We construct new many-body invariants for 2d Chern and 3d chiral hinge insulators, which are characterized by quantized pumping of dipole and quadrupole moments. The invariants that we devise are written entirely in terms of many-body ground state wavefunctions on a torus geometry with a set of unitary operators. We provide a number of supporting evidences for our invariants via topological field theory interpretation, adiabatic pumping argument, and direct mapping to free-fermion band indices. We finally confirm our invariants by numerical computations including infinite density matrix renormalization group on a quasi-one-dimensional system. The many-body invariants therefore explicitly encircle several different pillars of theoretical descriptions of topological phases.
Doped strong topological insulators are one of the most promising candidates to realize a fully gapped three-dimensional topological superconductor (TSC). In this letter, we revisit this system and reveal a possibility for higher-order topology which was previously missed. We find that over a finite-range of doping, the Fu-Berg superconducting pairing can give rise to both Majorana surface states, and nodal hinge states. Interestingly, we observe the coexistence of surface and hinge modes in the superconducting state only when there are both bulk and surface Fermi-surfaces in the normal state.…
When interacting spins in condensed matter order ferromagnetically, their ground state wave function is topologically trivial. Nonetheless, in two dimensions, the ferromagnetic state can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). Here, we theoretically unveil and numerically confirm a novel ferromagnetic state in three dimensions dubbed second-order TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken time-reversal symmetry, the hinge magnons are chiral, …
We propose and study a wave function describing an interacting three-dimensional fractional chiral hinge insulator (FCHI) constructed by Gutzwiller projection of two non-interacting second order topological insulators with chiral hinge modes at half filling. We use large-scale variational Monte Carlo computations to characterize the model states via the entanglement entropy and charge-spin-fluctuations. We show that the FCHI possesses fractional chiral hinge modes characterized by a central charge $c=1$ and Luttinger parameter $K=1/2$, like the edge modes of a Laughlin $1/2$ state. By changing…
Chern insulator ferromagnets are characterized by a quantized anomalous Hall effect, and have so far been identified experimentally in magnetically-doped topological insulator (MTI) thin films and in bilayer graphene moir{é} superlattices. We classify Chern insulator ferromagnets as either spin or orbital, depending on whether the orbital magnetization results from spontaneous spin-polarization combined with spin-orbit interactions, as in the MTI case, or directly from spontaneous orbital currents, as in the moir{é} superlattice case. We argue that in a given magnetic state, characterized for …
Recently, many numerical evidences of fractional Chern insulator, i.e. the fractional quantum Hall states on lattices, are proposed when a Chern band is partially filled. Some trial wavefunctions of fractional Chern insulators can be obtained by mapping the fractional quantum Hall wavefunctions defined in the continuum onto the lattice through the Wannier state representation (Phys. Rev. Lett. 107, 126803 (2011)) in which the single particle Landau orbits in the Landau levels are identified with the one dimensional Wannier states of the Chern bands with Chern number C = 1. However, this mappin…
Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation sy…
We analyse the topological transition and localization evolution of disordered two dimensional systems with non trivial topology based on bipartite lattices. Chern insulators with broken time reversal symmetry show non standard behavior for disorder realizations selectively distributed on only one of the sublattices. The Chern number survives to a much stronger disorder strength (one order of magnitude higher) than in the equally distributed disordered case and the final state in the strongly disordered case is metallic.
Even if a noninteracting system has zero Berry curvature everywhere in the Brillouin zone, it is possible to introduce interactions that stabilise a fractional Chern insulator. These interactions necessarily break time-reversal symmetry (either spontaneously or explicitly) and have the effect of altering the underlying band structure. We outline a number of ways in which this may be achieved, and show how similar interactions may also be used to create a (time-reversal symmetric) fractional topological insulator. While our approach is rigorous in the limit of long range interactions, we show n…
We study the localization properties of the equal-time electron Green's function in a Chern insulator in an arbitrary dimension and with an arbitrary number of bands. We prove that the Green's function cannot decay super-exponentially if the Hamiltonian is finite-range and the quantum Hall response is nonzero. For a general band Hamiltonian (possibly infinite-range), we prove that the Green's function cannot be finite-range if the quantum Hall response is nonzero. The proofs use methods of algebraic geometry.
