Three-dimensional topological insulators and bosonization

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

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“…In Ref. 60, an action is proposed that is a special case of Eq. (16) with r = 1 (therefore, t = N = K).…”

Section: Topological Quantum Field Theorymentioning

confidence: 99%

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

Cheng

Fradkin

2017

Phys. Rev. B

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“…because its quantized integral (10) extracts the topological charge enclosed within a sphere S d−1 in the same fashion as the integral I d treated to Stokes-Cartan theorem in (91). The topological current (75) similarly emerges by coarse-graining…”

Section: B Topological Term From Spinor Fieldsmentioning

confidence: 94%

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

Nikolić

2020

Phys. Rev. B

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Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. Homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups πn(S n ), n ≥ 1. These homotopies govern monopole and hedgehog topological defects in d = n + 1 dimensions, and enable the generalization of both weakly-interacting and fractional quantum Hall liquids of vortices to d > 2. Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and d = 3 topological materials. We predict novel topological orders in systems with U(1)×Spin(d) symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the RVB spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment. CONTENTS F. Braiding of multi-dimensional excitations 46 G. Fractional braiding statistics 48 H. The electromagnetic angular momentum of charges and monopoles 49 I. Dyon braiding 50 References 52

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“…Besides classical applications such as solving certain many body quantum models exactly [24,25] or overcoming sign problems in Monte Carlo studies [26][27][28], bosonization has been invoked in the study of problems in quantum computation [29,30] and topological phases of matter [31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Bosonization of Majorana modes and edge states

Bochniak,

Ruba,

Wosiek

2021

Preprint

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We present a bosonization procedure which replaces fermions with generalized spin variables subject to local constraints. It requires that the number of Majorana modes per lattice site matches the coordination number modulo two. If this condition is not obeyed, then bosonization introduces additional fermionic excitations not present in the original model. In the case of one Majorana mode per site on a honeycomb lattice, we recover a sector of Kitaev's model. We discuss also decagonal and rectangular geometries and present bosonization of the Hubbard model. For geometries with a boundary we find that certain fermionic edge modes naturally emerge. They are of different nature than edge modes encountered in topological phases of matter. Euclidean representation for the unconstrained version of a spin system of the type arising in our construction is derived and briefly studied by computing some exact averages for small volumes.

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Three-dimensional topological insulators and bosonization

Cited by 14 publications

References 85 publications

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

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

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

Bosonization of Majorana modes and edge states

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