Session A32: Topological Insulators: Quantum Hall Effect

Weak antilocalization in HgTe quantum wells and topological surface states: Massive versus massless Dirac fermions

HgTe quantum wells and surfaces of three-dimensional topological insulators support Dirac fermions with a single-valley band dispersion. In this work we conduct a comparative theoretical study of the weak antilocalization in HgTe quantum wells (QWs) and topological surface states. The difference between these two single-valley systems comes from a finite band gap (effective Dirac mass) in HgTe quantum wells in contrast to gapless (massless) surface states in topological insulators. The finite effective Dirac mass implies a broken internal symmetry, leading to suppression of the weak antilocalization in HgTe quantum wells and transition to the weak localization regime as a function of the gap or carrier density. In particular, we show how the difference in the behavior of the weak localization corrections for HgTe QWs allows to distinguish topological versus normal insulators. Further for the topological surface states we predict specific weak-antilocalization magnetoconductivity in a parallel magnetic field due to their exponential decay in the bulk. The relevant experiments will be discussed.   

Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized $\theta$-term

We investigate how a magnetic field induces one-dimensional edge channels when the two-dimensional surface states of three-dimensional topological insulators become gapped. The Hall effect, measured by contacting those channels, remains quantized even in situations, where the $\theta$-term in the bulk and the associated surface Hall conductivities, $\sigma_{xy}^S$, are not quantized due to the breaking of time-reversal symmetry. The quantization arises as the $\theta$-term changes by $\pm 2 \pi n$ along a loop around $n$ edge channels. Model calculations show how an interplay of orbital and Zeeman effects leads to quantum Hall transitions, where channels get redistributed along the edges of the crystal. The network of edges opens new possibilities to investigate the coupling of edge channels.   

The topological Hubbard model and its high-temperature quantum Hall e

The quintessential two-dimensional lattice model that describes the competition between the kinetic energy of electrons and their short-range repulsive interactions is the repulsive Hubbard model. We study a time-reversal symmetric variant of the repulsive Hubbard model defined on a planar lattice: Whereas the interaction is unchanged, any fully occupied band supports a quantized spin Hall effect. We show that at 1/2 filling of this band, the ground state develops spontaneously and simultaneously Ising ferromagnetic long-range order and a quantized charge Hall effect when the interaction is sufficiently strong. We ponder on the possible practical applications, beyond metrology, that the quantized charge Hall effect might have if it could be realized at high temperatures and without external magnetic fields in strongly correlated materials.   

Magnetic field dependence of the edge channels in HgTe quantum wells

In recent years much attention has been devoted to topological insulators, that is, materials which are insulating in the bulk, but have conducting states at their surface. This new class of topological states has first been observed experimentally in HgTe quantum wells. In such a two-dimensional topological insulator, which is also synonymously referred to as a quantum spin Hall insulator, these surface states are one-dimensional, helical edge channels. Here we study a HgTe quantum well in the presence of a constant perpendicular magnetic field. Using tight-binding calculations as well as deriving an analytical expression to determine the electron dispersion and states, we solve the effective low-energy Hamiltonian of this system in a finite-strip geometry. Our main focus is on the behavior of these solutions with increasing magnetic field. In particular, we describe the evolution of the edge states as the system crosses from the quantum spin Hall to the quantum Hall regime.   

Phase Diagram of Massive Dirac Fermions with Tunable Interactions in High Magnetic Fields

We study the strongly correlated states of massive fermions in two dimensions with Berry's phase $\pi$ and $2\pi$, in the limit of high magnetic fields. Due to the chiral band structure and massive carriers, the effective Coulomb interactions depend on the external magnetic field, and lead to a number of phases within a single low-lying Landau level. The tunability of the interactions allows the study of the transitions between phases in a more direct manner than in GaAs-based systems where the form of the interactions is independent of the magnetic field. We map the phase diagram at partial fillings $\nu = 1/3, 1/2, 3/5$ of the low-lying Landau levels, and find transitions between fractional quantum Hall states, compressible Fermi-liquid-like states, as well as charge-density-waves. We also find a new, broad regime of the effective interactions which favor the paired non-Abelian states. Our study identifies the strongly correlated phases expected in high-mobility graphene, bilayer graphene, topological insulators, and other materials with the non-trivial Berry phases, and provides a realistic method for studying the phase transitions between them.   

