Topological order

Topological order. A phase of matter distinct from conventional symmetry-breaking phases, characterized by long-range entanglement and global properties robust against local perturbations. It was first identified in the context of the fractional quantum Hall effect and lacks a local order parameter, being instead described by topological invariants. This order supports exotic excitations like anyons and has profound implications for quantum computation and the classification of quantum phases.

Definition and basic concepts

The concept was pioneered through the study of the fractional quantum Hall effect by Robert Laughlin and later formalized by Xiao-Gang Wen and Michael Levin. It is defined by patterns of long-range entanglement in a system's ground state wavefunction that cannot be adiabatically connected to a trivial product state. Unlike phases described by Landau's symmetry-breaking theory, it does not spontaneously break a symmetry and thus lacks a local order parameter. Key hallmarks include a ground state degeneracy that depends on the genus of the manifold and the existence of quasiparticles with fractional statistics. The stability of these phases arises from an energy gap to excitations and the topological nature of the ground state manifold, which is insensitive to local deformations.

Examples and classification

The most prominent experimental realization is the fractional quantum Hall state, observed in systems like those studied at the MIT and Bell Labs. Theoretical models include the toric code introduced by Alexei Kitaev and chiral spin liquids analogous to the Laughlin wavefunction. Classification efforts, involving tools from algebraic topology and category theory, distinguish between abelian states, like those in the toric code, and non-abelian states, which are predicted for certain quantum Hall plateaus like the ν=5/2 state. Other examples include Z₂ spin liquids and the Kitaev honeycomb model, with connections to work by Shoucheng Zhang on topological insulators.

Physical properties and detection

A definitive property is a ground state degeneracy that is robust and depends on the topology of the underlying space, as conceptualized in the toric code on a torus. These systems exhibit a gap to all local excitations but possess gapless edge states described by conformal field theory, as seen in the chiral edge modes of the quantum Hall effect. Detection methods often rely on probing non-local order: the fractional quantum Hall effect is signaled by quantized Hall conductance plateaus, while anyon statistics may be inferred from interference experiments like those proposed for the ν=5/2 state. Other signatures include topological entanglement entropy, a contribution to the von Neumann entropy scaling with system boundary, and quantized thermal Hall conductance.

Mathematical description

Mathematically, it is described by topological quantum field theories such as Chern-Simons theory, which captures the low-energy physics of the fractional quantum Hall effect. The universal properties are encoded in topological invariants like the ground state degeneracy and the braiding statistics of excitations, organized into a modular tensor category. Lattice models, notably the exactly solvable toric code, provide a discrete framework where the ground state subspace is defined by the homology of the lattice. The entanglement structure is quantified by measures like the topological entanglement entropy, introduced by Kitaev and John Preskill.

Applications and related phases

A major application is in topological quantum computation, where non-abelian anyons can be used to implement fault-tolerant quantum gates, as initially proposed by Kitaev. This connects to the study of Majorana fermions in systems like the Kitaev chain. Related phases of matter include symmetry-protected topological phases like the quantum spin Hall effect discovered in HgTe quantum wells, and topological insulators such as those studied in Bi₂Se₃. The concepts also influence the understanding of high-temperature superconductivity via resonating valence bond theory and are explored in engineered systems like Rydberg atom arrays and superconducting circuits. Category:Condensed matter physics Category:Quantum mechanics Category:Topology