Introduction

Exotic phases of matter that defy classification methods have revolutionised condensed matter physics in recent decades. In the classical Landau paradigm, symmetry breaking and local order parameters explain phases, but quantum phases are not fully explicable. One of the most remarkable examples is topological quantum order (TQO), which arises from the global, topological features of quantum many-body systems rather than symmetry.

Fault-tolerant quantum computing and uncommon phenomena like fractional quantum Hall states, quantum spin liquids, and topological insulators depend on topological quantum order. TQO is non-local, unlike traditional order. The many-body wavefunction's entanglement structure gives it topology-dependent ground state degeneracy, long-range quantum entanglement, and exotic quasiparticle excitations called anyons.

This page details TQO's philosophical foundations, mathematical framework, physical realisations, experimental signatures, and vital role in quantum computation.

Limitations of the Landau Paradigm

Historically, Landau's symmetry-breaking theory dominated matter's phase classification. In this structure:

Phases are characterised by order parameters that imply broken symmetry, such as ferromagnet magnetisation.

Shifting symmetries cause phase transitions.

Water freezing destroys crystal lattice rotational symmetry, and magnet cooling below its Curie temperature disrupts rotational spin symmetry. The 1980s discovery of the quantum Hall effect had a flaw: integer and fractional quantum Hall states were different phases with the same symmetries. They were identified using global topological invariants like the Chern number, not local order parameters. This showed Landau's framework was weak despite its strength.

Define Topological Quantum Order

In the early 1990s, Xiao-Gang Wen coined topological quantum order (TQO) to describe such unique phases. TQO highlights include:

Ground State Degeneration

On a torus or higher-genus manifold, TQO systems have degenerate ground states.

The spatial topology determines the degeneracy, not microscopic details. Long-Range Tangled

Local unitary transformations cannot erase wavefunction entanglement.

This distinguishes topological order from band insulators and other short-range entangled states.

Anyonic Excitations

Like anyons, quasiparticles interpolate between bosons and fermions in TQO phases.

The braiding statistics encode topological information.

Robust topology

Local disturbance-resistant features include degeneracy and braiding statistics.

This makes TQO desirable for fault-tolerant quantum processing.

Math Framework

Topological Field Theories

Topological quantum field theories (TQFTs) like Chern-Simons theory can describe various TQO systems. For instance:

A U(1) Chern-Simons theory at level mmm describes the fractional quantum Hall effect at filling fraction v=1/m\nu = 1/mv=1/m.

Wilson loop operators encode anyon braiding statistics.

Entanglement and Tensor Network Characterisation

Topological phase wavefunctions are often modelled using tensor networks, especially PEPS. Entanglement matters:

Topological entanglement entropy (TEE): S=αL−γ+…, where γ\gammaγ encodes topological order via universal correction.

The value of 𝛾γ affects the entire quantum dimension of anyonic theory.

Modular Tensor Categories

Modular tensor categories (MTCs) can formalise algebraic anyon fusing and braiding. Math formulas represent:

The laws of fusion (anyon mixing).

Braiding statistics stages.

Modular S and T matrices define topological order.

TQO Physicalizations

A fractional quantum hall effect

The fractional quantum Hall (FQH) states are still the classic TQO realisation. A filling fraction of ν=1/m yields the Laughlin wavefunction:

Quasiparticles with partial charges.

Anyonic braiding statistics.

Topological degeneracy on nontrivial manifolds.

Spin-quantum liquids

Anderson (1973) postulated that quantum spin liquids (QSLs) are magnetic systems without long-range magnetic order at zero temperature. QSLs like the Kitaev honeycomb model show:

Emerging gauge fields.

Majorana fermions—fractionalized excitations.

Some non-Abelian anyon regimes.

Superconductors, Topological Insulators

Topological insulators and superconductors have exotic topological boundary states, although these are called symmetry-protected topological (SPT) phases rather than intrinsic TQO. They can cause TQO phases with interactions.

Man-made systems

Cold atoms, photonic lattices, and superconducting qubits enable the toric code model and other TQO Hamiltonian-like constructed systems.

TQO in Quantum Computation

One of the most exciting TQO applications is topological quantum computation. The main ideas:

Qubit Encoding in Degenerate Ground States

Keeping information in non-local degrees of freedom prevents local errors.

Quantum gates with anyonic braiding

Move anyons to execute unitary operations on encoded qubits.

Kitaev's honeycomb model and the Moore Read Pfaffian state predict powerful non-Abelian anyons.

Design-Based Fault Tolerance

Local disturbances cannot destroy encoded data.

This integrated error correction method differs from quantum error correction codes.

Wen's toric coding model supports Abelian anyons, is totally solvable, and may be used to evaluate quantum error correction.

Test Signatures

Experimental detection of TQO is difficult due to its lack of local order. Instead, researchers use indirect indicators:

Hall conductivity in fractional quantum Hall states.

Anyonic statistics are studied by interference experiments.

Measurements of numerical simulation entanglement entropy.

