Topological BF field theory description of topological insulators

• Published in: Annals of physics Vol. 326; no. 6; pp. 1515 - 1535
• Main Authors: Cho, Gil Young, Moore, Joel E.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01-06-2011; Elsevier BV
• Subjects: ANGULAR MOMENTUM; BOSON EXPANSION; Bosonization; Chern–Simons theory; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ELECTRODYNAMICS; ELECTROMAGNETISM; ENERGY-LEVEL TRANSITIONS; EXCITATION; FERMIONS; FIELD THEORIES; Field theory; GAUGE INVARIANCE; HALL EFFECT; Halls; Insulators; Integers; INVARIANCE PRINCIPLES; MAGNETISM; MATHEMATICS; MATTER; PARTICLE PROPERTIES; Phases; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum Hall effect; Quantum physics; SPIN; STATISTICS; SURFACES; SYMMETRY; SYMMETRY BREAKING; Theory; Three dimensional; THREE-DIMENSIONAL CALCULATIONS; Topological insulator; Topological phase; TOPOLOGY; Two dimensional; TWO-DIMENSIONAL CALCULATIONS; Topological insulator; Quantum Hall effect; Bosonization; Chern–Simons theory; Topological phase
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2010.12.011

► We show that a BF theory is the effective theory of 2D and 3D topological insulators. ► The non-gauge-invariance of the bulk theory yields surface terms for a bosonized Dirac fermion. ► The “axion” term in electromagnetism is correctly obtained from gapped surfaces. ► Generalizations to possible fractional phases are discussed in closing.
Topological phases of matter are described universally by topological field theories in the same way that symmetry-breaking phases of matter are described by Landau–Ginzburg field theories. We propose that topological insulators in two and three dimensions are described by a version of abelian
BF theory. For the two-dimensional topological insulator or quantum spin Hall state, this description is essentially equivalent to a pair of Chern–Simons theories, consistent with the realization of this phase as paired integer quantum Hall effect states. The
BF description can be motivated from the local excitations produced when a
π flux is threaded through this state. For the three-dimensional topological insulator, the
BF description is less obvious but quite versatile: it contains a gapless surface Dirac fermion when time-reversal-symmetry is preserved and yields “axion electrodynamics”, i.e., an electromagnetic
E
·
B term, when time-reversal symmetry is broken and the surfaces are gapped. Just as changing the coefficients and charges of 2D Chern–Simons theory allows one to obtain fractional quantum Hall states starting from integer states,
BF theory could also describe (at a macroscopic level) fractional 3D topological insulators with fractional statistics of point-like and line-like objects.