Contents

1. Overview
2. Etymology
3. Cultural Impact

Anyon

For other uses, see Anyon (disambiguation) .

Not to be confused with anion , a negatively charged ion.

Statistical mechanics

Particle statistics

Thermodynamic ensembles

Models

Potentials

Scientists

In the grand, often tedious, theatre of physics , an anyon emerges as a particularly elusive and intriguing type of quasiparticle , whose existence has, so far, been definitively observed exclusively within two-dimensional systems . This is a stark contrast to the more conventional and, frankly, less imaginative three-dimensional realms we inhabit, where only two fundamental categories of elementary particles are known to roam: the standoffish fermions and the gregarious bosons . Anyons, with their peculiar intermediate statistical properties, refuse to conform to this binary classification, carving out their own unique niche in the quantum landscape. They are, in essence, the quantum rebels of the flatlands.

Typically, in quantum mechanics, the mere act of exchanging two identical particles , while potentially inducing a global phase shift across the system’s wavefunction, is fundamentally forbidden from altering any observables – those aspects of a system that can actually be measured. Anyons, however, challenge this seemingly immutable rule in their two-dimensional habitat. They are broadly categorized into two types: abelian and non-abelian. It was the abelian anyons, those somewhat simpler, yet still exotic, cousins, that finally yielded to direct detection in two distinct experiments in 2020. These abelian anyons are not just theoretical curiosities; they play a profoundly significant role in explaining the intricate and often counterintuitive phenomena observed in the fractional quantum Hall effect , a testament to their tangible, if subtle, influence on the real (or rather, quasi-real) world.

Introduction

The foundational tenets of statistical mechanics , particularly when applied to vast, many-body systems, are typically governed by the well-understood principles of Maxwell–Boltzmann statistics . However, delving into the realm of quantum statistics reveals a considerably more intricate picture, largely due to the inherently different behaviors exhibited by the two primary classes of particles: fermions and bosons . Yet, as if to remind us that reality is rarely so neatly categorized, two-dimensional systems introduce a third, entirely distinct type of particle, the aforementioned anyon, adding a layer of complexity that few initially anticipated.

Consider, if you must, the familiar three-dimensional world that we all, regrettably, call home. Here, the particle taxonomy is refreshingly straightforward, almost to the point of being dull: you have “fermions,” which possess an inherent aversion to sharing space with their identical counterparts, and “bosons,” which, conversely, seem to enjoy congregating. A rather common example of a fermion, crucial for the very electricity that powers your mundane devices, is the electron. And, of course, the ubiquitous photon, the carrier of light, serves as a prime illustration of a boson. But venture into the theoretical, yet increasingly experimentally accessible, two-dimensional world, and you encounter the anyon—a particle that steadfastly refuses to align itself with either the aloofness of fermions or the camaraderie of bosons. It simply operates by its own, rather peculiar, rules.

“Finally, anyons reveal their exotic quantum properties,” Aalto University press release, April 2020

Indeed. After decades of theoretical musings and experimental prodding, these elusive entities decided to grace us with their presence.

In this peculiar two-dimensional existence, the very act of two identical anyons swapping positions fundamentally alters their collective wavefunction in ways that are simply impossible within the confines of three-dimensional physics . It’s as if the universe decided to throw a curveball just to keep physicists on their toes:

“…in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles’ wavefunction after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their wavefunction, leaving us with only two possibilities: bosons, whose wavefunction remains the same even after a single exchange, and fermions, whose exchange only changes the sign of their wavefunction.”
— Kirill Shtengel, “A home for anyon?”, Nature Physics

This profound distinction arises from the topological differences between 2D and 3D space. In three dimensions, you can continuously deform any path taken by a particle around another into any other path, as long as the start and end points are the same. This means there’s only one “kind” of exchange. In two dimensions, however, particles can effectively “wind” around each other in distinct ways—clockwise or counterclockwise—and these winding paths cannot be smoothly deformed into one another. This topological inequivalence is the root cause of anyons’ exotic behavior.

This intricate process of repeatedly exchanging identical particles, or, more poetically, of one particle gracefully circling another, is formally known as “braiding ”. When two anyons engage in this quantum dance of braiding, they don’t just move; they create a lasting “historical record” of their interactions. Their wave functions, subtly yet fundamentally altered by each braid, meticulously encode the precise number and nature of these exchanges. This inherent “memory” of past interactions is what makes them so profoundly interesting, particularly for applications that demand robust information storage.

