11.5 Topological field theories and the quantum Hall effect

Topological field theories are quantum field theories that remain unchanged under smooth deformations of spacetime. They're crucial for understanding exotic states of matter like the quantum Hall effect, where electron behavior is governed by topology rather than geometry.

In the quantum Hall effect, conductance is quantized due to topological properties. This phenomenon is described by Chern-Simons theory, a type of topological field theory. Understanding these concepts is key to grasping advanced topics in quantum field theory.

Principles of Topological Field Theories

Fundamental Properties and Invariance

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• Topological field theories are quantum field theories invariant under smooth deformations of the spacetime manifold
  • Depend only on the topology of the manifold, not local geometry
• The action functional of a topological field theory is independent of the metric tensor
  • Results in a theory insensitive to local geometry
• Observables in topological field theories are topological invariants (linking number of loops, genus of surfaces)
  • Remain unchanged under continuous deformations

Partition Function and Hilbert Space

• The partition function of a topological field theory is a topological invariant
  • Encodes information about the global properties of the spacetime manifold
• Topological field theories have a finite-dimensional Hilbert space on a compact manifold
  • The dimension is given by the partition function
• Correlation functions of observables depend only on the topology of the spacetime manifold and the observables themselves
  • Not on the local details of the fields

Topological Field Theories and the Quantum Hall Effect

Quantum Hall Effect and Topological Properties

• The quantum Hall effect is a prime example of a physical system exhibiting topological properties
  • Can be described by a topological field theory
• In the quantum Hall effect, the Hall conductance is quantized in integer or fractional multiples of $e^2/h$
  • A topological invariant related to the Chern number of the filled Landau levels
• Edge states in the quantum Hall effect are chiral (propagate in a single direction along the boundary)
  • Their number is determined by the topological invariant (Chern number)

Effective Field Theory and Composite Fermions

• The bulk-boundary correspondence in topological field theories relates bulk properties to edge state behavior
  • Observed in the quantum Hall effect
• The effective field theory describing the low-energy excitations of the quantum Hall system is a Chern-Simons theory
  • A topological field theory in 2+1 dimensions
• The fractional quantum Hall effect can be understood in terms of composite fermions
  • Bound states of electrons and flux quanta
  • Their behavior is captured by a Chern-Simons theory

Topology and Geometric Phases in Condensed Matter

Topological Properties and Applications

• Topological properties of condensed matter systems (quantum Hall effect) are robust against local perturbations and disorder
  • Promising for applications in quantum computing and information processing
• The Berry phase is a geometric phase acquired by a quantum state evolving adiabatically along a closed path in parameter space
  • Plays a crucial role in describing topological properties in condensed matter systems
• The Chern number characterizes the topology of a band structure
  • Related to the Berry curvature (curl of the Berry connection, a gauge field associated with the Berry phase)

Bulk-Boundary Correspondence and Exotic States

• The bulk-boundary correspondence in topological insulators and superconductors relates bulk topological invariants to robust, gapless edge or surface states
• Classification of topological insulators and superconductors based on symmetries and dimensions has led to the discovery of exotic states of matter
  • Quantum spin Hall effect, topological superconductors
• Interplay between topology, geometry, and many-body interactions in condensed matter systems gives rise to emergent phenomena
  • Fractional statistics, anyonic excitations

Topological Field Theory for Exotic Matter States

Studying and Classifying Exotic States

• Topological field theories provide a powerful framework for studying and classifying exotic states of matter with topological properties
  • Topological insulators, topological superconductors, quantum spin liquids
• Effective field theories describing low-energy excitations of topological states often involve Chern-Simons terms
  • Capture topological properties and quasiparticle statistics
• Braiding statistics of anyonic excitations can be described using topological quantum field theory
  • Braiding matrices related to modular S and T matrices of the theory

Techniques and Applications

• Topological quantum field theories can compute ground state degeneracy and fusion rules of anyonic excitations in topological phases
• Bulk-boundary correspondence in topological field theories predicts and characterizes edge or surface states of topological insulators and superconductors
• Topological field theory techniques (surgery method, gluing formula) can be applied to study properties of topological states on various spacetime manifolds and compute topological invariants