Static Mean-Field Description of the KLM

The picture of composite quasiparticles from hybridized bands is further elucidated in a mean-field treatment of the KLM. Moreover, the static mean-field description serves

Figure 3.3: Schematic Doniach diagram as function of exchange couplingJ showing the competing energy scales ERKKY (dotted green) and TK (dashed blue). At the point where both energy scales cross, a QCP exists at zero temperature. It divides the antiferromag-netically ordered (AF) phase from the Fermi-liquid (FL) regime. Quite often the QCP is hidden by a dome of superconducting (SC) order above which a non-Fermi-liquid (NFL) regime exists. T^∗ denotes the coherence scale below which a FL is formed.

as a reference to which DMFT results can be compared to. As the derivation of the mean-field model for the Kondo lattice also includes some important physical insight, it will be outlined in the following.

The spin operators in the Kondo-lattice Hamiltonian in Eq. (3.2) can be rewritten in a fermionic representation

S_i^z = (n^f_i↑−n^f_i↓)/2 S_i⁺ =f_i↑^†fi↓

S_i⁺ =f_i↓^†fi↑

i.e., one uses auxiliary fermionic fields f. This mapping is exact only if one adds the constraint of single occupation of thef-levels at each site,n^f_i =n^f_i↑+n^f_i↓ = 1. Thus, the charge of the auxiliary fermions is conserved,Q^f_i = 1, and this representation of the f-shell

local moments has a localU(1) gauge symmetry. The KLM can then be written as

that is, the Kondo interaction now includes four-fermion terms. These can be decoupled so as to obtain a mean-field theory for the KLM. In the standard approach, a functional-integral representation of the KLM is introduced and a Hubbard-Stratonovich transform in the particle-hole channel is carried out, whereby additional bosonic fields Vi, V_i^∗ are introduced [Rea83]. Thereafter the interaction term reads¹

HV =^X

In addition, the constraint Q^f_i = 1 has to be dynamically enforced by a time-dependent bosonic field² λi that couples to (n^f_i −Q^f_i). The full Lagrangian L(τ) in the functional

This Lagrangian was shown to be invariant under a certain gauge transformation, the Read-Newns transformation [Rea83]. The gauge phase can be absorbed in the fieldλ and thenV becomes real. By this Anderson-Higgs-type mechanism the f-degrees of freedom are endowed with a physical charge and can be regarded as electrons. This process justifies the standard picture of two hybridized electronic bands once the heavy Fermi liquid has formed.

The above derivation may be interpreted as a sort of reverse Schrieffer-Wolff transformation of the KLM to the periodic Anderson model, though without a Coulomb repulsion among f-electrons. The philosophy is, however, quite different. The KLM serves as an effective model for the PAM in the Kondo-limit, with an effective spin-¹₂ per site originating from electrons in f-orbitals. From this perspective, the picture of additional electronic states, i.e., composite heavy fermions, appears to emerge naturally from the hybridization with f-electrons. However, the spins in the KLM could also be of nuclear origin. In this case,

1 Note that theVi are fullfluctuating bosonic fields for the time being.

2 Theλ-fields are actually dynamic Lagrange multipliers, see e.g. [Col13].

the mean-field decoupling implies the generation of additional electronic states near the Fermi energy from nuclear spins once the lattice Kondo effect takes place. In the single-ion Kondo model this leads to a Kondo resonance at the Fermi energy, while in the dense Kondo lattice the new states form the coherent heavy quasiparticle band.

