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arXiv:1701.08828 · Physical review. B./Physical review. B

Surfaces and slabs of fractional topological insulator heterostructures

Sharmistha Sahoo, Alexander Sirota, Gil Young Cho, Jeffrey C. Y. Teo

Published Oct 12, 2017• 7 citations
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Abstract

Fractional topological insulators (FTIs) are electronic topological phases in $(3+1)$ dimensions enriched by time reversal (TR) and charge $U(1)$ conservation symmetries. We focus on the simplest series of fermionic FTIs, whose bulk quasiparticles consist of deconfined partons that carry fractional electric charges in integral units of ${e}^{*}=e/(2n+1)$ and couple to a discrete ${\mathbb{Z}}_{2n+1}$ gauge theory. We propose massive symmetry preserving or breaking FTI surface states. Combining the long-ranged entangled bulk with these topological surface states, we deduce the novel topological order of quasi-$(2+1)$-dimensional FTI slabs as well as their corresponding edge conformal field theories.

DOI
10.1103/physrevb.96.161108

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Fractional topological insulators (FTIs) are electronic topological phases in (3+1)(3+1)(3+1) dimensions enriched by time reversal (TR) and charge U(1)U(1)U(1) conservation symmetries. We focus on the simplest series of fermionic FTIs, whose bulk quasiparticles consist of deconfined partons that carry fractional electric charges in integral units of e∗=e/(2n+1){e}^{*}=e/(2n+1)e∗=e/(2n+1) and couple to a discrete Z2n+1{\mathbb{Z}}_{2n+1}Z2n+1​ gauge theory. We propose massive symmetry preserving or breaking FTI surface states. Combining the long-ranged entangled bulk with these topological surface states, we deduce the novel topological order of quasi-(2+1)(2+1)(2+1)-dimensional FTI slabs as well as their corresponding edge conformal field theories.
In this paper, we propose a generalization of the SSS-duality of four-dimensional quantum electrodynamics (QED4{\text{QED}}_{4}QED4​) to QED4{\text{QED}}_{4}QED4​ with fractionally charged excitations, the fractional SSS-duality. Such QED4{\text{QED}}_{4}QED4​ can be obtained by gauging the U(1)\text{U(1)}U(1) symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\ensuremath{\theta}$) can take a nontrivial but still time-reversal-invariant value $\ensuremath{\pi}/{t}^{2}$ ($t\ensuremath{\in}\mathbb{Z}$). Here, 1/t1/t1/t specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and U(1)\text{U(1)}U(1) global symmetry (fermion number conservation) are fractional topological insulators (FTIs). We propose a topological quantum field theory description, which microscopically justifies the fractional SSS-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 ($t\ensuremath{\in}{\mathbb{Z}}_{\mathrm{odd}}$) can be obtained by directly stacking a noninteracting topological insulator and a fractionalized gapped fermionic state with minimal charge 1/t1/t1/t and vanishing $\ensuremath{\theta}$. But type II ($t\ensuremath{\in}{\mathbb{Z}}_{\mathrm{even}}$) cannot be realized through any stacking. Finally, we study the surface topological order of fractional topological insulators.
A long-standing problem in the study of topological phases of matter has been to understand the types of fractional topological insulator (FTI) phases possible in 3+1 dimensions. Unlike ordinary topological insulators of free fermions, FTI phases are characterized by fractional \Theta Θ -angles, long-range entanglement, and fractionalization. Starting from a simple family of \mathbb{Z}_N ℤ N lattice gauge theories due to Cardy and Rabinovici, we develop a class of FTI phases based on the physical mechanism of oblique confinement and the modern language of generalized global symmetries. We dub these phases oblique topological insulators. Oblique TIs arise when dyons—bound states of electric charges and monopoles—condense, leading to FTI phases characterized by topological order, emergent one-form symmetries, and gapped boundary states not realizable in 2+1-D alone. Based on the lattice gauge theory, we present continuum topological quantum field theories (TQFTs) for oblique TI phases involving fluctuating one-form and two-form gauge fields. We show explicitly that these TQFTs capture both the generalized global symmetries and topological orders seen in the lattice gauge theory. We also demonstrate that these theories exhibit a universal “generalized magnetoelectric effect” in the presence of two-form background gauge fields. Moreover, we characterize the possible boundary topological orders of oblique TIs, finding a new set of boundary states not studied previously for these kinds of TQFTs.
We propose a simple microscopic model to numerically investigate the stability of a two-dimensional fractional topological insulator (FTI). The simplest example of an FTI consists of two decoupled copies of a Laughlin state with opposite chiralities, or double-semion phase. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion-symmetry-breaking term that couples the copies at the single-particle level. Using exact-diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time-reversal-invariant system survives the introduction of interaction coupling on a larger scale than the time-reversal-symmetry-breaking one, stressing the importance of time-reversal symmetry in the FTI phase stability. We also discuss possible fractional phases beyond $\ensuremath{\nu}=1/2$.
We propose a simple microscopic model to numerically investigate the stability of a two dimensional fractional topological insulator (FTI). The simplest example of a FTI consists of two decoupled copies of a Laughlin state with opposite chiralities. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion symmetry breaking term that couples the copies at the single particle level. Using exact diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time reversal invariant system survives the introduction of interaction coupling on a larger scale than the time reversal symmetry breaking one, stressing the importance of time reversal symmetry in the FTI phase stability.
In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\ensuremath{\nu}$ (and the same quantized Hall conductance ${\ensuremath{\sigma}}_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\ensuremath{\sigma}}_{xy}=1/m\ifmmode\cdot\else\textperiodcentered\fi{}{e}^{2}/h$ FCI wave function can be either SU(m)\text{SU}(m)SU(m) or the discrete group Zm{Z}_{m}Zm​, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.
Time-reversal invariant three-dimensional topological insulators can be defined fundamentally by a topological field theory with a quantized axion angle $\ensuremath{\theta}$ of 0 or $\ensuremath{\pi}$. It was recently shown that fractional quantized values of $\ensuremath{\theta}$ are consistent with time-reversal invariance if deconfined, gapped, fractionally charged bulk excitations appear in the low-energy spectrum due to strong correlation effects, leading to the concept of a fractional topological insulator. These fractionally charged excitations are coupled to emergent gauge fields, which ensure that the microscopic degrees of freedom, the original electrons, are gauge-invariant objects. A first step towards the construction of microscopic models of fractional topological insulators is to understand the nature of these emergent gauge theories and their corresponding phases. In this work, we show that low-energy effective gauge theories of both Abelian or non-Abelian type are consistent with a fractional quantized axion angle if they admit a Coulomb phase or a Higgs phase with gauge group broken down to a discrete subgroup. The Coulomb phases support gapless but electrically neutral bulk excitations while the Higgs phases are fully gapped. The Higgs and non-Abelian Coulomb phases exhibit multiple ground states on boundaryless spatial three-manifolds with nontrivial first homology, while the Abelian Coulomb phase has a unique ground state. The ground-state degeneracy receives an additional contribution on manifolds with boundary due to the induced boundary Chern-Simons term.