🔬 condensed matter

Finite-temperature quantum topological order in three dimensions

This paper demonstrates that the three-dimensional fermionic toric code, through its anomalous 2-form symmetry, sustains long-range entanglement and a unique quantum topological order at finite temperatures, thereby establishing the existence of equilibrium phases of matter that only emerge at nonzero temperatures.

Original authors: Shu-Tong Zhou, Meng Cheng, Tibor Rakovszky, Curt von Keyserlingk, Tyler D. Ellison

Published 2026-07-16

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CC BY 4.0

Original authors: Shu-Tong Zhou, Meng Cheng, Tibor Rakovszky, Curt von Keyserlingk, Tyler D. Ellison

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of physics get a little weird, not because things are moving fast or are incredibly heavy, but because they are "entangled." In the realm of quantum physics, this is called topological order. Think of it like a giant, invisible knot tying a whole system together. If you have a piece of string tied in a complex knot, you can't untie it just by pulling on the ends; you have to cut the string or pass it through a loop. In these quantum materials, the "knot" is made of information shared between particles across the entire system. This is special because it's incredibly robust against small bumps or scratches, making it a holy grail for building super-powerful, error-proof quantum computers.

For a long time, scientists believed these magical knots could only exist in a perfect, frozen world at absolute zero temperature. The fear was that if you warmed up the system even a tiny bit, the heat would act like a chaotic crowd, tangling the string until the knot fell apart. It was like trying to keep a delicate origami crane folded while shaking it in a hurricane. Furthermore, experts thought that even if you could keep the knot, you'd need a universe with four or more dimensions to make it work, which is a bit of a problem since we only live in three. But what if there's a way to keep the knot tied even when things are warm? What if there's a special kind of material that stays "knotted" in our everyday 3D world, even when it's not frozen?

This is exactly the question tackled by Shu-Tong Zhou and their team in their new paper. They have discovered a theoretical model of a 3D material that manages to keep its quantum "knot" tied even at warm temperatures. They call this model the fermionic toric code. To understand how it works, imagine a 3D grid made of tiny switches (qubits). In most similar models, if you heat them up, the "knots" unravel because tiny, point-like glitches (excitations) start popping up everywhere and running wild. However, this new model has a secret weapon: a special kind of symmetry that acts like a rulebook for how these glitches can move.

The authors found that in their model, the point-like glitches are "fermions," which is a fancy way of saying they have a very specific, quirky personality: if you swap two of them, the whole system flips a sign, like a secret code changing from positive to negative. This personality creates a "2-form symmetry," which is a bit like a force field that only allows these glitches to move in specific loops. Crucially, this symmetry is "anomalous," meaning it's a bit broken or weird in a way that actually helps. The authors argue that this weirdness prevents the glitches from destroying the long-range entanglement (the knot) when the temperature is low but not zero.

Using a clever mathematical proof, they showed that below a certain temperature (specifically, below $T_0 \approx 1.24$ in their model's units), the thermal state of this material cannot be mimicked by any simple, un-entangled state. In other words, the "knot" is still there, and it's a genuine quantum knot, not just a messy classical pile of strings. This is a big deal because it proves, for the first time, that quantum topological order can exist in three dimensions at non-zero temperatures. It suggests that there might be a whole new phase of matter that only exists when things are warm, a phase that disappears if you freeze it to absolute zero.

The team also points out that while this material would be a terrible quantum computer at room temperature (it would act like a simple classical memory), it opens the door to a new way of thinking about quantum matter. They suggest that other materials with similar "anomalous" symmetries might also have this property. While they haven't built this material in a lab yet, their work proves that the laws of physics don't strictly forbid it. It's like finding a blueprint for a bridge that can float on water; just because we haven't built it yet doesn't mean it's impossible. This discovery challenges the old "no-go" theorems that said 3D quantum knots must melt in the heat, offering a glimmer of hope that we might one day harness these exotic states for real-world technology, provided we can find or engineer the right kind of "anomalous" symmetry to hold the knot together.

Technical Summary: Finite-Temperature Quantum Topological Order in Three Dimensions

Problem Statement
A central challenge in condensed matter physics is determining which topological phases of matter can persist in thermal states at nonzero temperatures. While zero-temperature topological orders are well-classified, it has long been established that in spatial dimensions $d < 4$ , standard topological orders (such as the 3D toric code) are destroyed at any nonzero temperature. This destruction occurs because point-like excitations proliferate, rendering the thermal states short-range entangled (SRE) and effectively classical. Previous "no-go" theorems suggested that finite-temperature quantum topological order—characterized by long-range entangled (LRE) thermal states—could only exist in four or more spatial dimensions. The specific question addressed by this work is whether a quantum topological order can exist in three dimensions at nonzero temperature, distinct from classical thermal phases.

