Kondo insulators are rare earth or actinide (typically cerium or uranium) compounds that exhibit a small gap in an otherwise continuum spectrum. Some examples of Kondo insulators are Ce_{3}Bi_{4}Pt_{3}, CeNiSn, UNiSn, U_{3}Sb_{4}Ni_{3}, SmB_{6} among others. These compounds, which were discovered fifteen years ago, present a very challenging problem in Condensed Matter Physics. The conduction electrons interact with localized electrons in the f-shell, making up a complicated many body system. The goal of my research is to provide a better understanding of the fundamental physics of these systems.
The fact that these materials have a gap in their energy spectrum reminds us of standard semiconductors such as silicon or germanium. However, Kondo insulators are fundamentally different. In standard semiconductors, the gap is a consequence of the interactions of the electrons with the periodic lattice. In Kondo insulators, the gap is formed due to strong correlations between extended conduction electrons and the localized f-electrons of the rare earth or actinide. A typical energy gap for a Kondo insulator is of the order of 100^{\circ} K, in contrast with 10,000^{\circ}K for silicon. Also, in Kondo insulators the gap for charge excitations (charge gap) is different than the one for magnetic excitations (spin gap).It is generally accepted that the Anderson lattice Hamiltonian is an appropriate model to describe Kondo insulators. The Anderson single impurity model was originally developed to explain the effect of low concentrations of magnetic impurities in metals. It considered a single magnetic impurity as a localized orbital that interacts with an extended band of conduction electrons. Double occupation of the localized orbital is penalized by a strong Coulomb repulsion between the two electrons. If one puts one impurity at each lattice site, one obtains the Anderson lattice model. Each ``impurity'' corresponds to one atom of cerium or uranium. We arrange the atoms in a linear chain, so we get a one-dimensional system, which is numerically easier to study. When there are two electrons per atom, the lower energy band is filled and the upper one is empty, making it a good candidate for the description of Kondo insulators.
This apparently simplified model is in itself a very difficult problem to solve. We use a recently developed numerical method, the density matrix renormalization group technique, to solve the problem. This method, developed by Prof. Steve White, of U. C. Irvine, has been very successful in describing the ground state and low energy excitations of one dimensional systems. It allows us to solve the problem for lattices of up to 24 sites, much larger than what is achievable by exact diagonalization (exact diagonalization is a standard numerical method but it can only do about 8 sites at most). In some cases, we are able to do reasonable extrapolations for the infinite length limit.
We concentrate on the low temperature properties of the system. For a wide range of parameters, we investigate the low energy excitations (charge gap and spin gap) and also the magnetic correlations in the ground state. We also calculate the magnetic susceptibility by including the effect of a magnetic field.
We find that the charge gap is larger than the spin gap for any choice of the parameters. This agrees with the experimental findings. Furthermore, for a certain range of the parameters, the ratio between the charge gap and the spin gap is roughly 2. This is consistent with what is found experimentally for Ce_{3}Bi_{4}Pt_{3}. Our results for the magnetic susceptibility help interpret the experimental measurements of the susceptibility of CeNiSn.
There is still a lot of work to be done in order to answer all the questions regarding the physics of Kondo insulators. The complete understanding of the physical mechanisms responsible for their unusual properties is crucial for the development of potential technological applications, such as cooling devices.