The Bloch wave functions have been playing a crucial role in the diagnosis of topological phases in non-interacting systems. However, the Bloch waves are no longer applicable in the presence of finite Coulomb interaction and alternative approaches are needed to identify the topological indices. In this paper, we focus on three-dimensional higher-order topological insulators protected by $C_4T$ symmetry and show that the topological index can be computed through eigenstates of inverse Green's function at zero frequency. If there is an additional $S_4$ rotoinversion symmetry, the topological ind…
We study the ground-state topology and quasiparticle properties in bosonic Mott insulators with two- dimensional spin-orbit couplings in cold atomic optical lattices. We show that the many-body Chern and spin-Chern number can be expressed as an integral of the quasihole Berry curvatures over the Brillouin zone. Using a strong-coupling perturbation theory, for an experimentally feasible spin-orbit coupling, we compute the Berry curvature and the spin Chern number and find that these quantities can be generated purely by interactions. We also compute the quasiparticle dispersions, spectral weigh…
Moir{é} superlattice realized in two-dimensional heterostructures offers an exciting platform to access strongly-correlated electronic states. In this work, we study transition metal dichalcogenides (TMD) Moir{é} superlattices with time-reversal-symmetry and nontrivial spin{/valley}-Chern numbers. Utilizing realistic material parameters and the method of exact diagonalization, we find that at a certain twisting angle and fractional filling, gapped fractional topological states, i.e., fractional Chern insulators, are naturally {stabilized} by simply introducing the Coulomb repulsion. In contras…
We propose to use generic Chern numbers for a characterization of topological insulators. It is suitable for a numerical characterization of low dimensional quantum liquids where strong quantum fluctuations prevent from developing conventional orders. By twisting parameters of boundary conditions, the non-Abelian Chern number are defined for a few low lying states near the ground state in a finite system, which is a ground state multiplet with a possible (topological) degeneracy. We define the system as a topological insulator when energies of the multiplet are well separated from the above. T…
A set of recent results indicates that fractionally filled bands of Chern insulators in two dimensions support fractional quantum Hall states analogous to those found in fractionally filled Landau levels. We provide an understanding of these results by examining the algebra of Chern band projected density operators. We find that this algebra closes at long wavelengths and for constant Berry curvature, whereupon it is isomorphic to the W-Infinity algebra of lowest Landau level projected densities first identified by Girvin, MacDonald and Platzman [Phys. Rev. B 33, 2481 (1986).] For Hamiltonians…
The Hofstadter model is a popular choice for theorists investigating the fractional quantum Hall effect on lattices, due to its simplicity, infinite selection of topological flat bands, and increasing applicability to real materials. In particular, fractional Chern insulators in bands with Chern number $|C|>1$ can demonstrate richer physical properties than continuum Landau level states and have recently been detected in experiments. Motivated by this, we examine the stability of fractional Chern insulators with higher Chern number in the Hofstadter model, using large-scale infinite density ma…
We study a generic model of a Chern insulator supplemented by a Hubbard interaction in arbitrary even dimension $D$ and demonstrate that the model remains well-defined and nontrivial in the $D \to \infty$ limit. Dynamical mean-field theory is applicable and predicts a phase diagram with a continuum of topologically different phases separating a correlated Mott insulator from the trivial band insulator. We discuss various features, such as the elusive distinction between insulating and semi-metal states, which are unconventional already in the non-interacting case. Topological phases are charac…
The phase transition driven by electron correlations in a Chern insulator is investigated within the dynamical mean-field theory. The Chern insulator is described by the Haldane model and the electron correlations are incorporated by introducing the short-range interaction between the itinerant electrons and localized fermions. In the preservation of the inversion symmetry, the electron correlations drive the system from the Chern insulator to a renormalized pseudogap metal, and then to the topologically trivial Mott insulator. When the inversion symmetry is broken, a charge ordering and a non…
We formulate a Chern-Simons composite fermion theory for Fractional Chern Insulators (FCIs), whereby bare fermions are mapped into composite fermions coupled to a lattice Chern-Simons gauge theory. We apply this construction to a Chern insulator model on the kagome lattice and identify a rich structure of gapped topological phases characterized by fractionalized excitations including states with unequal filling and Hall conductance. Gapped states with the same Hall conductance at different filling fractions are characterized as realizing distinct symmetry fractionalization classes.
We respond to the comments expressed by L. D'Alessio in arXiv:1412.3481 on our work "Floquet Fractional Chern Insulators" [Phys. Rev. Lett. 112, 156801 (2014)]. We confirm the central result that the ground state of the effective Hamiltonian is an interacting fractional Chern insulator.