It’s no surprise, then, that Microsoft has thrown considerable resources into the research surrounding anyons, viewing them as a tantalizing potential bedrock for topological quantum computing . The hope is that these quasiparticles could serve as a remarkably stable form of memory. The very act of anyons circling each other—their “braiding” interactions—could encode information in a manner far more resilient to environmental noise and decoherence than many other contemporary quantum computing technologies. This robustness stems from the topological nature of the braiding, meaning the information is encoded in the global properties of the paths, not local fluctuations. Despite this promising avenue, it’s worth noting that the lion’s share of current investment in quantum computing still gravitates towards methodologies that do not rely on the exotic properties of anyons, perhaps due to the inherent experimental challenges in wrangling these two-dimensional oddities.

History

Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to Dirac .
— Biedenharn et al., The Ancestry of the ‘Anyon’

Indeed, the foundations of what would eventually become the concept of anyons were laid by the likes of Dirac , whose profound insights often seemed to precede the experimental capabilities to verify them by decades. The journey from abstract theory to tangible observation is rarely straightforward, and the anyon’s story is a prime example of this protracted scientific pilgrimage.

The initial theoretical crack in the solid edifice of particle classification appeared in 1977. Two perceptive theoretical physicists laboring at the University of Oslo , Jon Magne Leinaas and Jan Myrheim , meticulously demonstrated that the entrenched, binary classification of particles as either fermions or bosons simply would not hold if these particles were confined to move solely within two dimensions . This groundbreaking work suggested the hypothetical existence of particles that defied conventional labels, expected to exhibit a diverse and previously unimaginable spectrum of properties. Their theoretical foresight paved the way for a new paradigm in particle statistics. Building upon this, in 1982, the visionary Frank Wilczek published two seminal papers delving into the concept of fractional statistics for these two-dimensional quasiparticles. He fittingly bestowed upon them the evocative name “anyons,” a moniker chosen to convey the profound idea that the phase shift incurred upon their permutation could, quite literally, take any value, not just the rigid ±1 of bosons and fermions.

The theoretical framework provided by Wilczek proved remarkably prescient and immediately useful. In the same year, 1982, Daniel Tsui and Horst Störmer made the pivotal discovery of the fractional quantum Hall effect . This complex phenomenon, observed in two-dimensional electron systems at extremely low temperatures and strong magnetic fields, found a compelling explanation in Wilczek’s mathematics. Bertrand Halperin at Harvard University quickly recognized the utility of anyons in deciphering various aspects of this effect. The connection was further solidified in 1985 when Frank Wilczek, alongside Dan Arovas and Robert Schrieffer , provided a rigorous verification. Their explicit calculations unequivocally predicted that the exotic particles responsible for the observed phenomena in these systems were, in fact, anyons, thereby providing a strong theoretical anchor for their existence. This convergence of theory and observation marked a critical turning point in the understanding of two-dimensional quantum systems.

Abelian Anyons

In the intricate world of quantum mechanics , and even in certain classical stochastic systems, a fundamental property of indistinguishable particles dictates that if you were to swap the quantum states of particle i with particle j (symbolically, ⁠ $\psi _{i}\leftrightarrow \psi _{j}{\text{ for }}i\neq j$ ⁠), the resulting many-body state would be measurably identical to the original. This principle underlies much of how we categorize particles.

To illustrate this, consider a quantum mechanical system comprising two indistinguishable particles. Let particle 1 occupy state ⁠ $\psi _{1}$ ⁠ and particle 2 reside in state ⁠ $\psi _{2}$ ⁠. In the convenient shorthand of Dirac notation , the system’s state is represented as ⁠ $\left|\psi _{1}\psi _{2}\right\rangle$ ⁠. Now, if we perform the conceptual exchange, swapping the states of these two particles, the system’s state would transform into ⁠ $\left|\psi _{2}\psi _{1}\right\rangle$ ⁠. Given that these particles are indistinguishable, these two resulting states must not exhibit any measurable difference. Consequently, they are, in essence, the same vector in Hilbert space , differing at most by a global phase factor :

$\left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle .$

Here, the term ⁠ $e^{i\theta }$ ⁠ represents this phase factor, a complex number of unit magnitude.