Spin liquid candidates utilising NMR and neutron scattering. Topological superconductors' Majorana zero modes are studied using tunnelling spectroscopy.

Advances in interferometry, quantum simulators, and ultracold atom settings make direct observation of TQO features conceivable.

Open Questions and Future Plans

Despite great advances, TQO research still faces many challenges:

The Classification Problem

Despite modular tensor classifications, 2D and 3D TQO phases are still unclassified.

Non-Abelian Anyons in Experiments

Non-Abelian anyons, needed for quantum computation, are hard to prove.

High Dimensions

Although little is known about TQO in 3D systems, it may reveal complicated structures like fractons with limited mobility.

Interaction with Symmetry

Symmetry-enriched topological phases create new opportunities by integrating TQO with global symmetries.

Quantum Simulators

Cold atoms, confined ions, and superconducting qubits can simulate TQO Hamiltonians, enabling new experiments.

In conclusion

Topological quantum order redefines phases and phase transitions, making it a major discovery in modern physics. It relies on quantum matter's global entanglement structure and transcends symmetry breakdown. Quantum computation, quantum spin liquids, and the fractional quantum Hall effect were inspired by TQO.

Topically Quantified Field Theories

New Quantum Realm Order: Innovative Models Promise Stronger Quantum Computing

A recent finding in theoretical physics has revealed a new class of models that display behaviours previously thought to be outside the typical descriptions of topological quantum field theories (TQFTs). This is challenging the conventional understanding of topological quantum order (TQO). This innovative discovery offers a compelling vision for quantum computing's future with quantum memories that are more thermally robust than surface codes.

Understanding TQFTs

Topological Quantum Field Theories (TQFTs) have long been thought to describe topological quantum order (TQO). Their properties, especially anyon excitations, are similar to TQO-exhibiting systems. It is generally known that a microscopic model with TQO renormalises to a specific TQFT at the low-energy limit, allowing ground-state degeneracies to be calculated. Essentially metric-independent, TQFTs describe a zero-energy subspace.

Category Theory nicely describes Michael Atiyah's rigorous axiomatisation of TQFTs, inspired by Edward Witten. They are functors that keep their monoidal structure from bordisms to vector spaces.

Bordisms (sometimes termed cobordisms) are higher-dimensional manifolds that describe “space” at a specific moment and are “morphisms” between lower-dimensional manifolds. Bordisms are orientated manifolds of one higher dimension with the disjoint union of Σ1 and Σ2 as their border. Closed, orientated (n-1)-manifolds are bordisms, and their equivalence classes are morphisms. Bordism's tensor product operation is disjoint union. Categories are built from “objects” (sets, vector spaces, topological manifolds) and “morphisms” (structure-preserving applications). Applications between categories that preserve composition and identity morphisms are factors. In an associative, commutative, unit-element category, a “tensor product” operation is called symmetric monoidal structure. The usual vector space tensor product. The TQFT factor preserves these features, making it “symmetric monoidal”.

Classification and comprehension are easier with TQFTs' geometric and topological manifold invariants. These theories physically tie (d+1)-dimensional manifolds, which describe spacetime processes, to operators between Hilbert spaces and d-dimensional manifolds, which represent space, to Hilbert spaces (quantum states). This idea motivates TQFTs to link general relativity with quantum theory. Z(Σ) has the same dimension as a TQFT Z applied to a product manifold Σ × S1.

Challenges to Common Purview: Topological Order Beyond TQFT

Contrary to common perception, 'Topological Order Beyond Quantum Field Theory' has proven that the lowest-energy excitations of these new theories are not invariably linked to a finite number of localised anyons of a TQFT. These gapped lowest-energy excitations are anyons that densely cover the system, not TQFTs.

These new models are distinguished by their distance-dependent interactions between anyons, which are expected in realistic experimental setups. However, axiomatically defined TQFTs are metrically independent. Even at infrared wavelengths, the new TQO systems' anyon excitations become metric dependent. The system obviously demonstrates TQO, but this metric dependency and the dense occupation of anyons prevent these theories' low-energy region from being related to a regular TQFT.

The researchers examined these models by performing exact dualities on traditional (Landau) ordering systems. This innovative method transfers Landau-type theories to dual models with topological order while maintaining spatial dimension. The star-plaquette product model (SPPM), a Zq (q≥2) extension of the Kitaev toric code (TC) stabiliser Hamiltonian, received special attention.

Dual high-dimensional classical systems with large free-energy barriers are produced by the SPPM and its extensions, which include stabiliser group elements beyond independent generators. This makes them more thermally stable than the standard Zq TC model, which is dual to decoupled 1D chains and thermally fragile Non-Abelian, string-net, and higher-dimensional models can use this approach, not simply SPPMs.

Quantum Computing Implications

The most important physical consequence of this research is that it may lead to heat-resistant quantum memory. Surface codes reduce errors well, but thermal noise and temperature increase error rates. Current research may be able to construct quantum memory structures that are less susceptible to heat disruptions, enabling more reliable and stable quantum information storage at higher temperatures. These models realise unstudied quantum codes.