We propose MnBi$_{2n}$Te$_{3n+1}$ as a magnetically tunable platform for realizing various symmetry-protected higher-order topology. Its canted antiferromagnetic phase can host exotic topological surface states with a Möbius twist that are protected by nonsymmorphic symmetry. Moreover, opposite surfaces hosting Möbius fermions are connected by one-dimensional chiral hinge modes, which offers the first material candidate of a higher-order topological Möbius insulator. We uncover a general mechanism to feasibly induce this exotic physics by applying a small in-plane magnetic field to the antifer…
We predict twisted double bilayer graphene to be a versatile platform for the realization of fractional Chern insulators readily targeted by tuning the gate potential and the twist angle. Remarkably, these topologically ordered states of matter, including spin singlet Halperin states and spin polarized states in Chern number $\mathcal C=1$ and $\mathcal{C}= 2$ bands, occur at high temperatures and without the need for an external magnetic field.
In this paper we provide analytical counting rules for the ground states and the quasiholes of fractional Chern insulators with an arbitrary Chern number. We first construct pseudopotential Hamiltonians for fractional Chern insulators. We achieve this by mapping the lattice problem to the lowest Landau level of a multicomponent continuum quantum Hall system with specially engineered boundary conditions. We then analyze the thin-torus limit of the pseudopotential Hamiltonians, and extract counting rules (generalized Pauli principles, or Haldane statistics) for the degeneracy of its zero modes i…
We show that the topologically nontrivial bands of Chern insulators are adiabatic cousins of the Landau bands of Hofstadter lattices. We demonstrate adiabatic connection also between several familiar fractional quantum Hall states on Hofstadter lattices and the fractional Chern insulator states in partially filled Chern bands, which implies that they are in fact different manifestations of the same phase. This adiabatic path provides a way of generating many more fractional Chern insulator states and helps clarify that nonuniformity in the distribution of the Berry curvature is responsible for…
Motivated by the recent work of QED$_3$-Chern-Simons quantum critical points of fractional Chern insulators (Phys. Rev. X \textbf{8}, 031015, (2018)), we study its non-Abelian generalizations, namely QCD$_3$-Chern-Simons quantum phase transitions of fractional Chern insulators. These phase transitions are described by Dirac fermions interacting with non-Abelian Chern-Simons gauge fields ($U(N)$, $SU(N)$, $USp(N)$, etc.). Utilizing the level-rank duality of Chern-Simons gauge theory and non-Abelian parton constructions, we discuss two types of QCD$_3$ quantum phase transitions. The first type h…
Mirror symmetric surfaces of a topological crystalline insulator host even number of Dirac surface states. A surface Zeeman field generically gaps these states leading to a quantized anomalous Hall effect. Varying the direction of Zeeman field induces transitions between different surface insulating states with any two Chern numbers between -4 and 4. In the crystal frame the phase boundaries occur for field orientations which are great circles with (111)-like normals on a sphere.
Studies on two-dimensional electron systems in a strong magnetic field first revealed the quantum Hall (QH) effect, a topological state of matter featuring a finite Chern number (C) and chiral edge states. Haldane later theorized that Chern insulators with integer QH effects could appear in lattice models with complex hopping parameters even at zero magnetic field. The ABC-trilayer graphene/hexagonal boron nitride (TLG/hBN) moiré superlattice provides an attractive platform to explore Chern insulators because it features nearly flat moiré minibands with a valley-dependent electrically tunable …
We provide a detailed study of the Abelian quasiholes of $ν=1/2$ bosonic fractional quantum Hall states on the torus geometry and in fractional Chern insulators. We find that the density distribution of a quasihole in a fractional Chern insulator can be related to that of the corresponding fractional quantum Hall state by choosing an appropriate length unit on the lattice. This length unit only depends on the lattice model that hosts the fractional Chern insulator. Therefore, the quasihole size in a fractional Chern insulator can be predicted for any lattice model once the quasihole size of th…
We propose to realize and observe Chern Kondo insulators in an optical superlattice with laser-assisted $s$ and $p$ orbital hybridization and synthetic gauge field, which can be engineered based on the recent cold atom experiments. Considering a double-well square optical lattice, the localized $s$ orbitals are decoupled from itinerant $p$ bands and are driven into a Mott insulator due to strong Hubbard interaction. Raman laser beams are then applied to induce tunnelings between $s$ and $p$ orbitals, and generate a staggered flux simultaneously. Due to the strong Hubbard interaction of $s$ orb…
We predict that a family of antiperovskite materials realize a higher order topological insulator phase, characterized by a previously introduced $\mathbb{Z}_4$ index. A tight binding model and a $k\cdot p$ model are used to capture the physics of the bulk, surface and hinge states of these materials. A phase diagram of the higher order and weak topological invariants is obtained for the tight binding model. The mirror Chern number is also discussed. In order to reveal the gapless hinge states in the presence of mirror Chern surface states, several ways of opening the surface gap are proposed …