In the familiar vastness of three or more spatial dimensions, the universe, in its infinite wisdom (or perhaps just its inherent simplicity), restricts this phase factor to precisely two values: ⁠ $1$ ⁠ or ⁠ $-1$ ⁠. This elegant limitation is why all known elementary particles are neatly classified as either fermions, whose exchange introduces a phase factor of ⁠ $-1$ ⁠, or bosons, which blissfully retain a phase factor of ⁠ $1$ ⁠. These two fundamental types, as you might expect, exhibit profoundly different statistical behaviour . Fermions, for instance, rigidly adhere to Fermi–Dirac statistics , while bosons cheerfully abide by Bose–Einstein statistics . The significance of this phase factor is perhaps most strikingly demonstrated by the Pauli exclusion principle , which fermions obey with an almost militant precision: if two fermions were to attempt to occupy precisely the same quantum state, the resulting mathematical expression would be:

$\left|\psi \psi \right\rangle =-\left|\psi \psi \right\rangle .$

This equation implies that the state vector must be zero, which is mathematically nonsensical for a physical state (it’s not normalizable, hence unphysical). Thus, the Pauli exclusion principle, a cornerstone of atomic structure, is a direct consequence of the fermion’s -1 phase factor.

However, when we confine our gaze to two-dimensional systems, the rules become, shall we say, more accommodating. Here, quasiparticles have been theoretically predicted and now experimentally observed to exhibit statistics that span a continuous range between the strictures of Fermi–Dirac and Bose–Einstein statistics. This groundbreaking insight was first articulated by Jon Magne Leinaas and Jan Myrheim of the University of Oslo back in 1977. For a system of two such particles, their exchange relation can be expressed as:

$\left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle ,$

where ⁠ $e^{i\theta }$ ⁠ is no longer constrained to the binary choice of ⁠ $-1$ ⁠ or ⁠ $1$ ⁠. It can, in theory, take any value on the unit circle in the complex plane. It is crucial to acknowledge a subtle abuse of notation here; this shorthand expression implies a simplicity that belies the reality that the wave function can, and often is, multi-valued. More precisely, this equation signifies that when particle 1 and particle 2 are interchanged through a specific process—one where each particle executes a counterclockwise half-revolution about the other—the two-particle system’s quantum wave function returns to its original form, but with an additional multiplication by the complex unit-norm phase factor $e^{i\theta}$. Conversely, if this intricate dance involves a clockwise half-revolution, the wave function is multiplied by $e^{-i\theta}$. Such a nuanced theory, with its explicit reliance on directionality, only truly makes physical sense in two dimensions, where the concepts of “clockwise” and “counterclockwise” are unambiguously defined and topologically distinct.

When the angle $\theta = \pi$, we neatly recover the familiar Fermi–Dirac statistics ($e^{i\pi} = -1$). When $\theta = 0$ (or, equivalently, $\theta = 2\pi$), we arrive back at the Bose–Einstein statistics ($e^{2\pi i} = 1$). But it is in the vast expanse between these two extremes that we encounter something entirely novel and, frankly, far more interesting. Frank Wilczek , in 1982, was the one who meticulously explored the behavior of these unique quasiparticles and, with a flair for the descriptive, coined the term “anyon” to perfectly encapsulate their ability to possess any phase when particles are interchanged. Unlike their bosonic and fermionic counterparts, anyons exhibit the rather peculiar characteristic that if they are interchanged twice in precisely the same manner (for instance, if anyon 1 and anyon 2 perform a counterclockwise half-revolution to swap positions, and then execute another counterclockwise half-revolution to return to their original locations), the resulting wave function is not necessarily identical to the initial one. Instead, it is generally multiplied by some complex phase, specifically $e^{2i\theta}$ in this illustrative example.

We can also express this phase in terms of the particle’s spin quantum number, $s$, by setting $\theta = 2\pi s$. For bosons, $s$ is an integer , and for fermions, $s$ is a half-integer . This leads to the relation:

$e^{i\theta } = e^{2i\pi s} = (-1)^{2s},$

which simplifies the exchange relation to:

$|\psi _{1}\psi _{2}\rangle =(-1)^{2s}|\psi _{2}\psi _{1}\rangle .$

This formulation highlights how the continuous phase of anyons interpolates between the discrete values of bosons and fermions.

It’s also worth noting that in specific contexts, such as at the edge of a system exhibiting the fractional quantum Hall effect , anyons find themselves confined to movement within a single spatial dimension. Mathematical models developed for these one-dimensional anyons provide a robust theoretical foundation for the commutation relations we’ve discussed, offering a simpler, albeit still profound, environment to study their unique statistics.

