Several actinide and lanthanide materials have a remarkable response to extreme pressure: they undergo a sudden structural phase transition in which they lose up to 20 percent of their volume. An archetypical material is cerium, in which the first-order phase transition is *isostructural,* resulting in about a 15 percent loss of volume. Although this volume collapse transition has been known for many years, and several competing theories exist, there has yet to be a definitive quantitative theoretical study.

Professor Richard T. Scalettar of UC Davis, Drs. Andrew K. McMahan and Roy Pollock of Lawrence Livermore National Laboratory, and I are studying this volume collapse in a unique collaboration which combines quantum Monte Carlo (UCD) and electronic structure calculation (Lawrence Livermore) techniques in a complementary way that exploits the strengths of each. This work is part of the U.S. Department of Energy's Accelerated Strategic Computing Initiative (ASCI). In addition to the intrinsic theoretical interest in understanding the volume collapse, there is also a practical interest in obtaining an accurate phase diagram that can be used to gain a quantitative understanding of the thermodynamic properties of a range of materials as part of the Department of Energy stockpile stewardship program.

The materials of interest have outlying f-orbitals and an inner d-orbital conduction band. The leading theory to explain the volume collapse involves hybridization between the outer f-orbitals and the inner d, conduction orbitals. Normally, the f-orbitals are relatively isolated and do not mix with the d-band electrons, except for quantum and thermal fluctuations. The f-band orbitals also have very little overlap with each other and do not contribute appreciably to the conductivity of these metals. The d-band orbitals, on the other hand, have a large overlap with neighboring d-band orbitals, and hence they form a conductive band which gives rise to metallic properties.

In the Kondo Volume Collapse Model, extreme pressure (about half the pressure at the center of the earth) causes the outer f-orbitals and the inner d-orbitals to overlap or *hybridize*. When this happens, the f and d electrons can mix. This mixing could occur in several ways, but experimental evidence indicates that when the mixing occurs the electron spins combine to form a singlet state, called a "Kondo singlet" by analogy with the Kondo impurity problem. When this singlet forms, the f-electrons are believed to spend more time in the inner region of the atoms. Hence, the atoms individually shrink and the material composed of many atoms shrinks.

This Kondo singlet is much like the singlet state studied in elementary quantum mechanics; it is the result of a linear combination of the wave functions of electrons on neighboring, hybridized sites. In elementary quantum mechanics one usually studies only two electrons, while in this three-dimensional material the singlet probably involves at least the six nearest neighbors. Nevertheless, the analogy is quite strong, and one can think of these Kondo singlets in terms of the singlets studied in undergraduate quantum mechanics. The volume collapse is characterized by strongly-correlated electrons, since it is associated precisely with this strong interaction between the f and d electrons when the Kondo singlets form.

Our approach is to first encapsulate the essential physics of this phenomenon in a model and identify the calculable quantities that will allow us to relate our theory to experiment. This model is the Anderson Lattice Model, which contains the following terms. The conduction, d-band is appropriately described as a weakly-interacting metallic conduction band. This part of the model has been extensively studied and is amenable to many-body perturbation calculations because of the weak interactions. The f-band is accurately described by a very weakly conducting set of orbitals dominated by a moderately strong on-site repulsion. Here, moderate repulsion means that the interaction is too strong to treat accurately with perturbation theory but not sufficiently strong that the electrons can be treated as localized spins. Finally, the f-d interaction is described by a term that allows f-electrons to hop to a d-orbital and vice versa. This f-d hybridization gives rise to the Kondo singlet formation that characterizes the Kondo Volume Collapse.

Our model involves various parameters corresponding to properties of the materials of interest. Electronic structure calculations provide us with precise values of these parameters. We then use quantum Monte Carlo simulations to calculate thermal averages of these quantities. We can relate these calculated quantities directly to the qualitative properties of the materials, such as the formation of the Kondo singlets as the f-d hybridization increases. We also can compare with quantitative measurements, using electronic structure calculations to relate our calculated quantities back to the physical properties of the materials.