Isotropic Landau levels of relativistic and non-relativistic fermions in 3D flat space

The usual Landau level quantization, as demonstrated in the 2D quantum Hall effect, is crucially based on the planar structure. In this talk, we explore its 3D counterpart possessing the full 3D rotational symmetry as well as the time reversal symmetry. We construct the Landau level Hamiltonians in 3 and higher dimensional flat space for both relativistic and non-relativistic fermions. The 3D cases with integer fillings are Z$_{2}$ topological insulators. The non-relativistic version describes spin-1/2 fermions coupling to the Aharonov-Casher SU(2) gauge field. This system exhibits flat Landau levels in which the orbital angular momentum and the spin are coupled with a fixed helicity. Each filled Landau level contributes one 2D helical Dirac Fermi surface at an open boundary, which demonstrates the Z$_{2}$ topological nature. A natural generalization to Dirac fermions is found as a square root problem of the above non-relativistic version, which can also be viewed as the Dirac equation defined on the phase space. All these Landau level problems can be generalized to arbitrary high dimensions systematically. \\[4pt] [1] Yi Li and Congjun Wu, arXiv:1103.5422.\\[0pt] [2] Yi Li, Ken Intriligator, Yue Yu and Congjun Wu, arXiv:1108.5650.   

Topological property for magnetic flux tubes in a two-dimenaionl electron gas

We have studied the energy spectrum in the presence of square magnetic flux-tubes in a two-dimensional electron gas. Our finding is that the Dirac-like dispersion can be formed out between the third and the forth magnetic subbands without including the Zeeman interaction. This Dirac-like dispersion is not band --inverted type with a global gap. This Dirac-like dispersion becomes band-inverted when the Zeeman interaction is included. We expect that the inverted magnetic subbands due to the Zeeman interaction can be recognized as the topological insulator. This topology property can be supported by the Chern number from the magnetic subbands.   

Pi-flux as a universal probe of two-dimensional topological insulators

We show that the existence of a Kramers pair of zero-energy modes bound to a vortex carrying $\pi$-flux is a generic feature of topologically nontrivial phases in the $M-B$ model, describing HgTe quantum wells, and therefore this vortex represents the bulk probe of the band topology [1]. We explicitly find the form of the zero-energy states by analytically solving Dirac equation which contains a momentum-dependent Schr\" odinger mass, besides the usual Dirac mass term. A particular regularization of the vortex potential yields the modes exponentially localized and regular at the origin that carry nontrivial charge or spin quantum number. \\[4pt] [1] V. Juricic, A. Mesaros, R.-J. Slager, and J. Zaanen, arXiv:1108.3337.   

Strongly driven Floquet topological insulator in semiconductor quantum wells

Floquet topological insulator in a weak-field driven semiconductor quantum well was proposed most recently. In this article we extend the situation to strongly driving field, which can generate high-order harmonic resonances. With appropriate form of driving field, it is found that whether topological transition can happen depends on the number of resonances N we can observe. If N is odd, topological transition can happen; if N is even, topological transition cannot happen. This phenomenon may be observed in semiconductor quantum wells by applying a strongly oscillating magnetic field. In addition, our discussion can be extended to other systems such as p-wave superconductors and spin chains.   

Quantum Hall effect in a one-dimensional dynamical system

We construct a periodically time-dependent Hamiltonian with a phase transition in the quantum Hall universality class [1]. This Hamiltonian is closely related to that of a discrete time quantum walker, but additionally it allows us to study effects of disorder. A particular choice for the form of the Hamiltonian enables us to determine the time evolution of the system in one of the dimensions exactly. Simulations of the system can thus be performed in one dimension, thereby reducing the computational effort required. We investigate the topological phase transition associated with tuning between different quantum Hall plateaux and determine the critical exponent for the divergence of the localisation length. Our scheme can in principle also be implemented in cold atoms experiments, opening the doors to investigating the quantum Hall phase transition in a one-dimensional cold atoms set up. \\[4pt] [1] J.~P. and Edge, J.~M. and Tworzydlo, J. and Beenakker, C.~W.~J., PRB 84 115133 (2011).   