In the familiar three-dimensional position space, the statistical operators for fermions and bosons (which are simply -1 and +1, respectively) can be understood as one-dimensional representations of the permutation group ($S_N$ for N indistinguishable particles) acting upon the space of wave functions. However, the situation transforms dramatically in two-dimensional position space. Here, the abelian anyonic statistics operators ($e^{i\theta}$) similarly act as one-dimensional representations, but not of the permutation group. Instead, they are representations of the braid group ($B_N$ for N indistinguishable particles), reflecting the topologically distinct ways particles can wind around each other. Furthermore, non-abelian anyonic statistics, which are even more complex, correspond to higher-dimensional representations of this same braid group. It is crucial to distinguish anyonic statistics from parastatistics , which, while also describing particles with more complex statistical behaviors, refers to wavefunctions that are higher-dimensional representations of the permutation group, rather than the braid group. The distinction, while subtle, is fundamental to understanding the topological nature of anyons.

Topological equivalence

The very fact that the homotopy classes of paths—essentially, the notion of equivalence when considering different braids —are so profoundly relevant to the behavior of anyons hints at a deeper, more subtle insight into the fabric of quantum reality. This understanding largely springs from the enigmatic Feynman path integral , a powerful theoretical construct in which all conceivable paths a particle could take from an initial to a final point in spacetime contribute, each weighted by an appropriate phase factor , to the overall quantum amplitude. The path integral itself can be rigorously derived and understood by employing a technique known as time-slicing, where time is conceptually discretized into infinitesimally small intervals, allowing for a step-by-step summation of contributions.

Within this framework, paths that belong to different homotopic classes are, by definition, those that cannot be continuously deformed into one another. This non-deformability means that one cannot transition from any point on one time slice to any other point on the very next time slice through paths belonging to different classes. Consequently, it becomes imperative to consider that these distinct homotopic equivalence classes of paths might inherently possess different weighting factors within the path integral formulation. This is not merely an arbitrary assignment but a reflection of the fundamental topological distinctions between these paths.

Therefore, it becomes abundantly clear that the topological notion of equivalence, far from being a mere mathematical abstraction, is an indispensable consequence derived directly from a meticulous study of the Feynman path integral . It’s a powerful demonstration of how deep mathematical structures underpin physical reality. For those seeking a more intuitively transparent illustration of why the homotopic notion of equivalence is indeed the “correct” one for describing these phenomena, one need look no further than the profound insights offered by the Aharonov–Bohm effect , where a charged particle is influenced by electromagnetic potentials even when it is in a region where both the electric and magnetic fields are zero, a phenomenon that is inherently topological.

Experiment

The pursuit of empirical validation for anyons, these elusive two-dimensional entities, culminated in a breakthrough year in 2020. Two independent teams of scientists—one operating out of Paris, the other situated in Purdue—each triumphantly announced compelling new experimental evidence for the existence of anyons. Such was the significance of these findings that both experiments were prominently featured in Discover Magazine ’s prestigious annual “state of science” issue for 2020, a clear indicator of their impact on the scientific community.

In April of 2020, researchers affiliated with the esteemed École normale supérieure (Paris) and the Centre for Nanosciences and Nanotechnologies (C2N) unveiled their findings. Their ingenious setup involved what they termed a “tiny particle collider” specifically engineered for anyons. This device, a marvel of nanoscale engineering, allowed them to precisely observe and measure the interactions of these quasiparticles. The critical outcome was that the properties they detected—the statistical signatures of the anyons—were in uncanny agreement with the theoretical predictions. This wasn’t a mere suggestion; it was a robust confirmation that theory had, once again, accurately foreseen a complex physical reality. The experiment involved creating a minuscule 2D anyon collider, so small that its inner workings necessitated observation via an electron microscope. This collider essentially contained a quantum Hall liquid, meticulously maintained within a powerful magnetic field, providing the ideal environment for anyon observation. The researchers measured fluctuations in the currents within this collider, finding that the anyons’ behavior precisely matched theoretical models, effectively demonstrating fractional statistics in anyon collisions.

A few months later, in July 2020, scientists at Purdue University published their own compelling evidence, employing an entirely different experimental methodology. Their team utilized a sophisticated interferometer, a device designed to precisely route electrons through an intricately etched, maze-like nanostructure composed of layers of gallium arsenide and aluminium gallium arsenide . This particular setup was meticulously designed to filter out any extraneous interactions that might otherwise obscure the subtle behavior of the anyons. The lead researcher reported that “In the case of our anyons the phase generated by braiding was 2π/3.” He emphasized the significance of this value, adding, “That’s different than what’s been seen in nature before.” This fractional phase change, a hallmark of anyonic statistics, provided independent and unambiguous confirmation of their existence. The experiment directly observed anyonic braiding statistics, showing a clear fractional statistical phase, a direct deviation from the integer or half-integer phases of bosons and fermions.