Electronic structure calculations provide an accurate method to relate experimentally measured physical characteristics of the materials to the parameters of our model. However, Electronic structure calculations are based on a single-electron ("mean field") treatment in which electron interactions are treated as coming from an average, approximate background. Therefore, they generally fail to accurately calculate the effects of the strong electron correlations that dominate the physics of the Kondo Volume Collapse. The missing correlated electron information is supplied by quantum Monte Carlo, in which a model Hamiltonian, including all correlated electron effects, is treated without approximation.

We use the auxiliary field quantum Monte Carlo (AFQMC) method, also known as

determinant quantum Monte Carlo. In this method, the electron-electron interaction terms in the Hamiltonian, which plague perturbation calculations, are replaced by an exact mathematical transformation that decouples the electron interactions by introducing an extra, "auxiliary," field. Using standard field-theory techniques, we then integrate out the fermionic degrees of freedom (i.e., the electrons) and we are left with a statistical mechanics problem involving the 2^{2NL} possible states of this auxiliary field, where *N* is the number of sites in the lattice (*N* = 64 for a

4 x 4 x 4 lattice) and *L* is related to the temperature (typically 2 £
*L* £
8). This phase space is obviously too big to explore exactly, so we use Monte Carlo techniques to efficiently sample the phase space and obtain thermal expectation values of interesting quantities.

AFQMC involves many linear algebra operations and is highly computationally intensive. In the past, the computational cost has prohibited extensive studies of three-dimensional systems and limited studies to a few hundred electrons, at most. With the advent of massively parallel supercomputers it is now feasible to undertake extensive studies. This is due partly to raw computing power, but also to the inherent parallel nature of the AFQMC algorithm: We obtain an almost linear speedup in simulation time as we use more and more processors of a massively parallel computer. That is fortunate, as these simulations are so big that they can be performed only on fast supercomputers. We perform the simulations for this project on the fastest computer in the world, the 9,000 processor ASCI-Red computer at Sandia National Laboratories.

Our research plan is to first explore the limit where the f-orbitals do not overlap with each other and the d conduction orbitals do not have any on-site repulsive interaction. This is a generalization of the Kondo impurity problem, in which a single impurity interacts with a lattice of conduction electrons. In our case, we have a lattice of f-orbitals instead of a single impurity, so we expect a richer phase diagram due to subtle correlation effects. Nevertheless, in the limit that the f-d hybridization is small, the f-electrons should behave much like isolated magnetic moments and one may expect that the extensively studied single impurity model to be a good approximation to the full Anderson lattice model which we use.

In the first extensive quantum Monte Carlo study of the three-dimensional Anderson lattice model, we find good agreement with past work on the impurity problem. We also have obtained the first phase diagram of the 3d Anderson lattice model in this regime. We see the formation of the Kondo singlets as the pressure on the system is increased. We also identify a low-temperature, intermediate hybridization regime in which the f-electrons are aligned anti-ferromagnetically. This is expected both experimentally and theoretically, since even though the f-orbitals do not overlap, the f-electrons can become ordered by communicating via the d-orbitals through f-d hybridization (this is known as the "RKKY" interaction).

Our work continues as we now focus on the actual materials. Here prior work becomes less applicable, as the strong correlations make analytic techniques less possible. This is an exciting regime in which magnetic RKKY ordering, f-d singlet formation, f-f hopping, strong f-band on-site repulsion, normal d-band conduction, and the screened d-band on-site repulsion sometimes cooperate and sometimes compete in subtle ways. We expect new physics and a rich magnetic phase diagram. This is a wonderful opportunity for me as a student and for our research program generally to combine the ASCI challenge to add to the understanding of the thermodynamic properties of these materials with the basic research goal of understanding the essential physics of this phenomenon. I expect a continued stream of exciting results as we proceed to explore these systems.

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