalar spatial and momentum variables x and k by vectors r and k respectively [167].

The labels ↑, +1 and ↓, −1 will be used interchangeably for σ; in formulas σ always denotes ± 1 as the spin is

, respectively, along the z-axis.

The 2D and 3D Fermi momenta take a different form:

is the 3D form in a box of volume L ³; k _F = (2π N _e/L ²)^1/2 is the 2D form on a square of area L ². In all cases the ground-state consists of a Fermi sea with momentum/spin states |kσ⟩ with |k| < k _F occupied, and those with |k|>k _F empty. Based on analogy with the 3D case, in all dimensions the set of points with |k| = k _F is called the Fermi surface.

For negative t the roles of electrons and holes are interchanged.

Similar considerations apply in calculating the representation for the non-interacting Hamiltonian.