Session Y10: Topological Insulators: Band Structures Theory

The Viscoelastic Response of Topological Tight Binding Models

The topological response to external perturbations is an effective probe to characterize different topological phases of matter. The Hall viscosity is an example of such a response which has been the subject of a great interest recently. So far, most of studies have focused on the continuum field theories. Here, we investigate this response for the tight binding (lattice) models. The presence of lattice breaks the continuous translational symmetry to a discrete symmetry and this causes two complications: it introduces a new length scale associated with lattice constant and makes the momentum a compact variable. We develop two different methods of how to implement a lattice deformation: (1) the lattice distortion is encoded in a U(1) phase acquired by a particle traversing a link between two sites; (2) a microscopic view is adopted and the lattice deformation appears in the gradient expansion of the hopping matrix elements. Consequently, we compute the Hall viscosity through the linear response (Kubo) formula. We examine these methods for three models: the Hofstadter model, the Chern insulator, and the surface of a 3D topological insulator. Our results in certain regimes of parameters, where the continuum limit is relevant, are in agreement with the field theory calculations.   

Direct Manifestation of the Band Topology via the Zak Shift of the Wannier-Stark Ladder

Topological phases of matter have been topics of intense interest in modern condensed matter physics. Numerous efforts have been devoted to investigating various exotic properties of materials with non-trivial band topology. The dissipationless transport via gapless helical edge or surface states is one of the defining properties of such materials, which, however, has been very difficult to realize in experiment due to various backscattering sources induced in the sample boundaries. In this work, we show that there is a fundamental connection between the non-trivial topology of the band structure and the Zak shift of the Wannier-Stark ladder emerging under a static electric field. As an application of this connection, we propose a novel spectroscopic method to directly manifest the band topology by counting the winding number of the Zak phase across the first Brillouin zone, which is shown to be robust against electron-impurity scattering.   

Strong orientation dependence of electronic properties of Antimony Selenide ($Sb_{2}Se_{3}$) nanostructures

Antimony Selenide has applications in thermoelectric, photovoltaic and optical storage. Recently, it was demonstrated that bulk material under high pressure becomes a topological insulator and further undergoes insulator to metal to superconducting transitions. The $Sb_{2}Se_{3}$ nanostructures reported so far exhibit direct bandgaps, whereas the bulk has an indirect gap. Considering different crystallographic orientations of synthesized nanostructures and the anisotropic nature of its structure, we have studied the influence of orientation on their electronic behavior. Using first principle methods, we explore the stability of nanowires in different orientations and its influence on electronic structure. We find confinement effects for the narrower nanostructures, whereas the [001] orientation showed a reduced bandgap. This anomalous behavior is discussed considering that bandgap reduction could be attributed to recent experimental findings of a insulator-metal transition, which is related to topological quantum transition. The surface reconstructions show similarities to the distortion of polyhedras occurring in bulk $Sb_{2}Se_{3}$ under high pressure, which are related to the insulator-metal transition and superconductivity at 8.0 K.   

Topological characterization and dynamics of a fermionic fractional Chern insulator

Using the density matrix renormalization group (DMRG) method on a cylinder geometry, we characterize the fractional Chern insulator (FCI) state in the Haldane honeycomb lattice model at $\nu = 1/3$ filling of the lowest band and check its stability. We investigate the chiral and topological properties of this state through (i) its Hall conductivity, (ii) the topological entanglement entropy, (iii) the U(1) charge spectral flow of the many body entanglement spectrum, and (iv) the charge of the anyons. As the interaction strength is lowered, we observe a direct transition from the FCI into a metallic phase and find indications for the transition to be of first order. Since our approach does not rely on any band or subspace projection, we are able to prove the stability of the fractional state in the presence of interactions exceeding the band gap, as has been suggested in the literature. Additionally, we investigate the characteristic dynamics of the FCI phase using time dependent DMRG.   

Disorder and inversion symmetry breaking effect in Chern insulators

We show that Chern insulators with and without inversion symmetry respond differently to strong disorder. We consider a Haldane model with Anderson disorder and and use a real-space numerical approach to calculate the conductivity tensor of a Haldane model in the presence of Anderson disorder. The inversion symmetry is explicitly broken by adding a sub-lattice potential $\Delta_{AB}$. While disorder closes the gap and destroys the Chern insulator in the system with inversion symmetry, the quantum anomalous Hall effect is insensitive to increasing Anderson disorder in the case of broken symmetry. In this case, the symmetry breaking works as a valley filter that protects the topological state from inter-valley scattering and strong disorder gives rise to a novel topological state that is similar to the topological Anderson insulator.   

