Composite Dirac liquids: parent states for symmetric surface topological order

TL;DR

The paper introduces composite Dirac liquids (CDLs) as highly interacting, symmetry-preserving surface states of 3D electronic topological insulators, featuring a charge gap yet hosting a neutral Dirac cone of emergent fermions. It develops extrinsic and intrinsic routes to realize the CDL, analyzes universal properties, and demonstrates how neutral-Dirac-cone pairing or magnetism drives symmetric gapped surface phases such as the T-Pfaffian and the Abelian 113 state, with a hierarchical relation to Pfaffian-antisemion and 331-antisemion descendants. A key contribution is showing that these CDL descendants correspond to paired or proximate 2D composite Fermi liquid phases, clarified via unimodular transformations of K-matrices and connections to 2D edge theories. The framework is then extended to bosonic topological insulators, yielding symmetric surface topological orders through domain-wall decoration with bosonic quantum Hall states and E8-based constructions. Overall, the work provides a unified, Hamiltonian-based route to a broad class of symmetric surface topological orders from a single, tunable gapless parent state, and highlights deep connections between 3D TI surfaces, 2D composite Fermi liquids, and bosonic topological phases.

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 non-interacting 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.

Figures (5)

• 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 worksBondersonTOWangTOChenTOMetlitskiTO.
• 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 $\sigma_{xy} = \pm e^2/(2h)$. The domain walls, however, bind an array of gapless 1D electron modes with staggered chirality. Importantly, the surface remains invariant under the symmetry $\tilde{\mathcal{T}}$ 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 $\nu = 1$ integer quantum Hall strips separated by insulators. Contrary to the surface in (a), here $\tilde{\mathcal{T}}$ symmetry is always explicitly broken.
• Figure 3: (a) Magnetic domains of a TI surface overlaid with alternating $\nu = +1/2$ and $-1/2$ fractional quantum Hall fluids. The quantum Hall strips cancel the half-integer Hall conductance from the magnetically gapped domains, so that $\sigma_{xy} = 0$ 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 $\nu = \pm 1/2$ 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 $\tilde{\mathcal{T}}$ symmetry yields the neutral Dirac cone characteristic of the CDL state.
• 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. \ref{['ChargeGapFig']}(b) are decomposed into a pair of Majorana modes $\gamma_{1,y}$ and $\gamma_{2,y}$. 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 $\nu = 1/2$ charge mode arising from the uppermost quantum Hall strip. (b) Interface between the $\tilde{\mathcal{T}}$-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 $\nu = 1/2$ charge mode; in the other a counterpropagating $\nu = 1$ neutral mode coexists.
• Figure 5: Connection between CDL descendants in the absence of $\tilde{\mathcal{T}}$ symmetry and strictly 2D topological orders. (a) With broken $\tilde{\mathcal{T}}$ 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 $q = 2$.