Composite Dirac Liquids: Parent States for Symmetric Surface Topological Order

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Composite Dirac Liquids: Parent States for Symmetric Surface Topological Order

David F. Mross, Andrew Essin, and Jason Alicea

Phys. Rev. X 5, 011011 – Published 5 February 2015

Abstract

We introduce exotic gapless states—“composite Dirac liquids”—that can appear at a strongly interacting surface of a three-dimensional electronic topological insulator. Composite Dirac liquids exhibit a gap to all charge excitations but nevertheless feature a single massless Dirac cone built from emergent electrically neutral fermions. These states thus comprise electrical insulators that, interestingly, retain thermal properties similar to those of the noninteracting topological insulator surface. A variety of novel fully gapped phases naturally descend from composite Dirac liquids. Most remarkably, we show that gapping the neutral fermions via Cooper pairing—which crucially does not violate charge conservation—yields symmetric non-Abelian topologically ordered surface phases captured in several recent works. Other (Abelian) topological orders emerge upon alternatively gapping the neutral Dirac cone with magnetism. We establish a hierarchical relationship between these descendant phases and expose an appealing connection to paired states of composite Fermi liquids arising in the half filled Landau level of two-dimensional electron gases. To controllably access these states we exploit a quasi-1D deformation of the original electronic Dirac cone that enables us to analytically address the fate of the strongly interacting surface. The algorithm we develop applies quite broadly and further allows the construction of symmetric surface topological orders for recently introduced bosonic topological insulators.

Received 22 October 2014

DOI:https://doi.org/10.1103/PhysRevX.5.011011

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

David F. Mross, Andrew Essin, and Jason Alicea

Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

Popular Summary

Experimentalists routinely identify three-dimensional topological insulators by detecting an odd number of massless Dirac cones formed by electrons at their boundaries. These peculiar surface states comprise a rich playground for exploring physics that is fundamentally impossible to realize in strictly two-dimensional media. Introducing magnetism or superconductivity, for instance, generates unconventional gapped surface phases that one cannot simply “peel away” from the topological insulator bulk. Strong electron-electron interactions, which often strikingly invalidate predictions based on free-particle descriptions of matter, constitute another potentially fruitful source of exotic topological-insulator surface physics. On the theoretical front, studies of this correlated arena aim to further elucidate the elaborate interplay between interactions and topology in quantum systems. At the same time, the advent of strongly correlated topological insulators underscores the need for theories that transcend the well-studied noninteracting band theory paradigm.

We uncover a new class of correlated topological-insulator surface states that we refer to as “composite Dirac liquids.” Here, strong interactions localize the surface electrons without breaking any symmetry or ruining the topological nature of the bulk material. This localization necessarily requires a highly unusual (fractionalized) surface featuring new emergent particles. In a composite Dirac liquid, these are electrically neutral fermions—not electrons—that form an odd number of Dirac cones. Consequently, the surface behaves like a conventional insulator for electrical current but nevertheless exhibits metallic heat transport—a striking experimental signature. Additional nontrivial surface states with still more exotic, non-Abelian particles arise simply by Cooper pairing the neutral fermions.

Our results reveal interesting connections to quantum Hall physics and suggest new strategies for accessing novel surface phases in various topological materials.

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Vol. 5, Iss. 1 — January - March 2015

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Figure 1
Executive summary of results. Stripping the electric charge off of the original surface Dirac cone (top left) yields a composite Dirac liquid (top center) that exhibits a charge gap but features a gapless Dirac cone formed by electrically neutral fermions. Analogously stripping a fictitious pseudocharge from the latter yields a nested composite Dirac liquid (top right) with a second-generation neutral Dirac cone. These composite Dirac liquids serve as parent states for topologically ordered surface phases (bottom row) obtained by gapping the neutral Dirac cones with pairing or magnetism. Importantly, pairing proceeds without breaking electric charge conservation—because the paired fermions are neutral—and hence produces symmetric surface topological orders captured by previous works [13, 14, 15, 16].
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Figure 2
(a) Strongly interacting coupled-chain setup analyzed in this paper. Ferromagnetic strips of alternating magnetization reside on a 3D topological insulator surface. Within each domain the electrons are gapped and exhibit a surface Hall conductance of $σ_{x y} = \pm e^{2} / (2 h)$ . The domain walls, however, bind an array of gapless 1D electron modes with staggered chirality. Importantly, the surface remains invariant under the symmetry $\tilde{T}$ corresponding to time reversal composed with a translation by half a unit cell. (b) The same set of gapless 1D modes arising in a strictly 2D setup consisting of $ν = 1$ integer quantum Hall strips separated by insulators. Contrary to the surface in (a), here $\tilde{T}$ symmetry is always explicitly broken.
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Figure 3
(a) Magnetic domains of a TI surface overlaid with alternating $ν = + 1 / 2$ and $- 1 / 2$ fractional quantum Hall fluids. The quantum Hall strips cancel the half-integer Hall conductance from the magnetically gapped domains, so that $σ_{x y} = 0$ everywhere. Each domain wall thus hosts a chiral electron native to the TI surface together with an opposing pair of quantum Hall charge modes. (“Spectator” neutral modes arising from the $ν = \pm 1 / 2$ states are not shown for simplicity.) When interactions open a charge gap, a set of alternating-chirality neutral fermions remains as in (b). Tunneling between these modes that preserves $\tilde{T}$ symmetry yields the neutral Dirac cone characteristic of the CDL state.
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Figure 4
(a) Interface between the symmetric T-Pfaffian surface topological order and a trivial magnetically gapped region in the extrinsic construction. To construct the T-Pfaffian the neutral fermions from Fig. 3 are decomposed into a pair of Majorana modes $γ_{1, y}$ and $γ_{2, y}$ . The Majoranas then gap out by hybridizing with counterpropagating modes as indicated by wavy lines. An unpaired chiral Majorana mode necessarily remains at the interface, together with a gapless $ν = 1 / 2$ charge mode arising from the uppermost quantum Hall strip. (b) Interface between the $\tilde{T}$ -breaking 113 surface topological order and a trivial magnetically gapped region. Dashed lines represent neutral fermions dimerized in two possible ways depending on whether $\tilde{m} = \tilde{t}$ or $\tilde{m} = - \tilde{t}$ . In one case the interface supports only a $ν = 1 / 2$ charge mode; in the other a counterpropagating $ν = 1$ neutral mode coexists.
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Figure 5
Connection between CDL descendants in the absence of $\tilde{T}$ symmetry and strictly 2D topological orders. (a) With broken $\tilde{T}$ the pairing-gapped CDL reduces to a trivial ferromagnetically gapped surface together with a 2D T-Pfaffian arising from a weakly paired composite Fermi liquid. (b) Conversely, the magnetically gapped CDL is equivalent to a ferromagnetic surface coexisting with a strongly paired composite-Fermi-liquid state with $K$ matrix $K = 8$ and charge vector $q = 2$ .
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