Session W37: Topological Insulators: Fractionalization

Zoology of Fractional Chern Insulators

We study four different models of Chern insulators in the presence of strong electronic repulsion at partial fillings. We observe that all cases exhibit a Laughlin-like phase at filling fraction $1/3$. We provide evidence of such a strongly correlated topological phase by studying both the energy and the entanglement spectra. In order to identify the key ingredients of the emergence of Laughlin physics in these systems, we show how they are affected when tuning the band structure. We also address the question of the relevance of the Berry curvature flatness in this problem. Using three-body interactions, we show that some models can also host a topological phase reminiscent of the $\nu=1/2$ Pfaffian Moore-Read state. Additionally, we identify the structures indicating cluster correlations in the entanglement spectra.   

Time-reversal symmetric hierarchy of fractional incompressible liquids

In this talk I shall establish a hierarchical construction for a class of BF theories and present it as a candidate topological field theory to describe fractional topological insulators in 2D. The structure of the K matrix and the universal properties that it describes will be discussed, and the properties of the time-reversal symmetric edge theory will be derived via the bulk-edge correspondence.   

Fractional Chern Insulator

Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translation symmetry. We conclusively show that a partially filled Chern insulator at $1/3$ filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: the system is incompressible and has a $3$-fold degenerate groundstate whose momenta can be computed by postulating an generalized Pauli principle with no more than $1$ particle in $3$ consecutive orbitals. The groundstate density is constant, and equal to $1/3$ in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit the identical counting of Laughlin $1/3$ quasiholes. The $3$ groundstates and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator.   

Fractional Chern Insulators from the nth Root of Bandstructure

I will describe some recent theoretical results pertaining to fractional Chern insulators. These are interacting lattice models of partially filled Chern bands which have been numerically shown to realize some universal aspects of fractional quantum Hall physics. We use parton/slave-particle techniques to provide model wavefunctions for these phases. We also provide a strong coupling expansion that gives new insights into the foundations of the parton approach. I will conclude by describing some of the practical uses of our results, including suggesting candidate models to realize non-Abelian fractional Chern insulators.   

Wannier states and pseudopotential Hamiltonians for fractional Chern insulators

Fractional Chern insulators are fractional quantum Hall states realized in lattice models with full lattice translational symmetry in the absence of an external magnetic field. In fractional quantum Hall systems, pseudopotential Hamiltonians have been constructed for which the ideal ground states such as Laughlin states are exact ground states. In this work, we constructed pseudopotential Hamiltonians for the fractional Chern insulators by making use of the Wannier function representation. The physical interaction Hamiltonians can be expanded in pseudopotentials, from which one can analyze the preferred fractional quantum Hall states in the corresponding systems. The results of this analysis are compared with exact diagonalization. Our approach may be generalized to studying nonabelian fractional Chern insulator states and time-reversal invariant fractional topological insulators.   

Topological insulators and fractional quantum Hall effect on the ruby lattice

We study a tight-binding model on the two-dimensional ruby lattice. This lattice supports several types of first and second neighbor spin-dependent hopping parameters in an $s$-band model that preserves time-reversal symmetry. We discuss the phase diagram of this model for various values of the hopping parameters and filling fractions, and note an interesting competition between spin-orbit terms that individually would drive the system to a $Z_2$ topological insulating phase. We also discuss a closely related spin-polarized model with only first and second neighbor hoppings and show that extremely flat bands with finite Chern numbers result, with a ratio of the band gap to the band width approximately 70. Such flat bands are an ideal platform to realize a fractional quantum Hall effect at appropriate filling fractions. The ruby lattice can be possibly engineered in optical lattices, and may open the door to studies of transitions between quantum spin liquids, topological insulators, and integer and fractional quantum Hall states.   

Charge and spin fractionalization in strongly correlated topological insulators

The recently discovered two-dimensional topological insulators (TI) with time-reversal symmetry are closely related to integer quantum Hall states in which electron spin plays the role of charge. The appearance of protected edge states in these systems can be understood by describing the spin-orbit coupling as the source of an SU(2) (spin-dependent) magnetic flux. However, the absence of spin conservation cripples the quantum spin-Hall effect. In this talk we will explore the possibility of obtaining strongly correlated TIs with fractional quasiparticles. Such states are the SU(2) analogues of fractional quantum Hall states, but with modified topological orders due to the spin non-conservation. We will discuss two heterostructure designs featuring a ``conventional'' TI quantum well that could host a fractional TI state of Cooper pairs or excitons. These devices exploit a quantum critical point for electron localization to provide a fragile spectrum that can be dramatically reshaped by the strong spin-orbit coupling. Then, we will present a topological spinor-field theory of fractional TIs and explain how the spin-orbit coupling can produce a combined charge and spin fractionalization despite the spin non-conservation.   

Symmetry protected fractional Chern insulators and fractional topological insulators

We construct fully symmetric wavefunctions for the spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant fractional topological insulators (FTI) 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 $\nu$ (and the same quantized Hall conductance $\sigma_{xy}$ in FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases with which are realized in a partially filled band with Chern number one and filling fraction $\nu=1/m$. Examples of FCI/FTI wavefunctions 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 two are discussed. Generic FTI wavefunctions preserving all lattice symmetries in the absence of spin conservation are also presented for filling fraction $\nu=2/m$.   

