Fractional Topological Insulators Research Articles

Dual quantum spin Hall insulator by density-tuned correlations in TaIrTe4.

The convergence of topology and correlations represents a highly coveted realm in the pursuit of new quantum states of matter1. Introducing electron correlations to a quantum spin Hall (QSH) insulator can lead to the emergence of a fractional topological insulator and other exotic time-reversal-symmetric topological order2-8, not possible in quantum Hall and Chern insulator systems. Here we report a new dual QSH insulator within the intrinsic monolayer crystal of TaIrTe4, arising from the interplay of its single-particle topology and density-tuned electron correlations. At charge neutrality, monolayer TaIrTe4 demonstrates the QSH insulator, manifesting enhanced nonlocal transport and quantized helical edge conductance. After introducing electrons from charge neutrality, TaIrTe4 shows metallic behaviour in onlya small range of charge densities but quickly goes into a new insulating state, entirely unexpected on the basis of the single-particle band structure of TaIrTe4. This insulating state could arise from a strong electronic instability near the van Hove singularities, probably leading to a charge density wave (CDW). Remarkably, within this correlated insulating gap, we observe a resurgence of the QSH state. The observation of helical edge conduction in a CDW gap could bridge spin physics and charge orders. The discovery of a dual QSH insulator introduces a new method for creating topological flat minibands through CDW superlattices, which offer a promising platform for exploring time-reversal-symmetric fractional phases and electromagnetism2-4,9,10.

Theory of oblique 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 \ThetaΘ-angles, long-range entanglement, and fractionalization. Starting from a simple family of \mathbb{Z}_NℤN 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.

Fractional second-order topological insulator from a three-dimensional coupled-wires construction

We construct a three-dimensional second-order topological insulator with gapless helical hinge states from an array of weakly tunnel-coupled Rashba nanowires. For suitably chosen interwire tunnelings, we demonstrate that the system has a fully gapped bulk as well as fully gapped surfaces, but hosts a Kramers pair of gapless helical hinge states propagating along a path of hinges that is determined by the hierarchy of interwire tunnelings and the boundary termination of the system. Furthermore, the coupled-wires approach allows us to incorporate electron-electron interactions into our description. At suitable filling factors of the individual wires, we show that sufficiently strong electron-electron interactions can drive the system into a fractional second-order topological insulator phase with hinge states carrying only a fraction $e/p$ of the electronic charge $e$ for an odd integer $p$.

Majorana and parafermion corner states from two coupled sheets of bilayer graphene

We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spin-orbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small in-plane magnetic field and weak proximity-induced superconductivity leads to the emergence of zero-energy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a second-order topological superconducting phase. The presence of strong electron-electron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional second-order topological superconducting phase exhibiting zero-energy $\mathbb{Z}_{2m}$ parafermion corner states, where $m$ is an odd integer determined by the position of the chemical potential.

Topological Quantum Optics Using Atomlike Emitter Arrays Coupled to Photonic Crystals.

We propose an experimentally feasible nanophotonic platform for exploring many-body physics in topological quantum optics. Our system is composed of a two-dimensional lattice of nonlinear quantum emitters with optical transitions embedded in a photonic crystal slab. The emitters interact through the guided modes of the photonic crystal, and a uniform magnetic field gives rise to large topological band gaps, robust edge states, and a nearly flat band with a nonzero Chern number. The presence of a topologically nontrivial nearly flat band paves the way for the realization of fractional quantum Hall states and fractional topological insulators in a topological quantum optical setting.

