This award supports theoretical research and education that aims to use recently developed concepts for describing topological states of electrons in materials to gain insight into how strongly interacting systems of electrons organize themselves in electrically conducting states. There are many different kinds of materials in the world, such as metals, insulators, and semiconductors. These materials make electronic devices, such as cell phones, computers, TV's, and even the internet possible. To design electronic devices with desired functions, theories are needed to describe and to predict the properties of metals, insulators, and semiconductors. As the theoretical physicist Landau discovered, states of matter such as liquids and crystals can be organized through the symmetry transformations performed on a given state that leaves it unchanged. For example, a rotation of 90 degrees around a principle axis of a crystal of common salt, can rotate new atoms into positions that were originally occupied by the very same kind of atom, so the crystal appears unchanged. The concept of symmetry also allows magnetic and other states to be organized by similar considerations. In recent years, it has been discovered that ideas from topology, the branch of mathematics concerned with geometric properties that are unchanged by deformations, twisting, and stretching objects, bring insight into new possible phases of matter, called topological phases. Topological insulators are a common example. They are fundamentally different from ordinary insulators in that while the bulk does not conduct electricity, their surfaces do, as if they belonged to a metal.
It is becoming understood that quantum entanglement is another underlying principle for materials, one which leads to a new class of quantum materials, also known as topological materials. Entanglement is a purely quantum property of a system that has no analog in our everyday experience. It reflects connections between the properties of a quantum system's parts, even if the parts become physically separated. The property is reflected in the structure of the many-electron state. The corresponding material theory -- the theory of topological order -- predicts a new class of insulators and semiconductors with new topological and quantum properties. These topological materials may play key roles in making quantum computers, just like silicon plays a key role in making commonly available conventional computers and cell phones of today.
This award supports theoretical research and education that aims to use modern fundamental concepts developed for topological states of matter to gain insight into how strongly interacting systems of electrons organized in gapless states. Research over the last 30 years reveals that Landau's symmetry breaking theory only describes a small set of possible phases that matter can have. The phases of matter can be much richer than has ever imagined before. The concepts of topological order and symmetry protected trivial (SPT) order to describe those new types of quantum phases. After much research, a systematic classification understanding of all topological orders and SPT orders for both bosonic and fermionic systems, in 1-, 2-, and 3-dimensional spaces has emerged. The time is now ripe to attack the next big problem: a systematic understanding of strongly correlated gapless states. This award supports the PI's research to engage this problem.
The PI takes the view that in general, an interacting system wants to be gapped, the most stable state. If an interacting system is gapless, the gapless state must be very special and highly organized, so that those gapless excitations can remain gapless even in the presence of interactions. This suggests that the general and systematic understanding of gapless quantum states is possible. First, the low energy part of a gapless state may become several decoupled sectors, where the interactions between different sectors flow to zero in the infrared limit under renormalization group flow. This appears to happen quite generally, such as the sectors with different velocities in 1d gapless system become decoupled at low energies. Consequently, in the low energy limit, there are often emergent symmetries and higher symmetries. Since each decoupled low energy sector is not a full system, each sector by itself is often anomalous. A sector by itself may have a gravitational anomaly or higher symmetry anomaly. It is well known that an anomaly can affect low energy dynamics, in particular, it can protect the low energy excitations with the result that they are gapless in some cases. The "gaplessness" of each decoupled sector may be understood via its anomaly. This may enable a systematic understanding of strongly correlated gapless states.
The strongly correlated gapless states should be much more complicated and much richer than strongly correlated gapped states. It may take some time to gain a fully systematic understanding of gapless states.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