High-temperature fractional quantum Hall states

Using a suitable combination of geometric frustration, ferromagnetism, and spin-orbit coupling in a hopping model on the kagome lattice, we obtain a flat band with nonzero Chern number. Partial filling of this band could give rise to the fractional quantum Hall effect in this system which, when considering realistic parameters, would persist at room temperature. Possible material realizations that indicate new directions for exploration and synthesis are discussed.   

Fractional Quantum Hall states in strongly correlated multi-orbital systems

For topologically nontrivial and very narrow bands, Coulomb repulsion between electrons has been predicted to give rise to a spontaneous fractional quantum-Hall (FQH) state in absence of magnetic fields. We will discuss how orbital degrees of freedom in frustrated lattice systems lead to a narrowing of topologically nontrivial bands [1]. This robust effect does not rely on fine-tuned long-range hopping parameters and is directly relevant to a wide class of transition metal compounds. In addition, we will show that strongly correlated electrons in a $t_{2g}$-orbital system on a triangular lattice self-organize into a spin-chiral magnetic ordering pattern that induces precisely the required topologically nontrivial and flat bands [2]. On top of a self-consistent mean-field approach, we use exact diagonalization to study an effective one-band model for the emerging flat band in the presence of longer-range interactions and establish the signatures of a spontaneous $\nu = \frac{1}{3}$ FQH state. \\[4pt] [1] J. W.F. Venderbos, M. Daghofer, J. van den Brink, PRL {\bf 107}, 076405 (2011) \\[0pt] [2] J. W.F. Venderbos, S. Kourtis, J. van den Brink, M. Daghofer, arXiv:1109.5955   

Hall Crystal States in Fractionally Filled Chern Bands

Two-dimensional time-reversal-invariant topological insulators can be thought of as a time-reversed pair of Chern bands. Numerical evidence shows the existence of states at fractional filling which are analogous to FQH states[1]. In [2] it was noted that at small momenta, the algebra of the density operators projected to the Chern band resembles the magnetic translation algebra. The authors have constructed a mapping[3] between Chern bands and a Landau level in a periodic potential which works at all momenta. This mapping is dynamically faithful, and reproduces the commutators of the projected density operator. There turn out to be Hall Crystal states, characterized by a Hall conductance, and another integer which described the charged dragged when the potential is adiabatically moved by a lattice unit. Using the Hamiltonian formalism developed by the authors some time ago for the FQHE[4], we calculate gaps and collective mode dispersions for such states. 1. D. N. Sheng et al, arxiv:1102.2568, N. Regnault and B. A. Bernevig, arxiv:1105.4867. 2. S. Parameswaran, R. Roy, and S. L. Sondhi, arxiv:1106.4025. 3. G. Murthy and R. Shankar, arxiv:1108.5501 4. G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101 (2003)   

Chiral Topological Phases and Fractional Domain Wall Excitations in One-Dimensional Chains and Wires

According to the general classification of topological insulators, there exist one-dimensional chirally (sublattice) symmetric systems that can support any number of topological phases. We introduce a zigzag fermion chain with spin-orbit coupling in magnetic field and identify three distinct topological phases. Zero-mode excitations, localized at the phase boundaries, are fractionalized: two of the phase boundaries support $\pm $e/2 charge states while one of the boundaries support $\pm $e and neutral excitations. In addition, a finite chain exhibits $\pm $e/2 edge states for two of the three phases. We explain how the studied system generalizes the Peierls-distorted polyacetylene model and discuss possible realizations in atomic chains and quantum spin Hall wires.   

Quantum Hall Effect and Bound Fractional Charge in Topological Insulator Magnetic Tunnel Junctions

Proximity coupling 2D and 3D time-reversal invariant topological insulators to ferromagnetic domain walls is known to lead to bound fractional charge and an integer quantum Hall effect respectively. Here we show that by correctly engineering the sample geometry these effects can appear in the presence of only a single magnet with no domain walls, thus reducing the experimental complexity. We will prove that a magnetic layer sandwiched between 3D topological insulator films will exhibit the quantum Hall effect, possibly leading to a room-temperature realization of the quantum Hall effect.   

Phase Diagram of a Simple Model for Fractional Topological Insulator

We study a simple model of two species of (or spin-1/2) fermions with short-range intra-species repulsion in the presence of opposite (effetive) magnetic field, each at filling factor 1/3. In the absence of inter-species interaction, the ground state is simply two copies of the 1/3 Laughlin state, with opposite chirality. Due to the overall time-reversal symmetry, this is a fractional topological insulator. We show this phase is stable against moderate inter-species interactions. However strong enough inter-species repulsion leads to phase separation, while strong enough inter-species attraction drives the system into a superfluid phase. We obtain the phase diagram through exact diagonalization caluclations. Nature of the fractional topological insluator-superfluid phase transition is discussed using an appropriate Chern-Simons-Ginsburg-Landau effective field theory.   

Fractional Chern Insulators and the $W_\infty$ Algebra

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 $W_\infty$ algebra of lowest Landau level projected densities first identified by Girvin, MacDonald and Platzman [Phys. Rev. B 33, 2481 (1986).] For Hamiltonians projected to the Chern band this provides a route to replicating lowest Landau level physics on the lattice.