Kondo Lattice Model
Introduction
In condensed matter physics, the Kondo Lattice Model (KLM) is used to describe the interaction between localized magnetic moments and conduction electrons in a metallic system. This model is particularly important for understanding the behavior of heavy fermion materials, where the interplay between localized f-electrons and delocalized conduction electrons leads to rich and complex phenomena such as Kondo screening, magnetic ordering, and unconventional superconductivity.
The Hamiltonian for the Kondo Lattice Model can be written as:
$$ H = H_{\text{band}} + H_{\text{Kondo}} $$
where:
$H_{\text{band}}$ describes the kinetic energy of the conduction electrons: $$ H_{\text{band}} = \sum_{k, \sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} $$ Here, $c_{k\sigma}^\dagger$ and $c_{k\sigma}$ are the creation and annihilation operators for conduction electrons with momentum $k$ and spin $\sigma$, and $\epsilon_k$ is the dispersion relation.
$H_{\text{Kondo}}$ represents the Kondo interaction between the localized spins $\mathbf{S}_i$ and the conduction electrons: $$ H_{\text{Kondo}} = J \sum_i \mathbf{S}_i \cdot \mathbf{s}_i $$ Here, $\mathbf{s}_i$ is the spin density of the conduction electrons at site $i$, and $J$ is the exchange coupling constant.
The above KLM describes three key components of the model:
Localized Magnetic Moments: In the KLM, localized magnetic moments (often represented by spins) are associated with atoms in a periodic lattice. These moments arise from partially filled f-orbitals in rare-earth or actinide compounds.
Conduction Electrons: The conduction electrons are delocalized and form a Fermi sea. They interact with the localized spins through an exchange coupling, typically described by the Kondo interaction.
Kondo Interaction: The interaction between the localized spins and the conduction electrons is modeled by an antiferromagnetic exchange coupling, often denoted as $J$. This coupling leads to the Kondo effect, where the localized spins are screened by the conduction electrons at low temperatures.
Phenomena
The Kondo Lattice Model exhibits a variety of intriguing physical phenomena, including:
Kondo Screening: At low temperatures, the localized spins are screened by the conduction electrons, forming a non-magnetic ground state known as the Kondo singlet.
Heavy Fermion Behavior: The screening of localized spins leads to the formation of quasiparticles with greatly enhanced effective masses, giving rise to heavy fermion behavior.
Magnetic Ordering: Depending on the strength of the Kondo interaction and the lattice structure, the system can exhibit magnetic ordering, such as antiferromagnetism or ferromagnetism.
Unconventional Superconductivity: In some heavy fermion materials, the Kondo lattice model can give rise to unconventional superconducting states with non-trivial pairing symmetries.