Recently, generalizations of fractional quantum Hall (FQH) states known as fractional quantum anomalous Hall or, equivalently, fractional Chern insulators states have been realized in lattice models. Ideal wavefunctions such as the Laughlin wavefunction, as well as the corresponding trial Hamiltonians for which the former are exact ground states, have been vital to characterizing FQH phases. The Wannier function representation of fractional Chern insulators proposed in [X.-L. Qi, Phys. Rev. Lett., 126803] defines an approach to generalize these concepts to fractional Chern insulators. In this …
We investigate a new class of topological antiferromagnetic (AF) Chern insulators driven by electronic interactions in two-dimensional systems without inversion symmetry. Despite the absence of a net magnetization, AF Chern insulators (AFCI) possess a nonzero Chern number $C$ and exhibit the quantum anomalous Hall effect (QAHE). Their existence is guaranteed by the bifurcation of the boundary line of Weyl points between a quantum spin Hall insulator and a topologically trivial phase with the emergence of AF long-range order. As a concrete example, we study the phase structure of the honeycomb …
Experiments demonstrating the controlled growth of oxide heterostructures have raised the prospect of realizing topologically nontrivial states of correlated electrons in low dimensions. Here, we study heterostructures consisting of {111}-bilayers of double perovskites separated by inert band insulators. In bulk, these double perovskites have well-defined local moments interacting with itinerant electrons leading to high temperature ferromagnetism. Incorporating spin-orbit coupling in the two-dimensional honeycomb geometry of a {111}-bilayer, we find a rich phase diagram with tunable ferromagn…
The discovery of magic angle twisted bilayer graphene (MATBG) has unveiled a rich variety of superconducting, magnetic and topologically nontrivial phases. The existence of all these phases in one material, and their tunability, has opened new pathways for the creation of unusual gate tunable junctions. However, the required conditions for their creation - gate induced transitions between phases in zero magnetic field - have so far not been achieved. Here, we report on the first experimental demonstration of a device that is both a zero-field Chern insulator and a superconductor. The Chern ins…
In the framework of Hofstadter`s approach we provide a detailed analysis of a realization of exotic topological states such as the Chern insulator with large Chern numbers. In a transverse homogeneous magnetic field a one-particle spectrum of fermions transforms to an intricate spectrum with a fine topological structure of the subbands. In a weak magnetic field $H$ for a rational magnetic flux, a topological phase with a large Chern number is realized near the half filling. There is an abnormal behavior of the Hall conductance ${σ_{xy}\simeq \frac{e}{2 H}}$. At half-filling, the number of chir…
One salient feature of systems with Moiré superlattice is that, Chern number of "minibands" originating from each valley of the original graphene Brillouin zone becomes a well-defined quantized number because the miniband from each valley can be isolated from the rest of the spectrum due to the Moiré potential. Then a Moiré system with a well-defined valley Chern number can become a nonchiral topological insulator with $U(1) \times Z_3 $ symmetry and a $\mathbb{Z}$ classification at the free fermion level. Here we demonstrate that the strongly interacting nature of the Moiré system reduces the…
Fractional Chern insulators are new realizations of fractional quantum Hall states in lattice systems without orbital magnetic field. These states can be mapped onto conventional fractional quantum Hall states through the Wannier state representation (Phys. Rev. Lett. 107, 126803 (2011)). In this paper, we use the Wannier state representation to construct the pseudopotential Hamiltonians for fractional Chern insulators, which are interaction Hamiltonians with certain ideal model wavefunctions as exact ground states. We show that these pseudopotential Hamiltonians can be approximated by short-r…
We report the first numerical observation of composite fermion (CF) states in fractional Chern insulators (FCI) using exact diagonalization. The ruby lattice Chern insulator model for both fermions and bosons exhibits a clear signature of CF states at filling factors 2/5 and 3/7 (2/3 and 3/4 for bosons). The topological properties of these states are studied through several approaches. Quasihole and quasielectron excitations in FCI display similar features as their fractional quantum hall (FQH) counterparts. The entanglement spectrum of FCI groundstates shows an identical fingerprint to its FQ…
We review various features of interacting Abelian topological phases of matter in two spatial dimensions, placing particular emphasis on fractional Chern insulators (FCIs) and fractional topological insulators (FTIs). We highlight aspects of these systems that challenge the intuition developed from quantum Hall physics - for instance, FCIs are stable in the limit where the interaction energy scale is much larger than the band gap, and FTIs can possess fractionalized excitations in the bulk despite the absence of gapless edge modes.
