# Solid-state physics: Probing the geometry of energy bands

June 01, 2016Scientists at Ludwig-Maximilians-Universitaet (LMU) in Munich and the Max Planck Institute for Quantum Optics (MPQ) have devised a new interferometer to probe the geometry of band structures.

The geometry and topology of electronic states in solids play a central role in a wide range of modern condensed-matter systems, including graphene and topological insulators. However, experimentally accessing this information has proven to be challenging, especially when the bands are not well isolated from one another. As reported by Tracy Li

*et al*. in last week's issue of

*Science*(

*Science*, May 27, 2016, DOI: 10.1126/science.aad5812), an international team of researchers led by Professor Immanuel Bloch and Dr. Ulrich Schneider at LMU Munich and the Max Planck Institute of Quantum Optics has devised a straightforward method with which to probe band geometry using ultracold atoms in an optical lattice. Their method, which combines the controlled transport of atoms through the energy bands with atom interferometry, is an important step in the endeavor to investigate geometric and topological phenomena in synthetic band structures.

A wide array of fundamental issues in condensed-matter physics, such as why some materials are insulators while others are metals, can be understood simply by examining the energies of the material's constituent electrons. Indeed, band theory, which describes these electron energies, was one of the earliest triumphs of quantum mechanics, and has driven many of the technological advances of our time, from the computer chips in our laptops to the liquid-crystal displays on our smartphones. We now know, however, that traditional band theory is incomplete.

Among the most surprising and fruitful developments in modern condensed-matter physics was the realization that band structure involves more than the just the electron energies - the geometric form of the bands also plays an important role. Indeed, this geometric contribution is responsible for much of the exotic physics in newly discovered materials such as graphene or topological insulators, and underlies a variety of exciting technological possibilities from spintronics to topological quantum computing. It is, however, notoriously difficult to access this information experimentally.

Now, an international team of researchers led by Immanuel Bloch (Professor of Experimental Physics at LMU Munich and a Director of the Max Planck Institute of Quantum Optics (MPQ)) has devised a straightforward method to probe band geometry using ultracold atoms in an optical lattice, a synthetic crystal formed from standing waves of light. Their method relies on creating a system that can be described by a quantity known as the Wilson line, and the experimental tests performed at LMU and the MPQ have verified that the technique allows one to explore the geometry of band structure.

Although originally formulated in the context of quantum chromodynamics, it turns out that Wilson lines also describe the evolution of degenerate quantum states, i.e., quantum states with the same energy. Applied to condensed-matter systems, the elements of the Wilson line directly encode the geometric structure of the bands. Therefore, to access the band geometry, the researchers need only to access the Wilson line elements.

The problem, however, is that the bands of a solid are generally not degenerate. However, the researchers realized that there was a way to get around this: When moved fast enough in momentum space, the atoms no longer feel the effect of the energy bands and their behavior is influenced only by the essential geometric information. In this regime, two bands with different energies behave like two bands with the same energy.

In their work, the researchers first cooled atoms to quantum degeneracy. The atoms were then placed into an optical lattice formed by laser beams to realize a system that mimics the behavior of electrons in a solid, but without the added complexities of real materials. In addition to being exceptionally clean, optical lattices are highly tunable -- different types of lattice structures can be created by changing the intensity or polarization of the light. In their experiment, the researchers interfered three laser beams to form a graphene-like honeycomb lattice.

Although spread out over all lattice sites the quantum degenerate atoms carry a well-defined momentum in the light crystal. The researchers then rapidly accelerated the atoms to a different momentum and measured the magnitude of the excitations they created. When the acceleration is fast enough, such that the system is described by the Wilson line, this straightforward measurement reveals how the electronic wave function at the higher momentum differs from the wave function at the initial momentum. Repeating the same experiment at many different crystal momenta would yield a complete map of how the wave functions change over the entire momentum space of the artificial solid.

The researchers not only confirmed that it was possible to move the atoms in such a fashion that the dynamics were described by two-band Wilson lines, the measurements at different momenta also revealed both the local, geometric properties and the global, topological structure of the bands. While the lowest two bands of the honeycomb lattice are known not to be topological, the results demonstrate that Wilson lines can indeed be experimentally used to probe and uncover the band geometry and topology in these novel synthetic settings.

-end-

Ludwig-Maximilians-Universität München

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