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Long-Range Toric Code: Finite-Temperature Topological Order in Two Dimensions

This paper proposes two-dimensional toric code models with long-range loop interactions that preserve topological ground states while confining anyons at finite temperatures, thereby enabling finite-temperature topological order and self-correcting quantum memory.

Original authors: Jacopo Gliozzi, Taylor L. Hughes, Kyung-Su Kim

Published 2026-10-30

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CC BY 4.0

Original authors: Jacopo Gliozzi, Taylor L. Hughes, Kyung-Su Kim

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

In the quest to build a quantum computer, scientists face a fundamental problem: the information stored in these machines is incredibly fragile. Unlike a classical computer bit, which is simply a zero or a one, a quantum bit exists in a delicate superposition of states. To protect this fragile information, researchers look to a special state of matter called topological order. Imagine a pattern woven into a fabric; you can stretch or twist the fabric, but the pattern itself remains unchanged unless you cut the threads. In two-dimensional materials, this "pattern" is a form of order that lives in the global connections of the system rather than in local parts. At absolute zero temperature, this order is robust, acting as a perfect vault for quantum data. However, as soon as the temperature rises even slightly, heat creates tiny, mobile defects that scramble the pattern, destroying the stored information. For decades, physicists believed that in two dimensions, this thermal destruction was inevitable, forcing them to look to higher dimensions or complex error-correction software to save the day.

A team of researchers has now proposed a way to keep this topological order alive in two dimensions, even at warm temperatures, by changing the rules of how the particles in the material talk to one another. In their study, they designed a new model of matter based on the famous "toric code," a theoretical framework for topological order. The standard version of this model only allows particles to interact with their immediate neighbors. The new model, however, introduces long-range interactions, where particles can influence one another across the entire grid, with the strength of this influence fading slowly as the distance grows. By carefully tuning how this long-range force behaves, the researchers found that they could trap the destructive defects in place, preventing them from wandering and scrambling the quantum memory.

The researchers, working with a model called the long-range toric code, discovered that the behavior of the system depends entirely on how quickly the long-range force weakens with distance. They identified two distinct regimes. In the first regime, where the force weakens moderately, the system acts as a "partially self-correcting" memory. Here, the long-range interactions are strong enough to keep the destructive defects confined, but only up to a certain size. If the system is small, the defects are trapped, and the memory holds. If the system grows larger than a specific threshold, the thermal energy eventually overcomes the trap, and the defects escape. However, this threshold is so large that for any practical size, the memory remains stable for an extraordinarily long time, effectively acting as a perfect vault for a human lifetime.

In a second, more extreme regime, where the long-range force weakens even more slowly, the researchers found a truly remarkable result: a "fully self-correcting" quantum memory. In this scenario, the energy required to separate the destructive defects grows without bound as they move apart. It becomes energetically impossible for the heat to pull the defects apart, no matter how large the system is. Consequently, the defects remain locked in tight pairs, unable to roam freely and destroy the topological pattern. This means the quantum information is protected indefinitely, even as the system grows larger, without the need for any active error correction or external intervention. The researchers confirmed these findings through extensive computer simulations, observing that the system undergoes a clear transition from a disordered, high-temperature state to an ordered, low-temperature state where the memory is preserved.

The study also revealed that the movement of these defects is heavily restricted by the new long-range rules. In the fully protected regime, individual defects are essentially immobile, behaving like particles stuck in a solid lattice. They can only move if they absorb other specific clusters of defects created by the heat, a process that is so rare at low temperatures that it effectively never happens. This kinetic constraint acts as a natural shield, ensuring that the quantum information encoded in the system's global structure remains intact. The researchers showed that the state of this system at finite temperatures possesses long-range entanglement, a deep quantum connection that persists across the entire material, confirming that the topological order is genuinely present and not just a temporary illusion.

This work challenges the long-held belief that two-dimensional topological order cannot survive heat. By extending the interactions between particles to be long-range, the researchers have created a theoretical framework where a quantum memory can correct its own errors simply by virtue of its physical laws. While the model relies on interactions that are difficult to realize perfectly in current physical materials, it provides a clear blueprint for how to design robust quantum memories. The findings suggest that if scientists can engineer materials with similar long-range properties, perhaps using programmable quantum devices to simulate the interactions, they could finally build a quantum computer that stores information reliably without constant human oversight. The study does not claim to have built such a device yet, but it proves that the laws of physics allow for such a state of matter to exist, opening a new path toward stable quantum technology.

Technical Summary: Long-Range Toric Code: Finite-Temperature Topological Order in Two Dimensions

Problem Statement
Two-dimensional (2D) topologically ordered phases, such as the standard toric code, serve as quantum memories at zero temperature by encoding information nonlocally in a degenerate ground state manifold. However, at any finite temperature ( $T > 0$ ), these systems are unstable. Thermally excited anyons proliferate because the energy cost to separate them ( $\Delta E$ ) is finite, while the entropy gain ( $\Delta S \sim \ln L$ ) grows logarithmically with system size $L$ . This results in a negative free energy change ( $\Delta F = \Delta E - T\Delta S$ ), leading to the rapid destruction of stored quantum information. While 4D toric codes and certain 3D fracton models offer finite-temperature stability, constructing a thermodynamically well-defined 2D system with finite-temperature topological order has remained an open challenge.

