Three-dimensional topological insulators and bosonization

Journal of High Energy Physics, May 2017

Massless excitations at the surface of three-dimensional time-reversal invariant topological insulators possess both fermionic and bosonic descriptions, originating from band theory and hydrodynamic BF theory, respectively. We analyze the corresponding field theories of the Dirac fermion and compactified boson and compute their partition functions on the three-dimensional torus geometry. We then find 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.

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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)


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Andrea Cappelli, Enrico Randellini, Jacopo Sisti. Three-dimensional topological insulators and bosonization, Journal of High Energy Physics, 2017, pp. 1-50, Volume 2017, Issue 5, DOI: 10.1007/JHEP05(2017)135