Classifying non-equilibrium phases of matter

Illinois physicists devise a new way to define and categorize non-equilibrium mixed-state phases of matter, providing an alternative definition that solves issues plaguing existing classification methods.

Most of us recognize basic phases of matter and their properties: solids maintain their shape, liquids flow freely with constant volume, and gases expand to fill their container. Quantum phases, on the other hand, are much harder to categorize, overturning traditional notions of what a phase even is and forcing physicists to rethink their definitions.

Even so, physicists have made substantial advances in classifying quantum phases for isolated systems. Yet categorizing those for open, non-equilibrium systems—those that freely interact with their environment—has presented numerous technical challenges, hindering our understanding of fundamentally new quantum behaviors and implementing them for quantum-computing applications. 

Now, in a new paper published in the journal Physical Review X on Oct. 2nd, 2026, Illinois physicists at the Anthony J. Leggett Institute for Condensed Matter Theory have invented a framework for classifying non-equilibrium quantum phases of matter, generalizing principles governing closed systems. Their work overcomes the shortcomings of current classification schemes, enabling scientists to distinguish between a wider range of quantum phenomena than ever before.

How do we classify phases?

Ordinary phases of matter are generally classified by their symmetries. Liquids, for instance, look roughly the same from any direction and have a high degree of symmetry, whereas crystalline solids possess symmetries only along well-defined axes, a difference showing that they’re distinct phases. This classification paradigm, pioneered by Lev Landau in the 1930s, has extraordinary explanatory power, describing everything from solids, liquids, and gases to magnets and even superconductors.

Since the 1980s, however, physicists began realizing that many phases can’t be explained by symmetry alone. They also require topology, a branch of mathematics that studies fundamental, global properties of shapes while ignoring their local, small-scale details. To give an oft-cited example, although seemingly different, a coffee cup can be mathematically “massaged,” or continuously deformed, into a donut like clay, showing that they’re globally identical and belong to the same class of shapes. Each class is characterized by so-called topological invariants, special numbers such as the number of holes a shape possesses, that don’t change under continuous deformations. On the other hand, making a new, distinct topological shape requires a discontinuous change such as tearing—a banned operation in topology.

Like the topological shapes they correspond to, topological phases are distinguished by their invariants too, maintaining their order when locally perturbed by external interactions. Physically, they arise when temperatures are brought so low that quantum fluctuations, once masked by thermal jostling, emerge to produce system-wide quantum entanglement.

In these phases, information gets stored nonlocally so that it's “smeared out” across the entire system. Particle-like excitations called anyons that don’t fall into the familiar boson-fermion paradigm can also arise, a phenomenon that physicists are only just beginning to understand. And because of their resilience, topological phases could form the basis for new quantum-computing technologies, which are notoriously vulnerable to environmental noise.

How are these phases classified? Phases in closed quantum systems, those isolated from their environment, are typically described using tractable quantum states called pure states. Classifying pure-state phases conventionally rests on the structure of their gapped Hamiltonians, mathematical objects that encode their states’ energies and impose energy gaps between the states.

Illinois Physics Professor Jong Yeon Lee emphasized, “The most important idea in defining a phase of matter is that it should be stable. It should be defined in such a way that if you perturb the state slightly, it still stays in the same phase.

“For pure states, this notion is well defined: two states belong to the same phase if their local parent Hamiltonians can be connected without closing the energy gap, while an unavoidable gap closing signals a phase transition.”

Sadly, pure states are often idealizations. In the real world, they interact with their environment and degrade, or decohere, turning into unpredictable statistical messes called mixed states. And unlike pure states, out-of-equilibrium mixed states don’t have Hamiltonians, so classifying their phases using the conventional approach doesn’t work.

Another way in: Bootstrapping

Luckily, there’s another classification scheme: entanglement bootstrapping. In this procedure, one concocts stability criteria that define fixed points, quantities of a system that look the same at different length scales. Physicists hunt for a system’s fixed points by zooming out to large length scales, or coarse-graining, to observe how the system’s parameters change.

“As you coarse-grain further and further,” Lee elaborated, “each point in the phase diagram converges to a fixed point. We can think of phases of matter as perturbations away from these fixed points.”

Such points are indicators of stability, serving as anchors for defining phases. Once identified, one can move away from fixed points and look for regions of stable quantum states, which can be collectively defined as phases.

This bootstrapping procedure—identifying fixed points through stability criteria then defining phases as sets of stable quantum states around fixed points—has found much success in classifying pure-state phases. Naturally, Lee’s team wondered whether the same bootstrap approach could be extended to mixed states, an effort that gained momentum when Illinois Physics Postdoctoral Fellow Bowen Shi joined Lee’s group.

“The entanglement-bootstrap program provides a complementary way of looking at the same physics,” Lee asserted. “But instead of starting from a Hamiltonian, it asks how much topological order we can reconstruct directly from the entanglement structure of a given quantum state.

“This philosophy is especially useful for our mixed-state problem, suggesting that we can identify a small set of information-theoretic properties that play a similar role as the Hamiltonian.”

Stability criteria & topological data

To find fixed points for mixed-state phases, the first step in bootstrapping, the researchers looked for ways to eliminate correlations, which suppress stability, and incorporate the right topological ingredients that such phases require. They devised three conditions to capture these ideas.

