Three-dimensional topological insulators and bosonization
Received: January
Three-dimensional topological insulators and
Andrea Cappelli 0 1 2 3
Enrico Randellini 0 1 2 3
Jacopo Sisti 0 1 3
0 Via Bonomea 265 , 34136 Trieste , Italy
1 Via G. Sansone 1, 50019 Sesto Fiorentino
2 INFN , Sezione di Firenze
3 Open Access , c The Authors
cScuola Internazionale Superiore di Studi Avanzati (SISSA), topological insulators possess both fermionic and bosonic descriptions, originating from band theory and hydrodynamic BF theory, respectively. We analyze the corresponding eld theories of the Dirac fermion and compacti ed boson and compute their partition functions on the three-dimensional torus geometry. We then nd some non-dynamic exact properties of bosonization in (2+1) dimensions, regarding fermion parity and spin sectors. Using these results, we extend the Fu-Kane-Mele stability argument to fractional topological insulators in three dimensions.
bDipartimento di Fisica e Astronomia; Universita di Firenze
-
Bosonic topological insulators
Hydrodynamic BF e ective action
Surface bosonic theory
Canonical quantization of the compacti ed boson in (2+1) dimensions
Bulk topological sectors and boundary observables
Bosonic partition functions
Spin sectors of the bosonic theory
Bosonization in (2+1) dimensions
Modular transformations
Dimensional reduction
Fermion parity in the bosonic theory
Bosonic Neveu-Schwarz and Ramond sectors in (2+1) dimensions
Neveu-Schwarz sector
Ramond sector
1 Introduction
Fermionic topological insulators
2.1 Introduction: surface states and anomaly cancellation
Torus partition functions
Flux insertions and stability argument
Neveu-Schwarz sector
Ramond sector
Modular transformations
Stability and modular invariance
Dimensional reduction
Massless case 2 = 0
Massive case 2 = 1=2
Stability of bosonic topological insulators
Modular transformations
The topological phases of matter [1{4] have been described by several approaches, such
as wavefunction modeling [5], band theory [6{10] and e ective
eld theory of boundary
excitations [11, 12], whose interplay has been extremely rich and fruitful. In this paper, we
analyze (3 + 1)-dimensional time-reversal invariant topological insulators using eld theory
The main motivation of our study is the success of the eld theory approach for
(2+1)-dimensional topological states, where exact methods are available for describing
the one-dimensional edge excitations, most notably those of conformal eld theory [13].
In several instances, these methods give access to strongly interacting dynamics and make
use of powerful symmetry principles. The rich modeling of quantum Hall states has been
applied to the quantum spin Hall e ect and then to time-reversal invariant topological
insulators [14{17].
In particular, the Z2 characterization of stability for topological insulators, originally
derived within band theory by Fu, Kane and Mele [18{23], has been reformulated in
eldtheory language and extended to interacting fermion models with Abelian [14{17] and
nonAbelian [24, 25] fractional statistics of excitations. The Z2 stability analysis also extends
to (3 + 1)-dimensional band insulators and it is interesting to
nd the corresponding eld
theory argument for analyzing interacting systems. In this paper, we shall present results
in this direction.
More generally, the theoretical methods in (3 + 1) dimensions are facing the problem
of bosonization, namely that of nding correspondences between two seemingly di erent
That of fermionic theories, dealing with band structures and topological e ects
related to Berry phases, and leading to the ten-fold classi cation of non-interacting
topological states [6{10].
That of bosonic theories, also called hydrodynamic approach, dealing with
topological gauge theories and their description of braiding relations and boundary
excitations [12, 26{31].
Bosonization is an exact map in (1 + 1)-dimensional eld theories that is very well
understood [13]; thus, the above interplay does not cause any problem for (2 + 1)-dimensional
topological states. The bosonic approach can provide exact results for interacting systems
and well as the methods for discussing bulk wavefunctions and braiding statistics [1, 5].
In this paper, we review and develop both the fermionic and bosonic eld theory
descriptions of massless surface states for time-reversal invariant topological insulators in
(3 + 1) dimensions. Our main method is the study of partition functions on the space-time
geometry of the three-torus and their behaviour under ux insertions and modular
transformations, namely for large gauge transformations of the electromagnetic and gravitational
backgrounds [24, 32]. In the fermionic theory, we study the free Dirac excitations at the
surface of topological insulators [33]. In the bosonic approach, we analyze the BF
topological gauge theory and the associated surface excitations, described by the compacti ed
boson eld in (2 + 1) dimensions [12, 34]. We then quantize this theory.
Althou (...truncated)