As of 2023, the study of anyons remains a vibrant and exceptionally active area of research, continually pushing the boundaries of quantum physics. Demonstrating the rapid pace of advancement, Google Quantum AI, utilizing a superconducting processor, reported in an arXiv article by Andersen et al. in October 2022, on the first experimental braiding of non-Abelian anyon-like particles. This significant work was subsequently published in the prestigious journal Nature, solidifying its place in the scientific record. Not to be outdone, in an arXiv article released in May 2023, Quantinuum, using a trapped-ion processor, also reported on the observation of non-abelian braiding, showcasing the diverse experimental platforms being brought to bear on this challenging problem.

Adding yet another layer to the narrative, researchers at the University of Innsbruck observed anyons in a one-dimensional quantum system in May 2025. This further expands the contexts in which these exotic particles can be studied, demonstrating that their peculiar statistics are not solely confined to strictly two-dimensional planes.

Non-abelian Anyons

Unsolved problem in physics

Is topological order stable at non-zero temperature ?

Fusion of Anyons

In much the same way that two individual fermions (for instance, two particles each possessing a spin of 1/2) can be collectively regarded as a composite boson (with their total spin existing in a superposition of 0 and 1), a collection of two or more anyons, when brought together, can form a composite anyon. This composite entity might, surprisingly, even exhibit the statistics of a boson or a fermion, depending on the specifics of its constituent parts. This process, where individual anyons combine to form a larger, unified entity, is formally referred to as the fusion of its components. It’s a fundamental interaction that dictates how these exotic particles combine and interact.

If, for example, we consider a system where $N$ identical abelian anyons, each characterized by an individual statistical phase $\alpha$ (meaning the system acquires a phase $e^{i\alpha}$ when any two of these individual anyons undergo an adiabatic counterclockwise exchange), all decide to fuse together, the resulting composite entity will possess a collective statistical phase of $N^2\alpha$. This can be intuitively understood by considering the interactions upon a counterclockwise rotation of two such composite anyons about each other. In this scenario, there are precisely $N^2$ distinct pairs of individual anyons—one from the first composite anyon and one from the second—that each contribute their characteristic phase $e^{i\alpha}$ to the total. A similar, analogous analysis can be applied to the fusion of non-identical abelian anyons, where the statistics of the resulting composite anyon are uniquely and deterministically derived from the statistics of its constituent components.

Non-abelian anyons, as one might expect from their name, exhibit significantly more intricate and complex fusion relations. As a general rule, within a system populated by non-abelian anyons, the statistical label of a composite particle is not uniquely determined by the statistics labels of its individual components. Instead, the composite particle’s state exists as a quantum superposition of various possible statistical outcomes. This phenomenon is entirely analogous to how two fermions, each known to possess spin 1/2, can collectively exist in a quantum superposition of total spin 1 and total spin 0. Furthermore, even if the overall statistics of the fusion of an entire collection of anyons is known, there can still be considerable ambiguity regarding the precise fusion outcome for subsets of those anyons. Each of these unresolved possibilities corresponds to a unique quantum state. These multiple, distinct quantum states within a degenerate subspace are not merely theoretical curiosities; they provide a natural and robust Hilbert space upon which the operations of quantum computation can be performed, forming the very basis of topological quantum computing.

Topological basis

Anticlockwise rotation

Clockwise rotation

Exchange of two particles in 2+1 spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).

The visual representation above, depicting the exchange of two particles in 2+1 spacetime (two spatial dimensions plus one time dimension) via rotation, starkly illustrates the core topological distinction: anticlockwise and clockwise rotations are fundamentally inequivalent. One cannot be smoothly deformed into the other without forcing the particles’ worldlines to exit the two-dimensional plane, an act that is, by definition, impossible in a 2D space. This topological constraint is the bedrock upon which anyonic statistics are built.