Quarter-Filled Honeycomb Lattice with a Quantized Hall Conductance

We study a generic two-dimensional hopping model on a honeycomb lattice with strong spin-orbit coupling, without the requirement that the half-filled lattice be a Topological Insulator. For quarter-(or three-quarter) filling, we show that a state with a quantized Hall conductance generically arises in the presence of a Zeeman field of sufficient strength. We discuss the influence of Hubbard interactions and argue that spontaneous ferromagnetism (which breaks time-reversal) will occur, leading to a quantized anomalous Hall effect. G. Murthy, E. Shimshoni, R. Shankar, and H. A. Fertig, arxiv:1108.2010[cond-mat.mes-hall]   

Berry Curvature and Phonon Hall Effect

Abstract We establish the general phonon dynamics of magnetic solids by incorporating the Mead-Truhlar correction in the Born-Oppenheimer approximation. The effective magnetic-field acting on the phonons naturally emerges, giving rise to the phonon Hall effect. A general formula of the intrinsic phonon Hall conductivity is obtained by using the corrected Kubo formula with the energy magnetization contribution properly incorporated. The resulting phonon Hall conductivity is fully determined by the phonon Berry curvature and the dispersions. Based on the formula, the topological phonon system could be rigorously defined. In the low temperature regime, we predict that the phonon Hall conductivity is proportional to $T^{3}$ for the ordinary phonon systems, while that for the topological phonon systems has the linear $T$ dependence with the quantized temperature coefficient. \\[4pt] [1] Tao Qin and Junren Shi, arXiv:1111.1322 (2011) \\[0pt] [2] Tao Qin, Qian Niu and Junren Shi, Phys. Rev. Lett., Accepted, (2011).   

Dissipationless Phonon Hall Viscosity

We study the acoustic phonon response of crystals hosting a gapped time-reversal symmetry breaking electronic state. The phonon effective action can in general acquire a dissipationless ``Hall'' viscosity, which is determined by the adiabatic Berry curvature of the electron wave function. This Hall viscosity endows the system with a characteristic frequency, $\omega_v$; for phonons of frequency $\omega$, it shifts the phonon spectrum by an amount of order $(\omega/\omega_v)^2$ and it mixes the longitudinal and transverse sound waves with a relative amplitude ratio of $\omega/\omega_v$ and with a phase shift of $\pm \pi/2$, to lowest order in $\omega/\omega_v$. We study several examples, including the integer quantum Hall states, the quantum anomalous Hall state in Hg$_{1-y}$Mn$_{y}$Te quantum wells, and a mean-field model for $p_x + i p_y$ superconductors. We discuss situations in which the phonon response is directly related to the gravitational response, for which striking predictions have been made. When the electron-phonon system is viewed as a whole, this provides an example where measurements of Goldstone modes may serve as a probe of adiabatic curvature of the gapped sector of a system.   

3D Topological Insulators in the Continuum with Nearly Flat Bands

We propose a three-dimensional topological insulating state in the harmonic potential with a strong spin-orbit coupling breaking the inversion symmetry. The system gives rise to Landau-level like quantization with the full 3D rotational symmetry and time-reversal symmetry. The radial quantization generates the energy gap between neighboring bands. The states inside each band are characterized by their angular momentum over which the dispersions are suppressed by spin-orbit coupling and thus nearly flat and without Bloch-wave states. Surface states exhibit helical Dirac Fermi surfaces which are described by the Z2 index. Similar analysis in 2D shows the existence of topological insulators in the harmonic potential with strong Rashba spin-orbit coupling. These topological insulating states can be achieved from the dimensional reduction of the quantum Hall states in 3D and 4D flat space.   

Photo-induced quantum Hall insulators without Landau levels

In this talk, we demonstrate that non-equilibrium transport of graphene under the application of off-resonant light displays a near quantization of Hall conductance. While previous studies of topological phenomena under coherent drives focused on the effective topological band structure, electron occupations in the non-equilibrium systems, which are so far neglected, play a crucial role to determine the transport properties. Using the formalism that consistently take this into account, we show that the topological band structures can directly manifest themselves in the transport properties in Landauer-type configurations under the application of off-resonant light. We give an intuitive explanation of the induction of Chern numbers in the band structure, by showing that the virtual photon absorption/emissions of electrons produces an effective second-order hopping with phase accumulation, leading to the effective Haldane model. Our proposal opens the perspective to realize so-called quantum Hall systems without Landau levels in materials such as graphene and three dimensional topological insulators under coherent drives.