Fractional topological insulator precursors in spin-orbit fermion ladders

We study precursor states of fractional topological insulators (FTIs) in interacting fermionic ladders with spin-orbit coupling. Within a microscopically motivated bosonization approach, we investigate different competing phases depending on same-spin and interspin interactions at fractional effective filling $\nu=1/3$ per spin. In the spin-decoupled limit, we find that strong repulsive interactions of already moderate range may lead to a partially gapped state with two time-reversed copies of a quasi-one dimensional Laughlin phase. This FTI precursor competes with an interleg partially gapped phase displaying quasi long-range density wave order, however it may be stabilized if interactions have suitable anisotropy, or are sufficiently near SU($2$) symmetry, in leg space. When the FTI phase is present, it is moderately robust to small interspin interactions; these introduce competing partially gapped phases of orbital antiferromagnetic and bond density wave character. Performing a strong coupling analysis of the FTI precursor regime, we find that the main effect of interspin interactions is to induce correlated quasiparticle backscattering between the precursor FTI edge modes. Although this process competes with the topological phase, we show, by considering an array of ladders, that its influence may disappear upon approaching the two dimensional case. Considering time-reversal symmetry breaking perturbations, we also describe a protocol that adiabatically pumps $1/6$ charge per half-cycle, thus providing a quantized FTI signature arising already in the single ladder regime.

Paired parton quantum Hall states: A coupled wire construction

The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a new family of incompressible paired parton FQH states at filling $\nu=1/6$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\nu=1/3$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-like anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principle existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a new notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimension.

Spin separation in the half-filled fractional topological insulator

Motivated by the close analogy with the fractional quantum Hall states (FQHSs), fractional Chern insulators (FCIs) are envisioned as strongly correlated, incompressible states emerging in a fractionally filled, (nearly) flat band with non-trivial Chern number. Built upon this vision, fractional topological insulators (FTIs) have been proposed as being composed of two independent copies of the FCI with opposite Chern numbers for different spins, preserving the time-reversal symmetry as a whole. An important question is if the correlation between electrons with different spins can be really ignored. To address this question, we investigate the effects of correlation in the presence of spin-dependent holomorphicity, i.e., electrons of one spin species reside in the holomorphic lowest Landau level, while those of the other in the antiholomorphic counterpart. By constructing and performing exact diagonalization of an appropriate model Hamiltonian, here, we show that generic, strongly correlated, fractionally filled states with spin-dependent holomorphicity cannot be described as two independent copies of the FQHS, suggesting that FTIs in the lattice cannot be described as those of the FCI either. Fractionally filled states in this system are generally compressible except at half filling, where an insulating state called the half-filled spin-holomorphic FTI occurs. It is predicted that the half-filled spin-holomorphic FTI is susceptible to an inherent spontaneous symmetry breaking, leading to the spatial separation of spins.

Higher angular momentum band inversions in two dimensions

We study a special class of topological phase transitions in two dimensions described by the inversion of bands with relative angular momentum higher than 1. A band inversion of this kind, which is protected by rotation symmetry, separates the trivial insulator from a Chern insulating phase with higher Chern number, and thus generalizes the quantum Hall transition described by a Dirac fermion. Higher angular momentum band inversions are of special interest, as the non-vanishing density of states at the transition can give rise to interesting many-body effects. Here we introduce a series of minimal lattice models which realize higher angular momentum band inversions. We then consider the effect of interactions, focusing on the possibility of electron-hole exciton condensation, which breaks rotational symmetry. An analysis of the excitonic insulator mean field theory further reveals that the ground state of the Chern insulating phase with higher Chern number has the structure of a multicomponent integer quantum Hall state. We conclude by generalizing the notion of higher angular momentum band inversions to the class time-reversal invariant systems, following the scheme of Bernevig-Hughes-Zhang (BHZ). Such band inversions can be viewed as transitions to a topological insulator protected by rotation and inversion symmetry, and provide a promising venue for realizing correlated topological phases such as fractional topological insulators.