In the familiar realms of more than two spatial dimensions, the spin–statistics theorem stands as a fundamental pillar, dictating that any multiparticle state composed of indistinguishable particles must adhere strictly to either Bose–Einstein or Fermi–Dirac statistics. This rigid dichotomy arises from the mathematical properties of the underlying symmetry groups. For any spatial dimension $d > 2$, the relevant Lie groups SO($d$,1) (which serves as the generalization of the Lorentz group that governs relativistic symmetries) and Poincaré($d$,1) (the full spacetime symmetry group) share a crucial characteristic: their first homotopy group is isomorphic to $Z_2$. Since this cyclic group $Z_2$ contains only two elements, it leaves precisely two possibilities for particle statistics, elegantly explaining the fermion-boson binary. (While the full mathematical details are, as always, considerably more involved, this topological property is the critical takeaway.)

However, the situation undergoes a profound transformation in two dimensions. Here, the first homotopy group of SO(2,1), and similarly for Poincaré(2,1), is isomorphic to $Z$ (the infinite cyclic group). This seemingly minor difference has colossal implications: it means that Spin(2,1), the spin group that typically serves as the double cover of the special orthogonal group SO(2,1), is not the universal cover . In simpler terms, the space is not simply connected , allowing for an infinite number of topologically distinct paths when particles are exchanged. Specifically, there exist projective representations of SO(2,1) that cannot be derived from ordinary linear representations of SO(2,1) or even its double cover, Spin(2,1). These additional representations are precisely what anyons embody, representing a richer spectrum of spin polarization for charged particles than previously thought possible.

This concept isn’t confined solely to relativistic quantum field theory; it extends, with equal validity, to nonrelativistic systems. The pertinent point here is that the spatial rotation group SO(2)—the group describing rotations in a two-dimensional plane—also possesses an infinite first homotopy group. This topological property, allowing for indefinite winding, is the fundamental reason why anyonic statistics can exist even in nonrelativistic contexts.

This profound mathematical fact is also intimately connected to the braid groups , a concept well-established within the mathematical discipline of knot theory . The relationship becomes clear when one considers that in two spatial dimensions, the group describing permutations of two particles is no longer the simple symmetric group $S_2$ (which only contains two elements, representing a single swap or no swap). Instead, it is the far more complex braid group $B_2$, which possesses an infinite number of elements. The essential distinction lies in the fact that in two dimensions, one particle can effectively wind around another, an operation that can be performed an infinite number of times, in both clockwise and counterclockwise directions, each distinct path contributing uniquely to the system’s quantum state. This topological winding is what fundamentally differentiates anyons from their conventional bosonic and fermionic counterparts.

A completely different, yet highly promising, approach to tackling the notorious stability and decoherence challenges inherent in conventional quantum computing involves the development of a topological quantum computer . This innovative paradigm leverages the unique properties of anyons, which are envisioned as “threads” that encode quantum information through their braiding patterns. By relying on braid theory —the mathematical framework governing these topological interactions—this approach aims to create inherently stable quantum logic gates that are robust against local perturbations, offering a potentially revolutionary path towards truly fault-tolerant quantum computation.

Generalization to higher dimensions

While fractionalized excitations, when considered as point particles, are strictly limited to being bosons, fermions, or anyons exclusively in 2+1 spacetime dimensions (two spatial dimensions plus one time dimension), the situation becomes more constrained in higher dimensions. It is a well-established principle that point particles can only ever be either bosons or fermions in 3+1 and all higher spacetime dimensions. The topological constraints simply don’t allow for the “in-between” statistics of anyons for point-like excitations in these higher-dimensional spaces.

However, this restriction on point particles does not apply to extended objects. Loop- (or string-) or membrane-like excitations, which are inherently extended rather than point-like, possess the fascinating potential to exhibit fractionalized statistics even in higher dimensions. These extended objects introduce new topological possibilities that are absent for point particles.

Current cutting-edge research indicates that loop- and string-like excitations indeed exist for certain topological orders within the 3+1 dimensional spacetime. Crucially, their multi-loop/string-braiding statistics serve as the definitive key signatures for identifying and characterizing these complex 3+1-dimensional topological orders. This means that while point particles are confined to the binary of bosons and fermions, these extended quantum entities can, through their intricate braiding, reveal a richer statistical landscape. The multi-loop/string-braiding statistics associated with these 3+1-dimensional topological orders can be elegantly captured by the sophisticated link invariants of particular topological quantum field theories defined in 4 spacetime dimensions. To put it in more accessible terms, the extended objects—be they loops, strings, or even membranes—can, in principle, exhibit anyonic behavior in 3+1 and higher spacetime dimensions, particularly within long-range entangled systems . This opens up entirely new avenues for exploring exotic phases of matter beyond the conventional classifications.