In the following the hybridized-band picture is elucidated in a bit more detail. The decoupling introduced above motivates a saddle-point approximation, that is, the fluctuating fields V(τ) and λ(τ) are replaced by their static values at the saddle-point of the partition function. The resulting mean-field Hamiltonian for the KLM reads

H_KLM^MF =^X

Since λ effectively constitutes an on-site energy, it is renamed by εf. Both V and εf in principle have to be determined self-consistently. After a Fourier transform the Hamiltonian can be rewritten in a matrix form (µ= 0 in the following)

H_KLM^MF =^X and the quasiparticle energies can be easily obtained by diagonalizing this matrix. For a finite hybridization V the resulting energies split in two branches, also referred to as hybridized bandstructure of the KLM:

E_k^± = εf +εk±^q(εk−εf)²+ 4V²

2 (3.7)

In (Fig. 3.4) the typical qualitative shape of quasiparticle bands in the paramagnetic KLM according to Eq.(3.7) is shown. Essential quantities are a direct gap of width 2Veff, an indirect hybridization gap ∆g and the renormalized chemical potential εf of the f-level.

The eigenvectors of the mean-field Hamiltonian Eq. (3.6) are mixtures ofc- and f-degrees of freedom, the weight of which is indicated in (Fig. 3.4) by color: The strongly dispersive parts of the upper and lower band correspond to the conductionc-electrons (blue) while the flat parts belong tof-electrons (green). Although thef-level itself is non-dispersive in its origin, it gains a slight dispersion by hybridization. These flat and almost dispersionless segments of the quasiparticle bands close toEk ≈εf correspond to the quasiparticles with a strongly renormalized mass, i.e., heavy fermions.

In the hybridized-band picture, the insertion of electronic states in the conduction band can be understood by noting that the intersection of the lower band with the Fermi energy between theΓ- and M-point shifts towardsk= (π,π). It thus yields a large Fermi surface. A rigorous proof of this fact is far from trivial. While Luttinger’s theorem [Lut60]

Figure 3.4: Visualization of hybridization in a typical renormalized quasiparticle band-structure of the KLM in the mean-field picture. Left: Weight of c-states. Right: Weight of f-states. The dashed line indicates the position ofε_f.

states that the Fermi volume of a metal is proportional to the fermionic particle density, V^f ∝ nc +nf, which certainly applies to the PAM with a hybridization [Lan66], spin degrees of freedom are not included. The localized spins in the Kondo lattice do, however, contribute to the Fermi surface provided that Kondo screening is intact [Mar82]. Another version of Luttinger’s theorem [Osh00] based on topological arguments proves this fact. A large hole-like Fermi surface is thus always present in the paramagnetic phase of the KLM with a finite antiferromagnetic Kondo coupling.

However, in the antiferromagnetic phase of the KLM a small Fermi surface is possible. For weak coupling, the RKKY interaction prevails such that the local moments are effectively decoupled from the conduction band electrons. Thereby, a transition from a particle-like to a hole-like Fermi surface is induced upon increasingJ. Approaching this transition from the strong coupling regime, it is often called “Kondo breakdown”. From a more general perspective it constitutes an example for an orbital-selective Mott transition [Voj10]. It has been proposed, that an additional energy scale E_loc^∗ in the Doniach diagram separates the two regimes [Geg08]. At zero temperature it marks a (f-) moment-localization phase transition while at elevated temperature it is more to be understood as a crossover regime.

To the left of E_loc^∗ Kondo screening is incomplete and the local moments do not deliver a crucial contribution to the Fermi volume. On the right of theE_loc^∗ line the local moments delocalize and do contribute to a large Fermi volume. At T = 0 the position of E_loc^∗ determines the nature of the QCP: If E_loc^∗ hits ERKKY at zero temperature, i.e., the AF phase boundary falls together with the breakdown of the Kondo effect, the quantum-critical fluctuations do include both the quantum fluctuations of the magnetic order parameter and the ones of the setting-in or destruction of Kondo screening. This case is termed a

local QCP. In the other case E_loc^∗ crosses ERKKY and the quantum-critical fluctuations at the antiferromagnetic QCP are only the ones of the order parameter. The local moments are delocalized in the part of the AF phase close to the QCP and the magnetic groundstate of the system can be described as a spin-density-wave state. The QCP is thus called spin-density-wave-type QCP.