Methodology
The authors investigate the fermionic toric code (fTC), a variant of the 3D toric code defined on a cubic lattice with qubits on edges. The model is distinguished from the standard (bosonic) toric code (bTC) by two key features:

Fermionic Point-like Excitations: The violations of vertex terms ( $A_v$ ) correspond to point-like excitations that obey fermionic exchange statistics.
Anomalous 2-form Symmetry: The model possesses a 2-form symmetry associated with space-like Wilson loops of these fermionic excitations. Crucially, this symmetry is "anomalous" because the string operators inheriting the fermionic self-statistics prevent the symmetry from being gauged without condensing the fermions.

The authors employ a proof strategy based on Quasi-Local Channels (QLCs) to analyze the entanglement structure of the thermal state $\rho_{fTC}$ at low temperatures ( $T < T_0$ ). The proof proceeds in three logical steps:

Loop Cleaning: They construct a QLC, $\mathcal{C}$ , composed of five layers of local operations. This channel systematically removes small loop-like excitations (violations of plaquette terms $B_p$ ) from the thermal state. The channel partitions the system into blocks of size $\ln(L)^2$ and applies unitary corrections to eliminate loops contained within or crossing specific seams.
Approximation to a Loop-less State: The authors demonstrate that at sufficiently low temperatures, the thermal state $\rho_{fTC}$ is mapped by $\mathcal{C}$ to a state $\rho_\emptyset$ (a state with exactly no loop excitations) with high fidelity ( $1 - O(L^{-\infty})$ ).
Symmetry Argument: They argue that the loop-less state $\rho_\emptyset$ possesses a "strong" anomalous 2-form symmetry. Citing recent results on symmetry-protected phases, they show that an SRE state cannot approximate a state with such an anomalous symmetry. Since QLCs map SRE states to SRE states, if the original thermal state $\rho_{fTC}$ were SRE, its image under $\mathcal{C}$ would also be SRE, leading to a contradiction.

Key Contributions and Results

Existence of 3D Finite-Temperature Topological Order: The paper provides the first example of a system in fewer than four spatial dimensions that exhibits quantum topological order at nonzero temperature.
Theorem 1: The authors prove that below a critical temperature $T_0 \approx 1.24$ (which is below the phase transition temperature $T_c \approx 1.31$ ), the fidelity between the thermal state of the fTC, $\rho_{fTC}$ , and any arbitrary SRE state $\sigma_{SRE}$ decays faster than any polynomial in the system size $L$ :
$F(\rho_{fTC}, \sigma_{SRE}) = O(L^{-\infty})$
This implies the trace distance approaches 1, confirming the state is not a classical mixture of SRE pure states.
Role of Anomalous Symmetry: The work identifies the anomalous 2-form symmetry as the specific mechanism protecting the long-range entanglement against thermal fluctuations. This contrasts with the bosonic toric code, where the 2-form symmetry is anomaly-free, allowing the thermal state to be SRE.
Distinction from Quantum Memory: The authors clarify that while the fTC acts as a quantum memory at zero temperature, it behaves as a classical memory at nonzero temperatures below the phase transition. The topological order here is a property of the mixed thermal state itself, not a protection of quantum information against local errors in the traditional memory sense.

Significance and Claims
The paper claims to resolve the question of whether 3D quantum topological order can exist at nonzero temperatures, answering in the affirmative. The significance lies in identifying a phase of matter that, within the context of equilibrium phases, only exists at nonzero temperatures.

The authors conjecture that this phenomenon is not unique to the fTC but extends to a broader class of discrete gauge theories possessing anomalous 2-form symmetries, such as Walker-Wang models based on premodular categories with transparent fermions. They suggest that these systems represent a new class of "mixed-state topological orders" that are distinct from pure-state topological orders.

The work opens the door to studying quantum topological order in physically realistic dimensions (3D) at finite temperatures, moving beyond the theoretical requirement of 4D or higher dimensions previously thought necessary for such stability. The authors note that while their specific temperature bound $T_0$ is derived from their specific cleaning algorithm, they expect the LRE phase to persist up to the critical temperature $T_c$ . They also identify the need for future work to develop efficiently computable diagnostics for this order and to identify physical systems (such as Kitaev-like models with 2-body interactions) that could realize this phase.

Technical Summary: Finite-Temperature Quantum Topological Order in Three Dimensions

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