Methodology
The authors introduce a class of Long-Range Toric Code (LRTC) models defined on a 2D Euclidean lattice. Unlike the standard toric code, which utilizes only local star and plaquette stabilizers, the LRTC incorporates additional stabilizers supported on large loops (specifically square loops) whose coupling strengths decay algebraically with the loop perimeter $\ell$ as $\ell^{-\alpha}$ .

The Hamiltonian is given by:
$H_{sq} = -\sum_{s \in S} \frac{J_B}{\ell(s)^\alpha} B_s - \sum_{\tilde{s} \in \tilde{S}} \frac{J_A}{\ell(\tilde{s})^\alpha} A_{\tilde{s}}$
where $B_s$ and $A_{\tilde{s}}$ are products of Pauli operators around square loops on the direct and dual lattices, respectively. Crucially, because these large-loop operators are products of elementary stabilizers, the ground state manifold remains identical to that of the ordinary toric code. The authors analyze the system through:

Analytical Energetics: Calculating the energy cost $\Delta E(d)$ of separating anyons by a distance $d$ at $T=0$ and $T>0$ , accounting for thermal screening by background anyon pairs or clusters.
Monte Carlo Simulations: Performing classical Monte Carlo simulations on the magnetic sector (due to the CSS structure of the Hamiltonian) to compute specific heat, logical error probabilities, and memory lifetimes across various system sizes $L$ and interaction ranges $\alpha$ .
Renormalization Group (RG) Decoding: Employing an RG decoder to test for long-range entanglement (LRE) in the Gibbs state by tracking the decay of anyon density under coarse-graining.

Key Contributions and Results

Zero-Temperature Confining Topological Order:
The authors demonstrate that for specific ranges of the interaction exponent $\alpha$ , the energy cost to separate anyons diverges with distance, confining them despite the topological ground state.
- Regime $2 < \alpha \le 3$ : Anyons are confined with a sublinear power-law ( $\Delta E \sim d^{3-\alpha}$ ) or logarithmic ( $\Delta E \sim \ln d$ ) energy cost.
- Regime $1 < \alpha \le 2$ : The system exhibits "super-confinement." Isolated anyon dipoles have diverging energy costs ( $\Delta E \sim d L^{2-\alpha}$ ), rendering them non-finite-energy excitations. Instead, the fundamental finite-energy excitations are compact anyon quadrupoles (four-anyon clusters). This regime also features an emergent subsystem symmetry where anyon parity is conserved on every row and column, rendering individual anyons immobile (fracton-like behavior).
Finite-Temperature Stability and Screening:
The paper analyzes how thermal backgrounds affect confinement:
- For $2 < \alpha \le 3$ : A thermal background of tightly bound anyon dipoles screens the confining potential beyond a large but finite length scale $L_{conf} \sim e^{\beta \Delta_2}$ . Consequently, topological order is only partially preserved up to this scale, yielding a partially self-correcting quantum memory.
- For $1 < \alpha \le 2$ : The thermal background consists of finite-energy quadrupoles. Crucially, these quadrupoles do not screen the confining potential at large distances because they insert an even number of anyons into large loops, preserving the loop sign. Thus, confinement persists at arbitrary scales, supporting finite-temperature topological order in the thermodynamic limit.
Quantum Memory Performance:
- Partially Self-Correcting ( $2 < \alpha \le 3$ ): The memory lifetime $t_{mem}$ grows with system size $L$ only up to $L_{conf}$ , after which it saturates. However, the saturation value is super-exponentially large in $\beta$ ( $t_{mem} \sim \exp[\exp(\beta)]$ ), offering extremely long storage times for accessible system sizes.
- Fully Self-Correcting ( $1 < \alpha \le 2$ ): The memory lifetime diverges as $L \to \infty$ ( $t_{mem} \sim \exp[\beta L^{3-\alpha}]$ ). The authors argue this is due to kinetic constraints: individual anyons cannot move without creating infinite-energy excitations, and logical errors (which require moving an anyon around a non-contractible loop) are dynamically suppressed.
Long-Range Entanglement (LRE):
Using an RG decoder, the authors show that for $1 < \alpha \le 2$ , the low-temperature Gibbs state is long-range entangled. The decoder successfully cleans up anyon defects at a quasi-local depth, mapping the thermal state to the ground state subspace. For $2 < \alpha \le 3$ , LRE signatures exist only up to the scale $L_{conf}$ .

Significance and Claims
The paper claims to present the first example of a thermodynamically well-defined 2D system that hosts finite-temperature topological order. By introducing long-range loop interactions, the authors circumvent the standard 2D locality obstruction that prevents self-correcting quantum memories.

The work establishes a direct analogy between the LRTC and the long-range Ising model, where long-range interactions stabilize order against thermal fluctuations. Furthermore, the authors highlight a striking parallel between the anyon dynamics in the $1 < \alpha \le 2$ regime and defect dynamics in 2D crystals (fracton-elasticity duality), where individual anyons behave as immobile fractons, dipoles as lineons, and quadrupoles as mobile excitations.

The authors modestly note that while they provide strong analytical and numerical evidence for a fully self-correcting memory in the $1 < \alpha \le 2$ regime, a rigorous proof of the diverging memory time remains challenging due to the non-local nature of the interactions and the complexity of entropic factors. They also leave the precise characterization of the mixed-state phase and the nature of the phase transition as directions for future research.

Technical Summary: Long-Range Toric Code: Finite-Temperature Topological Order in Two Dimensions

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