Illinois Physicists Build a Stability Test That Sorts Quantum Phases an Older Method Lumps Together

Physicists at the University of Illinois Urbana-Champaign have developed a way to classify quantum phases of matter in systems that interact with their surroundings, a problem that has blocked progress for years. The framework, published in Physical Review X by researchers at the Anthony J. Leggett Institute for Condensed Matter Theory, can tell apart phases that an older classification method wrongly groups together.

Ordinary phases such as solids, liquids and gases are sorted by symmetry, a paradigm pioneered by Lev Landau in the 1930s. Since the 1980s, physicists have known that many phases also need topology, the mathematics of global shape. A coffee cup can be smoothly reshaped into a doughnut, so the two share the same class, and each class is labeled by numbers called topological invariants. Topological phases have invariants too, and they hold up when local details are disturbed. In these phases, information is stored nonlocally, spread across the whole system through entanglement, and they can host exotic particle-like excitations called anyons. Their resilience makes them candidates for quantum computers, which are notoriously sensitive to noise.

For isolated systems, called closed systems, physicists have good tools. Their quantum states are tractable pure states, and two states belong to the same phase if their parent Hamiltonians can be connected without closing the energy gap. "The most important idea in defining a phase of matter is that it should be stable," said Illinois physics professor Jong Yeon Lee. "If you perturb the state slightly, it still stays in the same phase."

Real systems are not isolated. They decohere, turning into mixed states, which are statistical blends. Out-of-equilibrium mixed states have no Hamiltonian, so the standard recipe cannot be applied. The Illinois team turned to a different approach called entanglement bootstrapping, which defines stability through fixed points, states that look the same at every length scale. Researchers find them by coarse-graining, or zooming out, and watching how a system's parameters flow. Lee's group, joined by postdoctoral fellow Bowen Shi, asked whether that method could be extended to mixed states.

They devised three conditions. The first, called M0, requires that two separated regions of matter do not much affect each other, so a disturbance cannot spread across the system like a highway pileup. The second, P0, requires that if information in a small region is lost or corrupted, the surrounding neighborhood can recover it, reflecting the nonlocal storage of topological information. The third, M1, adds technical requirements that keep the states mathematically workable. Together, holding at all length scales, they define a fixed point. Lee noted that for pure states the three conditions reduce to those used in existing pure-state bootstrapping, so the new scheme includes the old one.

The team also derived quantities, collectively called topological data, that distinguish fixed points and serve as invariants for labeling phases. To define whole phases, they relaxed the stability conditions so they only need to hold approximately near a fixed point. Two mixed states belong to the same phase if an intermediate boundary can be built at which the deviation from the criteria falls off exponentially fast as one coarse-grains. Put simply, if two states look the same when you zoom out, apart from an exponentially small difference, they are the same phase.

The researchers confirmed their definition satisfactory by implementing coarse-graining numerically and measuring how far the stability criteria deviate from the fixed points. The upshot, according to the Illinois summary, is that scientists can now distinguish a wider range of quantum phenomena in open systems than before, which matters for building topological quantum devices that must work in noisy, real-world conditions.