Fractional S -duality, classification of fractional topological insulators, and surface topological order

Abstract

In this paper, we propose a generalization of the S-duality of four-dimensional quantum electrodynamics (QED 4 ) to QED 4 with fractionally charged excitations, the fractional S-duality. Such QED 4 can be obtained by gauging the U(1) symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle (θ) can take a nontrivial but still time-reversal invariant value π/t 2 (t ∈ Z). Here, 1/t specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and U(1) global symmetry (fermion number conservation) are fractional topological insulators (FTI). We propose a topological quantum field theory description, which microscopically justifies the fractional S-duality. Then, we consider stacking operations (i.e., a direct sum of Hamiltonians) among FTIs. We find that there are two topologically distinct classes of FTIs: type-I and type-II. Type-I (t ∈ Z odd ) can be obtained by directly stacking a non-interacting topological insulator and a fractionalized gapped fermionic state with minimal charge 1/t and vanishing θ. But type-II (t ∈ Zeven) cannot be realized through any stacking. Finally, we study the Surface Topological Order of fractional topological insulators. CONTENTS

Quantization of a self-dual conformal theory in (2 + 1) dimensions

“…This theta term is consistent with Dirac quantization condition provided that the minimal electric charge of the system is [14]:…”

Surfaces and slabs of fractional topological insulator heterostructures

“…Alternatively it can acquire a many-body interacting mass while preserving both symmetries, and exhibit long-ranged entangled surface topological order [5][6][7][8] . On the other hand, fractional topological insulators (FTI) [9][10][11][12][13][14][15] are long-range entangled topologically ordered electronic phases in (3 + 1) dimensions outside of the single-body mean-field band theory description. They carry TR and charge U (1) symmetries, which enrich its topological order (TO) in the sense that a symmetric surface must be anomalous and cannot be realized non-holographically by a true (2 + 1)-D system.…”

Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks

“…We reveal various interesting properties of these topological orders related to quantum entanglement, and their notable survival at finite temperatures (in contrast to fractional quantum Hall states). Earlier studies have focused on the attachment of charge to U(1) monopoles, giving rise to dyons in high energy physics [42][43][44][45] and magnetoelectric effect in condensed matter physics [20][21][22][23][24][25][26][27][28][29][30]46 ; we reproduce some of their results here for completeness. However, we stress that hedgehogs are more physically accessible than monopoles since the spin-orbit coupling in topological materials naturally tends to stimulate their existence.…”

“…The incompressible quantum liquids of monopoles are more constrained than those of hedgehogs due to the fact that charge attached to a monopole nucleates a quantized angular momentum in the surrounding electromagnetic field 54 . Nevertheless, we are not restricted by time-reversal symmetry and hence the monopole liquids we discuss are less constrained than those considered in the recent literature 30 . If the U(1) symmetry emerges at low energies in a purely magnetic system, the obtained fractionalized states are chiral spin liquids with different topological orders than the more familiar resonant-valence-bond (RVB) spin liquids.…”

“…On the purely theoretical front, the topological field theory we construct is a variant of the "background field" (BF) theory 20,29,30,46 with antisymmetric tensor gauge fields. Here we emphasize a new ingredient of such theories, the "linking" Lagrangian terms.…”