A Passage to Topological Matter: Colloquium

Topological matter has become one of the most important subjects in contemporary condensed matter physics. Here, I would like to provide a pedagogical review explaining some of the main ideas, which were pivotal in establishing topological matter as such an important subject. Specifically, I explain how the integer quantum Hall state played the role as a prototype for topological insulator, eventually leading to the concept of topological matter in general. The topological nature of the integer quantum Hall state is best represented by the Thouless-Kohmoto-Nightingale-den Nijs, or so-called TKNN formula, which connects between the Berry phase and the Hall conductivity. The topological non-triviality of topological insulator stems from the existence of a Dirac monopole in an appropriate, but often hidden Hamiltonian parameter space. Interestingly, having the identical Dirac monopole structure, the Hamiltonian describing the Rabi oscillation bears the essence of topological insulator. The concept of topological matter has expanded to include topological semimetals such as Weyl and Dirac semimetals. A final frontier in the research of topological matter is the interaction-induced topological phases of matter, namely, the fractional Chern and topological insulators. The existence of the fractional Chern and topological insulators has been proposed theoretically by drawing an analogy from the fractional quantum Hall states. The gist of this proposal is explained along with some of its issues. I conclude this review by discussing some of the future directions in the research of topological matter.

Three-Boson Spectrum in the Presence of 1D Spin-Orbit Coupling: Efimov’s Generalized Radial Scaling Law

Spin-orbit coupled cold atom systems, governed by Hamiltonians that contain quadratic kinetic energy terms typical for a particle's motion in the usual Schr\"odinger equation and linear kinetic energy terms typical for a particle's motion in the usual Dirac equation, have attracted a great deal of attention recently since they provide an alternative route for realizing fractional quantum Hall physics, topological insulators, and spintronics physics. The present work focuses on the three-boson system in the presence of 1D spin-orbit coupling, which is most relevant to ongoing cold atom experiments. In the absence of spin-orbit coupling terms, the three-boson system exibits the Efimov effect: the entire energy spectrum is uniquely determined by the $s$-wave scattering length and a single three-body parameter, i.e., using one of the energy levels as input, the other energy levels can be obtained via Efimov's radial scaling law, which is intimately tied to a discrete scaling symmetry. It is demonstrated that the discrete scaling symmetry persists in the presence of 1D spin-orbit coupling, implying the validity of a generalized radial scaling law in five-dimensional space. The dependence of the energy levels on the scattering length, spin-orbit coupling parameters, and center-of-mass momentum is discussed. It is conjectured that three-body systems with other types of spin-orbit coupling terms are also governed by generalized radial scaling laws, provided the system exhibits the Efimov effect in the absence of spin-orbit coupling.

Anomaly manifestation of Lieb-Schultz-Mattis theorem and topological phases

We construct the symmetric-gapped surface states of a fractional topological insulator with electromagnetic $\theta$-angle $\theta_{em} = \frac{\pi}{3}$ and a discrete $\mathbb{Z}_3$ gauge field. They are the proper generalizations of the T-pfaffian state and pfaffian/anti-semion state and feature an extended periodicity compared with their of "integer" topological band insulators counterparts. We demonstrate that the surface states have the correct anomalies associated with time-reversal symmetry and charge conservation.

Surfaces and slabs of fractional topological insulator heterostructures

Fractional topological insulators (FTI) 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 FTI, whose bulk quasiparticles consist of deconfined partons that carry fractional electric charges in integral units of $e^\ast=e/(2n+1)$ 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.

Symmetric-gapped surface states of fractional topological insulators

We construct the symmetric-gapped surface states of a fractional topological insulator with electromagnetic $\theta$-angle $\theta_{em} = \frac{\pi}{3}$ and a discrete $\mathbb{Z}_3$ gauge field. They are the proper generalizations of the T-pfaffian state and pfaffian/anti-semion state and feature an extended periodicity compared with their of integer topological band insulators counterparts. We demonstrate that the surface states have the correct anomalies associated with time-reversal symmetry and charge conservation.