Fractional Chern insulators are theoretically predicted states of electronic matter with emergent topological order. They exhibit the same universal properties as the fractional quantum Hall effect, but dispose of the need to apply a strong magnetic field. However, despite intense theoretical work, an experimental realization for these exotic states of matter is still lacking. Here we show that doped graphene turns into a fractional Chern insulator, when irradiated with high-intensity circularly polarized light. We derive the effective steady state band structure of light-driven graphene using…
Effects of an open surface on a magnetic Chern insulator are investigated in comparison with those of an interface to a capping magnetic layer. In magnets, an open surface often perturbs the magnetic order by a reconstruction of the magnetic moment directions near the surface. On the other hand, in topological insulators, it leads to the formation of topologically protected surface states. These two contrasting effects may coexist in magnetic Chern insulators, which give rise to nontrivial surface reconstruction. For instance, the chiral edge current is largely enhanced by the edge reconstruct…
The interplay between topology and correlation lies at the forefront of modern condensed matter physics. In this paper, we study the extended fermion-Hubbard model, including the on-site as well as the nearest-neighbor repulsive interactions, on a topological square lattice that supports the Chern insulator. Within the mean-field method, we find that the spontaneous symmetry breaking (SSB) charge-density wave or antiferromagnetic insulator dominates the system when the onsite or nearest-neighbor interactions are strong enough. It is interesting that the antiferromagnetic Chern insulator will a…
We use entanglement entropy signatures to establish non-Abelian topological order in projected Chern-insulator wavefunctions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number C=2, whose wavefunction may also be viewed as the square of the Slater determinant of a band insulator. We demonstrate that this wavefunction is captured by the $SU(2)_2$ Chern Simons theory coupled to fermions. This is established most persuasively by calculating the modular S-matrix from the candidate ground state wavefunctions, following a recent entanglement entropy based appr…
Long-range interactions drive some of the rich phenomenology of quasiparticle collective states in the fractional quantum Hall (FQH) regime. We test for analogues in models of fractional Chern insulators (FCIs) derived from a screened Coulomb interaction. We find that the uniform FCI liquid is surprisingly robust to long-range interactions but gives way to a unidirectional charge density wave (CDW) of fractionally charged quasiparticles with increased screening length. Our results show that FCIs offer a robust and important platform for studying quasiparticles collective states.
The recent discovery of topological Kondo insulators has triggered renewed interest in the well-known Kondo insulator samarium hexaboride, which is hypothesized to belong to this family. In this Letter, we study the spin texture of the topologically protected surface states in such a topological Kondo insulator. In particular, we derive close relationships between (i) the form of the hybridization matrix at certain high-symmetry points, (ii) the mirror Chern numbers of the system, and (iii) the observable spin texture of the topological surface states. In this way, a robust classification of t…
Fractional Chern insulator with higher Chern number in optical lattice
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；CAS Key Laboratory of Quantum Information, University of Science and Technology of China（中国科学技术大学量子信息重点实验室）；School of Physics, Sun Yat-Sen University（中山大学物理学院）
A monolayer of transition metal trichalcogenides has received a lot of attention as potential two dimensional magnetic materials. The system has a honeycomb structure of transition metal ions, where both spin-orbit coupling and electron correlation effect play an important role. Here, motivated by these transition metal series with effective doping or mixed valence case, we propose the possible realization of magnetic Chern insulators at quarter filled honeycomb lattice. We show that the interplay of intrinsic spin-orbit coupling and electron correlation opens a wide region of ferromagnetic Ch…