Topological phase transition in the Hofstadter-Hubbard model

We study the interplay between topological and conventional long-range order of attractive fermions in a time-reversal-symmetric Hofstadter lattice using quantum Monte Carlo simulations, focusing on the case of one-third flux quantum per plaquette. At one-third filling, the noninteracting system is a topological insulator, and a nonzero critical interaction strength is needed to drive a transition from the quantum spin Hall insulator to a superfluid. We probe the topological signature of the phase transition by threading a magnetic flux through a cylinder and observe quantized topological charge pumping.   

Non-trivial edge physics in a featureless Mott insulator

While the Lieb-Schultz-Mattis theorem forbids the existence of fully symmetric quantum paramagnetic phases on lattices with fractional filling of particles per unit cell, such a phase is in principle allowed with certain fractional numbers of particles per site on non-Bravais lattices, including half-filling on the honeycomb lattice. It has been shown that a non-interacting Hamiltonian of spinless fermions or bosons cannot have such a symmetric insulating ground state, and an explicit construction using interactions is challenging. Recently, Kimchi et al. constructed a wavefunction for bosons at half-filling that does not break any symmetries and is not topologically ordered--and in this sense is a featureless insulator in the bulk. Here, however, we reveal that this wavefunction exhibits non-trivial structure at the edge. We apply recently developed techniques based on a tensor network representation of the wavefunction to demonstrate the presence of a gapless entanglement spectrum and a non-trivial action of combined charge-conservation and spatial symmetries on the edge. We will also discuss the possibility of finding a parent Hamiltonian and analyzing the existence of a symmetry-protected topological phase around this state.   

Bound states of three fermions forming symmetry-protected topological phases

We propose a simple theoretical construction of certain short-range entangled phases of interacting fermions, by putting the bound states of three fermions (which we refer to as clustons) into topological bands. We give examples in two and three dimensions, and show that they are distinct from any free fermion state. We further argue that these states can be viewed as combinations of certain free fermion topological states and bosonic symmetry-protected topological (SPT) states. This provides a conceptually simple understanding of various SPT phases.   

Generalized Dirac points and topological surface states in a three-dimensional nonsymmorphic photonic crystal

In condensed matter physics, the three-dimensional~(3D) Dirac equation describes low-energy excitations of spin-$\frac{1}{2}$ electrons in quantum materials ranging from topological insulators to Dirac semimetals. Here we discover, in photonic crystals, the existence of robust 3D linear point degeneracies between two pairs of band crossings with different velocities. Based on symmetry considerations, we demonstrate that such dispersion is governed by a generalization of 3D Dirac equation for {\it spin-1} photons propagating in a periodic medium. This 3D Dirac phase in photonic crystals represents a new topological phase of matter protected by nonsymmorphic crystal symmetry and exhibits novel two-dimensional surface states, which we characterize. Time-reversal symmetry is preserved but not required in our photonic system.   

Accurate band gaps of semiconductors and insulators from Quantum Monte Carlo calculations

Ab initio calculations are useful tools in developing materials with targeted band gaps for semiconductor industry. Unfortunately, the main workhorse of ab initio calculations -- density functional theory (DFT) in local density approximation (LDA) or generalized gradient approximation (GGA) underestimates band gaps. Several approaches have been proposed starting from empirical corrections to more elaborate exchange-correlation functionals to deal with this problem. But none of these work well for the entire range of semiconductors and insulators. Deficiencies of DFT as a mean field method can be overcome using many-body techniques. Quantum Monte Carlo (QMC) methods can obtain a nearly exact numerical solutions of both total energies and spectral properties. Diffusion Monte Carlo (DMC), the most widely used QMC method, has been shown to provide gold standard results for different material properties, including spectroscopic constants of dimers and clusters, equation of state for solids, accurate descriptions of defects in metals and insulators. To test DMC's accuracy in a wider range of semiconductors and insulators we have computed band gaps of several semiconductors and insulators. We show that DMC can provide superior agreement with experiment compared with more traditional DFT approaches including high level exchange-correlation functionals (e.g. HSE).