Majorana fermions in the Kitaev quantum spin system α-RuCl3

α-RuCl3 has recently attracted great interest as a possible experimental realization of the Kitaev model. Neutron scattering measurements of a single crystal of this material reveal signatures of Majorana excitations, consistent with Kitaev’s predictions. Geometrical constraints to the electronic degrees of freedom within condensed-matter systems often give rise to topological quantum states of matter such as fractional quantum Hall states, topological insulators, and Weyl semimetals1,2,3. In magnetism, theoretical studies predict an entangled magnetic quantum state with topological ordering and fractionalized spin excitations, the quantum spin liquid4. In particular, the so-called Kitaev spin model5, consisting of a network of spins on a honeycomb lattice, is predicted to host Majorana fermions as its excitations. By means of a combination of specific heat measurements and inelastic neutron scattering experiments, we demonstrate the emergence of Majorana fermions in single crystals of α-RuCl3, an experimental realization of the Kitaev spin lattice. The specific heat data unveils a two-stage release of magnetic entropy that is characteristic of localized and itinerant Majorana fermions. The neutron scattering results corroborate this picture by revealing quasielastic excitations at low energies around the Brillouin zone centre and an hour-glass-like magnetic continuum at high energies. Our results confirm the presence of Majorana fermions in the Kitaev quantum spin liquid and provide an opportunity to build a unified conceptual framework for investigating fractionalized excitations in condensed matter1,6,7,8.

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

In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, the fractional $S$-duality. Such $\text{QED}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\theta$) can take a nontrivial but still time-reversal invariant value $\pi/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and $\text{U(1)}$ 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 ($t\in\mathbb{Z}_{\rm odd}$) can be obtained by directly stacking a non-interacting topological insulator and a fractionalized gapped fermionic state with minimal charge $1/t$ and vanishing $\theta$. But type-II ($t\in\mathbb{Z}_{\rm even}$) cannot be realized through any stacking. Finally, we study the Surface Topological Order of fractional topological insulators.

Three-dimensional fractional topological insulators in coupled Rashba layers

We propose a model of three-dimensional topological insulators consisting of weakly coupled electron- and hole-gas layers with Rashba spin-orbit interaction stacked along a given axis. We show that in the presence of strong electron-electron interactions the system realizes a fractional strong topological insulator, where the rotational symmetry and condensation energy arguments still allow us to treat the problem as quasi-one-dimensional with bosonization techniques. We also show that if Rashba and Dresselhaus spin-orbit interaction terms are equally strong, by doping the system with magnetic impurities, one can bring it into the Weyl semimetal phase.

Three-dimensional topological insulators and bosonization

Massless excitations at the surface of three-dimensional time-reversal invariant topological insulators possess both fermionic and bosonic descriptions, originating from band theory and hydrodynamic BF theory, respectively. We analyze the corresponding field theories of the Dirac fermion and compactified boson and compute their partition functions on the three-dimensional torus geometry. We then find some non-dynamic exact properties of bosonization in (2+1) dimensions, regarding fermion parity and spin sectors. Using these results, we extend the Fu-Kane-Mele stability argument to fractional topological insulators in three dimensions.

The D3-probe-D7 brane holographic fractional topological insulator

The D3-probe-D7 brane system, oriented so as to have 2+1-dimensional Poincare symmetry, is argued to be the holographic representation of a strongly correlated fractional topological insulator which exhibits a zero-field quantized Hall effect with half-units of Hall conductivity. The phase diagram of the system with charge density and external magnetic field is found and, as well as charge gapped quantum Hall states, it exhibits metallic and semi-metallic phases with interesting behaviours. The relationship of this to other models of fractional topological insulators is discussed.

Composite particle theory of three-dimensional gapped fermionic phases: Fractional topological insulators and charge-loop excitation symmetry

Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e. the deconfined phase of $\mathbb{Z}_2$ gauge theory) is a $U(1)\times U(1)$ Chern-Simons theory in which gauge charges (i.e., $e$ and $m$ particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a new route to constructing exotic 3D gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $U(1)\times U(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $U(1)\times U(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel-like screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\Theta\neq \pi$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry ($\mathsf{Charles}$) of 3D Abelian topological phases. We show that line twist defects, which realize $\mathsf{Charles}$ transformations, exhibit non-Abelian fusion properties.