The first condition, M0, ensures stability by requiring that any two physically separated regions of matter, A and C, don’t affect each other much ((b) in diagram). Otherwise, perturbations could use the regions’ long-range correlations to propagate throughout the system, destabilizing the phase, similar to how a car crash on a highway can propagate from car to car, causing a pile up if they're too jam-packed.

Another condition, P0, demands that if the information encoded in a local region C is lost or corrupted, its surrounding neighborhood B can recover this lost information ((a) in diagram). This condition accounts for the notion that topological information is stored nonlocally, independent of local details, so that the information is spread across the system E through long-range entanglement.

Finally, condition M1 imposes mathematical technicalities on quantum states to ensure they’re tractable enough to talk about the right topological physics.

These three conditions hold at all length scales and together define fixed points, so that any quantum state satisfying all three simultaneously is stable. Moreover, Lee added, “If we limit our scheme to pure states, then our three conditions become equivalent to those used in pure-state bootstrapping. So our framework includes pure-state classification as well.” 

The researchers also derived several important quantities, collectively known as topological data, to distinguish and characterize different fixed points. Not only do these data provide measures of the topological information content contained in the fixed points, they’re also topological invariants, providing a way to label phases once they’re defined.

Defining mixed-state phases

To finally define whole phases as part of bootstrapping’s second step and move away from fixed points, the researchers allowed their stability criteria to relax. States near fixed points can’t be expected to be as stable as the fixed points themselves, but they should be stable enough. That is, the stability conditions need not hold exactly, just approximately.

With this in mind, the researchers made the following definition: if one can build an intermediate boundary between two mixed states at which the deviation from the stability criteria drops off exponentially fast as you coarse-grain, then the states belong to the same phase. Simply put, if two phases look the same when you zoom out—except for perhaps an exponentially small difference—then they’re essentially the same phase.

To confirm this definition is satisfactory, the researchers implemented coarse-graining numerically, quantifying the degree to which the stability criteria deviate from the fixed points as one coarse-grains away from them. They discovered that the deviation indeed decays exponentially with coarse-graining, showing that the states are, in fact, stable enough to define phases.

 

Significantly, this definition for mixed-state phases is compatible with the topological data: as a result of topological invariance, if two mixed states have different values of a topological quantity, then they belong to different phases—a quick, surefire test to identify and label distinct phases without having to establish the stability criteria from scratch.

Comparing approaches

How does this scheme compare to others? The leading method for classifying mixed-state phases relies on the idea of a finite-depth local channel (FDLC), a kind of communication link between two quantum states. According to this approach, two states belong to the same phase if there exists an FDLC that maps one state to the other, and another FDLC that maps that state back to the first.

Upon closer inspection, however, this FDLC approach breaks down: In their paper, Lee’s team gives an example of two states—a generic product state and the maximally dephased toric code—that physically belong to different phases even though the FDLC definition incorrectly says they should belong to the same phase.

“The FDLC approach doesn’t necessarily preserve topological structure,” Lee noted. “It seems researchers forgot this subtlety and naively extended this idea to describe mixed states.”

Mixed-state bootstrapping, by contrast, correctly categorizes the product state and toric code into different, distinct phases, in exact agreement with what is observed physically.

Lee continued, “Our method diagnoses what goes wrong: The channel changes the underlying topological structure, violating one of our stability criteria. Instead, bootstrapping gives us intrinsic diagnostics rather than defining a phase solely through the existence of a particular preparation protocol.”

On to new topological data, phases, and transitions

With this new classification scheme, the researchers have surmounted major hurdles in frontier condensed matter theory, most notably the perennial inability to distinguish clearly different quantum phases that have hampered even the most inventive methods.

And they’re not done yet. Lee’s team is actively hunting for more topological data to complete their framework. Computing certain topological invariants exactly, for instance, isn’t always easy or even possible. And whereas states having different values of a given invariant denote different phases, the converse is not true in general, so that states sharing the same value could still belong to different phases. A comprehensive lineup of topological data would give physicists additional tools to constrain and better distinguish between phases.

Armed with this new method, Lee’s team is looking to experimentally incorporate mixed-state topological order into actual devices. They’re also exploring specific kinds of phases and phase transitions, particularly those near critical points, where different types of correlations blow up.

“What excites me most is that this work points toward a new way of defining phases of matter directly from how quantum information is organized in the state, both in and out of equilibrium,” Lee said. “At the same time, the present bootstrap framework is not yet universal—for example, it does not naturally capture fracton phases. I see that limitation as an important clue. Ultimately, I hope to develop a broader framework that can encompass these more exotic forms of quantum matter as well.”

This research was primarily supported by faculty startup funds from the University of Illinois Urbana-Champaign. Additional support was provided by the Taiwan-UIUC Scholarship Program and the Elite Dream Project Grant of the Veterans and Dependents Foundation; the Perimeter Institute for Theoretical Physics; the National Science Foundation under Award No. PHY-2337931; and the IBM-Illinois Discovery Accelerator Institute. This work also used the Illinois Campus Cluster, a computing resource operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois Urbana-Champaign.